Understanding Macroeconomic Theory

Understanding MacroeconomicTheory

At each point in time individuals in an economy are making choices with respectto the acquisition sale andor use of a variety of different goods Such activitycan be summarized by aggregate variables such as an economyrsquos total productionof various goods and services the aggregate level of unemployment the generallevel of interest rates and the overall level of prices Macroeconomics is the studyof movements in such economy-wide variables as output employment and prices

The focus of this book is on developing simple theoretical models that provideinsight into the reasons for fluctuations in such aggregate variables These modelsexplore how shocks or ldquoimpulsesrdquo to the economy impact individualsrsquo behaviorin specific markets and the resulting implications in terms of changes in aggregatevariables

Understanding Macroeconomic Theory will provide the reader with anin-depth understanding of standard theoretical models Walrasian Keynesian andneoclassical It is written in a concise accessible style and will be an indispensabletool for all students who wish to gain firm grounding in the complexities ofmacroeconomic theories

John M Barron is the Loeb Professor of Economics in the Krannert School ofManagement at Purdue University

Bradley T Ewing is the Jerry S Rawls Endowed Professor in OperationsManagement in the Rawls College of Business at Texas Tech University

Gerald J Lynch is Professor of Economics in the Krannert School of Managementat Purdue University

Routledge Advanced Texts in Economics and Finance

Financial EconometricsPeijie Wang

Macroeconomics for Developing Countries 2nd editionRaghbendra Jha

Advanced Mathematical EconomicsRakesh V Vohra

Advanced Econometric TheoryJohn S Chipman

Understanding Macroeconomic TheoryJohn M Barron Bradley T Ewing and Gerald J Lynch

John M Barron Bradley T Ewingand Gerald J Lynch

First published 2006by Routledge270 Madison Ave New York NY 10016

Simultaneously published in the UKby Routledge2 Park Square Milton Park Abingdon Oxon OX14 4RN

Routledge is an imprint of the Taylor amp Francis Group an informa business

copy 2006 John M Barron Bradley T Ewing and Gerald J Lynch

All rights reserved No part of this book may be reprinted orreproduced or utilized in any form or by any electronicmechanical or other means now known or hereafterinvented including photocopying and recording or in anyinformation storage or retrieval system without permission inwriting from the publishers

Library of Congress Cataloging in Publication DataBarron John M

Understanding macroeconomic theory John M BarronBradley T Ewing and Gerald J Lynch

p cmIncludes bibliographical references and index1 Macroeconomics I Ewing Bradley T II Lynch Gerald J III Title

HB1725B3753 2006339ndashdc22 2005026372

British Library Cataloguing in Publication DataA catalogue record for this book is available from the British Library

ISBN10 0ndash415ndash70195ndash3 ISBN13 978ndash0ndash415ndash70195ndash2 (hbk)ISBN10 0ndash415ndash70196ndash1 ISBN13 978ndash0ndash415ndash70196ndash9 (pbk)ISBN10 0ndash203ndash08822ndash0 ISBN13 978ndash0ndash203ndash08822ndash7 (ebk)

This edition published in the Taylor amp Francis e-Library 2006

ldquoTo purchase your own copy of this or any of Taylor amp Francis or Routledgersquoscollection of thousands of eBooks please go to wwweBookstoretandfcoukrdquo

Contents

1 Introduction 1

2 Walrasian economy 7

3 Firms as market participants 18

4 Households as market participants 39

5 Summarizing the behavior and constraints of firms andhouseholds 64

6 The simple neoclassical macroeconomic model (withoutgovernment or depository institutions) 78

7 Empirical macroeconomics traditional approaches and timeseries models 96

8 The neoclassical model 116

9 The ldquoKeynesian modelrdquo with fixed money wage modifyingthe neoclassical model 130

10 The Lucas model 146

11 Policy 167

12 Open economy 186

Notes 202References 223Index 228

1 Introduction

The topics of macroeconomics

At each point in time individuals in an economy are making choices with respectto the acquisition sale andor use of a variety of different goods Such activity canbe summarized by aggregate variables such as an economyrsquos total production ofvarious goods and services the aggregate level of employment and unemploymentthe general level of interest rates and the overall level of prices1 Macroeconomicsis the study of movements in such economy-wide variables as output employmentand prices

The focus of this book will be on developing simple theoretical models thatprovide insight into the reasons for fluctuations in such aggregate variables Thesemodels explore how shocks or ldquoimpulsesrdquo to the economy (eg changes to tech-nology the money supply or government policy) impact individualsrsquo behavior inspecific markets and the resulting implications in terms of changes in aggregatevariables

An overview of some facets of theoreticalmacroeconomic analysis

Given the breadth of economic activity in an economy the study of macroeco-nomics must involve an examination of a variety of different markets For instanceit is common for macroeconomic analysis to consider exchanges of labor servicesin the labor markets of consumption and capital goods in the output marketsand of financial assets in the financial markets The fact that macroeconomicssimultaneously analyses exchanges of different goods in different markets meansthat macroeconomic theory is a general equilibrium theory That is macro-economic theory must by necessity incorporate the links across markets that arefundamental to general equilibrium analysis As we will see throughout this booka key reflection of the links across markets is Walrasrsquo law named in honor ofthe nineteenth-century French economist Leon Walras2 Simply put Walrasrsquo lawnotes that the budget constraints faced by individual agents in the economy sug-gest that if n minus 1 of the n markets in the economy are in equilibrium then the nthmarket must be in equilibrium We will repeatedly rely on Walrasrsquo law or variantsof it to simplify macroeconomic analysis

2 Introduction

While macroeconomic theories have in common (a) an attempt to explainfluctuations in aggregate variables and (b) a general equilibrium character thereremain wide differences among macroeconomic models Below we break downthese differences across macroeconomic models in several ways in order to makesome sense of what passes for simple theoretical macroeconomic analysis

Static dynamic and stationary analysis

One way of breaking down macroeconomic analyses is into static models dynamicmodels and stationary analysis of dynamic models Static macroeconomic modelsanalyze the economy at a point in time They consider the determination of pro-duction exchange and prices of various goods only for the markets that currentlyexist John Hicks (1939) sketched out an analysis of ldquospotrdquo or ldquotemporaryrdquo equi-librium The advantage to such an approach is that it provides for rather simpleldquocomparative staticrdquo analysis of the effects of changes in a variety of exogenousvariables on the endogenous variables3 Such static analysis is useful in providinginsight into a variety of questions of interest

Static macroeconomic analysis can be viewed as a modification of a Walrasiangeneral equilibrium analysis or what is commonly referred to as ldquoArrowndashDebreutheoryrdquo (Arrow and Hahn 1971 Debreu 1959) In ArrowndashDebreu theory eachcommodity is described by its physical characteristics its location and its dateof availability It is assumed there are a complete set of spot and forwardmarkets Prices adjust to clear all markets However if one restricts attentionto just spot markets then one moves from traditional Walrasian generalequilibrium to an analysis of ldquotemporary equilibriumrdquo a phrase coined by Hicks(1939) This restriction to spot markets is one element of static macroeconomicanalysis4

A second element of static models is that if there is a future then staticmacroeconomic analysis simply assumes given expectations of future prices andenvironment How expectations of future events are formed is left unspecified sothat expectations of future prices become simply an element in the set of exogenousvariables5

While static analysis provides insights there are several disadvantages of staticanalysis severe enough that it alone does not provide an adequate grounding inmacroeconomic analysis The key disadvantage of static analysis is that it breaksties between current events and future events To show the limitations of staticanalysis let us suppose that underlying a simple static macroeconomic analy-sis of current markets is a microeconomic analysis of individualsrsquo decisions thatidentifies the anticipated future level of prices as one of the exogenous variablesaffecting current behavior As we have seen static analysis takes expectations ofsuch variables as future prices as exogenous variables Doing so however resultsin (a) an incomplete enumeration of exogenous variables that can impact currenteconomic activity and (b) a potentially incomplete accounting of the effects of theimpact on current economic activity of a change in those exogenous variables thatare identified by the analysis

Introduction 3

To illustrate the first point of an incomplete listing of exogenous variables letus suppose that the static model identifies changes in the current money supplyas one factor that influences current prices This suggests that if we replicate thestatic analysis in future periods changes in the money supply in the future would beshown to affect prices at that time It seems natural to then presume that individualsrsquoanticipation of future prices would incorporate this link between changes in thefuture money supply and future prices in forming their expectations of futureprice levels so that the anticipated future money supply becomes a determinantof current activity6 Yet static analysis since it does not analyze markets beyondthe current period will not identify the potential impact of future changes in themoney supply on current activity7

To illustrate the second point of an incomplete accounting of the effects of achange in an exogenous variable let us suppose that underlying the static macro-economic analysis of current markets is a microeconomic analysis of firmsrsquo currentinvestment behavior that identifies the anticipated future tax levels as well as futureprices as two exogenous variables affecting investment decisions Thus staticanalysis would suggest that a change in future tax levels will impact current activitythrough the direct effect on current investment It is not hard to see however that(a) the current change in investment means a different future capital stock and(b) the change in future tax levels could affect future as well as current investmentEither or both of these changes would likely impact future prices and if such animpact were anticipated be a second way that future tax changes impact currentactivity8

An obvious way to avoid the above problems is to introduce forward marketsso that the macroeconomic analysis determines the prices of goods to be tradedin the future along with the prices of goods traded at the current time9 In doingso we have moved from static to dynamic analysis That is the macroeconomicmodels now determine the paths of variables (such as prices) over time rather thanprices (and other variables) at only one point in time

In a deterministic setting this expansion of dynamic analysis incorporates thenotion of ldquoperfect foresightrdquo in which individuals correctly anticipate all futureprices If there were uncertainty the analysis indexes goods by both the date oftrade and the ldquostate of naturerdquo with trades contingent on the realized state ofnature10 The result is that at each date there is a distribution of potential prices atwhich trade for a good could occur and given common knowledge of likelihood ofthe states of nature expectations of future prices would be defined by the analysis(ldquorational expectationsrdquo)

Once dynamic analysis is introduced we can consider a special limiting form ofdynamic analysis termed stationary analysis The aim of stationary analysis is toidentify in the context of a dynamic model the limiting tendencies of endogenousvariables such as the capital stock or the rate of growth in prices given that theexogenous variables remain constant or stationary over time11

While stationary analysis is distinct from static analysis in some cases one canthink of static analysis as a form of stationary analysis That is static analysis insome cases can be viewed as the outcome that would emerge each period given

4 Introduction

that the exogenous variables remain constant (or in some cases grow at a steadyrate over time) and given that one picks the correct fixed level of certain keyexogenous variables (eg the capital stock and the rate of change in the moneysupply) Note however that this implies that for static analysis to perfectly mimicstationary analysis one must to all intents and purposes have first executed theunderlying dynamic analysis

Period (discrete) versus continuous-time analysis

Macroeconomic analysis can be broken down into period or discrete-time macro-economic models and continuous-time macroeconomic models Substantivedifferences in terms of theoretical predictions do not exist between these two typesof analyses if one is careful to assure identical underlying assumptions Yet the twoanalyses do differ in the analytical techniques used For instance while discretemacroeconomics relies on the techniques of dynamic programming and differenceequations to characterize elements of the model in similar circumstances con-tinuous macroeconomic analysis turns to the techniques of optimal control anddifferential equations

Although substantive issues are not raised by the discrete- versus continuous-time dichotomy it is sometimes argued that one is preferred to the other Forinstance an attractive feature of the continuous-time analysis is that it highlightsquite clearly the distinctions between stocks and flows something that is not soclearly discernable in discrete analysis On the other hand an attractive feature ofdiscrete analysis is that it makes more transparent the link between the theoreticalanalysis and empirical testing since such analysis coincides with the obvious factthat empirical data on macroeconomic variables is discrete

New classical economics versus non-market-clearing

Classical analysis refers to the widely adopted view of how the macroeconomyshould be modeled that existed prior to the experience of the Great Depressionand John Maynard Keynesrsquo General Theory of Employment Interest and Money(1936) In classical theory the real side of the economy was separate from themoney side Classical analysis of the ldquorealrdquo side of the economy is aimed atdetermining such variables as total production relative prices the real rate ofinterest and the distribution of output Classical analysis of the money side ofthe economy meant analysis is aimed at determining money prices and nominalinterest rates

The separation of the monetary side from the real side in classical or neoclassicalanalysis led to the prediction that monetary changes do not have any effect onreal variables such as total output12 A similar prediction is often obtained bymore recent macroeconomic analysis and this is one reason why this more recentanalysis is referred to as the new classical economics13 Alternative labels ofthese new classical models include rational expectations models with market

Introduction 5

clearing neoclassical models of business fluctuations and equilibrium businesscycle models

A common feature of the analyses of new classical economics besides thefact that it suggests a divorcement of monetary changes from the real side ofthe economy is that prices are determined in the analysis so as to clear marketsThis view that prices serve to equate demands and supplies a view common tomicroeconomics is taken as an important strength of the analysis for it meansthat the models have consistent ldquomicroeconomic foundationsrdquo One implicationof the market-clearing assumption is the same as in microeconomics ndash the analysissuggests that all gains to exchange have been extracted

Contrasting the new classical economics with what preceded it helps one putthis rebirth of classical analysis into perspective Following the Great Depressionmacroeconomic analysis took as its main premise the idea that markets did notclear ndash in particular that prices did not adjust In this context the business cyclewas defined by ldquomarket failurerdquo and the role of government to stabilize the eco-nomy was clear There are a number of different types of non-market-clearing orKeynesian models One version of such Keynesian models that popularized byPatinkin (1965 chapters 13ndash14) Clower (1965) and Barro and Grossman (1971)takes as given output prices such that the output market fails to clear A secondmodel popularized by Fischer (1977) Phelps and Taylor (1977) and Sargent(1987a) as an alternative formalization of the Keynesian model takes as given theprice of labor such that the labor market fails to clear

The common theme of these non-market-clearing analyses is that for variousreasons prices do not clear markets and concepts such as excess demand and supplyplay a role in the analysis Yet no concise reason is given as to why there is marketfailure other than suggesting such items as ldquocoordination problemsrdquo and ldquotrans-action costsrdquo14 The result is that such analysis is challenged by the new classicaleconomics as lacking the microeconomic foundations for price determination AsHowitt (1986 108) suggests such a view ldquoforces the proponent of active stabi-lization policy to explain the precise nature of the impediments of transacting andcommunicating that prevent private arrangements from exhausting all gains fromtraderdquo15 This is not an easy task according to Howitt since ldquoimpediments to com-munication in a model simple enough for an economist to understand will typicallyalso be simple enough that the economist can think of institutional changes thatwould overcome themrdquo (1989 108)

Microeconomic foundations and aggregation issues

An important feature of macroeconomic analysis is that it reflects the aggre-gation of individual decisions A common approach to such aggregation is toassume ldquorepresentativerdquo agents characterize their optimal behavior then use suchbehavioral specifications in building the macroeconomic model Thus much ofmacroeconomic analysis entails looking at individualsrsquo decisions such as house-holdsrsquo decisions to work consume and save or firmsrsquo decisions to produceborrow and invest in capital

6 Introduction

These characterizations of optimizing individual behavior make up part of thebuilding blocks or ldquomicroeconomic foundationsrdquo of macroeconomic analysisYet microeconomic foundations of macroeconomics are not restricted to suchanalysis For instance such foundations also include a characterization of howprices in individual markets are determined as we saw in our discussion of newclassical economics

In developing the microeconomic foundations of macroeconomic models wewill often be struck by the extent to which the analysis restricts any role forheterogeneity or diversity among the individual agents in the economy Yet suchdiversity can in certain instances be critical to the analysis One attempt to introducediverse or heterogeneous agents into macroeconomic analysis is represented by theoverlapping generations models These models also have the advantage of beinggenuinely dynamic in nature and as such represent one area of macroeconomicsthat has recently received significant attention

Deterministic versus stochastic

In recent years an important element to macroeconomic models has been to intro-duce stochastic elements The rationale is clear the presence of uncertainty as tofuture events is real As noted by Lucas (1981 286)

the idea that speculative elements play a key role in business cycles that theseevents seem to involve agents reacting to imperfect signals in a way whichafter the fact appears inappropriate has been commonplace in the verbaltradition of business cycle theory at least since Mitchell It is now entirelypractical to view price and quantity paths that follow complicated stochasticprocesses as equilibrium ldquopointsrdquo in an appropriately specified space

As the quote suggests in dynamic models especially for new classical economicswhere market clearing is presumed stochastic elements are incorporated into theanalysis so that the role played by shocks to an economy in a dynamic setting canbe well defined

2 Walrasian economy

Introduction

This chapter develops a competitive model of the economy The key assumptionsneeded for this model are spelled out in detail One important aspect of this modelis the ldquonumerairerdquo or the commodity price that is used as a reference in the modelA distinction is made between accounting prices and relative prices and it isseen that traditional general equilibrium analysis does not determine the levelof accounting prices but rather simply relative prices A number of modificationsto the model are mentioned and add a sense of ldquorealismrdquo to the framework Thesemodifications include the introduction of futures markets quantity constraintsand the costs associated with carrying out a transaction

The chapter continues by considering individual decision-making and the theoryof the consumer and how this relates to the determination of market demand Thegeneral equilibrium conditions are stated in terms of relative prices and allocationsA theme carried throughout this book is emphasized and that is the use of theaggregate budget constraint Finally it is shown that explicitly excluding moneyas a ldquomarketrdquo allows one to understand how Sayrsquos conclusion that ldquosupply createsits own demandrdquo is arrived at in an economy composed of a single aggregatecommodity

A simple Walrasian model

As discussed previously the idea that prices adjust to clear markets is commonto much of new classical macroeconomics Thus we begin our examination ofmacroeconomic analysis by considering an economy consisting of perfectly com-petitive markets This means that individuals take prices at which exchanges canbe made as parametric and prices adjust to eliminate excess demands so that indi-vidualsrsquo plans at given prices are feasible As the title to this section suggests sucha characterization is sometimes referred to as being indicative of a ldquoWalrasianrdquoeconomy Walras described the process by which prices adjust to excess demand orsupply as a groping or tatonnement process (see Walras 1954)1 A fictitious auc-tioneer calls out different prices for the various markets and no exchange occursuntil equilibrium prices are reached2

8 Walrasian economy

Our analysis also begins at a very simple level The term ldquosimplerdquo reflects atleast the following three characteristics of the economy that we consider

1 It is a barter economy That is any commodity can be freely traded for anyother commodity There is no role for a medium of exchange (money) inreducing the costs of arranging exchanges

2 It is an exchange economy That is there is no production Instead individualshave initial fixed endowments of various commodities

3 It is a timeless economy Goods are indexed by physical characteristics andlocation but not dated according to availability This rules out futures mar-kets or the formation of expectations of future events and planning Such aneconomy was suggested by Hicks (1939) Patinkin (1965 Chapter 1) andHansen (1970 Chapter 4) provide a more detailed view of such an economyAn actual example of a pure barter exchange economy is offered by Radford(1945) The simple model of the economy developed below is useful in high-lighting such concepts as relative prices the numeraire individual versusmarket experiments aggregation issues conditions for general equilibriumand Walrasrsquo and Sayrsquos laws

The first model developed below also takes an approach to modeling the econ-omy that is in vogue in current theoretical macroeconomics As Sargent (1987b)states the ldquoattraction of (such) general equilibrium models is their internal con-sistency one is assured the agentsrsquo choices are derived from a common set ofassumptionsrdquo

Yet this advantage of general equilibrium analysis is not fully exploited until theelements of time money and production are introduced and so we will expand thediscussion in subsequent sections by introducing such features Below we intro-duce in more detail some of the key assumptions underlying the simple Walrasianmodel we start with

Key assumptions underlying a simple Walrasian model

As noted by Debreu (1959 74) ldquoan economy is defined by m consumers (charac-terized by their consumption sets and their preferences) n producers (characterizedby their production sets) and the total resources (the available quantities of thevarious commodities which are a priori given)rdquo As discussed above we considera special case of Debreursquos ldquoconcept of an economyrdquo one in which productionis absent As the following set of assumptions makes clear we also restrict ouranalysis to private ownership economies with a price system In particular assume(partial listing)

Assumption 21 There are m individuals (agents) in the economy indexed bya = 1 m There are T commodities indexed by i = 1 T 3 Agent arsquos initial

Walrasian economy 9

endowment of commodity i is denoted by cai cai ge 0 i = 1 T Naturally

msuma=1

cai = ci gt 0

Note that this assumption reflects an exchange economy in which private propertyrights exist ldquoPrivate property rightsrdquo means that for each unit of each good theexclusive right to determine use has been assigned to a particular individual

Assumption 22 All exchanges occur at a single point in time

Assumption 23 Each individual confronts the same known set of prices at whichexchange can occur4 A relative (purchase) price of commodity i indicates the unitsof commodity j required to purchase one unit of commodity i A relative (sale)price of commodity i indicates units of commodity j received when one unit ofcommodity i is sold

Assumption 24 Purchase and sale prices are identical for each commodity Thismeans that there are no ldquoprice spreadsrdquo which would suggest either a gain to anindividual buying and selling the same commodity or the presence of costs tomaking an exchange5 Thus for the T commodities there are T 2 exchange ratesor relative prices taking two commodities at a time

The numeraire

While there are T 2 exchange rates the complete set of exchange rates can bededuced directly or indirectly by the set of T minus 1 relative prices

(π1j πjminus1 j πj+1 j πTj)

where the πij denotes the price of commodity i in terms of commodity j6 In thelisting of relative prices (π1j πjminus1 j πj+1 j πTj) commodity j is referredto as the ldquonumerairerdquo

To see how the set of relative prices reduces to T minus 1 we rely on the factthat πhh = πhjπhj for all h j and k Let us see what this means for a simpleexample of three commodities h j and k There are then T 2 or nine differentrelative prices which are πhh πjj πkk πhj πjh πhk πkh πjk and πkj But we canuse the relationship πhh = πhjπhj to reduce this to T minus 1 = 2 relative prices withinformational content In particular we know that

1 πhh πhkπhj and similarly for πjj and πkk when h = j = k In other wordsthe exchange rate of a commodity with itself is unity This reduces from nineto six the number of relative prices for which information is required

2 πjh = 1πhj when k = j (such that πkj = πkk = 1) Similarly πhk = 1πkhand πjk = 1πkj For example if the jth commodity is pears and the hth

10 Walrasian economy

commodity is oranges then if πjh = 3 (3 oranges = 1 pear) πhj = 13(13 pear = 1 orange) This reduces from six to three the number of relativeprices for which information is required to reconstruct the complete set ofrelative prices

3 Finally πkh = πkjπhj (for k = j = h) For example if the jth commodity ispears the kth commodity is apples and the hth commodity is oranges thenif πkj = 3 (3 pears = 1 apple) and πhj = 13 (13 pears = 1 orange) thenπkh = 9 (9 oranges = 1 apple) That is with 9 oranges you can get 3 pearswhich in turn will purchase 1 apple This drops us from three to two relativeprices required to reconstruct the complete set of relative prices Since thenumber of commodities T = 3 we have shown how the T 2 relative pricescan be constructed from T minus 1 relative prices

In subsequent discussions we will arbitrarily let commodity T be the numerairesuch that the set of relative prices can be summarized by

(π1T πTminus1T )

For simplicity let us change notation such that πiT = πi i = 1 T Thus theset of relative prices can be rewritten as

(π1 πTminus1)

Note that πT = 1 since we are assuming the T th good is the numeraireIn traditional general equilibrium theory there is a concept of ldquoaccounting pricesrdquo

as well as the concept of relative prices Accounting prices can be represented bya set of real numbers (say pi i = 1 T ) attached to the T commodities7 Therelationship between these accounting prices and the set of relative prices that doimpinge on behavior is that πi = pipT i = 1 T (for Pi = 0) As we will seetraditional general equilibrium analysis does not determine the level of accountingprices but rather simply relative prices8

Anticipating future modifications

Before continuing it might be useful to anticipate some of the subsequent changeswe will make in the characterization of the economy Besides the introduction ofproduction we will

bull introduce time implying either forward (futures) markets or an important rolefor expectations of future spot prices

bull introduce quantity constraints that can arise if prices are fixed at non-market-clearing levels (ie depart from a Walrasian framework) and

bull introduce the cost of carrying out an exchange

Walrasian economy 11

With respect to point (c) such costs have been characterized by Coase (1960) asldquotransaction costsrdquo arising from the costs ldquonecessary to discover who it is that onewishes to deal with and to inform people that one wishes to dealrdquo the costs ofldquoconduct[ing] negotiations leading up to a bargain and to draw up a contractrdquoand the costs ldquoto undertake the inspection needed to make sure that the terms ofthe contract are being observedrdquo9 Such costs will alter the nature of contracts(exchange agreements) formed and may provide the reason for ldquoprice rigiditiesrdquoSuch costs also suggest a role for money

Transaction costs are assumed to be zero in the simple Walrasian sys-tem outlined above Sometimes this is referred to as a situation wherethere is ldquoperfect informationrdquo or where markets are complete and ldquoperfectlycompetitiverdquo

Individual experiments

General equilibrium analysis can be divided into what Patinkin (1965) refers to asldquoindividual experimentsrdquo and ldquomarket experimentsrdquo In the context of the simpleWalrasian barter exchange economy individual experiments consider the behaviorof individual agents given an initial endowment and preferences when confrontedwith a set of prices Market experiments consider the resulting determination ofprices

In the simple Walrasian model under consideration individual experimentsreplicate standard microeconomic analysis of consumer behavior In particularassume

Assumption 25 Individual arsquos preferences are described by his utility functionua(ca1 caT ) where ca1 caT denote agent arsquos consumption bundle cai ge 0i = 1 T ua maps the set of all T -tuples of non-negative numbers into the setof all real numbers (ua RT+ rarr R) We make the appropriate assumptions withrespect to individualsrsquo preferences such that a utility function exists and is wellbehaved10

Assumption 26 Individual a will choose the most preferred consumption bundlefrom the set of feasible alternatives (rationality) Given the possibility of cost-less exchange at the set of relative prices represented by (π1 πTminus1) feasibleconsumption bundles or sets (ca1 caT ) are defined by

Tsumi=1

πi cai minusTsum

πicai ge 0

wheresumT

i=1 πi cai denotes the initial endowment of individual a in terms ofcommodity T The above expression defines the budget set

The consumer problem

From Assumptions 25 and 26 individual arsquos optimum consumption bundle is thesolution to the problem

maxCa1CaT

ua(ca1 caT )

subject to

Tsumi=1

πi cai minusTsum

πicai ge 0 cai ge 0 i = 1 T

The constrained maximization problem can be translated into the unconstrainedLagrangian expression

maxca1caT λ

L(ca1 caT λ) = ua(ca1 caT ) + λ

πi cai minusTsum

πicai

with first-order (necessary) conditions being11

partcaile 0 i = 1 T

partcaicai = 0 i = 1 T

cai ge 0 i = 1 T

partλge 0

λpartL

partλ= 0

λ ge 0

maxca1caT λmicro1microT

L(ca1 caT λ micro1 microT ) = ua(ca1 caT )

πi cai minusTsum

πicai

with the necessary conditions being

partcai= 0 i = 1 T

partλge 0

λpartL

partλ= 0

partmicroige 0 i = 1 T

microipartL

partmicroi= 0 i = 1 T

λ ge 0

microi ge 0 i = 1 T

Individual demands and excess demands

The optimal consumption bundle for agent a is defined by the above first-orderconditions and will be denoted by the (demand) set

(ca1 caT ) cai ge 0 i = 1 T

Individual arsquos demand functions will be of the form

(π1 πTminus1

Tsumi=1

πi cai

) i = 1 T

That is individual arsquos demand (consumption) of commodity i depends on the T minus1relative prices of commodities and the initial endowment Note that the form ofthe utility function implies the utility-maximizing consumption bundle meets thebudget constraint with equality Thus at the optimal bundle we have

partλ=

Tsumi=1

πi cai minusTsum

πicdai = 0

An important point to note about demand functions is that they are homogeneousof degree zero in what might be called accounting prices12 Accounting prices aredefined such that pi = πi middot pT i = 1 T so it is clear that if all prices increaseby the multiple θ relative prices and the initial endowment are unchanged

Individual arsquos excess demand function for commodity i is defined byzai = cd

ai minus cai If zai is positive agent a is a net buyer of commodity i whileif zai is negative the agent is a net seller of commodity i The market value

(in terms of the numeraire) of the quantity of the ith commodity that individuala seeks to exchange (buy or sell) is then given by πizai From the budget con-straint we know that

sumTi=1 πi(cd

ai minus cai) = 0 orsumT

i=1 πizai = 0 In other words foreach individual the market value (in terms of commodity T ) of individual excessdemands must sum to zero This rather obvious finding generates what is referredto as Walrasrsquo law as we will see

Market experiments

In the previous section we reviewed the nature of individual demand functions andindividual excess demand functions Now consider the collection of m individualsAggregating or summing individual demand functions we obtain an aggregate orldquomarketrdquo demand function for commodity i of the form

(π1 πTminus1

Tsumi=1

πi ci1 Tsum

)equiv

msuma=1

cdai(middot)

Similarly summing agentsrsquo excess demand functions for commodity i gives usthe aggregate or ldquomarketrdquo excess demand function for commodity i of the form

zi equivmsum

(zai) equivmsum

(cdai minus cai)

Note that a zero aggregate excess demand for commodity i does not imply thatno exchange of commodity i occurs among the m agents However as we haveseen a zero individual excess demand for commodity i does imply no exchangeof commodity i by that particular individual

Aggregation issues

So far our aggregations have remained true to the underlying microeconomicanalysis Yet this is rarely the case in macroeconomic analysis which typicallyabstracts from what might be termed ldquodistributionalrdquo effects An example of thisin the above context as we will see later is to ignore the effects of the distributionof initial endowments across individuals on market demands such that the marketdemand function for commodity i is assumed to be of the form

(π1 πTminus1

Tsumi=1

πi ci

Tsumi=1

πicai equivmsum

πi cai

As you can see with heterogeneity (either in initial endowments or preferences)such a posited aggregate market demand function is unlikely to follow exactlyfrom the underlying microeconomic analysis One should keep in mind suchapproximations when interpreting macroeconomic analysis

Equilibrium an isolated market

With respect to a single market equilibrium is characterized by an accounting pricepi and implied relative price πi = pipT such that cd

i = ci (demand equals fixedendowment) or equivalently zi = 0 (excess demand equals zero) The tatonnementprocess is the description of how prices change to clear the market In the Walrasianmodel movement toward equilibrium the tatonnement process involves twofacets

1 The Walrasian excess demand hypothesis which indicates that the accountingprice of commodity i rises if there is excess demand and falls if there is excesssupply That is

dpi = fi(zi) i = 1 T fi(0) = 0dfidzi

In terms of relative prices

dπi = dpi

pT= f (zi)

pT i = 1 T minus 1

Note that the change in price is not across time since each market is assumedto clear instantaneously at the same point in time

2 The recontracting assumption which states that offers to buy or sell at var-ious relative prices are not binding unless market(s) clear Only when theequilibrium price (or price vector) is obtained are contracts then made final

General equilibrium (conditions)

A general equilibrium will be characterized by a set of T minus 1 relative prices(πlowast

1 πlowasttminus1) and allocations (clowast

a1 clowastaT ) for individual a a = 1 m such

clowastai = cd

(πlowast

1 πlowasttminus1

Tsumi=1

πi cai

) i = 1 T a = 1 m (21)

rArr clowasti equiv

msuma=1

clowastai =

msuma=1

cdai equiv cd

clowasti = ci i = 1 T (22)

Equation (21) indicates that the equilibrium allocation must be optimal in thatit must satisfy all demands for commodities at the specified set of pricesEquation (22) indicates that such an allocation must be feasible that is sumto total resource endowment Together these two conditions imply a set of relativeprices such that excess demands are zero or

cdi = ci i = 1 T

For questions concerning the existence uniqueness and stability of generalequilibrium consult Varian (1992) Debreu (1959) and Arrow and Hahn (1971)

Walrasrsquo law and Sayrsquos law

Note that the above statement of general equilibrium involves setting T excessdemand equations equal to zero but there are only T minus 1 unknowns (relativeprices) Walras solved this problem by showing that one of the equations arbitrarilychosen can be deduced from the other T minus 1 equations In other words there areonly T minus 1 independent equations To show the dependency remember that thebudget constraint for each individual is given by

Tsumi=1

πicdai minus

Tsumi=1

πi cai = 0

Summing across all individuals it must then be the case that

msuma=1

πi cai minusTsum

πicdai

Rearranging and substituting in cdi for

summaminus1 cd

ai and ci forsumm

aminus1 cai we obtain

Tsumi=1

πi(cdai minus cai) = 0 or

Tsumi=1

πizi = 0

The above is an explicit statement of Walrasrsquo law Walrasrsquo law states that the sumof the excess demands across all markets must be zero Note that in summing theexcess demand of each commodity is weighted by its relative price so that we aresumming common units (ie all excess demands are in units of the numeraire)The above aggregate budget constraint is sometimes referred to as Sayrsquos law orSayrsquos identity If there is a distinction between the two it is that Sayrsquos law explicitlyexcluded money as a ldquomarketrdquo In this setting one can understand how Sayrsquosconclusion that ldquosupply creates its own demandrdquo is arrived at in an economycomposed of a single aggregate commodity

Conclusion

This chapter has developed a general equilibrium framework that sets the stagefor a thorough understanding of how the macroeconomy works Particular atten-tion has been paid to the development of relative prices and the development ofaggregate demand through a process of many individual consumers operating in anenvironment in which they set out to maximize their own utility This frameworkand the tool of constrained optimization is used throughout this book

3 Firms as market participants

Introduction

In this chapter the simple Walrasian model is discussed in the context of moneyfinancial assets and production The chapter clearly illustrates the firmrsquos objectivethat is to maximize profits However the firm is constrained in that it must financepurchases of capital and equipment as well as pay its workers Moreover attentionis paid to all the costs faced by the firm not just the obvious ones The investmentand financing decisions of firms are discussed and issues related to Tobinrsquos Q anddebt-to-equity are explored This chapter provides a detailed examination of therole that firms play in the macroeconomy

A simple Walrasian model with money financialassets and production

In the exchange economy we just considered endowments of the commodity goodwere magically bestowed on individuals each period We now introduce productionas the source of commodities At the start of each period there now exists a ldquolaborrdquomarket in which labor services are exchanged New agents denoted ldquofirmsrdquo hirethe labor services provided by households During the period firms combine thelabor services with an existing capital stock to produce output (commodities)which is sold in the output market net of output retained to replace capital usedup during production Revenues from the sale of output are distributed to ldquohouse-holdsrdquo in the form of wages during the period At the end of the period interestpayments and dividends are made to households out of revenues Each period firmsalso enter the output market to augment their capital stock with such purchasesfinanced by the issue of bonds and a new financial asset denoted ldquoequity sharesrdquoIn particular we assume

Assumption 31 There are new agents in the economy denoted as ldquofirmsrdquo Theseagents are initially endowed at time t with a capital stock K and a technologyfor transforming capital services from a capital stock and labor services into a

Firms as market participants 19

single ldquocompositerdquo commodity The technology is summarized by the productionfunction

yt = f (Nt K)

where K denotes the capital stock the firm inherits at time t Nt denotes the employ-ment of labor services arranged at time t for period t (from time t to time t + 1)and yt denotes the constant rate of production of the commodity for period t (fromtime t to time t + 1)1 Similarly for period i (i = t + 1 t + 2 ) which runsfrom time i to time i + 1 output produced is given by2

yi = f (Ni Ki)

Assumption 32 During each period firms sell the output produced in the outputmarket For output produced during period t (from time t to time t+1) let pt denoteits price when it is sold during the period up to and including time t + 1 Let pt+1denote the price of output produced during period t + 1 that is sold beyond timet +1 up to and including time t +2 and so on At time t the price level associatedwith the prior period is denoted by p

Assumption 33 At the start of each period households rent their labor servicesto firms for the period At the start of period t agreements to exchange the Ntlabor services during period t (from time t to t + 1) are entered into at the moneywage rate denoted by wt Similarly at the start of period t + 1 Nt+1 labor servicesare exchanged at the money wage rate wt+1 and so on3

Assumption 34 At the end of each period two types of financial assets areexchanged bonds (in the form of perpetuities) and equity shares Bonds promiseto pay a fixed money (coupon) payment z each future period in perpetuity Let pbpbt and pbt+1 denote the money price of such bonds in markets at the end of periodstminus1 t and t+1 respectively the gross (nominal) interest rates over period t (fromtime t to time t + 1) and over period t + 1 (from time t + 1 to time t + 2) are thengiven by4

1 + r = (z + pbt)pb

1 + rt = (z + pbt+1)pbt

Note that if ri = rt i = t + 1 then successive substitution for the priceof bonds in future periods will result in the following expression for the price ofbonds at time t

pb = 1

1 + r(z + pbt) = 1

infinsumi=1

(1 + rt)i

(z + z

where pbt = zrt The number of previously issued bonds outstanding at time t isdenoted by B5

Assumption 35 Equity shares are the second type of financial asset exchangedat the end of each period Equity shares (ldquostocksrdquo) are contracts that obligate theissuer (firms) to pay to bearers at the end of each future period the income from thesale of output produced during the period net of other contractual obligations ofthe firms (eg wage payments to suppliers of labor services and interest paymentsto holders of bonds issued by firms) Let S denote the number of previously issuedequity shares outstanding at time t Holders of these equity shares are the ldquoownersof the firmsrdquo Let pe pet and pet+1 denote the money price of equity sharesexchanged in markets at the end of periods t minus 1 t and t + 1 respectively Thegross (nominal) rate of return on equity shares over period t (from time t to timet + 1) and period t + 1 (from time t + 1 to time t + 2) is then given by6

1 + re = [( ptdtS) + pet)]pe

1 + ret = [( pt+1dt+1St) + pet+1)]pet

where ptdt and pt+1dt+1 denote total nominal dividend payments made at the endof periods t and t + 1 (at time t + 1 and time t + 2) respectively St denotesthe anticipated number of equity shares outstanding after the equity market at theend of period t7 Note that the price of an equity share indicates the fact that thepurchase of an equity share entitles the holder to a portion of future (not current)dividends

Assumption 36 Households view bonds and equity shares as perfect substitutesldquoPerfect substitutesrdquo means that if equality in yields did not hold households wouldrefuse to purchase the asset with the lower yield forcing an adjustment in its pricethat would result in equivalent yields8 With bonds and equity shares as perfectsubstitutes we can speak of a single ldquofinancial asset marketrdquo that incorporatesboth bonds and equity shares and determines a single ldquointerest raterdquo

Assumption 37 There are incomplete markets Let pei i = t + 1 t + 2

denote the expectation formed in period t concerning the price of the consumptiongood over period i9 Similarly in period t we have pe

ei i = t + 1 and wei

i = t + 1 Such expectations are assumed to be held with subjective certaintyallowing us to abstract from risk considerations for the moment

Assumption 38 There are positive transaction costs to arranging exchanges ofthe consumption commodity during each period t Money holdings serve to reducethe transaction costs of arranging exchanges during a period10

Assumption 39 It is prohibitively costly for individuals to directly store theldquocompositerdquo commodity for consumption in future periods However output notconsumed during the period can be transformed (by ldquofirmsrdquo) into output in thesubsequent periods through the augmentation of the capital stock which permitshigher rates of production of output in future periods

Individual experiments firms

As always we start our analysis at the individual level The behavior of two typesof agents must now be considered ndash firms and households We start with firmsIn doing so we consider a ldquorepresentativerdquo agent a unit whose behavior exceptfor scale is identical to the behavior of the aggregate of such units Thus the samenotation will be used to represent both the individual unit and the aggregate of allunits In addition we consider an infinite time horizon

You should now recognize that an analysis of a representative unit neglectscertain potentially important ldquodistributionalrdquo aspects of the problem For instancefor households the ldquoreal indebtedness effectsrdquo of a price change on demand maynot be offsetting in the aggregate but that potential impact is ignored11 For firmsthe distribution of the initial capital stock can given adjustment costs affect totalemployment and output but that too is ignored12

To consider the behavior of a firm (or more specifically the manager whodirects production for the representative firm) we assume

Assumption 310 Technology is represented by the concave production function

yt = f (Nt K)

where yt denotes the firmrsquos planned (at time t) constant rate of output for the timeperiod from time t to t +1 to be sold during the period and at time t +1 Nt denotesthe firmrsquos planned (at time t) rate of employment of labor during period (t t + 1)with labor services purchased in the labor market at time t and K denotes thefirmrsquos planned capital stock for period t Recall that to simplify matters we takethe capital stock for the current period K as fixed at the individual firm levelThis would be the case given appropriate capital adjustment costs13

Assumption 311 The representative firm will choose the most preferred inputcombination and implied output given technology and prices (both current andanticipated future prices) At time t the objective of the firm is to maximize theexpected real market value of the S equity shares

Vt = peS

where pe is the price of equity shares at the end of period t minus 1 (at time t)14

A restatement of the firmrsquos objective

We have indicated that the objective of the firm at time t is to maximize the realmarket value of the S equity shares as given by

Vt = peS

To understand what underlies this market value of the firm we have to definethe elements underlying the price of equity shares and dividends We start byexamining what lies behind the price of equity shares

The assumption that equity shares and bonds are perfect substitutes means thatthe price of an equity share can be expressed as

pe = [ptdtS + pet](1 + r)

where dt denotes real dividends at the end of period t so that ptdt denotes nominaldividends S denotes the number of equity shares outstanding at time t pet is theprice of equity shares at the end of period t and r denotes the interest rate overperiod t (ie from time t to t + 1)

By successively substituting in a similar expression for the price of equity sharesin the next period we obtain for an infinite horizon that15

pe = [1(1 + r)]⎡⎣ptdtS +

infinsumk=1

[(pt+kdt+k)St+kminus1]kprod

(1 + rt+jminus1)

⎤⎦

That is the price of an equity share at the end of period t minus1 (at time t) pe reflectsthe anticipated discounted future stream of nominal dividends per share16

Since the real value of the firm is given by Vt = peSp we can now expressthe value of the firm as

Vt = [Sp(1 + r)]⎡⎣ptdtS +

infinsumk=1

(1 + rt+jminus1)

⎤⎦

which means that the objective of the firm can be stated in terms of maximizingthe discounted stream of current and future dividends Before examining whatdetermines dividends each period let us simplify the above expression for Vtby putting it in terms of real dividends each period To do so note that bydefinition

pt equiv p(1 + π)

pt+k equiv p(1 + π)

⎛⎝ kprod

(1 + rt+jminus1)

⎞⎠ k = 1 2 3

where πt+j denotes the rate of change in the price level between period t + j andt + j + 1 Thus we have

Vt = (SR)

⎡⎣dtS +

infinsumk=1

[dt+kSt+kminus1]kprod

Rt+jminus1

⎤⎦

where R = (1 + r)(1 + π) and Ri = (1 + ri)(1 + πi) denotes the real gross rateof interest for period i (from time i + 1 to time i + 2 i = t t + 1 ) Our nextstep in outlining the firmrsquos problem is to obtain an expression for real dividendsin each period

Dividends and the firm distribution constraint

In general we may denote real dividends at the end of any period as the differencebetween a firmrsquos total real revenues during the period and the total costs incurredduring the period Total revenues derive from the sale of output produced duringthe period In addition one could add revenues from a change in the numberof equity shares outstanding or a change in the number of bonds outstanding atthe end of the period17 Total costs to the firm in any period include the agreedupon wage payments to the labor hired during the period payments to replacedepreciated capital plus coupon payments at the end of the period to holders ofpreviously issued bonds Payments at the end of the period for the purchase ofcapital and associated adjustment costs could be counted as well18 That is firmsare constrained to have

dividends = revenue from sale of output

minus wages

minus interest payments

+ funds from change in outstanding bonds and stocks

minus costs to replace depreciated capital add new capital

and capital adjustment costs

The above constraint is typically divided into two separate constraints Oneconstraint earmarks funds raised from the change in outstanding bonds andequity shares at the end of the period to pay for or ldquofinancerdquo the installationof new capital stock during the period plus any capital adjustment costs Thisis the ldquofirm financing constraintrdquo The remaining revenues minus expendituresthen determine the level of dividends This part is called the ldquofirm distributionconstraintrdquo

The firm distribution constraint simply states that the revenues from the saleof output that exceed expenditures to meet wage payments purchases of capitalto replace that used up in the production process and interest payments to bondholders at the end of the period are distributed at the end of the period to householdsas dividends Thus real dividends at the end of period t are given by

dt = yt minus (wtpt)Nt minus zBpt minus δK

According to this expression real dividends at the end of period t equal realrevenues derived from the sale of output yt produced during period t minus costs

to the firm during period t that reflect the real wage wtpt times the quantity oflabor hired Nt less real coupon payments at the end of the period on previouslyissued bonds zBpt plus purchases of capital during the prior period to replacethat used up in the production process δK 19

In similar manner real dividends for periods t + 1 and t + 2 (paid at the end ofeach period) are given by

dt+1 = yt+1 minus (wt+1pt+1)Nt+1 minus zBtpt+1 minus δKt+1

dt+2 = yt+2 minus (wt+2pt+2)Nt+2 minus zBt+1pt+2 minus δKt+2

We can rewrite the above definition of dividends to see more clearly why itis also termed the ldquofirm distribution constraintrdquo This constraint simply says thatrevenue from the sale of output net of that retained to replace capital used up inthe production process (ie ldquonet productrdquo) is distributed to households either aswage payments interest payments or dividends That is

dt + (wtpt)Nt + zBpt = yt minus δK

dt+1 + (wt+1pt+1)Nt+1 + zBtpt+1 = yt+1 minus δKt+1

The firm financing constraint

The second part of the general constraint that firmsrsquo total expenditures equalrevenues is that changes in the firmrsquos holdings of capital as well as any asso-ciated adjustment costs are financed by a change in outstanding equity sharesandor bonds This linking of funding for capital purchases to the issuing of equityshares and bonds is denoted the ldquofirm financing constraintrdquo For instance at theend of periods t + 1 t + 2 we have the following firm financing constraints

Kt+1 minus K + ψ(Int) = [ pet middot (St minus S) + pbt middot (Bt minus B)]pt

Kt+2 minus Kt+1 + ψ(Int+1) = [ pet+1 middot (St+1 minus St) + pbt+1 middot (Bt+1 minus Bt)]pt+1

where ψ(Ii) denotes the costs of installing new capital at rate Ii during period i(between time i and time i + 1) a cost that depends directly on the rate of netinvestment (Ini = Ki+1 minusKi) planned at time t to occur between time i and i+120

Recall that we assume that firmsrsquo plans with respect to the number of stocks andbonds that will be outstanding following the financial market at the end of period i(time i + 1) mirror householdsrsquo expectations concerning the number of bonds andstocks that will be outstanding so we do not distinguish between firmsrsquo plans andhouseholdsrsquo expectations with respect to these variables

Since bonds and equity shares are perfect substitutes we can rewrite the abovefirm financing constraints in the simpler form

Int + ψ(Int) = At minus At = net At

Int+1 + ψ(Int+1) = At+1 minus At+1 = net At+1

where Ini denotes net real investment (ie Ki+1 minusKi) Ai denotes the real planned(at time t) value of total equity shares and bonds to be outstanding after the financialmarket at the end of period i and Ai denotes the initial real value of equity sharesand bonds for period i reflecting financing and capital decisions in prior periodsbut period i prices For instance

At = [ petSt + pbtBt]pt and At = [ petS + pbtB]pt

At+1 = [ pet+1St+1 + pbt+1Bt+1]pt+1 and

At+1 = [ pet+1St + pbt+1Bt]pt+1

There are several aspects of interest with respect to the firm financingconstraints First note that net capital purchases planned for period i to be installedbetween time i and time i + 1 are paid for at the end of period i when completelyinstalled from the sale of financial assets at that time

Second note that firms purchase the output of the composite good to augmentthe capital stock That is we have a ldquoone-sectorrdquo model in which the same goodserves both households (for consumption) and firms (for investment) There isonly a single commodity price A typical extension is a two-sector model in whichtwo goods are produced a consumption good and a capital good In such casesa new variable the relative price of the capital good in terms of the consumptiongood is introduced

Third note that the firm financing constraint holds whether the firm financescapital with new bonds new equity shares or ldquoretained earningsrdquo Suppose forinstance that a firm plans to add 100 units to its capital stock by buying a newpiece of machinery If the firm issues a bond with real value of 100 to pay for themachinery then there is a direct 100 unit increase (the new bond) in the value of thefinancial assets issued by the firm Note that the value of the current shareholdersrsquostock is unchanged in this case of bond financed investment While it is true thatthe tangible assets of the firm have increased by the 100 addition to capital thisbenefit to shareholders is exactly offset by the fact that the firmrsquos debt has alsoincreased by 10021

If the firm finances the 100 net investment by issuing new shares of stock equalto 100 again there is a 100 unit increase (the new equity shares) in the real valueof the financial assets issued by the firm As with bond financing however thereal value of the initial shareholdersrsquo stock is unchanged when the firm finances itscapital purchases by issuing new equity shares The new shares do not dilute thevalue of the shares of the initial shareholders since the capital purchase increases

the firmrsquos tangible assets by 100 which is exactly the real value of the new equityshares issued

Finally if the firm finances the 100 net investment through retained earningsthere is in essence a 100 unit increase in the value of the financial assets issued bythe firm for the following reason When a firm retains earnings in order to financea capital purchase the current stockholders own the right to the income generatedfrom the additional capital As a consequence the value of their equity shares risesto reflect the value of the new capital owned by the firm We could equivalentlyview this as the firm paying out 100 units in dividends to its initial shareholderswho then use the dividends to buy additional ldquoconstant valuerdquo equity shares equalto the value of the capital purchased by the firm In other words when the firmuses retained earnings to finance its investment spending it is implicitly issuingnew financial assets ndash equity shares

The nature of capital adjustment costs

An important aspect of the above financing constraint is that it incorporatespotential adjustment costs to purchases of capital as captured by the terms ψ(middot)which depend on Int Int+1 where Ini denotes the planned (at time t) net rateof investment for period i22 The total cost of capital purchases is thus the sum of(a) the real payments (or receipts if negative) involved in the purchase (or sale ifnegative) of capital in the output market and (b) potential real payments denotedldquoinstallationrdquo or adjustment costs associated with new capital acquisitions Forperiod t adjustment costs are given by ψ(Int) where Int denotes net investmentbetween time t and t + 1 Gross investment for period t is given by Int + δK Notethat we can thus decompose gross investment over the period into the change inthe capital stock Kt+1 minus K which is termed ldquoplanned net investmentrdquo and thereplacement of capital used up in the production process δK which is termedldquodepreciationrdquo23

To understand the conversion of the above analysis to continuous time we notethat in general adjustment costs over period t of length h are given by hψ(Inth)where the limit of the term Inth defines the rate of net investment That is incontinuous time the planned rate of investment would be defined by the rate ofgross investment

it = limhrarr0

h= lim

hrarr0

Kt+h minus K + hδK

h= Kt + δKt

and the adjustment cost function in continuous time would be ψ(int)For the aggregate rate of gross investment (a flow) to be defined by the above

expression K must equal K To achieve this one of two approaches is typicallytaken One approach assumes zero adjustment costs in that ψ equiv 0 This situationsometimes referred to as the case of ldquoperfect malleabilityrdquo means that the rateof investment may not be defined at the level of the individual firm That is ifthe existing capital stock were higher or lower than the planned level investment

would be infinitely positive or negative However it can be shown that withzero adjustment costs the output market in a continuous-time model at a pointin time is simply a ldquocapital marketrdquo and the expression Kt = K emerges as anequilibrium condition with respect to the capital market at time t24 Thus wecan apply LrsquoHospitalrsquos rule to define the aggregate rate of (net) investment in thecontinuous-time model with zero adjustment costs as25

it = limhrarr0

Kt+h minus K

h= limhrarr0 d(Kt+h minus K)dh

limhrarr0 dhdh= K

where it is rate of investmentIn contrast to the case of zero adjustment cost one can assume adjustment costs

that take the following form

ψ(0) = 0

ψ(β) gt 0 if β gt 0 ψ primeprime gt 0 and limβrarrinfin ψ(β) = infin

ψ(β) lt 0 if β lt 0

The above set of assumptions reflects the presumption that adjustment costsincrease at an increasing rate with the rate of change in capital and that it isinfinitely costly to change the capital stock arbitrarily fast The result of suchadjustment costs in both discrete-time analysis and continuous-time analysis isthat at time t the firm chooses Kt = K That is at time t the firm views the inher-ited capital stock as optimal since it is prohibitively costly to change the capitalstock at a point in time given such adjustment costs26

As we will see with ldquocosts of installing a unit of new capitalrdquo there will be adifference between the market value of capital goods in place and their replacementcost In particular the ratio of these two values known as ldquoTobinrsquos Qrdquo will exceedone In addition adjustment costs will mean that the firmrsquos decision with respectto investment will not be myopic (ie plans will not be based on forecasts thatextend only one period into the future) Rather the firm will consider all futureperiods in making current investment decisions

The firm problem a general statement

One way to state the optimization problem faced by the firm is to say that attime t the firm makes plans with respect to current and future employment oflabor (Nt Nt+1 ) the future employment of capital (Kt+1 Kt+2 ) the stockof outstanding bonds (Bt Bt+1 ) and the stock of outstanding equity shares(St St+1 ) in order to maximize the real value of the previously issued equityshares outstanding at time t with that real value given in general form by

Vt = (SR)

⎡⎣dtS +

infinsumk=1

Rt+jminus1

⎤⎦

Such plans are subject to the combined distribution and financing constraints listedabove as well as to the production function That is in general the firmrsquos problemcan be stated as27

max(SR)

⎡⎣dtS +

infinsumk=1

Rt+jminus1

⎤⎦

subject to the financing constraints

minus [Kt+1 minus K + ψ(Kt+1 minus K)] + (petpt) middot [Bt minus B]+ (pbtpt) middot (St minus S)] = 0

minus [Kt+2 minus Kt+1 + ψ(Kt+2 minus Kt+1)] + (pet+1pt+1) middot [Bt+1 minus Bt]+ (pbt+1pt+1) middot (St+1 minus St)] = 0

and the distribution constraints

and given the production functions

yt = f (Nt K)

yt+1 = f (Nt+1 Kt+1)

for i = t t + 1 The above problem has a recursive nature to it At the start of any given period

the firm inherits a stock of capital an outstanding stock of bonds and an outstand-ing stock of equity shares28 These variables are state variables Each period thefirm chooses a set of the ldquocontrolrdquo variables ndash employment investment Thesechoices in conjunction with the production function result in outcomes in termsof (a) a one-period return (dividends) at the end of the period and (b) a new setof ldquostaterdquo variables ndash capital stock and stock of financial assets (equity shares andbonds) ndash inherited in the subsequent period Further note that the objective of thefirm is additive in these one-period returns (dividends) Thus the problem is oneto which we can apply Bellmanrsquos dynamic programming technique

To reformulate the problem facing the firm as a dynamic programming prob-lem we use the conventional notation of dynamic programming problems29

Specifically the above problem can be viewed as involving

(a) A set of ldquocontrolrdquo variables each period

zt = Nt Int zt+1 = Nt+1 Int+1 etc

(b) A set of ldquostaterdquo variables each period

xt = K B S xt+1 = Kt+1 Bt St etc

(c) ldquoTransition functionsrdquo that link the choices specifically the choice of netinvestment during the period to the capital stock available at the start ofthe next period as well as the stock of financial assets outstanding in thesubsequent period For instance the choice of the net investment rate Int during period t dictates Kt+1 given K since

Kt+1 = Int + K

From the firm financing constraint we know that the choice of the investmentrate also determines the real stock of financial assets

( petpt)[Bt minus B] + ( pbtpt)[St minus S] = Int + ψ(Int)

(d) A set of one-period return functions (evaluated at the end of period t)30

rt(K B S) = dt rt+1(Kt+1 Bt St) = Sdt+1StRt etc

Since bonds and equity shares are perfect substitutes the above problem cannotbe solved for a unique optimal number of bonds or equity shares to have outstandingeach period Thus without any loss of generality we may restrict our focus toeither bond or equity share financing That is we can hold constant either bonds(ie Bi = B i = t t + 1 ) or equity shares (ie Si = S i = t t + 1 )Alternatively we can combine the distribution and financing constraints into asingle expression for dividends and hold constant both equity shares and bondsIn this case we have ldquoretained earningsrdquo financing of changes in the capital stockIt is this case that we consider below31

Simplifying the firm problem ldquoretained earnings financingrdquo

If we assume that capital expenditures are financed from ldquoretained earningsrdquo thereis a single state variable the stock of capital The problem facing the firm thencan be simply stated as follows The Bellman equation for period t given inheritedcapital stock K is

W (K) = maxNt Int Kt+1

dt + W (Kt+1)

subject to the transition function

Kt+1 = Int + K

and given the following definitions for real dividends and output for period t

dt = yt minus (wtpt)Nt minus δK minus zBpt minus Int minus ψ(Int)

yt = f (Nt K)

Substituting the above definitions for real dividends and output into the Bellmanequation and substituting in the transition function the first-order conditions are

partftpartNt

minus wt

pt= 0 (31)

minus (1 + ψ primet ) + dW (Kt+1)

dKt+1= 0 where ψ prime

t = dψ(Int)

dInt (32)

Equation (31) is the standard condition that labor is employed up to the pointwhere the real marginal gain for an additional unit of labor in terms of the increasein output attained in the current period (ie the marginal product of labor partftpartNt)equals the real marginal cost as reflected by the real wage wtpt Equation (32)indicating the optimal choice of investment is discussed in the next section

Optimal investment (and the future capital stock) zeroadjustment costs

To express the optimal condition for investment and thus the future capital stockin a more transparent form we need to expand upon the effect of an increase inthe capital stock on the value function for period t + 1 In other words we need toclarify the nature of the term dW (Kt+1)dKt+1 in Equation (32) To do so let usconsider the Bellman equation for period t + 1 To simplify matters we initiallyfocus on the case of zero adjustment costs (ie that ψ equiv 0 implying that ψ prime

i = 0i = t t +1 ) Given the inherited capital stock Kt+1 for period t +1 and a fixedstock of equity shares the Bellman equation for period t + 1 is

W (Kt+1) = maxNt+1 Int+1 Kt+2

dt+1(Rt)minus1 + W (Kt+2)

Kt+2 = Int+1 + Kt+1

and (assuming retained earnings financing of capital changes and zero adjustmentcosts) the following definitions for real dividends and output for period t + 1

dt+1 = yt+1 minus (wt+1pt+1)Nt+1 minus δKt+1 minus zBtpt+1 minus Int+1

yt+1 = f (Nt+1 Kt+1)

Thus we have32

dW (Kt+1)

dKt+1= 1

(partft+1

partKt+1 minus δ

)+ dW (Kt+2)

dKt+2 (33)

We can use first-order conditions for the Bellman equation for period t + 1 toclarify the nature of dW (Kt+2)dKt+2 in Equation (33) In particular we havethat the optimal choice of investment in period t + 1 satisfies

minus 1

Rt+ dW (Kt+2)

dKt+2= 0 (34)

Substituting (34) into (33) we obtain the following expression for the effect of achange in the inherited capital stock on the value function for period t + 1

dW (Kt+1)

dKt+1= 1

(partft+1

partKt+1 minus δ

Rt (35)

Substituting (35) into equation (32) and recalling our assumption that ψ prime = 0we thus have that the optimal choice of investment in period t satisfies

minus1 + 1

(partft+1

partKt+1 minus δ

Rt= 0 (36)

The above expression can be rearranged to obtain

partft+1

partKt+1= mt minus δ (37)

where mt Fisherrsquos expected real rate of interest equals Rt minus1 or (rt minusπt)(1+πt)The interpretation of (37) is fairly straightforward Each period the firm chooseslabor and capital such that the marginal gain in the subsequent period in terms ofincreased output equals the real marginal cost For capital the marginal cost is therate of depreciation plus the expected real rate of interest An explanation of thisreal ldquouserrdquo or ldquorentalrdquo cost of capital follows

Over period t the firm pays for one unit of capital at price pt Since the firmcould have instead used these funds to reduce the outstanding stock of bonds by pt the cost of this capital (reduced dividends) in nominal terms is pt(1 + rt) In realterms the cost one period later is anticipated to be pt(1 + rt)pt+1 After oneperiod 1 minus δ of the capital remains so that the sale of the remaining capital afterone period of use (or the reduced purchases of new capital) reaps a nominal returnof (1 minus δ)pt+1 and real return (1 minus δ) The real rental cost of the unit of capital isthus33

pt(1 + rt)pt+1 minus (1 minus δ) = (1 + rt)(1 + πt+1) minus 1 + δ

= (rt minus πt)(1 + πt) + δ

= mt + δ

Summarizing our discussion in the case of zero adjustment costs the optimalbehavior of firms in periods t and t +1 is given by the following demand functionsfor period t and t + 1

N dt = N d(wtpt K)

I dnt = I d

n (mt + δ wt+1pt+1 K)

where I dnt = Kd

t+1minusK and Kdt+1 = Kd

t (mt+δ wt+1pt+1) Note that the anticipatedreal wage next period affects Id

nt since changes in the real wage affect the employ-ment of labor and thus assuming part2f partNpartK does not equal zero the marginalproduct of capital A similar statement explains why K enters as an argument inthe labor demand function

An important feature of the above is that planned investment demand when thereare zero adjustment costs simply depends on adjacent expected real user costs ofcapital and real wages This reflects the fact that with zero adjustment cost capitaldemand is a function of the expected real user cost of capital and the real wageover the next period alone

Financing choices and different debt-to-equityratios a digression

We have characterized the above choice of the capital stock under the presumptionthat the firm finances capital purchases through retained earnings Yet we can showthat the planned (at time t) choice of the optimal capital stock at time t+1 t+2 is independent of the method of financing given that (a) bonds and equity sharesare assumed to be perfect substitutes and (b) there is no cost to arranging theexchange of financial assets (otherwise the retained earnings financing method ispreferred) This result is sometimes referred to as the ldquoModiglianindashMiller theoremrdquowhich states that the total value of the firm is independent of its financial structureThat is the present value of the stream of dividends to the initial owners is inde-pendent of how liabilities are divided between bonds and equity shares The resultis that the capital structure is indeterminant

The view that the method of financing capital purchases is largely irrelevant isa very useful simplification for macroeconomic analysis However you should beaware of some complicating factors that we are ignoring factors that can causefirms to care about the method by which they finance their capital purchases

When a firm issues bonds the value of its outstanding debt rises When it issuesstocks the value of its outstanding equity shares increases Thus the method offinancing capital purchases affects what is known as the firmrsquos debt-to-equityratio34 Financing capital purchases with bonds will increase the firmrsquos debt-to-equity ratio while financing capital purchases with equity shares (eitherexplicitly or implicitly by using retained earnings) will reduce the firmrsquos debt-to-equity ratio Two factors that can influence a firmrsquos desired ldquocapital structurerdquoare tax considerations and bankruptcy costs35

The corporate taxes that firms pay are calculated as a percentage of earningsFor tax purposes corporate earnings are equal to total revenue net of costs wherecosts are calculated as including not only wages and payments for raw materi-als and intermediate goods but also interest payments to bondholders If a firmfinances its purchases of capital using bonds the interest it pays in the futurewill reduce its taxable earnings and thus the taxes that it has to pay This meansthat a firm can lower its future tax liability by raising its debt-to-equity ratio ndashthat is by financing new capital purchases with new bonds rather than equityshares

Raising the debt-to-equity ratio however is generally not without costs whichare typically referred to as ldquobankruptcy costsrdquo Unlike equity shares which promiseshareholders dividend payments if profits are sufficiently high bonds promise fixedpayments to their holders Greater debt thus increases the fixed obligations thatfirms must meet in the future This means that a fall in future revenues is morelikely to force the firm into bankruptcy

Bankruptcy occurs when a firmrsquos revenues do not cover its costs and it isforced to default on its obligations to bondholders Associated with bankruptcyare bankruptcy costs the most obvious being the hefty legal costs associated witheither reorganizing or undergoing a court-supervised liquidation The existence ofbankruptcy costs serves to limit the amount of borrowing a firm will undertake Itwill hesitate to increase its debt-to-equity ratio beyond some level since the gainin tax savings will be offset by the costs associated with an increased likelihoodof incurring bankruptcy costs

To summarize a firm can be viewed as having an optimal debt-to-equity ratiothat reflects a tradeoff of tax and bankruptcy cost considerations Table 31 liststhe general level of debt-to-equity for a sample of industries in US manufacturingNote that the debt-to-equity ratios vary widely among the industries in the sampleranging from significantly over one to significantly less than one The ratio ishighest in the steel industry where the value of debt is close to 17 times the

Table 31 Debt-to-equity ratios across select industries

Book value Market valueof equity of equity

Steel 1973 1665Petroleum refining 1548 1117Textiles 1405 1296Motor vehicles 0922 0594Plastics 0843 0792Machine tools 0472 0425Pharmaceuticals 0194 0079

Source Kester (1986) The book value of equity is computed fromaccounting sources while the market value of equity is obtained bymultiplying the number of outstanding shares by the current marketprice of the outstanding shares

market value of outstanding market shares In contrast for pharmaceuticals thedebt-to-equity ratio is only 0079 indicating that the industry uses bond financingvery little instead financing its investment activities almost exclusively throughthe issuance of equity

Adjustment costs for capital and Tobinrsquos Q

Let us now consider the choice of capital when there exist adjustment costs Tokeep the maximization problem simple we shall continue to assume ldquoretainedearningsrdquo financing of changes in the capital stock we would however obtainidentical results with bond or equity share financing As we have seen in the caseof retained earnings financing the problem facing the firm is

dt + W (Kt+1)

Kt+1 = Int + K

and given the following definitions for real dividends and output for period t36

yt = f (Nt K)

As before substituting the above definitions for real dividends and output intothe Bellman equation and substituting in the transition function the first-orderconditions are

partftpartNt

minus wt

pt= 0 (31)

t = dψ(Int)

dInt (32)

The problem facing the firm in period t+1 assuming retained earnings financingof capital changes is

dt+1Rt + W (Kt+2)

Kt+2 = Int+1 + Kt+1

dt+1 = yt+1 minus (wt+1pt+1)Nt+1 minus δKt+1 minus zBpt+1 minus Int+1 minus ψ(Int+1)

yt+1 = f (Nt+1 Kt+1)

Again we can substitute the above definitions for real dividends and output intothe Bellman equation for period t+1 and substituting in the transition function (inparticular note the fact that dKt+2dInt+1 = 1) obtain the following first-orderconditions

partft+1

partNt+1minus wt+1

pt+1= 0 (38)

minus (1 + ψ primet+1) + dW (Kt+2)

t+1 = dψ(Int+1)

dInt+1 (39)

Finally from our expression for the value function at time t + 1 W (Kt+1) wehave that

dW (Kt+1)

dKt+1= 1

(partft+1

partKt+1 minus δ

)+ dW (Kt+2)

dKt+2 (310)

Substituting (39) into (310) and then substituting the resulting expression fordW (Kt+1)dKt+1 into the first-order condition for Int (equation (32)) we obtain

minus(1 + ψ primet ) + 1

(partft+1

partKt+1 minus δd

)+ 1 + ψ prime

Rt= 0 (311)

Rearranging and simplifying we have that

partft+1

partKt+1= mt + δ + (mt + 1)ψ prime

t minus ψ primet+1

where ψ primet = ψ prime(I d

nt) and ψ primet+1 = ψ prime(I d

nt+1) An important feature of adjustmentcosts that is highlighted by the above equation is that the choice of investment inperiod t is now linked to the optimal choice of investment next period Since thisholds for each period in the future the choice of investment today is linked toinvestment decisions over all subsequent periods

We can simplify and rearrange the above first-order condition for investment inperiod t to obtain what is known as ldquoTobinrsquos Qrdquo To do so let us first assume anidentical real rate of return over time in particular we then have Ri = Rt = 1+mi = t + 1 t + 2 where (1 + m) equiv (1 + r)(1 + π) Let us also assume thefirm has attained its optimal capital stock so that Kd

t+1 = K and I dnt = 0 If the

production function is separable into capital and labor then the assumption of aninvariant real interest rate implies that I d

ni i = t +1 t +2 equal zero as well38

In this case ψ primet and ψ prime

t+1 can be replaced by a common ψ prime(0) gt 0 Then we maywrite the first-order condition as

partft+1partKt+1 minus m minus δ = ψ primem

Dividing by m and adding one to both sides we have

Tobinrsquos ldquomarginalrdquo Q equiv 1 + [partft+1partKt+1 minus m minus δ]m = 1 + ψ prime gt 1(312)

The above provides the definition for Tobinrsquos Q39 More precisely we have Tobinrsquosldquomarginalrdquo Q for it represents the ratio of the market value of an additional unitof capital to its replacement cost

Given adjustment costs the market value of an additional unit of capital exceedsits replacement cost so Tobinrsquos marginal Q is greater than one Since investmentdemand planned over the coming period determines marginal adjustment costsψ prime(I d

nt) we see that investment can be written in terms of Tobinrsquos Q40 The optimalrate of investment is that rate for which Q minus 1 is equal to the marginal cost ofinstallation Thus net investment demand is sometimes expressed as

Idnt = I d

nt(Q minus 1)dId

d(Q minus 1)gt 0

The Q theory of investment is not operational as long as Q is not observableWhile marginal Q is not typically apparent with some additional assumptionswe can show that the expression known as Tobinrsquos ldquoaveragerdquo Q is identical toldquomarginalrdquo Q Tobinrsquos ldquoaveragerdquo Q is defined as the ratio of the total value of thefirmrsquos existing capital (the market value of its equity shares) to its total replacementcost and these variables are more easily measured41 In particular for a one-sectormodel in which the cost of capital and output are identical42

Tobinrsquos ldquoaveragerdquo Q equiv V

Hayashi (1982) has shown that if the firm is a price-taker if the production func-tion is linear homogeneous in K and N and if expectations of future real interestrates and real wages are static then the marginal and average Q are identical43 Tosee why this is the case note that if the real interest rate and real wage are invariantand the firm is at the optimal level of capital so that the capital stock is invariantover time then omitting time identifiers we have

V =infinsum

where (since ψ(Idnt) = ψ(0) = 0)

d = y minus (wp)N minus δK

V = [y minus (wp)N minus δK]m

where m equiv R minus 1 By Eulerrsquos theorem for a linear homogeneous production func-tion (ie K(partf partK) = y minus N (partf partN )) and the marginal productivity condition

for labor (ie wp = partf partN ) which reflects the price-taker assumption the firsttwo terms in the expression become K(partf partK)44 Dividing by K we thus obtain

Tobinrsquos ldquoaveragerdquo Q equiv V K = [partf partK minus m minus δ]m + 1

which as we can see is the same expression as that for Tobinrsquos marginal QAlternatively we can write the above as

V = [partf partK minus m minus δ]m + 1 = (Q minus 1) ([partf partK minus m minus δ]m)

Adjustment costs for labor labor as a ldquoquasi-fixed factorrdquo

Our prior characterization of the optimal choice of labor reflects an underlyingproduction process that incorporates a very simple view of labor For instancelabor markets are restricted to be only spot markets That is we rule out multiperiodlabor contracts Yet there is an extensive body of literature that investigates variousrationales for and the implications of such multiperiod (implicit) labor contractsOne reason why long-term contracts might emerge is an attempt by firms who areless risk-averse than workers to smooth out income over time45

A second reason why multiperiod labor contracts might emerge is if there areldquoadjustment costsrdquo to changes in the size of the labor force Adjustment costs couldreflect the fact that in order to hire new workers firms must incur hiring and trainingcosts Adjustment costs mean that a firm would view potential new hires andpreviously employed workers as imperfect substitutes and this would provide animpetus for multiperiod labor contracts The absence of adjustment costs simplifiesthe analysis by eliminating a rationale for multiperiod labor contracts and a choiceof labor given adjustment costs It also simplifies the analysis in two other ways

First the absence of adjustment costs suggests that we can measure labor ser-vices as the product of the fraction of the period each labor supplier works andthe number of individuals hired That is given no adjustment costs differences inldquohours workedrdquo and ldquonumber employedrdquo that leave total work hours unchanged areviewed by the firm as equivalent in terms of production In contrast with positiveadjustment costs firms would have a preference for meeting temporary changesin output by changing hours rather than by changing the number employed

A second implication of the absence of adjustment costs is that the employerviews as equivalent two workers working at ldquohalf-speedrdquo (and receiving ldquohalf-wagesrdquo) and one worker working at ldquofull-speedrdquo for ldquofull-wagesrdquo In contrastgiven adjustment costs firms would have a preference for meeting temporaryoutput changes by altering not only hours per worker but also the intensity that eachemployee was asked to work In fact output per work hour or ldquolabor productivityrdquodoes typically increase more rapidly during a recovery suggesting a more intensiveuse of labor

In contrast labor productivity growth is typically less rapid when the growth intotal output slackens This phenomenon is due in part to employersrsquo hoarding laborin slack times so as not to lose trained employees whom they will want when there

is an upturn in demand (That is to reduce subsequent adjustment costs given afuture upturn in production) Hoarding labor means that employers keep on moreworkers than necessary to produce the current output so that each worker has lesswork to do than normal The labor hoarding phenomenon also referred to as thelabor reserve hypothesis is the formal term for changes in the ldquointensityrdquo at whichlabor is used and explains lower output per work hour during periods of slackdemand46

Conclusion

The nature of the firm was discussed with the emphasis on profit maximizationDecisions of firm owners facing a variety of constraints and costs were analyzedwith particular attention paid to how financing constraints and adjustment costsaffected firm profits and the ability to adjust production levels A link was madebetween households and firms that will lead us into the next chapter Firms hireworkers who of course constitute households in the economy Workers are paidwages as costs to the firm but are an integral part of the production processMoreover firms may face costs of adjustment and other costly phenomena asso-ciated with decisions to alter the use of workers in the production process Takentogether then we see a link between firms and households as firm decisions havethe propensity to affect consumer income

4 Households as marketparticipants

Introduction

This chapter brings the household into our model of the macroeconomySpecifically the householdrsquos ability to obtain utility through consumption andthe labor supply decision is modeled within a choice framework The solutionis then used to formulate predictions about labor supply The concept of time iscritical to a thorough understanding of household behavior in the marketplace anda good deal of this chapter is spent analyzing intertemporal choices

The life-cycle and permanent income hypotheses are introduced and a theory ofportfolio choice is developed Finally the chapter ties many of these issues togetherand addresses the macroeconomic questions of absence of money illusion the realbalance effect and the real indebtedness effect

Individual experiments households

Two agents inhabit our expanded macroeconomic model with production firmsand households Having just discussed the nature of decisions confronting firmswe turn now to those confronting households These decisions can be broken downinto three types consumptionsaving portfolio and labor supply Consider nowthe first two decisions which should be familiar

The ldquoconsumptionsavingrdquo decision The representative household must deter-mine at time t the consumption purchases over each period at the implied ratect ct+1 We term this problem the ldquoFisherianrdquo problem

The ldquoportfoliordquo decision The individual must determine at time t the collectionof assets to hold at the end of each period In our expanded economy there areostensibly three types of assets

1 nominal money balances planned at time t to be held at the end of periodi Mi i = t t + 1

2 the nominal value of bonds planned at time t to be held at the end ofperiod i pbiBi i = t t + 1 where Bi denotes the planned number ofbonds held and pbi denotes the money price of bonds at the end of periodi and

40 Households as market participants

3 the nominal value of equity shares planned at time t to be held at the endof period i peiSi i = t t + 1 where pei denotes the money price ofan equity share at the end of period i and Si denotes the number of equityshares planned at time t to be held at the end of period i

Since bonds and equity shares are perfect substitutes we can consider them asa single entity with respect to householdsrsquo portfolio decisions We will let Aidenote the real holdings of financial assets planned at time t to be held at theend of period i i = t t + 1 That is for period t

At = [petSt + pbtBt]pt

and so on Excluding current dividend and interest payments the real value ofinherited financial assets at the end of period i reflecting portfolio decisions inthe prior period will be denoted by Ai i = t t + 1 For instance

At = [petS + pbtB]pt

At+1 = [pet+1St + pbt+1Bt]pt+1

The household problem

We start our analysis of the representative household as usual by discussing thehouseholdrsquos preferences constraints and objectives In particular we make thefollowing assumptions

Assumption 41 The representative householdrsquos preferences are described bythe utility function

u(ct ct+1 Mtpt Mt+1pt+1 1minusNt 1minusNt+1 )

where ci denotes the householdrsquos planned (at time t) rate of consumption dur-ing period i (from time i to time i + 1) Mi denotes the representative householdrsquosplanned (at time t) nominal holdings of money at the end of period i and Ni denotesthe householdrsquos planned (at time t) rate of supply of labor services during period i(from time i to time i +1) such that 1minusNi denotes the planned rate of leisure dur-ing period i It is assumed that partupartci gt 0 partupart(Mipi) gt 0 and partupart(1 minus Ni)

gt 0 i = t t + 1 Macroeconomics often assumes a time-separable utilityfunction a form of the utility function that ensures ldquotime consistencyrdquo1 In par-ticular following the tradition of macroeconomics we will assume that the totalutility for the representative household at time t with an infinite planning horizonis given by

infinsumi=t

β iminustu(ci Mipi 1 minus Ni)

Households as market participants 41

where β denotes the fixed personal or ldquoutilityrdquo discount factor with 0 lt β lt 12

In the above note that the one-period utility function for period t (time t to t + 1)is u(ct Mtpt 1minusNt) for period t +1 (time t +1 to t +2) it is u(ct+1 Mt+1pt+11 minus Nt+1) and so on Further note the infinite time horizon

Assumption 42 Individuals will choose the most preferred sequence of con-sumption money holdings and labor supply from the set of feasible alternatives(rationality) The feasible set of consumption money holdings and leisure

(ct ct+1 Mtpt Mt+1pt+1 1 minus Nt 1 minus Nt+1 )

is defined by the set of equalities

(wtpt)Nt + zBpt + At + Mpt minus [ct + Mtpt + At] = 0

(wt+1pt+1)Nt+1 + zBtpt+1 + At+1 + Mtpt+1

minus [ct+1 + Mt+1pt+1 + At+1] = 0

Note that we assume the budget constraints are met with equality Further note thatthe sum of dividends wage payments and interest payments equals total outputminus depreciation For instance for period t

(wtpt)Nt + dt + z middot Bpt = yt minus δK

and so on This is simply the firm distribution constraint for period t

Several aspects of the above problem deserve further elaboration First a wordon notation for future variables It is common in macroeconomics to derive themicroeconomic theoretical restrictions for the aggregate model under the conditionof certainty even though the analysis is then applied to situations that involvepotential stochastic elements One obvious way to eliminate considerations ofuncertainty from the analysis is to assume perfect foresight A second way is toassume that individual expectations of future events are point estimates held withsubjective certainty Note that either approach simplifies the analysis and in manycases this simplification gives us results that are not overturned if risk were to besystematically incorporated into the analysis

In the analysis of individual behavior below we will often for notationalsimplicity not distinguish between future prices and the expectations of futureprices Assuming expectations are held with subjective certainty this lack of dis-tinction will not be serious in discussing the result of the optimization problemsThat is the findings for perfect foresight can be made identical to those withoutthe assumption of perfect foresight by switching actual future prices for expectedprices Sometimes for clarity we will explicitly denote expected future prices(point estimates held with subjective certainty) by the superscript ldquoerdquo

A second aspect of the above analysis that may initially appear odd concerningthe above one-period utility function for period i (from time i to time i+1) is that itseems that we are mixing money balances that occur at one time with consumptionand leisure that occur at an earlier time The reason for this is that money balancesare a stock variable and we are recording their value at the end of each period ithat is at time i + 13 The following scenario for the discrete-time analysis mayhelp clarify what is going on

At time t the labor market takes place and agreements are made to exchangelabor services at rate Nt over the period (t t + 1) for the money wage wt Duringthe period an output market operates in which firms sell output produced at rate yt During the period households receive money wages wt At the end of the period(time t + 1) households anticipate real interest payments zBpt from their priorpurchases of bonds (B) and real dividends dt from their prior purchase of equityshares stock (S)4 Given the above income sources as well as inherited nominalholdings of money (M ) and the anticipated value of inherited financial asset At atthe end of the period households plan an average rate of consumption ct duringthe period

At the end of period t after all income is received and final planned pur-chases of consumption goods are made the remainder reflects householdsrsquo planned(at time t) end-of-the-period change in real money balances ((Mt minus M )pt) andplanned changes in real financial assets holdings (At minus At) For financial assetsthe real price for bonds at the end of period t is pbtpt and the real price for equityshares is petpt

According to the above scenario the sale of labor services at rate Nt and rateof consumption ct over the period from time t to t + 1 tend to coincide whilereal money balances Mtpt and real financial asset holdings At can be viewed asthe planned (at time t) real stocks of such assets to be held at the end of period tIn continuous-time analysis as the length of the period h goes to zero the rate atwhich leisure is lost from supplying labor services during the period (Nt) the rateof consumption (ct) and the stocks of real money and real financial asset holdingswould coincide

A third aspect of the above analysis is that we have interpreted 1 minus Ni as theportion of the period of length 1 that the individual spends at leisure given thesupply of labor at rate Ni This is a simplification however for at the same timewe have suggested that the ldquoutility yieldrdquo of money is derived from its ability toreduce the transaction costs in arranging exchanges with such transaction costsreflecting at least in part a loss of leisure To explicitly incorporate such a viewof money leisure during a period i of length h given the sale of labor services Niand the real money balances Mipi held at the end of the period would be givenby h(1 minus Ni minus (Mipi)) where the function (Mipi) reflects transactions costsin terms of the loss of leisure The fact that prime lt 0 indicates that increased moneyholdings raise utility by reducing leisure lost in arranging transactions In this casethe one-period utility function would formally be given by u(ci 1minusNiminus(Mipi))with partupartci gt 0 and partupart(1 minus Ni minus (Mipi)) gt 0

The general solution to the household problem

We can express the household problem in terms of a set of Bellman equationsAssuming perfect foresight (or equivalently interpreting future prices dividendetc as expectations of such variables held with subjective certainty) we thus havefor period t (time t to t + 1)

W (xt) = maxct Nt

Mtpt xt+1

[u(ct Mtpt 1 minus Nt) + W (xt+1)]

xt+1 = Rt[xt + (wtpt)Nt minus ct] minus [Rt minus Rmt] Mtpt

and given xt Rmt is the real gross rate of interest on money that is the gross realrate of return on money and equals one divided by one plus the rate of inflation(1(1+πt)) The term xt is the total value in period t derived from the ldquoinheritedrdquoholdings of money bonds and stocks This total value is the sum of currentdividends and interest (received at the end of period t) on stock and bond holdingsacquired previously the real value of these financial assets at the end of period texclusive of these current interest and dividend payments and the real value ofpreviously acquired money holdings

xt equiv dt + zBpt + At + Mpt

At equiv [petS + pbtB]pt

The difference between the total real value derived in period t from inheritedbonds stocks and money balances plus real wage income xt + (wtpt)Nt andconsumption in period t ct reflects the acquisition of bonds equity shares andmoney holdings at the end of period t by the representative household Letting Atdenote the planned holdings of financial assets at the end of period t we thus havefrom the household budget constraint that

At + Mtpt = xt + (wtpt)Nt minus ct

At equiv [petSt + pbtBt]pt

Recall that Rt the gross real rate of return on financial assets equals one plus thenominal interest rate divided by one plus the rate of inflation ((1 + rt)(1 + πt))Thus in period t + 1 and given our definition of Rmt the inherited real value of

bonds stocks and money balances including dividends and interest payments isgiven by

xt+1 = RtAt + RmtMtpt

Substituting in the expression for At derived from the household budget constraint(ie At = xt + (wtpt)Nt minus ct minus Mtpt) and rearranging we obtain the transitionfunction

xt+1 = Rt[xt + (wtpt)Nt minus ct] minus [Rt minus Rmt]Mtpt

Substituting the transition function into Bellmanrsquos equation for period t wehave the following first-order conditions for ct Nt and Mtpt assuming interiorsolutions (ie ct gt 0 1 gt Nt gt 0 and Mtpt gt 0)

partutpartct minus (partW (xt+1)partxt+1)Rt = 0 (41)

partutpart(1 minus Nt) + (partW (xt+1)partxt+1)Rt(wtpt) = 0 (42)

partutpart(Mtpt) minus (partW (xt+1)partxt+1)(Rt minus Rmt) = 0 (43)

The above conditions indicate that for period t (time t to time t + 1) we havefrom equations (41) and (42) that

partutpart(1 minus Nt)

partutpartct= wt

pt (44)

In words the optimal choice of leisure is such that the marginal value of leisurein terms of consumption that is the marginal rate of substitution between leisureand consumption as given by

partutpartct

equals the marginal cost of leisure in terms of consumption forgone in the currentperiod as given by the real wage wtpt

From equations (41) and (43) we have for period t that

partutpart(Mtpt)

partutpartct= (Rt)

minus1(Rt minus Rmt)

In words the optimal choice of real money balances is such that the marginal rateof substitution between real money balances and consumption as given by

partutpart(Mtpt)

partutpartct

equals the marginal cost in terms of the present value of the loss in interest incomein the subsequent period due to the holding of money balances instead of financialassets as given by the expression (Rt)

minus1(Rt minus Rmt) Recall that Rt minus Rmt equals(1 + rt)(1 + πt) minus (1(1 + πt)) which is simply rt(1 + πt) or essentially theanticipated nominal rate of interest Given that Rt = (1 + rt)(1 + πt) we thushave that (Rt)

minus1(Rt minus Rmt) = rt(1 + rt)We can expand upon the above discussion of the first-order conditions for

period t by first noting that W (xt+1) is defined by

W (xt+1) = maxct+1Nt+1

Mt+1pt+1xt+2

[βu(ct+1 Mt+1pt+1 1 minus Nt+1) + W (xt+2)]

xt+2 = Rt+1[xt+1 + (wt+1pt+1)Nt+1 minus ct+1] minus [Rt+1 minus Rmt+1]Mt+1pt+1

xt+1 equiv dt+1 + zBtpt+1 + At+1 + Mtpt+1

At+1 equiv [pet+1St + pbt+1Bt]pt+1

Again substituting the transition function into the Bellman equation for periodt + 1 we have the following first-order conditions for period t + 1

βpartut+1partct+1 minus (partW (xt+2)partxt+2)Rt+1 = 0 (41prime)minus βpartut+1part(1 minus Nt+1) + (partW (xt+2)partxt+2)Rt+1wt+1pt+1 = 0 (42prime)βpartut+1part(Mt+1pt+1) minus (partW (xt+2)partxt+2)(Rt+1 minus Rmt+1) = 0 (43prime)

Now consider the impact of the change in xt+1 on the value function W (xt+1)The above first-order conditions imply that the indirect effects of the change inxt+1 on W (xt+1) through the effect of such a change on the choice of the optimalvalues of consumption labor supply and real money balances for period t + 1 arezero5 This is simply an application of the envelope theorem which states thatthe change in the objective function adjusting the choice variables optimally isequal to the change in the objective function when one does not adjust the choicevariables This fact along with the transition function for xt+2 gives us

partW (xt+1)partxt+1 = (partW (ct+2)partxt+2)Rt+1 (45)

Substituting equation (41prime) into (45) we obtain

partW (xt+1)partxt+1 = βpartut+1partct+1 (46)

Alternatively by substituting in (42prime) we can obtain6

partW (xt+1)partxt+1 = β[partut+1part(1 minus Nt+1)]pt+1wt+1 (47)

Combining equations (41) and (46) gives us the standard Fisherian solutionfor the optimal allocation of consumption between period t (time t to t + 1) andperiod t + 1 (time t + 1 to t + 2)

partutpartct

βpartut+1partct+1= Rt

Combining equations (43) and (46) we obtain the standard expression for theoptimal portfolio choice of money

partutpart(Mtpt)

βpartut+1partct+1= Rt minus Rmt

Combining equations (42) and (47) gives us an expression for the optimalallocation of labor supply over time

βpartut+1part(1 minus Nt+1)= pt+1

wt+1Rt

pt (48)

Having discussed the ldquoFisherian problemrdquo and the ldquoportfolio problemrdquo con-fronting the household we turn our attention in the next section to the ldquolaborsupply problemrdquo as captured by equations (44) and (48) Before doing so how-ever a general comment should be made with respect to the discussions to followas well as the preceding discussions of the Fisherian problem and the portfoliodecision

In focusing on first-order conditions with respect to the particular variablesat issue (ie first-order conditions for consumption now and next period forthe Fisherian problem first-order conditions for real money holdings and futureconsumption for the portfolio problem and first-order conditions for labor sup-ply now and next period for the labor supply decision) one has a tendency toforget that the optimizing problem involves the simultaneous choice of consump-tion portfolio and leisure In general this means that the analysis is often notas straightforward as it may first appear For instance a change in the currentreal wage or an expected real interest rate can affect the first-order conditionconcerning the choice of labor supply through its impact on the choice of consump-tion if part2upartcpartN = 0 for then the change in consumption alters the ldquomarginalutilityrdquo of leisure One simple way to abstract from these ldquoindirectrdquo effects is toassume that the utility function is separable not only across time but also withrespect to consumption leisure and real money balances each period such thatpart2upartcpartN = part2upartcpart(Mp) = part2upartNpart(Mp) = 0

The choice of hours within a period

Equation (44) indicates that at the optimal labor supply the household cannot bemade better off by trading consumption for leisure within periods at the expectedreal wage for the period Equation (48) indicates that at the optimum the householdcannot be made better off by trading leisure across periods given the relevant realwages and the real interest rate7 Figure 41 captures the first situation that is theoptimal choice of consumption and leisure within a period

For the moment let us hold anticipated real interest rates constant Further letus assume unit elastic expectations with respect to wages as well as prices so thata change in the current wage or price level that alters the current real wage changesfuture expected real wages as well so that there is no change in the current realwage relative to future expected real wages In addition let us hold constant for themoment the effect on current real money balances of a change in the current realwage These assumptions help us to mimic the traditional ldquostaticrdquo or single-periodanalysis (eg Patinkin) of the effect of a change in the current periodrsquos anticipatedreal wage on individualsrsquo labor supply during period t Under such circumstancesan increase in the real wage for period t can have ambiguous effects As Patinkin(1965) states ldquofor simplicity it is assumed that [labor] supply is an increasingfunction of the real wage though there are well known reservations on this scorerdquo

To understand what Patinkin is referring to consider an increase in the real wagedue to a rise in the money wage wt Consider one possible result on the householdrsquoslabor supply decision The increase in the net real wage means a steeper budgetline as the householdrsquos optimal leisurendashconsumption combination changes from1 minus N s

t and cdt (call this choice A) to (1 minus N s

t )prime and (cdt )prime (choice C)

An increase in the real wage has two conceptually distinct effects on the house-holdrsquos labor supply decision an income effect and a substitution effect The incomeeffect refers to the fact that an increase in the real wage makes the household betteroff because it leads to an increase in the householdrsquos feasible consumption set in

Leisure

Figure 41 Consumption and leisure

the current period The higher real wage means that the household if it so desirescan increase both its leisure and consumption Thus the household is able to reacha higher indifference curve which has associated with it a higher utility levelThe substitution effect refers to the fact that the increase in the real wage makesan hour of leisure relatively more expensive in terms of consumption that must beforgone

To isolate the substitution and income effects of the change in the net real wagesuppose that after the real wage increases we temporarily take away just enoughof the householdrsquos nonlabor income so that it is just able to attain its originalindifference curve The householdrsquos choice of leisure and income would then be(1 minus N s

t )primeprime and (cdt )primeprime (call this choice B) Thus if we hold the householdrsquos utility

level constant the increase in the real wage leads unambiguously to a lower levelof leisure since an hour of leisure is relatively more expensive the householdsubstitutes away from leisure choosing to work more hours The movement fromchoice A to choice B constitutes the pure ldquosubstitution effectrdquo of the higher netreal wage8

Now suppose that we give the household back the nonlabor income that wetemporarily took away This causes an outward shift in the budget line and thehouseholdrsquos new choice of leisure and consumption would be (1 minus N s

t )prime and (cdt )prime

(choice C) The movement from B to C constitutes the ldquoincome effectrdquo of thechange in the net real wage In the present case the income effect on the choiceof leisure is positive reflecting the assumption that leisure is a normal good

Note that the substitution and income effects on leisure work in opposite direc-tions The substitution effect of the higher real wage causes leisure to fall andhours worked to rise while the income effect causes leisure to rise and hoursworked to fall The net effect on leisure and hours worked depends on whicheffect dominates9 If the substitution effect dominates then a higher real wageresults in a decrease in desired leisure and an increase in desired working hoursIf the income effect dominates the opposite is true and the individualrsquos laborsupply curve is ldquobackward bendingrdquo when plotted against the real wage

The available evidence suggests that for many workers the income effect tendsto dominate slightly Estimates are that for men an increase of 10 percent in the realwage results in approximately a 15 percent reduction in the hours worked Thisreduction in hours worked reflects an income effect of approximately minus25 percentand a substitution effect of about 1 percent Other evidence suggests a similarpattern for working women10

The choice of participation within a period

Thus far we have considered a household representative of those who are in thelabor force working a positive number of hours Yet this masks the unambiguouseffect of a higher real wage on the labor supply of those households not in the laborforce To show this we consider a corner solution with respect to labor supply inparticular a household denoted a that has chosen not to participate in the labormarket For such a household let us return to the Bellman equation for period t and

introduce explicitly the nonnegativity constraint for labor supply (ie Nt ge 0)Letting micron denote the multiplier associated with this constraint we have as first-order conditions for household arsquos consumption and labor supply11

partuatpartcat minus (partW (xat+1)partxat+1)Rt = 0

minus partuatpart(1 minus Nat) + micron + (partW (xat+1)partxat+1)Rtwtpt = 0

Nat ge 0 micronNat = 0

Substituting the first equation into the second and rearranging gives

minuspartuatpart(1 minus Nat)

partuatpartcat= wt

pt+ micron

partuatpartcat

For individuals not participating in the labor force micron ge 0 The ldquocorner solutionrdquo isa case in which the marginal rate of substitution of leisure in terms of consumptionis greater than the real wage In other words the absolute value of the indifferencecurve is equal to or greater than the absolute value of the budget line at the pointwhere N s

at = 0Note that at the optimal choice the corresponding indifference curve is more

steeply sloped than the budget line Thus even when all hours are devoted toleisure and none to work the individualrsquos marginal rate of substitution of leisurein terms of consumption still exceeds the real wage rate In other words theindividualrsquos valuation of leisure exceeds the marketrsquos valuation of leisure As aresult individual a does not find it worthwhile to participate in the labor market

The greater the real wage the greater is the probability that a given individualwill choose to participate in the labor market A higher real wage rotates thebudget line outward Since the individual is not working an increase in the netreal wage does not make him better off and thus has no income effect There isonly a substitution effect Thus if the real wage rises sufficiently the individualcan be induced to enter the labor market

According to the above analysis the economy-wide labor supply response toan increase in the current real wage is a combination of an ambiguous effecton the labor supply of those currently working but an unambiguous increase inthe labor supply among those not working12 It is the net of these two effects theldquohoursrdquo decision and the ldquoparticipationrdquo decision that is captured by the aggregatelabor supply function it is commonly assumed that this net effect is such that theaggregate quantity of labor supplied is an increasing function of the current realwage

The labor supply intertemporal substitution hypothesis

Our discussion has yet to consider ldquothe labor market intertemporal substitutionhypothesis (ISH) which states that labor supply responds positively to transitoryincreases in real wages and increases in the real interest rate a central hypoth-esis of modern competitive models of the business cyclerdquo (Alogoskoufis 1987)

To do so we simply expand our focus to the inherent intertemporal decisionconfronting the household In particular recall our expression (48) for the optimalallocation of labor supply over time The view of labor supply embedded in (48)has life-cycle as well as business-cycle implications With respect to life-cycleimplications the theory predicts that workers will concentrate their labor supplyin years of peak earnings consuming leisure in larger than average amounts duringchildhood and old age

With respect to the business cycle the above helps explain an apparent contra-diction in the static theory of labor supply ndash the observed wage inelasticity of laborsupply in the long run with short-run fluctuations in employment which requirean elastic labor supply if one takes a ldquomarket-clearingrdquo approach with respect tothe labor market13 It does so by introducing a distinction between a permanentchange in the real wage and a temporary or transitory change in the real wage

To show the intertemporal substitution effect with respect to labor supply spec-ify wlowastplowast as the permanent or ldquonormalrdquo real wage with the anticipated real wagenext period equal to this value such that

wipi = wlowastplowast i = t + 1 t + 2

Further we assume that

Ri = Rlowast i = t t + 1 t + 2

In this case equation (48) becomes

βpartut+1part(1 minus N lowast)= Rlowast wtpt

wlowastplowast (48prime)

where N lowast denotes the ldquolong-runrdquo supply of labor at ldquonormalrdquo wages Further letus assume that for the representative household βRlowast = 1 Then equation (48prime)becomes

partut+1part(1 minus N lowast)= wtpt

wlowastplowast (48primeprime)

Equation (48primeprime) indicates that if the current real wage is higher than the normal realwage then ldquomore labor is supplied than would be implied by the long-run laborsupply functionrdquo That is this theory views suppliers of labor as reacting primarilyto three variables an anticipated ldquonormalrdquo or ldquopermanentrdquo real wage rate whichcorresponds to the wage rate in the usual one-period analysis of the laborndashleisurechoice and has a negligible effect on labor supply the deviation of the current realwage from this normal rate which has a strong positive effect on labor supplyand the expected real rate of interest (Lucas and Rapping 1970 284ndash285)14

The above theory provides the underlying microtheoretical basis for the fol-lowing statement ldquomeasured unemployment (more exactly its nonfrictionalcomponent) is then viewed as consisting of persons who regard the wage ratesat which they can currently be employed as temporarily low and who therefore

choose to wait or search for improved conditions rather than invest in moving oroccupational changerdquo (Lucas and Rapping 1970 285)

Empirical tests seem to provide some support for this intertemporal substitutionhypothesis15 Alogoskoufis (1987 950) finds that for measures of the total numberof employees the real wage and interest rate elasticities are high and relativelywell determined ldquoThe elasticity of labor supply to transitory changes in real wagesis around unity and is statistically significant at conventional significance levelswith one exception The real interest rate always has a significant independentinfluencerdquo Note that as suggested by equation (48) Alogoskoufis finds that arise in the real interest rate increases current labor supply

Note that equation (48) does not explicitly identify the initial asset holdings asa variable that affects the relative choice of labor supply across periods Similarlythe condition for the optimal choice of consumption across periods did not have theinitial value of assets affecting the relative consumption purchases across periodsThis is a property of time-separable preferences

Special topics in intertemporal choices

As we have seen the intertemporal problem confronting the individual involvessimultaneous decisions with respect to consumption versus saving and with respectto the composition of the asset portfolio To make some sense of what is involvedwe start by considering what is known as the ldquoFisherianrdquo problem which focuseson the individualrsquos choice of consumption across time

Fisherian analysis

The Fisherian problem typically deals with the allocation of consumption acrosstime when there is a single means by which income can be allocated across time16

To restrict our analysis to such a case we can simply omit real money holdingsfrom the utility function Further we consider only interior solutions with respectto consumption (ie cai gt 0 i = t t + T )17

Thus the maximization problem becomes18

maxcat cat+T

xat+1xat+T+1

t+Tsumi=t

β iminustua(cai)

subject to

minus xat+1 + Rt[xat + cat minus cat] = 0

minus xat+2 + Rt+1[xat+1 + cat+1 minus cat+1] = 0

minus xat+T+1 + Rt+T [xat+T + cat+T minus cat+T ] = 0

xat+T+1 ge 0

Recall that Ri = (1 + ri)(1 + πi) i = t t + T denotes the ldquogross real rateof returnrdquo on agent arsquos portfolio between the end of period i and the end of periodi + 1 and asset holdings are solely in the form of bonds such that

xat equiv (1 + r)pbBapt

xat+1 equiv (1 + rt)pbtBatpt+1 = [(1 + rt)(1 + πt)] pbtBatpt

xat+i equiv (1 + rt+iminus1)pbt+iminus1Bat+iminus1pt+i

= [(1 + rt+iminus1)(1 + πt+iminus1)] pbt+iminus1Bat+iminus1pt+iminus1

i = 2 T + 1

Let λi i = t t +T denote the Lagrange multipliers for the constraints linkingthe total value of real asset holdings at the end of period i with the total value ofreal asset holdings at the end of period i + 1 Let microT denote the multiplier for thenonnegativity condition that xat+T+1 ge 0 Then the first-order conditions for theimplied Lagrangian L include

partLpartcai = β iminustduai dcai minus λiRi = 0 i = t t + T

partLpartxai+1 = minusλi + λi+1Ri+1 = 0 i = t t + T minus 1

partLpartxat+T+1 = minusλt+T + microT = 0

partLpartλi = minusxai+1 + Ri(xai + cai minus cai) = 0 i = t t + T 19

partLpartmicroT = xat+T+1 ge 0

microT ge 0

where uai = ua(cai) i = t t + T 20 Note that the above set of first-order

conditions consist of 3(T + 1) + 1 equations to determine 3(T + 1) + 1 variablesThe variables to be determined are cai i = t t + T xai i = t t + T + 1and microT Assuming continuity and strict concavity in ua(middot) and given a convex setof constraints

xai+1 xai cai| minus xai+1 + Ri[xai + cai minus cai] ge 0there is a unique solution to the problem

There are several implications of the above first-order conditions First theconditions imply that the desired total real value of assets (bonds) inherited at timet + T + 1 xd

at+T+1 will equal zero if duat+T dcat+T gt 0 In particular from the

condition

partLpartcat+T = βT (duat+T dcat+T ) minus λt+T Rt+T = 0

we see that if duat+T dcat+T gt 0 then λt+T gt 0 From the condition

we thus have that microT gt 0 This in turn implies from the condition

microT partLpartmicroT = microT xat+T+1 = 0

that xat+T+1 = 0 This finding should not be surprising With a time horizonof t + T agent a perceives no gain (utility) from acquiring assets at time t + Tto finance consumption in period t + T + 1 and a clear loss from doing so attime t + T in terms of consumption forgone given the assumption of nonsatiation(dua

t+T dcat+T gt 0) Second from the first set of equations we know that betweenany two periods i and i + 1

β iminustduai dcai

β iminust+1duai+1dcai+1

= λiRi

λi+1Ri+1 i = t t + T minus 1

This expression can be simplified to obtain

duai dcai

βduai+1dcai+1

= Ri i = t t + T minus 1

where Ri the real gross return between the end of periods i and i + 1 is given by(1+ri)(1+πi) Ri has been called Fisherrsquos ldquo(gross) real interest raterdquo since he wasone of the first to provide a lucid account of its role in determining consumptionacross time

Fisherrsquos ldquo(net) real interest raterdquo denoted by mi is then defined by

1 + mi equiv Ri = (1 + ri)(1 + πi)

Subtracting one from both sides and rearranging we have

mi = (ri minus πi)(1 + πi)

Thus for small expected rates of inflation we have the approximation21

mi = ri minus πi

In words the real interest rate is approximately equal to the nominal interest rateminus the rate of inflation The expected real interest rate is then the nominalinterest rate minus the expected rate of inflation

There are several features of the above that should be noted First if an individ-ualrsquos discount factor (β lt 1) equals the reciprocal of the real gross rate of interest([Ri]minus1 = (1 + πi)(1 + ri)) between periods i and i + 1 so that

βRi = 1

then the above expression of the first-order conditions for cai and cai+1 indicatesthat dua

i dcai = duai+1dcai+1 Given the concavity of the single-period utility

function (ua(middot)) and our assumption of a time-invariant one-period utility function

it then follows that the individual will choose the same rate of consumption acrossperiods i and i + 1 in such a case22 This constant path of consumption betweenthe two periods can be said to emerge if an individualrsquos ldquorate of time preferencerdquoequals the (real) interest rate

If the expected gross return between two periods were higher (or for an individualwith a higher discount factor) the fact that β gt (Ri)

minus1 or equivalently βRi gt 1means from the first-order conditions that dua

i dcai gt duai+1dcai+1 Given the

assumed concavity of the one-period utility function the implication is that agentarsquos consumption during period i would be less than during period i + 1 That isβRi gt 1 rArr cd

ai lt cdai+1 Conversely if the expected real gross return were to be

lower (or for an individual with a lower discount factor) the fact that β lt (Ri)minus1

or equivalently βRi lt 1 means that the individualrsquos consumption during period iwould be greater than during period i + 1 That is βRi lt 1 rArr cd

ai gt cdai+1

For the two-period case (say periods t and t + 1) the optimal consumption ineach of the two periods can be shown graphically by the point of tangency betweenan indifference curve with slope minus(dua

i dcai)β(duai+1dcai+1) and a budget line

with slope minusRt 23 If one were to place cat+1 on the vertical axis and cat on thehorizontal axis then it would be possible to determine whether an individual is aborrower or a lender in any period For example if disposable income were lessthan consumption in the first period t then the individual is a lender at time tNote that points on the same indifference curve are such that

d[ua(cat) + βua(cat+1)

] = (duat dcat+1)dcat + β(dua

t+1dcat+1)dcat+1

Rearranging we have the slope of an indifference curve given by

dcat+1

dcat= minus dua

t dcat

βduat+1dcat+1

Note that points on the budget line satisfy the present value constraint

cat + (Rt)minus1cat+1 = cat + xat + (Rt)

minus1cat+1

Rearranging we have

cat+1 = Rt[cat + xat minus cat] + cat+1

such that the slope of the budget line is given by

dcat+1dcat = minusRt

Thus at the point of tangency between the budget line and an indifference curve

duat dcat

βduat+1dcat+1

which is the expression we obtained previously concerning the optimal choice ofconsumption between periods i and i + 1

Note that the above analysis is for an individual consumer Thus aggregate con-sumption need not behave as that predicted above for the individual For instancean aging population could lead to variations in aggregate consumption that reflectthe aggregation at different times across agents with differing characteristics

Life-cycle and permanent income hypotheses

We have seen how a householdrsquos consumption in any period is not constrained bythe income it receives during that period but rather that the discounted value oflifetime consumption is constrained by the discounted stream of income accruingto the household over its lifetime plus initial asset holdings While income tendsto rise and fall during the lifetime of an individual through appropriate saving andborrowing the individual can maintain a smooth or constant rate of consumptionover his lifetime This smoothing of consumption across time plays a critical rolein Franco Modiglianirsquos ldquolife-cycle hypothesisrdquo of consumption24

A stylized pattern of income and consumption expenditures over an individ-ualrsquos lifetime is the following Prior to retirement income exceeds consumptionand saving is positive During this period saving increases household wealth Onretirement consumption is financed by dissaving During the retirement periodhousehold wealth falls as people draw on their accumulated savings to financeconsumption Implied in this discussion is an inverted U-shape wealthndashage pro-file (save during pre-retirement years and dissave in years following retirementrunning down the stock of accumulated wealth) A number of studies of aggregatehousehold consumption and saving behavior support this wealthndashage pattern25

To make clear the implications of consumption smoothing for the demand forthe consumption good at time t let us make the simplifying assumptions that

(a) for any period i = t t + T Ri = R and(b) the individualrsquos personal discount rate β equals the constant real ldquomarketrdquo

discount rate Rminus126

From the first-order conditions we thus have the result that agent a willcompletely smooth out consumption spending across time so that consumptioncd

ai = cda i = t t + T In this case we can use the prior combined budget

constraint to obtain

cdat =

t+Tsumi=t

(caiRiminust)

where which equals 1sumt+T

i=t (1Riminust) is what Modigliani has calledthe ldquoproportionality factorrdquo and indicates the proportion of householdsrsquo totalresources ndash consisting of initial assets current income and anticipated futureincome ndash devoted to consumption each year

An important implication of Modiglianirsquos life-cycle hypothesis is that thefraction of an increase in current income (cat) that goes toward increased cur-rent consumption (the ldquomarginal propensity to consumerdquo) will vary depending onwhether the increase in current disposable income is accompanied by an equivalentincrease in anticipated future income (cat i = t + 1 t + T )27 If a change incurrent income is viewed as ldquotransitoryrdquo most of the increase in income will goto saving in order to finance increased consumption during future years

This idea that the effect of a change in current disposable income on consump-tion demand depends on the degree to which the change in income is viewed astemporary or permanent lies at the heart of Milton Friedmanrsquos permanent incomehypothesis The permanent income hypothesis is like the life-cycle hypothesis inthat it emphasizes the fact that consumption demand in period t depends not onlyon current income but also on anticipated income in the future periods Permanentincome is that income which if received each year over a householdrsquos time hori-zon would yield an income stream with present value exactly equal to the presentvalue of the householdrsquos anticipated income stream That is permanent income cpis defined by the following equation

t+Tsumi=t

(cpRiminust) =t+Tsumi=t

(caiRiminust) + xat (49)

Factoring out cp on the left-hand side of (49) and rearranging we have

t+Tsumi=t

(caiRiminust)

Equation (410) indicates that permanent income is simply a weighted average ofcurrent and future incomes but in this case income in the more distant future isweighted less heavily since it is discounted more highly

Comparing permanent income to agent arsquos consumption demand in period tcd

at if there is complete smoothing of consumption spending across time then weobtain

cdat = cp

The implication of this equation is that the marginal propensity to consume out ofa change in current income that is perceived as permanent is equal to one whilethe marginal propensity to consume out of a change in current disposable incomethat is perceived as entirely transitory (having little impact on permanent income)is small Changes in transitory components of income are almost entirely saved ifpositive or borrowed if negative28

Our discussion so far of the impact of changes in income on current consump-tion demand has been restricted to what might be referred to as the effects ofldquotransitoryrdquo versus ldquopermanentrdquo income changes In doing so we have assumed adeterministic world in which individuals have perfect foresight concerning future

income streams But what happens if individuals do not have perfect foresight Inparticular what if we introduce stochastic elements so that realized future incomeis a random variable Then the above theories suggest a difference in the responseof consumption demand to income changes that are anticipated or expected versusunanticipated changes In particular the life-cyclepermanent income hypothesiswould predict that previously anticipated (or expected) changes in income wouldhave no effect on consumption demand since consumption plans have alreadyincorporated this income29

Portfolio choice

Now let us consider the more general case in which agent a chooses not onlyconsumption across time but the portfolio of assets (money and bonds) That isconsider the following problem

maxcat cat+T

xat+1 xat+T+1Matpt Mat+T pt+T

t+Tsumi=t

β iminustua(cai Maipi)

subject to

minus xat+1 + Rt[xat + cat minus cat] minus [Rt minus Rmt]Matpt = 0

minus xat+2 + Rt+1[xat+1 + cat+1 minus cat+1]minus [Rt+1 minus Rmt+1]Mat+1pt+1 = 0

minus xat+T+1 + Rt+T [xat+T + cat+T minus cat+T ]minus [Rt+T minus Rmt+T ]Mat+T pt+T = 0

xat+T+1 ge 0

As before to simplify the problem we assume interior solutions in this case notonly with respect to the consumption good but also with respect to money holdingsAgain let λi i = t t + T denote the Lagrange multipliers for the constraintslinking the real asset holdings at the end of period i with their real value at the endof period i + 1 and let microT denote the multiplier for the nonnegativity conditionthat xat+T+1 ge 0 The first-order conditions are

partLpart(Maipi) = β iminustpartuai part(Maipi) minus λi[Ri minus Rmi] = 0 i = t t + T

partLpartλi = minusxai+1 + Ri(xai + cai minus cai)

minus [Ri minus Rmi]Maipi = 0 i = t t + T 30

λi ge 0 i = t t + T

microT ge 0

where uai = ua(cai Maipi) i = t t + T To isolate the portfolio choice

consider the portfolio choice of money and bond holdings for a given level ofcurrent consumption That is let us look at the optimal choice of Maipi given thatcai is held constant From the second set of conditions

β iminustpartuai part(Maipi) minus λi[Ri minus Rmi] = 0

Substituting the condition for the optimal choice of the total value of assets in thatperiod

λi = λi+1Ri+1

we obtain

β iminustpartuai part(Maipi) minus λi+1Ri+1[Ri minus Rmi] = 0

Now substituting the condition for the optimal choice of consumption next period(i + 1) as given by

β iminust+1duai+1dcai+1 minus λi+1Ri+1 = 0

we obtain

β iminustpartuai part(Maipi) minus β iminust+1(dua

i+1dcai+1)[Ri minus Rmi] = 0

Rearranging gives

duai d(Maipi)

βduai+1dcai+1

= Ri minus Rmi

Recalling that the expected gross real return on bonds in period i Ri equals(1 + ri)(1 + πi) and that the expected gross real return on money Rmi equals1(1 + πi) the above expression can be written as

duai d(Maipi)

βduai+1dcai+1

1 minus πi

Note that in the limit (as the length of the period goes to zero) the expected rate ofinflation term would vanish The implication is that the optimal division of assetsbetween money and bonds depends primarily on the money interest rate aloneWhat is essentially being shown is that an increase in money holdings with nochange in current consumption means a reduction in bond holdings and thus theloss of nominal interest income ri or real interest income ri(1 + πi)

Absence of money illusion real balance effects and realindebtedness effects

There are two aspects of agent arsquos demand function that should be noted Firstagent arsquos demand functions at time t can be shown to be homogeneous of degree 0 inthe current price level pt initial money balances M a and initial bond holdings BaIn this economy this is said to reflect the ldquoabsence of money illusionrdquo One criticalreason for this is the assumption of unit elastic expectations with respect to futureprices so that changes in the current price level leave unchanged the expectedrates of change in the price level in subsequent periods Also note that the currentand expected future money payments attached to bonds are being held constantso that given the fixed money payment x on maturity interest rates are unchangedAlternatively one could have money prices and the fixed future money paymentattached to one-period bonds rise by the same proportion

The above implies that individual arsquos demand for the consumption good andreal money balances at time t can be represented by

cdat = cd

at(rt rt+1 rt+Tminus1 πt πt+Tminus1 Wat)

M datpt = M d

atpt(rt rt+1 rt+Tminus1 πt πt+Tminus1 Wat)

where xat is the individualrsquos real wealth at the end of period t as given by

Wat = xat + cat +t+Tminus1sum

⎡⎣ jprod

⎤⎦

minus1

(caj+1)

From the budget constraint for period t we know that the above two demandconditions imply a real demand for bonds of a similar form since

pbtBdatpt = cat + xat minus

at + M datpt

Note that the above demand functions do not depend solely on wealth and thepattern of expected real (gross) rates of interest since given the portfolio choice agiven pattern of expected real (gross) interest rates could alter demand dependingon the underlying values of the money interest rate As before individual arsquos excessdemand function for the consumption good and money in period t are defined byzat = cd

at minus cat zam = M datpt minus M apt and zab = pbtBd

atpt 31

The ldquoreal balance effectrdquo indicates the effect of a change in real balances (M apt)

on individual demand for goods other than money As before there is a real balanceeffect that reflects a wealth effect That is a decrease in initial real balances leadsto a reduction in real money demand M d

atpt and a reduction in the real demandfor bonds pbtBd

atpt There is also what might be referred to as a ldquoreal indebtednesseffectrdquo in that a change in prices alters not only real money balances but also realinitial debt If Ba gt 0 an increase in pt reduces wealth while if Ba lt 0 anincrease in pt increases wealth This is why the characterization of the absence ofmoney illusion has been expanded to include changes in Ba

It is typical in macroeconomics to adopt the convention of the ldquorepresentativeagentrdquo to reduce notational clutter Recall that the ldquorepresentative agentrdquo is essen-tially the average agent For instance if we let cd

at denote the demand for theconsumption good in period t by representative agent a and cd

t market demandat the time then cd

at = cdt Thus depending on the context we can interpret cd

tas demand by the representative agent or market demand Recall that in doingso we essentially ignore distribution effects such as effects on market demandof changes in the distribution of initial endowments of commodities or moneybalances or of changes in the distribution of future endowments In the context ofthe real indebtedness effect since in the aggregate B = 0 this effect is removedfrom our analysis

Intertemporal substitution the evidence

We have focused above on the behavior of an individual with respect to the plannedpath of consumption across time As Robert Hall (1988 340) indicates

The essential idea is that consumers plan to change their consumptionfrom one year to the next by an amount that depends on their expectationsof real interest rates Actual movements of consumption differ from plannedmovements by a completely unpredictable random variable that indexes allinformation available next year that was not incorporated in the planningprocess the year before If expectations of real interest rates shift then thereshould be a corresponding shift in the rate of change of consumption Themagnitude of the response of consumption to a change in real interest rateexpectations measures the intertemporal elasticity of substitution32

Hall (1988 340ndash341) goes on to state that

the basic model of the joint distribution of consumption and the return earnedby one asset that has emerged is the following The joint distribution ofthe log of consumption in period t log ct and the (real) return earned by theassets from period t minus 1 to period t mtminus1 is normal with a covariance matrixthat is unchanging over time The means obey the linear relation

E(log ct) = k + ctminus1 + σE(mtminus1) (411)

That is the expected change in the log of consumption is a parameter σ times the expected real return plus a constant If the expected real interestrate E(mtminus1) is observed directly then the key parameter σ can be estimatedsimply by regressing the change in the log of consumption on the expectedreal rate That regression also has the property that no other variable knownin period t minus 1 belongs in the regression

Hall proceeds to estimate the parameter using aggregate data on consumptionand finds that there is ldquolittle basis for a conclusion that the behavior of aggregateconsumption in the United States in the twentieth century reveals an importantpositive value of the intertemporal elasticity of substitutionrdquo (1988 356)

Hallrsquos empirical finding is of importance to macroeconomic analysis Thework by Hall and others is also of interest as an example of how theoreticalmacroeconomic analysis specifically Fisherian analysis can be tested To seethe link between the theory developed in the prior section and the proposed test(equation (411)) assume the following

Assumption 43 The path of aggregate consumption reflects agent arsquos decisionsconcerning the optimal allocation of consumption across time That is we treatagent a as the ldquorepresentative consumerrdquo Thus cai i = t t+T which denotesconsumption in period i of the representative agent a differs only in scale from cii = t t + T which denotes the aggregate level of consumption This assump-tion that an aggregate variable can be viewed as reflecting decisions of a representa-tive agent is not innocuous For instance the actual path of aggregate consumptioncould well differ from that predicted by an analysis of individualsrsquo optimaldecisions due to changes across time in the composition of individuals in theeconomy33

Assumption 44 Individualsrsquo expectations in period t minus 1 of future real interestrates incorporate all information available as of period t minus 1 New informationoccurring in period t that alters consumption from what was planned results inthe distribution of consumption being ldquolog normal conditional on informationavailable last period that is log ct is normal with mean E(ct)rdquo (Hall 1988 342)

Assumption 45 In period t minus 1 the representative agentrsquos utility function takesthe following form

t+Tsumi=tminus1

expminusδi + ((δ minus 1)δ) log(ci)

where c gt 0 σ gt 0 and δ gt 0 This exponential utility function has the followingdesired properties

1 It is time-separable2 If consumption were equal across any two periods the individual would place

greater value on an increase in consumption in the earlier period ndash that is if

ctminus1 = ct then

expminusδ(t minus 1) + ((δ minus 1)δ) log(ctminus1)gt expminusδt + ((δ minus 1)δ) log(ct)

3 Given 1 gt σ ge 0 utility increases in any period with increased consumptionbut at a decreasing rate ndash that is

du(ctminus1) = dctminus1

= [(1 minus δ)(δctminus1)] middot expminusδt(t minus 1)

+ ((δ minus 1)δ) log(ctminus1)gt 0 d2u(ctminus1)dc2

tminus1 lt 0

Note that the ldquointertemporal elasticity of substitutionrdquo will be given by σ As σ approaches 0 substitution of consumption across time in response tochanges in the real interest rate will approach zero as well

Assumption 46 Individualsrsquo forecasts of future variables are held with subjec-tive certainty This last assumption is a departure from Hallrsquos analysis that allowsus for the moment to maintain the ldquodeterministicrdquo aspect of the prior optimizationproblem That is we continue to assume that individualrsquos expectations of futurevariables such as expected rates of inflation and future interest rates are held withsubjective certainty

Given the above assumptions we know from our prior discussion that thechoice of consumption for periods t minus1 and t must satisfy the following first-ordercondition

dudctminus1

dudct= Rtminus1

Substituting in the appropriate expressions for the marginal utility of consumptionin periods t minus 1 and t we obtain

[(1 minus δ)δctminus1] middot expminusδ(t minus 1) + ((δ minus 1)δ) log(ctminus1)[(1 minus δ)δct] middot expδ + ((δ minus 1)δ) log(ct)

(ctctminus1) expδ + ((δ minus 1)δ) log(ctminus1ct) = Rtminus1

Taking the logarithm of both sides of the above expression and rearranging theabove first-order condition becomes

log(ctctminus1) + δ minus ((δ minus 1)δ) log(ctminus1ct) = log(Rtminus1)

which simplifies to

δ + (1δ) log(ctminus1ct) = log(Rtminus1)

log ct = minusδσ + log ctminus1 + σ log(Rtminus1) (412)

which is similar in form to (411) Note that the ldquointertemporal elasticity ofsubstitutionrdquo is given by

σ = (dctdRtminus1)ctRtminus1

The form of equation (42) can be made closer to that of equation (411) if wenote that we can define the ldquoinstantaneous real rate of interestrdquo associated withcontinuous compounding mtminus1 by the expression

exp(mtminus1) = Rtminus1

Then (412) becomes

log ct = minusδσ + log ctminus1 + σmtminus1

Conclusion

This chapter has developed an in-depth understanding of household behaviorA good deal of the discussion has dealt with intertemporal choices and the tradeoffsinherent in consuming today versus consuming in the future Many policy-relatedissues in macroeconomics are related to decisions made today that are not indepen-dent of future states or activities This issue will arise again and again throughoutthis book and it is imperative that one comprehend the nature of decision-makingand time

5 Summarizing the behavior andconstraints of firms andhouseholds

Introduction

In this chapter we summarize our discussion of the behavior of firms andhouseholds in the simple Walrasian model with money and production In doingso we consider first the nature of constraints faced by the participants in the econ-omy with respect to decisions during period t and then their behavior in termsof demand andor supply Along the way we will try to simplify the notationand introduce various expectations and assumptions of different macroeconomicmodels We start our discussion with firms

Summarizing firmsrsquo constraints

We have seen how we can divide the general constraint facing firms that totalrevenues from all sources just exhausts expenditures each period into two sepa-rate constraints One is the ldquofirm financing constraintrdquo which states that desiredchanges in the capital stock as well as any capital adjustment costs are financedby issuing new bonds or equity shares That is for period t

I dnt + ψ(I d

nt) minus net Ast = 0 (51)

net Ast equiv As

t minus At

Ast equiv [

pbtBst + petS

At equiv [pbtB + petS

Idnt equiv Kd

t+1 minus K

Note that we implicitly assume that firmsrsquo plans for purchasing capital during theperiod correctly anticipate the price of output (capital) during the period and theprices of bonds and equity shares to be issued at the end of the period to financesuch purchases

Behavior and constraints 65

The second constraint the ldquofirm distribution constraintrdquo is that all revenuesfrom the sale of output net of that required to replace capital used up in the produc-tion process during the period be distributed to households either as wages interestpayments or dividends At time t firmsrsquo anticipated distribution constraint isgiven by

dt + (wtpt)Ndt + zBpt minus (ys

t minus δK) = 0 (52)

where z is the coupon payment and planned output supply is related to labordemand by the production function

yst = f (N d

t K) (53)

Equation (52) implicitly assumes that firms at time t have perfect foresight withrespect to the price of output during period t Alternatively the firm financingconstraint would take the above form if we presumed there were futures marketsat time t for the exchange of output during period t and financial assets at the endof period t

The labor market at time t determines employment for the period at a level N lowastt

and an associated rate of output denoted by ylowastt At the realized price of output the

firm distribution constraint for period t will turn out to be

dt + (wtpt)Nlowastt + zBpt minus ( ylowast

As (54) indicates actual real output during the period net of that used to replacedepreciated capital will be distributed to households1

Summarizing householdsrsquo constraints

With respect to households there is a single budget constraint for period t Like thatof the firm distribution constraint its form changes depending on what is assumedconcerning the correctness of expectations or the timing of markets As we haveseen at time t households make plans with respect to labor supply consumptiondemand and desired additions to their real holdings of financial assets and moneybalances based on a perceived constraint of the form

cdt + (M d

t minus M )pet + net Ad

t minus (wtpt)eN s

t minus (dt + zBpt) = 0 (55)

net Adt equiv Ad

t minus At

Adt equiv

dt + petS

66 Behavior and constraints

We presume that households have perfect foresight at time t with respect to thereal value of financial assets and the real value of dividends plus interest paymentsreceived from firms at the end of period t We leave open the possibility of errorsin expectations (held with subjective certainty) concerning the price level as itaffects real money balances and the real wage The term pt would replace pe

t if wepresumed perfect foresight on the part of households concerning the price level orequivalently presumed that there were futures markets at time t for the exchangeof output during period t

The labor market at time t determines employment and output for the periodAs before we let N lowast

t and ylowastt denote the actual rate of employment and production

of output during period t In such a case the actual firm distribution constraint(54) can be substituted into the household budget constraint Since prices will beknown by households at this point we may also replace the expected prices ofoutput bonds and equity shares by their actual prices Thus during period t thehousehold budget constraint for period t can then be expressed as

cdt + (M d

t minus M )pt + net Adt minus ( ylowast

As discussed below householdsrsquo behavior in the output and financial markets willbe based on realized income and prices and will differ from their plans made attime t based on expected prices and a labor supply decision unless they possessperfect foresight at time t concerning the prices that will prevail over period t andthe labor market clears with actual employment equal to labor supply2

Walrasrsquo law labor market and other markets at time t

Recall that Walrasrsquo law reflects the summing up of the constraints faced by individ-ual agents in the economy Since our preceding analysis concerned the constraintsof the ldquorepresentativerdquo firm and household we need only sum the constraints ofsuch representative agents to obtain Walrasrsquo law There still remains a potentialproblem however as to which of the different versions of the constraints enumer-ated above for firms and households to use The choice as one would suspectdepends on whether the market for labor effectively occurs at the same time as themarkets for output and financial assets or at different times

One version of Walrasrsquo law essentially combines the market for labor at time twith a futures market at time t for the exchange of output during period t andfinancial assets at the end of period t Equivalently this version of Walrasrsquo lawassumes limited perfect foresight at time t by both firms and households withrespect to prices for the period3 In such a case we would sum constraints (51)(52) and (55) (with pt replacing pe

t in (55)) to obtain

t + I dnt + δK + ψ(I d

nt) minus yst

]+[net Ad

t minus net Ast

]+ (wtpt)

t minus N st

]+ [M d

t minus M ]pt = 0 (57)

Thus we have that the sum of excess demand across four markets ndash the laboroutput financial and money markets ndash must equal zero4 Note that one of thesefour markets the money ldquomarketrdquo reflects the equality between the demand forand supply of money The money ldquomarketrdquo is not of course like other marketsof an economy ndash which is why the word ldquomarketrdquo is in quotes That is unlike theother markets the money ldquomarketrdquo is not a place where the exchange of goods(eg labor financial assets or output) takes place5

Walrasrsquo law sequential markets and potential lackof perfect foresight

There is a modification to make with respect to the above that is required if marketsoccur sequentially and if there is not perfect foresight on the part of all agents attime t concerning prices for period t The sequential nature of markets is clearin that we have the labor market occurring at time t while the markets for outputand financial assets occur during the period In addition households in particularmay not correctly foresee at time t the price of output for period t Under suchcircumstances the prior version of Walrasrsquo law no longer holds for it would thensum constraints that are only anticipated not realized Instead given the sequentialnature of the markets we must break the analysis down into an analysis of thelabor market and an analysis of the other three markets

At time t the labor market occurs Assuming a competitive equilibrium for thelabor market we have a money wage determined at time t such that

N st = N d

Underlying the supply of labor at time t are householdsrsquo plans with respect toconsumption demand and saving (either in the form of financial assets or money)during the period These plans are influenced at time t by the anticipated pricelevel for commodities pe

t among other variables and as such these plans may notbe feasible given realized prices during period t

Once the labor market ends employment and output are determined for theperiod at levels N lowast

t and ylowastt respectively At that point households make plans with

respect to consumption and saving in light of the realized prices and the resultingeffective household budget constraint That realized household budget constraintis simply equation (56) which incorporates the actual firm distribution constraintAdding the firm financing constraint (51) we obtain a modified Walrasrsquo law forthe markets during period t of the form

nt) minus ylowastt

]+[net Ad

t minus net Ast

]+ [M d

t minus M ]pt = 0

In the absence of perfect foresight the demands for consumption money balancesand financial assets during the period expressed in the above equation can differfrom the plans made at time t

Summarizing firm behavior with limited perfect foresight

Consider firmsrsquo optimal plans at time t Note that at time t firms have an expectationof the price of output for the period We shall continue to assume that firms haveperfect foresight at time t with respect to the price of output over the periodWith respect to labor demand a diminishing marginal product of labor implies aninverse relationship between the real wage and labor demand

N st = N d

t (wtpt K)

It is typical to assume that the labor market is such that firms achieve employ-ment equal to that demanded In this case given the production function and thenature of the labor demand function we have an output supply function for periodt of the form6

yst = yd

t (wtpt K)

An increase in the real wage reduces labor demand and thus output supply so thatwe have

partN dt part(wtpt) lt 0 and partys

t part(wtpt) lt 0

Now consider firmsrsquo behavior with respect to investment and consequent finan-cial asset supply Given a diminishing marginal product of capital capital demandis inversely related to the expected real user cost of capital7

Assuming labor and capital are complements in the production process(part2f partKpartN gt 0) capital demand will be inversely related to the expected realwage in the subsequent period as well8 In particular in the absence of adjustmentcosts (for both capital and labor) we have the following capital demand function

Kdt+1 = Kd

t+1(met + δ we

t+1pet+1) (58)

partKdt+1part(me

t + δ) lt 0 and partKdt+1part(we

t+1pet+1) lt 09

The above demand for capital stock at the end of period t (in place at time t +1)implies a net investment demand function for period t of the form

I dnt = I d

nt(met + δ we

t+1pet+1 K)

where net investment demand is inversely related to the expected real user cost ofcapital me

t + δ the anticipated real wage in the next period wet+1pe

t+1 and theexisting capital stock at time t K

Recall that the firm financing constraint in the absence of capital adjustmentcosts equates firmsrsquo net real financial asset supply to net investment demandThus given the nature of the net investment demand function the net real financial

asset supply function for firms at the end of period t (at time t + 1) is identical tonet investment demand or

net Ast = net As

t (met + δ we

t+1pet+1 K)

where net real financial asset supply for period t like net investment demandduring period t is inversely related to the expected real user cost of capital theanticipated real wage in the next period and the existing capital stock at time t

With convex adjustment costs the optimal capital stock (as well as investmentdemand) depends on the entire future path of the expected real user cost of capitaland real wages That is with adjustment costs

Kdt+1 = Kd

t+1(met + δ me

t+1 + δ wet+1pe

t+1 wet+2pe

t+2 ) (59)

We can rewrite the above demand function for capital given adjustment costs soas to collapse future periods into essentially a single subsequent period10 To doso recall that the expected real rate of interest for period i me

i equals (ri minus πei )

(1+πei ) Let us assume static expectations concerning future interest rates so that

ri = rt i = t + 1 t + 2 Further let us assume static expectations concerningfuture expected inflation so that πe

i = πet i = t + 1 t + 2 The result is that

the expected constant real rate of interest between periods t and t + 1 is expectedto prevail in the future so that me

i = met i = t + 1 t + 2 We can then rewrite

the demand function for capital accumulated at the end of period t in the simplerform

Kdt+1 = Kd

t+1(met + δ we

t+1pet+1 we

t+2pet+2 )

Now note that we can decompose the anticipated wage for period i and theexpected price level for period i into two components the wage or price levelin period i minus 1 and the expected rate of change in wages or prices respectivelyIn particular the anticipated money wage and price level for period t + 2 can beexpressed by

wet+2 equiv we

t+1(1 + πewt+1) and pe

t+2 equiv pet+1(1 + πe

Let us now assume static expectations with respect to the rate of change in wagesbeyond the next period so that πe

wi = πewt+1 i = t + 2 t + 3 Recall that

we have already assumed a constant rate of price inflation in subsequent periodsIf we then add the assumption that beyond the next period the (constant) rate ofinflation in wages (πe

wt+1) equals the expected rate of inflation in prices (πet+1)

we have that wei pe

i = wet+1pe

t+1 i = t + 2 t + 3 In words the real wagein the subsequent period we

t+1pet+1 i = t + 2 t + 3 is anticipated to persist

indefinitely11 We can now rewrite the capital demand function given adjustmentcost (equation (59) as

Kdt+1 = Kd

t+1(met + δ we

t+1pet+1) (510)

Note that given our expectation assumptions the capital demand function withadjustment costs (equation (54)) has the same form as the capital demand functionin the absence of capital adjustment costs However the actual response to changesin the real user cost of capital or in the anticipated real wage in the subsequentperiod would typically be less with adjustment costs as such costs would leadfirms to only gradually move toward a new optimal capital stock

Equation (510) captures the idea that the demand for capital to be in placeat the end of period t (at time t + 1) depends inversely on the expected realuser cost of capital and that assuming capital and labor are complements capitaldemand depends inversely on the anticipated future real wage Similarly sincenet investment demand during period t simply reflects the difference betweencapital demand at the end of the period and the initial capital stock we haveinvestment demand being inversely related to the expected real user (or rental) costof capital and to the anticipated real wage for the subsequent period Finally fromthe firm financing constraint that relates firmsrsquo net real financial asset supply tonet investment demand we have firmsrsquo net financial asset supply being inverselyrelated to the expected real user cost of capital and the expected future real wageThus in summary we have

partKdt+1part(me

t + δ) lt 0 partI dntpart(me

t + δ) lt 0

partKdt+1part(we

t+1pet+1) lt 0 partI d

ntpart(wet+1pe

t+1) lt 0

partnet Ast part(me

t + δ) gt 0 partnet Ast part(we

t+1pet+1) lt 0

where mei equiv (rt minus πe

t )(1 + πet ) Note that gross investment demand is given by

Idt equiv Kd

t+1 minus K + δK = I dnt + δK

Summarizing household behavior with limitedperfect foresight

Householdsrsquo plans at time t concerning the labor supply choice the consump-tionsaving choice and the portfolio choice for period t are constrained by theanticipated household budget constraint as given by equation (55) From ourintertemporal analysis of these optimal choices we know that in general at time twe thus have labor supply at time t

N st = N s

t (wtpet we

t+1pet+1 rt rt+1 πe

t πet+1 At Mpe

+ zBpt)

consumption demand planned at time t

cdt = cd

t (wtpet we

t πet+1 At Mpe

+ zBpt)

and money demand planned at time t

Ldt = Ld

t (wtpet we

t πet+1 At Mpe

+ zBpt)

where for notational ease we have defined planned real money demand by the termLd

t From the anticipated budget constraint as given by

cdt + (M d

t minus (wtpet )N

st minus (dt + zBpt) = 0

we can infer net real financial asset demand planned at time t net Adt from

householdsrsquo plans concerning money demand consumption demand and laborsupply

With perfect foresight at time t concerning the prices for period t house-holdsrsquo plans made at time t concerning consumption money demand andnet real financial asset demand during period t will be fulfilled Thus lettingxt = dt + zBpt + At + M tpt we can express the above behavior conditions asfollows labor supply at time t is

N st = N s

t (wtpt wet+1pe

t+1 rt rt+1 πet πe

t+1 xt)

consumption demand for period t is

cdt = cd

t (wtpt wet+1pe

t+1 xt)

and money demand for period t is

Ldt = Ld

t (wtpt wet+1pe

t+1 xt)

where Ldt equiv M d

t pt As we did with respect to firmsrsquo capital demand function given adjustment

costs we now introduce expectation assumptions that essentially collapse theentire sequence of future periods into a single future period First we assume staticexpectations concerning future interest rates so that ri = rt i = t + 1 t + 2 Next we assume static expectations with respect to future rates of inflation suchthat πe

i = πet i = t + 1 t + 2 Finally we assume that the expected rate of

wage inflation beyond the next period is constant and equal to the expected rate ofinflation (ie πe

wi = πewt+1 i = t + 2 t + 3 and πe

wt+1 = πet+1) so that the

real wage anticipated for the next period is expected to prevail indefinitely Giventhe above restrictive expectation assumptions we can rewrite the householdsrsquobehavioral functions in the following form the labor supply at time t becomes

N st = N s

t (wtpt wet+1pe

t+1 rt πet xt)

consumption demand for period t becomes

cdt = cd

t (wtpt wet+1pe

t+1 rt πet xt)

and money demand for period t becomes

Ldt = Ld

t (wtpt wet+1pe

t+1 rt πet xt)

where real money demand at the end of period t is given by Ldt equiv M d

t pt Our discussion of the labor supply decision suggests that labor supply is directly

related to the real wage and inversely related to the real wage next period (whichreflects the real wage in all subsequent periods given our expectation assumptions)This response to changes in the real wages incorporates the intertemporal substitu-tion hypothesis The intertemporal substitution hypothesis also suggests that laborsupply is directly related to the interest rate and inversely related to the expectedrate of inflation in that either change implies a higher expected real rate of inter-est and thus a substitution from leisure to increased labor supply today Finallyhigher real initial holdings of financial assets real money balances or income inthe current period from bond and equity shares holdings will reduce labor supplyif leisure is a normal good In summary we thus have

partN st part(wtpt) gt 0 partN s

t part(wet+1pe

t+1) lt 0 partN st partrt gt 0

partN st partπe

t lt 0 partN st part xt lt 0

Our discussion of the consumptionsaving decision suggests that consumptiondemand is directly related to both the current and anticipated future real wagesince an increase in either implies an increase in the discounted stream of incomeFocusing on the Fisherian analysis of the allocation of consumption across timeconsumption demand would be inversely related to the money interest rate anddirectly related to the expected rate of inflation since either change implies ahigher expected real rate of interest Finally an increase in xt (reflecting sayhigher real initial holdings of financial assets increased real money balancesor higher income in the current period from bond and equity share holdings) willraise consumption demand in the current period In summary we thus have

partcdt part(wtpt) gt 0 partcd

t part(wet+1pe

t+1) gt 0 partcdt partrt lt 0

partcdt partπe

t gt 0 partcdt part xt gt 0

Our discussion of the portfolio decision suggests that real money demand isdirectly related to both the current and anticipated future real wage since an increasein either implies an increase in the discounted stream of income Focusing on theportfolio analysis money demand would be inversely related to the money interestrate as households shift from money to financial asset holdings Any effect of achange in the expected rate of inflation is indirect and will be ignored Finallyan increase in xt (reflecting say higher real initial holdings of financial assetsincreased real money balances or higher income in the current period from bondand equity share holdings) will likely raise money demand in the current period

In summary we thus have

partLdt part(wtpt) gt 0 partLd

t part(wet+1pe

t+1) gt 0 partLdt partrt lt 0

partLdt part xt gt 0

So far our discussion has not included householdsrsquo real net financial assetdemand To see what determines this we can simply use the budget constraintand money labor and commodity demand functions Specifically rewriting thebudget constraint for period t

net Adt = (wtpt)N

st + dt + zBpt minus cd

t minus (M dt minus M )pt

An increase in the current real wage initial real money balances or dividendand interest payments for the current period is presumed to increase not onlycurrent consumption and money demand but also future consumption and moneydemand and thus increase net financial asset demand by households From theintertemporal substitution hypothesis an increase in the expected future real wagewill reduce current labor supply as well as increase current consumption demandboth changes imply a fall in net real financial asset demand for households

From the Fisherian analysis a higher expected rate of inflation will reducethe expected real rate of interest the above constraint indicates that the resultingincrease in current consumption demand will reduce householdsrsquo acquisition offinancial assets Similarly an increase in real initial financial asset holdings byraising both current consumption and money demand will lead to a fall in real netfinancial asset demand On the other hand a higher money interest rate due toboth the Fisherian effect on current consumption demand and the portfolio effecton money demand implies a higher net financial asset demand In summary wethus have

partnet Adt part(wtpt) lt (or gt or =) 0 partnet Ad

t part(wet+1pe

t+1) lt 0

partnet Adt partrt gt 0

partnet Adt partπe

t lt 0 partnet Adt partAt lt 0 partnet Ad

t partMpt) gt 0

partnet Adt part(dt + zBpt) lt (or gt or =) 0

where there are ambiguous effects on net financial asset demand of a change in thereal wage and of a change in anticipated dividend and interest payments becausean increase in either raises both consumption demand and money demand Notethat in the limit as the length of the period goes to zero the household budgetconstraint at time t becomes

net Adt + (M d

t minus M )pt = 0

as all the flow terms go to zero at a point in time In this case assuming real moneydemand is directly related to income we obtain the unambiguous effect of

partnet Adt part(dt + zBpt) lt 0 partnet Ad

t part(wtpt) lt 0

However in the period analysis net Adt reflects net real financial asset demand at the

end of the period Thus assuming any increase in income is not fully reflected inan increased rate of consumption we have an offsetting effect and thus ambiguity

Summarizing household behavior without perfectforesight

If we presume that households learn of prices that will exist for period t aftertime t and they differ from what was expected then the actual demands for outputmoney and financial assets can differ from those reflecting plans made at time tThe key reason for this is that the actual constraint faced by households will differfrom that anticipated In particular using the actual firm distribution constraintwe replace anticipated real income from wages dividends and interest paymentsfrom firms with the actual real income net of depreciation ( ylowast

t minusδK) The resultingrealized household budget constraint after time t is then

cdt + (M d

t minus ( ylowastt minus δK) = 0

If anticipations by households were incorrect at time t concerning prices or div-idends during the period then revisions in plans for consumption and saving willbe made in light of the actual budget constraint faced In this case the actual house-hold demand functions for output and money are written as follows consumptiondemand during period t is

cdt = cd

t (wet+1pe

t+1 rt πet At Mpt ylowast

money demand during period t is

Ldt = Ld

t (wet+1pe

and we replace partcdt part(wtpt) gt 0 and partcd

t part(dt + zBpt) gt 0 with

partcdt partylowast

t gt 0

Note that the term partcdt partylowast

t is referred to as the ldquomarginal propensity toconsumerdquo12 Similarly we replace partLd

t part(wtpt) gt 0 and partLdt part(dt + zBpt) gt 0

partLdt partylowast

t gt 0

Money illusion and the real balance effect

Let us consider (sufficient) assumptions under which demands and supplies arehomogeneous of degree 0 in current wages prices and the nominal stock ofmoney ndash that is there is the absence of money illusion Note that in consideringwhether or not there exists money illusion we must now look at the behavior notonly of households but also of firms Consider firms first

Assuming perfect foresight on the part of firms it is clear that current labordemand is homogeneous of degree 0 in current prices (the wage rate wt and the pricelevel pt) Thus so also current output supply Assuming unit elastic expectationswith respect to wages and prices in all future periods and expectations of futureinterest rates that are invariant to changes in current prices and wages we havethat capital demand and thus also investment demand and firmsrsquo net real financialasset supply are homogeneous of degree 0 in current prices

We already assumed static expectations concerning future interest rates and unitelastic expectations with respect to prices beyond period t + 1 in terms of nextperiodrsquos price to obtain a simple form for the demand functions for investmentThus we need only add (a) the assumption of unit elastic expectations concerningthe price of output between period t and t+1 (implying an expected rate of inflationπe

t and thus an expected real rate of interest that is independent of a change in theprice level pt)

13 and (b) unit elastic expectations concerning next periodrsquos wageand price level (implying an anticipated real wage next period independent of anequiproportionate change in wages and prices in the current period) to obtain theabsence of money illusion with respect to firmsrsquo investment demand14

To obtain the absence of money illusion with respect to households we mustshow that each of the arguments in their demand and supply functions is invariantto an equiproportional change in current prices wages and the nominal stockof money Given perfect foresight the current real wage and initial real moneybalances meet this condition But what about the expected real wage next periodthe expected rate of inflation current dividends and interest payments and the realvalue of initial financial assets holdings As it turns out the assumption of unitelastic expectations with respect to all future prices and wages is again critical inshowing these to be invariant as it was in deriving a capital demand homogeneousof degree 0 in current wages and prices

First it is clear that the assumption of unit elastic expectations with respectto future wages and prices makes future real wages invariant to equiproportionalchanges in the current wages and prices But note that in so doing we have elimi-nated the ldquointertemporal substitution hypothesisrdquo effect on labor supply of a changein the current real wage Similarly the assumption of unit elastic expectations withrespect to the future price level eliminates any effect of a change in the price levelon the expected rate of inflation Finally assuming that expectations of nominalfuture interest rates are invariant to an equiproportionate change in current priceswages and the money supply the expected real rate of interest will not be affectedby equiproportionate changes in wages prices and the money supply But notethat the ldquointertemporal substitution hypothesisrdquo impact on labor supply of a change

in the expected real rate of interest initiated by a change in the current price levelis now absent as well15

What is left in order to obtain the absence of money illusion for households isto show that changes in current prices leave current dividend payments and thereal value of initial bond and stock holdings unchanged Once again as we seebelow the assumption of unit elastic expectations concerning next periodrsquos wagesand prices will be invoked to achieve this What we are looking for are sufficientassumptions that will result in the terms dt + zBpt and At being homogeneous ofdegree 0 in wt and pt

From the firm distribution constraint (52) and assuming firmsrsquo labor demandis satisfied (ie N lowast

t = N dt and thus ylowast

t = yst ) we know that

dt + zBpt = yst minus δK minus (wtpt)N

Since N dt and ys

t are homogeneous of degree 0 in prices it is clear that the sumof current dividends plus interest payments is not affected by equiproportionatechanges in both the current wage and price level A higher price level does alterthe composition of payments however as real dividends rise and real interestpayments fall

Now consider At To show that this can be homogeneous of degree 0 in thecurrent wage and price level note from the firm distribution constraint and fromthe assumption that firmsrsquo demand for labor is satisfied in subsequent periods that

dt + zBpet+1 = f (N d

t+1 Kdt+1) minus (we

t+1pet+1)N

dt+1 minus δKd

A similar equation holds for future periods as well At simply reflects the presentvalue of such future real payments using the appropriate expected real interest ratesfor discounting The assumption of unit elastic expectations concerning prices inall future periods coupled with expectations of future nominal interest rates thatare unaffected by an equiproportionate change in money prices and the moneysupply thus means that A is homogeneous of degree 0 with regard to a change inprices (price level and wages) and the money supply in period t16

A special case of the above is if we assume static expectations concerning futureinterest rates (ie ri = rt i = t + 1 t + 2 ) and zero adjustment costs In thiscase Kd

i i = t + 1 t + 2 would be the same in each future period Therewould be a constant labor demand (N d

i = N dt+1 i = t + 2 t + 3 ) as well given

the invariant real wage in conjunction with no change in their capital stock Nowrecall that At is defined by

At equiv [pbtB + petS]pt

where pbtBt is the present value of future interest payments and petS is the presentvalue of future dividends Since At is simply the present value of the now constantfuture stream of dividends and interest payments discounted using an invariant

expected real rate of interest we thus have

At = [dt+1 + zBtpet+1]me

Since dt+1 +zBtpet+1 = f (N d

t+1 Kdt+1)minus (we

t+1pet+1)N

dt+1 minusδKd

t+1 from the firmdistribution constraint we can rewrite the initial real holdings of financial assetsas

At = [ f (N dt+1 Kd

t+1) minus (wet+1pe

t+1)Ndt+1 minus δKd

t+1]met

which is homogeneous of degree 0 in current wages and prices if we assume that theanticipated real wage next period is independent of an equiproportionate changein the current wages and prices In other words to obtain the absence of moneyillusion for households we assume as we did with firms that there are unit elasticexpectations concerning wages and prices in the next period

Note that an increase in the money interest rate and the expected rate of inflationthat leaves the expected real rate of interest unchanged will alter the compositionof financial asset holdings although not the total To see this recall that with staticexpectations concerning the interest rate the price of bonds at the end of period tis given by

pbt =infinsum

z(1 + rt)i = zrt

An increase in the money interest rate and expected rate of inflation such that thereal rate of interest is unchanged means a fall in the price of bonds but an offsettingrise in the price of stock as over time real dividend payments will be rising morerapidly while real interest payments will be falling more rapidly Similarly anincrease in the current level of prices although leaving At unaffected wouldreduce the real value of bond holdings and lead to an exactly offsetting increasein the real value of equity share holdings

The real balance effect is apparent from the nature of the demand and supplyfunctions In particular an increase in nominal money balances or fall in prices withno change in nominal money balances will increase initial real money holdingsand in general lead to an increase in consumption demand real money demandand real net financial asset demand In general labor supply would fall

6 The simple neoclassicalmacroeconomic model (withoutgovernment or depositoryinstitutions)

Introduction

We have now covered a substantial part of the underlying structure for a simpleaggregate model of an economy with production The specific elements of theldquomicroeconomic foundationsrdquo of this aggregate model developed so far have dealtwith the optimizing behavior of individuals (ldquorepresentativerdquo firms and house-holds) in a setting in which individuals take prices as given Implicit in thesediscussions is another part of the microeconomic foundations the way in whichindividual markets operate We have been assuming that prices adjust so that thepresumption that buyers and sellers are price-takers is justified In other wordsequilibrium within individual markets entails price adjustment to equate suppliesand demands In addition we will assume that all individuals in the economy cor-rectly foresee period trsquos output prices when input supply and production decisionsare made at the start of the period As discussed below however there are otheroptions

Static macroeconomic models the options

Grandmont (1977 542) notes that

one way to look at the evolution of an economic system is to view it as asuccession of temporary or short-run competitive equilibrium That is onepostulates that at each date prices move fast enough to match supply anddemand Although one assumes equilibrium in each period the economicsystem displays a disequilibrium feature along a sequence of temporary com-petitive equilibrium at each date the plans of the agents for the future arenot coordinated and thus will be in general incompatible this is to becontrasted with the perfect foresight approach where by definition such adisequilibrium phenomenon cannot occur

This temporary equilibrium view of the economy is characteristic of the simplestatic neoclassical model a model in which all prices adjust to maintainequilibrium1 The second key assumption of the neoclassical model is that agents

Simple neoclassical macroeconomic model 79

are informed about prices within the period In particular when making their laborsupply and demand decisions at the start of the period households and firms areassumed to correctly anticipate the prices they will have to pay to purchase outputduring the period As it turns out this element of ldquoperfect foresightrdquo with respectto markets through time t +1 is a critical feature of the analysis Before examiningthe implications of these assumptions however some history on the origin of theneoclassical model might be helpful

The phrase ldquoneoclassical macroeconomic modelrdquo is a descendant of ldquoclassicalrdquoeconomic theory as reflected in the work of Sir William Petty during the 1600sIn Das Kapital Karl Marx (1976 85) stated that ldquoby classical Political EconomyI understand that economy which since the time of W Petty has investigated thereal relations of production in bourgeois societyrdquo As Marx suggested early classi-cal economists focused on the determinants of the economyrsquos productive capacityThe neoclassical macroeconomic model shares this focus on the productive capac-ity of the economy as the determinant of total output It also turns out to be thestatic precursor to much of the current analysis in the macroeconomic literaturethat falls under the heading of ldquoreal business cycle theoryrdquo

While the simple static neoclassical model along with its dynamic and stochas-tic counterparts is one popular approach to macroeconomic analysis there areother approaches In fact even though static analysis is restricted to markets in thecurrent period there remains enough flexibility to introduce at least four ways ofcharacterizing macroeconomic analysis

1 ldquoNeoclassical modelrdquo competitive equilibria are assumed to exist in cur-rent and future markets and limited perfect foresight is assumed for allparticipants

2 ldquoIllusion modelrdquo competitive equilibria are assumed to exist in current andfuture markets but imperfect foresight is assumed on the part of some agentsThe result is like the ldquoLucas supply functionrdquo popularized by Lucas (1973)in which output can respond directly to increases in the actual output prices

3 ldquoKeynesian modelrdquo a competitive equilibrium is assumed not to exist inthe labor market as the money wage is fixed and employment is demand-determined However other prices in particular the prices of output arepresumed to reflect competitive equilibria A rational expectation version ofthis model is developed by Fisher

4 ldquoNon-market-clearing modelrdquo a competitive equilibrium is assumed not toexist in the output market as the price of output is fixed above the competitiveequilibrium level This model forms the basis of much of what appears inundergraduate macroeconomic analysis including the IS-LM model

The neoclassical model with its assumptions of flexible prices and informedagents provides a benchmark against which we can compare the predictions ofother (static) macroeconomic models2 It also provides insight into the nature of thestationary states for dynamic macroeconomic models that presume market-clearingprices and accurate forecasts of prices

80 Simple neoclassical macroeconomic model

Hicksian temporary equilibrium and Walrasrsquo law

For the market for any good ldquocompetitiverdquo equilibrium is defined by equalitybetween market demand and supply3 A temporary competitive equilibrium for theeconomy during period t will be characterized by a money wage for labor (wlowast

t )a money price for the consumption commodity ( plowast

t ) a money price for bonds( plowast

bt) a money price for equity shares ( plowastet) allocations to households in terms

of employment consumption bond holdings equity share holdings and moneybalances (N lowast

t clowastt Blowast

t Slowastt and M lowast

t ) and allocations to firms in terms of employmentoutput investment bond issues and equity share issues (N lowast

t ylowastt Ilowast

t Blowastt Slowast

t ) suchthat

bull these allocations are in the demand (supply) set of each agentbull these allocations are feasible

Together these two conditions imply prices determined in the labor output bondand equity shares markets for period t that result in zero excess aggregate demandfor labor output bonds equity shares and money Thus we may rewrite theconditions for a general equilibrium as a money wage a price of output a price ofbonds and a price of equity shares such that

N st = N d

yst = cd

Bst = Bd

Sst = Sd

Mpt = M dt pt

I dnt = Kd

t+1 minus K

yst = f (N d

Given our assumption that bonds and equity shares are perfect substitutes ingeneral there will not be a unique equilibrium in terms of the number of bondsand equity shares supplied or demanded although the total value of financialassets supplied or demanded will be determinant Thus we replace the bond andequity share markets with a single market the financial market The equilibriumconditions then become

N st = N d

yst = cd

Ast = Ad

Mpt = M dt pt

Ast equiv [ pbtB

st + petS

st ]pt

Adt equiv [ pbtB

dt + petS

dt ]pt

We can view the financial market as simultaneously determining the price ofbonds pbt the price of equity shares pet and the interest rate rt That is once oneof these is known the other two are implied For instance from our definition ofthe interest rate

1 + rt equiv [z + pbt+1]pt

we see that given the coupon rate and expected price of bonds in the subsequentperiod the interest rate rt implies a price of bonds pbt As perfect substitutesbonds and equity shares must offer the identical expected gross return Thus wehave that

1 + rt = 1 + ret equiv [dt+1 + pet+1]pet

As you can see given expectations of future dividends and the future price ofequity shares an interest rate rt also implies a price of equity shares pet We oftentalk of the financial market in terms of an equilibrium interest rate The aboveshould make it clear that associated with such an equilibrium interest rate areprices of bonds and equity shares And a rise (fall) in the interest rate means a fall(rise) in the prices of bonds and equity shares

We will make one additional change in the characterization of the financialmarket to put it in terms of additional demands and supplies that is put it in netrather than gross terms The reason for this is that net financial asset demands andsupplies are what correspond to household saving in the form of financial assetsand firm investment Thus the equilibrium conditions become

N st = N d

yst = cd

net Ast = net Ad

Mpt = M dt pt

net Ast equiv As

t minus At

net Adt equiv Ad

t minus At

According to the above we have four equilibrium conditions but only threeprices ndash the money wage rate the level of output prices and the interest rate ndashto be determined As usual Walrasrsquo law is invoked to show that only n minus 1 ofthe n equilibrium conditions are independent However the nature of Walrasrsquo lawdepends on whether we assume limited perfect foresight or not

Walrasrsquo law for limited perfect foresight sums up the constraints faced by theindividual agents in the economy at time t to obtain

[cdt + I d

nt + δK + ψ(I dnt) minus ys

t ] + [net Adt minus net As

t ]+ (wtpt)[N d

t minus N st ] + M d

t pt minus Mpt = 0

Thus the sum of excess demands for output financial assets labor and moneymust equal zero

When there is not perfect foresight at time t concerning prices for period t inthe output and financial markets we have the equilibrium condition for the labormarket

N st minus N d

and the modified Walrasrsquo law based on the resulting employment and output ofthe form

[cdt + I d

nt + δK + ψ(I dnt) minus ylowast

t ] + M dt pt minus Mpt = 0

In this case the money wage employment and thus output are determined inthe labor market and the modified Walrasrsquo law indicates that the price level andinterest rate are determined by any two of the remaining three markets

As it turns out most macroeconomic analysis takes this second approach tosolving for equilibrium That is the analysis focuses on the labor (and other input)markets and determines the effect of changes in output price (and potentially othervariables such as the interest rate) on equilibrium employment and thus outputThis generates an ldquoaggregate supply equationrdquo which is then combined with twoof the remaining three equilibrium equations ndash typically the commodity and moneymarket equilibrium conditions ndash to determine the equilibrium price level interestrate and output The modified Walrasrsquo law is invoked to ensure equilibrium inthe financial market at this point Working backward one can infer from theaggregate supply equation the equilibrium money wage and employment impliedby the analysis

The advantage of the above approach is that it can be used whether or not thereexists limited perfect foresight at time t with respect to the price level and whetheror not prices adjust to clear markets A disadvantage of the analysis is that inthe case of the neoclassical model with limited perfect foresight and competitiveequilibrium it arbitrarily breaks up the analysis of markets In doing so it requiresthat demand functions for such goods as money and commodities be specifiedwith income as an argument This form of the demand functions obscures the fact

that as we have seen in standard general equilibrium analysis demand functionsdepend on prices (including the real wage) and income is a variable determinedby the choice of labor supply

With these qualifications in mind we take the standard approach of macroeco-nomics and separate out the analysis of the labor (and other input) markets (theldquoaggregate supplyrdquo part) from the other markets (the ldquoaggregate demandrdquo part)The next section considers the labor market at time t

Labor market equilibrium

At the start of period t the labor market takes place and a rate of production of outputis determined From our analysis of firm behavior at time t we know that behindlabor demand is an expected price of output over the period and associated expectedreal wage an existing capital stock and existing technology as incorporated in theproduction function We shall assume that firms correctly anticipate at time t theprice of output for the period so that firms confront the actual real wage wtpt

From our analysis of household decision-making at time t we know that behindlabor supply at time t is not only the expected price of output and implied expectedreal wage but also such factors as the relationship of the anticipated current realwage to anticipated future real wages the expected real rate of interest anticipatedwealth in the form of financial asset and real money holdings and anticipated cur-rent nonlabor income Like firms we will assume households have limited perfectforesight in that at time t they correctly foresee the price level for period tAssuming a Walrasian or ldquocompetitive equilibriumrdquo view of the labor marketsthe money wage wt adjusts to achieve equilibrium in the labor market under thesecircumstances

We have already seen how static expectations concerning future rates of wageand price inflation along with the assumption of unit elastic expectations concern-ing wages and prices next period simplify the labor supply function by removingexpected future real wages as explicit arguments Patinkin (1965) among oth-ers goes several steps beyond these simplifying assumptions and assumes that allother variables excepting the real wage do not have a significant impact on laborsupply4 Thus equilibrium in the labor market is given by a level of employmentNt and money wage wt such that

Nt = N dt (wtpt K) and Nt = N s

t (wtpt)

As Patinkin (1965 264) notes

it will immediately be recognized that we have greatly oversimplified theanalysis of this market Both the demand and supply functions for labor shouldactually be presented as dependent on the real value of bond and moneyholdings as well as on the real wage rate5 Further if we were to permit thefirm to vary its input of capital its demand for labor would depend also onthe rate of interest Finally a full utility analysis of individual behavior wouldshow the supply of labor also to depend on this rate6

Besides a competitive labor market and the above simplified labor supplyfunction the neoclassical model assumes that suppliers correctly anticipate theaggregate price level As we will see the assumption of the neoclassical modelthat suppliers like firms have perfect foresight at time t with respect to the priceof output for the period means that changes in the price of output have no effect onoutput supply This characteristic of the neoclassical model implies an underlyingldquoblock recursiverdquo nature to the analysis as described below

At the start of the period the labor market occurs with employment and thusoutput and the money wage being determined for the period Employment andoutput are determined based on individualsrsquo expectations of subsequent variablessuch as the price of output And the level of employment and output influencethe remaining variables to be determined But in the neoclassical model the levelof employment and output are not influenced by the remaining variables to bedetermined We can thus solve the equilibrium sequentially looking first at thelabor market and the determination of employment and output then looking atthe output financial and money markets and the determination of the price leveland interest rate

Financial market equilibrium

With regard to the financial market Patinkin (1965 215) states

a decrease in the price of bonds (and equity shares) decreases the amountdemanded of consumption commodities it will also be assumed that itdecreases the amount demanded of money balances hence by the house-holdrsquos budget constraint their total expenditures on [net] bond holdings mustincrease

Thus Patinkin invokes a ldquoFisherian effectrdquo and a ldquoportfolio effectrdquo of a change inthe interest rate to obtain

part(net Adt )partpbt lt 0

From the firm financing constraint and the fact that a firmrsquos net investment demandis inversely related to the interest rate we have

part(net Ast )partpbt gt 0

The above analysis differs slightly from that found in Patinkin (1965 214) Forinstance Patinkin uses the assumption of static expectations concerning interestrates and a coupon rate z equal to 1 to express the price of bonds as the reciprocalof the interest rate that is

pbt = 1rt

Further Patinkin assumes no equity shares so his financial market consists only ofbonds Finally Patinkin does not graph net real bond demand and supply Ratherhe graphs the total number of bonds demanded and supplied Thus Patinkin hasdemands and supplies of bonds of the form

Bs = rtptAst equiv Bs

Bd = rtptAdt equiv Bd

As might be expected Patinkinrsquos bond demand and supply curves differ in naturefrom net real financial asset demand and supply curves For instance it is clearfrom the analysis of investment demand that a rise in the interest rate (a fall inthe price of bonds) will lead to reduced investment demand and thus reduced netreal financial asset supply This relationship between investment and firmsrsquo net realfinancial asset supply follows directly from the firm financing constraint whichgiven pbt = 1rt is of the form

net At equiv 1ptrt[Bst minus B] = I d

nt + ψ(I dbt)

Naturally if net investment were initially zero and there were zero adjustment costs(so that Bs

t = B) the fall in investment that results from a rise in the interest ratewould imply a similar decrease in the number of bonds supplied (Bs

t ) Howeverif initially Bs

t gt B then a higher interest rate even though it decreases invest-ment could at the same time increase the number of bonds supplied As Patinkin(1965 217) observes

consider the effect of an increase in the rate of interest (fall in 1rt) The inter-nal consistency of our model requires that this decrease the amount of realbonds supplied

However this need not reduce the number of bonds supplied As Patinkin(1965 217) continues

a rise in the interest rate ( pbt falls) has lowered the price received for bondsand so may increase the number of bonds necessary to finance the firmrsquosexpenditures on investment commodities (Bs

t gt B initially) even thoughthese expenditures have decreased

A similar analysis would apply to a comparison of householdsrsquo demand for bondsin terms of numbers with household net real financial asset demand

Money ldquomarketrdquo equilibrium

As we know from Walrasrsquo law having depicted equilibrium in the labor andfinancial markets we need only look at one of the other two remaining markets

the output market or the money ldquomarketrdquo Consider a money market in whichnominal money supply M and nominal money demand M d

t equiv ptLdt are plotted

against the reciprocal of the price level which indicates the ldquorelativerdquo price of oneunit of money in terms of output If there is no real balance effect with respect toreal money holdings then the money demand curve is invariant to a change in theprice level (elasticity of demand equals 1) If there is a real balance effect withrespect to money demand then a fall in the price level (a rise in the relative price ofmoney (1pt)) would lead to a less than proportionate decrease in nominal moneydemand (elasticity of demand less than 1)

Aggregate supply and demand an introduction

Temporary equilibrium for the economy can be characterized in several ways Aswe alluded to above one way common in journal articles and textbooks is to dividethe analysis under the headings of ldquoaggregate supplyrdquo and ldquoaggregate demandrdquo

Under the heading of ldquoaggregate supplyrdquo is an analysis of the input markets inparticular the labor market One aim is to determine input prices (in particular themoney wage) the employment of inputs and the implied production of output thatoccur at different levels of output prices (and potentially different levels of interestrates) The term ldquoaggregate supplyrdquo is applied to this analysis for the simple reasonthat it determines the ldquosupplyrdquo of total output at different prices

Under the heading of ldquoaggregate demandrdquo is an analysis of the other marketsin the economy during period t in particular the output financial and moneymarkets The aim is to determine the level of output prices and the interest ratethat occur at different levels of output The term ldquoaggregate demandrdquo attached tothis analysis reflects the fact that the analysis determines how the price level andinterest rate adjust to equate the ldquodemandrdquo for total output to different levels ofproduction

Combining the aggregate supply and demand analysis we can determine theoutput price level and interest rate associated with temporary equilibrium as wellas the underlying equilibrium money wage real wage employment consumptioninvestment and real money balances To understand more clearly what is involvedin aggregate supply and demand analysis and how they can be combined we con-sider below the specific case of the neoclassical model starting with the aggregatesupply

Equilibrium and aggregate supply

As we have said behind aggregate supply is an analysis of various input marketsto determine the response of total output to changes in such variables as theprice level7 In the neoclassical model changes in prices lead to equiproportionatechanges in money wages with no change in the equilibrium level of employmentand thus no change in aggregate supply To formally show this let us start withthe following statement of equilibrium in the labor market in terms of a money

wage wt and level of employment Nt such that

N dt (wtpt K) minus Nt = 0

N st (wtpt) minus Nt = 0

A critical aspect of the above is the fact that suppliers in particular suppliers oflabor correctly anticipate the price level that will exist with respect to output sothat wtpt replaces wtpe

t in the labor supply functionTotally differentiating the above two equations with respect to wt Nt and pt

one obtains[(partN d

t part(wtpt))(1pt)

(partN st part(wtpt))(1pt)

minus1minus1

] [dwtdNt

]=[(partN d

t part(wtpt))(wtdpt( pt)2)

(partN st part(wtpt))(wtdpt( pt)

Applying Cramerrsquos rule one obtains

dwtdpt = wtpt and dNtdpt = 0

Thus in the neoclassical model the real wage and employment level determinedin the labor market are independent of changes in the price level8

The ldquoaggregate supply equationrdquo combines the analysis of the labor market (andother input markets) and resulting determination of employment of various inputswith the production function to determine the resulting output supplied For theneoclassical model the aggregate supply equation is of the form

ylowastt = ylowast

t (K ) (61)

What is important about this equation is that the price level and interest rateare not arguments in the supply equation Of course changes in the capital stockchanges in technology or changes in the supply of other inputs (eg changes inthe oil supply) can affect output Similarly a change in the composition of thelabor force or government policies that affect labor supply can affect equilibriumemployment and thus output9

We may summarize the above findings graphically Consider an increase in pt Given perfect foresight both firms and households at time t would anticipate thishigher price level In the labor market the result would be an increase in theequilibrium money wage in the same proportion as the increase in the expectedprice level so that the anticipated real wage would remain the same as wouldemployment and output This outcome for the labor market is shown in Figure 61Note that the result of no change in employment or output in light of a higher outputprice simply requires that both labor demand and supply curves shift vertically bythe same amount Such equal shifts reflect the fact that the same increase in themoney wage leaves both firms and households anticipating the same real wage asbefore

In undergraduate textbooks the fact that changes in the price of output leaveemployment and real output unaffected in the neoclassical model is often shown

N demand

N supply

Figure 61 Labor market equilibrium

in ( pt yt) space by a vertical ldquoaggregate supply curverdquo Such a curve summarizesthe underlying behavior for the economy-wide labor market Later we will see thatunder other assumptions such as embedded in Lucasrsquos model (ldquomoney illusionrdquo)and the Keynesian model (fixed nominal wage) the aggregate supply curve willbe upward sloping

It is important to realize that the aggregate supply shown in Figure 62 is not thetypical market supply curve of microeconomics In microeconomics a higher priceof good x and consequent increase in the demand for inputs by firms producing thatgood draws inputs away from the production of other goods so that the higherprice of good x induces increased output of that good and associated increasedemployment of inputs (such as labor) in the production of the good Implicit inthis analysis is that there is reduced production of other goods in the economyHowever as the above analysis makes clear if the focus is on the aggregation ofall commodity markets a higher price level no longer induces increased aggregateoutput unless the quantity of total inputs supplies rises which will not be the caseunder neoclassical assumptions10

The natural rate of unemployment

The key feature of the above analysis of aggregate supply is the assumption thatprices adjust to continuously maintain equilibrium in the various markets and thatindividuals are perfectly informed concerning prices The result is a level of realoutput sometimes called the ldquofull employment levelrdquo So far missing from theanalysis however is any mention of unemployment If one expands the model tointroduce unemployment the rate of unemployment is called the ldquonaturalrdquo rate11

By ldquonaturalrdquo is meant that it is the rate of unemployment that the economy will

Output

Aggregate supply

Figure 62 Aggregate supply

gravitate to as prices adjust to clear markets and individuals become fully informedconcerning prices (ie under the assumptions of the neoclassical model)

To introduce unemployment into the analysis when the labor market is in equilib-rium at full employment requires recognition of two facts First the labor marketis in a constant state of flux Not only do individuals enter and leave the laborforce continually but labor demand varies continuously across firms as they expe-rience variations in the relative demand for their output This instability of jobsthemselves has been estimated to account for roughly one-quarter of the averageunemployment rate as in an average year one in every nine jobs disappears and onein every eight is newly created (Leonard 1988) Second information is costly Ittakes time for new workers entering the labor force and for workers who have beenlaid off or have quit previous jobs to discover which employers have vacanciesand how wages vary across employers

When we take into account the continuous flows to unemployment together withworkersrsquo imperfect information about job vacancies we see that unemploymentis no longer inconsistent with the neoclassical model At any moment there existnew entrants into the labor market who are spending time searching for acceptablejobs There also exist laid-off workers who are either searching for alternativejobs or awaiting recall And there are workers who have quit their jobs and aresearching for other jobs This kind of unemployment is generally referred to asfrictional unemployment

Sometimes part of frictional unemployment is called structural unemploymentwith structural unemployment occurring because of a change in the compositionor ldquostructurerdquo of aggregate output across firms For example the replacement ofsteel with plastic in automobiles led to a shift in employment from steel factoriesto firms making plastic During this transition some steel workers experiencedstructural unemployment

To summarize when the labor market is in equilibrium there exists a positiveunemployment rate because workers continuously move into and out of the laborforce and between jobs12 To signify that this unemployment rate is ldquonaturalrdquo orconsistent with equilibrium in the labor market it is generally called the naturalrate of unemployment13 The corresponding level of employment is then oftenreferred to as the full employment level

Like equilibrium output and employment in the neoclassical model ldquosupplyfactorsrdquo determine the natural rate of unemployment as well14 Among these isthe demographic composition of the labor force but also unemployment insuran-ce and minimum wage legislation In recent years a large body of literature hasanalyzed labor markets and the sources of unemployment with the focus on searchand labor contracts Note that an understanding of the natural rate of unemploy-ment is important in determining the effects of government policies aimed at theunemployed

Equilibrium and aggregate demand

The aggregate demand side of macroeconomic models considers the equilibriumconditions of two of the remaining three markets in particular the output market(reflected by an ldquoISrdquo equation) and the money market (reflected by an ldquoLMrdquo orldquoportfoliordquo equation) The ldquoISrdquo equation since it is simply the expression forequilibrium during period t with respect to the output market is given by15

cdt + I d

t = 0 (62)

Note that the equation is termed the ldquoISrdquo equation because we can rewrite it toobtain

Idnt + δK + ψ(I d

nt) = ylowastt minus cd

indicating that equilibrium in the output market is equivalent to the equal-ity between Investment expenditures (the left-hand side of the equation) andhousehold Saving (the right-hand side of the equation)16

The assumption of unit elastic expectations concerning wages and prices givesthe following simple form for householdsrsquo consumption demand and firmsrsquoinvestment demand functions during period t17

cdt = cd

t (rt πet At Mpt ylowast

Idnt = I d

nt(met + δ K)

The ldquoLMrdquo equation since it is simply the expression for equilibrium in period twith respect to the ldquomoneyrdquo market is given by

Ldt = Mpt (63)

This equation is termed the ldquoLMrdquo equation for it reflects the equality betweenLiquidity preference or demand and the Money supply We have seen that ourassumption of unit elastic expectations concerning wages and prices gives us thefollowing simple form for the real money demand function

Ldt = Ld

From the modified Walrasrsquo law it follows that a price level and interest rate thatsatisfy the ldquoISrdquo equation and ldquoLMrdquo equation for a given level of output ylowast

t willalso result in equilibrium in the financial market

We thus have three equations (the aggregate supply equation (61) the ldquoISrdquoequation (62) and the ldquoLMrdquo equation (63)) that can be solved for the equilib-rium levels of output price and interest rate Looking at what underlies thesethree equations we can then infer the changes in money wages and employment(the labor market) as well as changes in the components of output demand (invest-ment and consumption demand) Equilibrium can be depicted in terms of a pricelevel and interest rate (Patinkinrsquos CCndashLLndashBB curves) or in terms of a price leveland output (ie aggregate demand and supply curves) We start with Patinkinrsquosdepiction of equilibrium

Depiction of equilibrium the Patinkin analysis

The neoclassical model aggregate supply is independent of changes in the pricelevel and interest rate18 This means that for any given level of output we canfocus on equations (62) and (63) to determine the equilibrium interest rate andprice level This is a reflection of the ldquoblock recursiverdquo nature of the solution tothe neoclassical model mentioned earlier

Patinkin (1965) suggests a graphical way of showing such an equilibrium com-bination of price level and interest rate using any two of three curves denoted theCC LL and BB curves The CC curve depicts combinations of the price level andinterest rate that satisfy the equilibrium condition for output (62) the LL curvedepicts combinations that satisfy the equilibrium condition for money (63) Wherethese two lines so constructed intersect it must then be the case that this price com-bination satisfies two of the three equilibrium conditions simultaneously It followsfrom the modified Walrasrsquo law that the curve indicating various combinations ofthe price level and interest rate that satisfy the equilibrium condition with respectto the financial market (the BB curve) goes through this point as well19

For the market for output equilibrium in period t is characterized by (62) Recallthat real output ylowast

t denotes that output reflecting the capital stock K technologyand the employment of labor determined in the labor market at time t From theneoclassical assumptions with respect to labor demand and supply equilibriumemployment and thus output are unchanged for any change in the price level orinterest rate

Let us presume that the pair ( plowastt rlowast

t ) is associated with equilibrium in the outputmarket In (price interest rate) space this combination is identified by a unique

point on the CC curve that identifies combinations of pt and rt associated withequilibrium in the commodity market (Figure 63)

To understand what lies behind the shape of the CC curve depicted above totallydifferentiate the market-clearing condition for the commodity market with respectto pt and rt Doing so we obtain20

[partcdt part(Mpt)] minus [Mp2

t ]dpt + [partcdt partrt + (1 + ψ prime)partI d

ntpartrt]drt = 0

Rearranging we have that the slope of the CC curve is given by

drtdpt | ydt minus ylowast

t = 0= [partcd

t part(Mpt)][Mp2t ][partcd

t partrt + (1 + ψ prime)partI dntpartrt] lt 0

The negative slope of the CC curve can be explained in the following way Thepositive numerator of the expression for the slope reflects the real balance effectwith respect to commodities this term indicates the fall in consumption demandthat would accompany a rise in the price level for such a rise reduces agentsrsquowealth in the form of initial real money balances21 The negative denominatorindicates the effect of a change in the interest rate on consumption and investmentdemand

Now let us consider the money market As before there is a unique point thatindicates an interest rate and a price of output at which there is equilibrium inthe economy and this point on the LL curve identifies combinations of pt andrt associated with equilibrium in the money market (Figure 64) Recall that theLL curve identifies combinations of pt and rt associated with equilibrium in themoney ldquomarketrdquo as given by (63)

The slope of the LL curve is given by totally differentiating the zero excessdemand condition with respect to money Doing so and rearranging one obtains

drtdpt |Ldt minus Mpt = 0 = [1 minus partLd

t part(Mpt)][Mp2t ][minuspartLd

t partrt] gt 0

Figure 63 Commodity market

Figure 64 Money market

Note that the numerator of this term is positive The change in real money balanceswill be greater than the consequent change in real money demand given real balanceeffects with respect to financial assets andor commodities The denominator ispositive as well reflecting the fact that an increase in the interest causes householdsto shift their portfolio out of money holdings into bonds

Putting together the CC and LL curves we have equilibrium in the economyat the point where these two curves intersect From the modified Walrasrsquo law weknow that at this point demand for financial assets equals supply as well In factthere is a corresponding BB curve that goes through this same intersection

As with the CC curve to understand what lies behind the shape of the BB curvewe can totally differentiate the market-clearing condition for the financial marketwith respect to pt and rt That market-clearing condition is

net Adt = net As

where in simplest form

net Ast = net As

t (met + δ K)

net Adt = net Ad

Differentiating we obtain

[partnet Adt part(Mpt)] minus [Mp2

t ]dpt + [partnet Adt partrt minus partnet As

t partrt]drt = 0

Rearranging we have that the slope of the BB curve is given by

drtdpt |net Adt minus net As

t = 0= [partnet Ad

t part(Mpt)][Mp2t ][partnet Ad

t rt minus partnet Ast partrt] gt 0

The positive slope of the BB curve can be explained in the following way Thepositive numerator of the slope expression reflects the real balance effect withrespect to financial assets The term indicates the fall in net real financial assetdemand (and planned decrease in future consumption) that would accompany arise in the price level for such a rise reduces agentsrsquo wealth in the form of initialreal money balances The positive denominator indicates the effect of a change inthe interest rate on household lending and firm borrowing A rise in the interestrate would tend to decrease current consumption demand (ldquoFisherianrdquo effect) andmoney demand (ldquoportfoliordquo effect) and thus increase household net financial assetdemand On the other hand a rise in the interest rate would tend to reduce firminvestment demand by raising the expected real user cost of capital and thusreduce firm net real financial asset supply

While both the slope of the BB curve and the LL curve are positive it is thecase that the LL curve is steeper

A depiction of equilibrium aggregate demand andsupply curves

Using Patinkinrsquos analysis a higher output would require a lower price level tomaintain equilibrium in the output financial and money markets Alternativelyone can show the effect of a change in ylowast

t on pt and rt by totally differentiatingequations (62) and (63) with respect to pt rt and ylowast

t The aggregate demand curve summarizes this inverse relationship between the

price level and output arising from an analysis of the output financial and moneymarkets Specifically such a curve depicts combinations of output and price levelassociated with equilibrium in the output financial and money markets It isdownward sloping indicating that an increase in output requires a lower pricelevel to clear the output financial and money markets Behind a movement downthe aggregate demand curve are larger real balances that stimulate output demandeither directly (through a real balance effect on consumption demand) or indirectly(the resulting increase in real money balances leads to an increase in net realfinancial asset demand by households and thus a lower interest rate)22

It is important to realize that the phrase ldquoequilibrium in the output mar-ketrdquo in this context abstracts from supply-side considerations At each pricelevel the aggregate demand curve indicates the output that if produced wouldequal output demand (along with satisfying the equilibrium conditions withrespect to the financial and money markets) Production of output equal tothat demanded would occur if firms sought simply to produce to meet mar-ket demand and if workers were readily available for employment so thatfirms could hire to achieve production equal to what was demanded The termldquoequilibrium in the output marketrdquo does not imply equality between outputdemand and the output that our analysis of the labor market suggests would besupplied

Output

Aggregate supply

Aggregate demand

Figure 65 Macroeconomic equilibrium

Figure 65 combines the aggregate demand and aggregate supply curves Wherethey intersect we know by construction that the resulting price level ( pt)0 andoutput ( yt)0 are such that

bull the labor market is in equilibrium (the economy is at a point on the aggregatesupply curve) and

bull the output financial and money markets are in equilibrium (the economy isat a point on the aggregate demand curve)

Conclusion

Using a very simple framework this chapter has developed a powerful macroeco-nomic model of the economy This model is grounded in the general equilibriumtheory set forth in earlier chapters Moreover this model highlights that degree ofconnectedness that exists among the output labor financial and money marketsThe microfoundations of macroeconomics have been emphasized and it has beenshown how market forces work in such a way as to lead to aggregate supply anddemand two key elements in any macroeconomic model

7 Empirical macroeconomicsTraditional approaches and timeseries models

Introduction

Quantitative approaches to analyzing economic data provide meaningful anduseful insight for understanding how variables interact and how they might beexpected to behave in a variety of circumstances including the future This chapteroutlines the traditional econometric-based method and the relatively simple butoften more elegant time series method for analyzing economic data in the timedomain The stochastic nature of economic data is discussed and the now com-mon ARIMA model found in much of the empirical macroeconomic literature isdeveloped in parts The chapter provides a solid background for understandingldquomacroeconometricsrdquo and time series analysis

Traditional approaches

Empirical macroeconomics can be roughly divided into two approaches ndash a tradi-tional approach that draws heavily on macroeconomic theory and a more recentapproach advocated for forecasting that does not rely to any great extent on the-ory To consider the former we start by presenting a simple theoretical model ofthe macroeconomy The model is used to illustrate traditional empirical analysesreflecting (a) tests of behavioral hypotheses (b) tests of reduced-form expressionsfor various economic aggregates and (c) the construction of econometric modelsfor policy simulations and forecasting The more recent empirical macroeconomicsapproach of using time series models for forecasting aggregate variables whichplaces less reliance on macroeconomic theory is then briefly considered

A simple theoretical macroeconomic model

Consider the following linear approximation of the simple static classical model(closed economy) such as that popularized by Sargent (1987a 20)1 The economyis divided into four markets a labor market (where wages w and total employ-ment n are determined) an output market (where total output y and the pricelevel p are determined) a financial market (where the interest rate r is deter-mined) and a money ldquomarketrdquo Our goal is to construct a model that will determine

Empirical macroeconomics 97

such endogenous variables as the level of employment wages output prices andthe interest rate

From Walrasrsquo law (as we will see more clearly later on) we need only explicitlyconsider three of the above four markets in the analysis For the labor market letus assume the following linear approximations for the key behavioral relations2

nd = f minus g(wp) (71)

ns = h + j(wp) (72)

These two linear equations indicate that firmsrsquo labor demand nd is inversely relatedto the real wage (wp) and householdsrsquo labor supply ns is directly related to thereal wage3

For the output market the key underlying behavioral and technological relationsare in linear form

cd = a + b( y minus T ) minus cr (73)

id = d minus er (74)

ys = f (n K) (75)

Equation (73) indicates that householdsrsquo consumption demand cd is directlyrelated to real disposable income (income y minus lump-sum taxes T ) andinversely related to the interest rate r4 Equation (74) indicates that firmsrsquo invest-ment demand id is inversely related to the interest rate r Equation (75) is theaggregate production function relating employment n and capital stock k tototal output supplied ys

For the money ldquomarketrdquo the key underlying behavioral relation is

Ld = l middot y minus m middot r (76)

Equation (76) indicates that householdsrsquo real money demand Ld is directly relatedto real income and inversely related to the interest rate Nominal money supplyM s is exogenous

General equilibrium in this economy means an equilibrium level of employmentnlowast wage wlowast price level plowast output ylowast and interest rate rlowast such that the labor marketis in equilibrium

ns minus n = 0 (77)

nd minus n = 0 (78)

the output market is in equilibrium

ys minus y = 0 (79)

cd + id + gd minus y = 0 (710)

98 Empirical macroeconomics

and the money ldquomarketrdquo is in equilibrium

Ld minus M sp = 0 (711)

Note that the component of output demand gd in (710) reflects exogenous realgovernment demand for output

The above macroeconomic model consists of 11 equations Macroeconomicmodels often can be represented by a system of equations This system of equationsis sometimes said to be a ldquostructural modelrdquo because the form is given from theunderlying theory As we will see below we can solve this system of equationsfor each of the ldquoendogenousrdquo variables as a function solely of the predeter-mined or ldquoexogenousrdquo variables5 Such solutions can be called the ldquoreduced-formrdquosolutions

By substituting the behavioral equations (71)ndash(76) into the equilibrium condi-tions we obtain the following set of five equilibrium conditions that can be solvedfor the five variables nlowast wlowast plowast ylowast and rlowast6

f minus g(wp) minus n = 0

h + j(wp) minus n = 0

f (n K) minus y = 0

a + by minus bT minus cr + d minus er + gd minus y = 0

ly minus mr minus M sp = 0

Note that the first three equations in the system (712) can be solved to obtainwlowast nlowast and ylowastas a function of the price level plowast In particular we obtain7

wlowast = [(f minus h)(g + j)]plowast (713)

nlowast = [1( j + g)][ jf + gh] (714)

ylowast = f (nlowast K) = f ([1( j + g)][ jf + gh] K) (715)

Equation (715) is an example of a reduced-form expression for output for itexpresses output solely in terms of the exogenous variables It is a special caseof what is called the ldquoaggregate supply equationrdquo in which the price level doesnot affect the level of output produced8 Note that equation (713) indicates thatchanges in the price level lead to equiproportionate changes in the equilibriummoney wage so that changes in the price level do not lead to changes in theequilibrium real wage

To obtain a reduced-form expression for the price level we can use the reduced-form expression for output in conjunction with the last two equations in the system(712) These last two equations are sometimes termed the ldquoIS equationrdquo andthe ldquoLM equationrdquo respectively9 Solving the LM equation (the last equation ofthe system (712)) for r and substituting both this expression for r and the priorexpression for equilibrium output ylowast (715) into the IS equation (the penultimate

equation of the system (712)) we obtain the reduced-form expression for theequilibrium price level

plowast = M scprimem

[1 minus b + cprimem] f ([1( j + g)][ jf + gh] K) minus aprime (716)

where aprime = a + d + gd minus b middot T and cprime = c + eChanges in the variable aprime indicate changes in the autonomous component of

consumption (the term a in (73)) autonomous investment demand (d in (74))and government spending or taxing (gd or T respectively) By ldquoautonomousrdquo wemean that part of householdsrsquo and firmsrsquo output demand that is independent of thevariables to be determined by the analysis particularly income and the interestrate The variable cprime indicates the combined response of consumption demand (cin (73)) and investment demand (d in (74)) to a unit change in the interest rate

Note that the procedure to obtain equation (716) could be alternatively describedas follows First combine the last two equations in the system (712) to eliminatethe interest rate r The result is termed the ldquoaggregate demand equationrdquo Thisrelates the price level to the level of output at which the money and output markets(and thus by a modified version of Walrasrsquo law the financial market) are inequilibrium In this context equilibrium in the output market is defined as thelevel of production that if produced would equal output demand This aggregatedemand equation would then be combined with the aggregate supply equation(715) to determine the equilibrium price level or output10

Tests of behavioral hypotheses

Theoretical macroeconomic models embody predictions concerning factors thatinfluence the behavior of various groups in the economy For instance considerthe behavioral equations (71)ndash(76) in the prior simple macroeconomic modelWe could test the prediction that investment is inversely related to the interest rate(see (74)) or that money demand depends inversely on the interest rate (see (76))Or we could expand our theory of consumption behavior (equation (73)) to testa particular behavioral relationship between aggregate household consumptionand permanent income11 Or we could expand our view of labor supply behavior(equation (72)) as Stuart (1981) did in an examination of Swedish data to test theprediction that sufficiently high marginal tax rates will reduce the economy-widelabor supply

Tests of reduced-form hypotheses

Theoretical macroeconomic models also generate predictions concerning thereasons for fluctuations in such aggregate variables as real output unemploy-ment and the level of prices These predictions typically reflect the reduced-formsolutions of macroeconomic models Examples of reduced forms in our simplemacroeconomic model are equations (715) for output and (716) for price

One example of an empirical test of macroeconomic theory that focuses ontesting the reduced-form predictions is Taylor (1979) Specifically Taylor teststhe reduced-form relationships between the money supply and the logarithmof income from trend and between the money supply and the rate of infla-tion using quarterly US data from 1953 to 197512 Similarly Christopher Sims(1972) uses quarterly US data on nominal output and the money supply to testwhether changes in the money supply lead to changes in the current dollar valueof national income (see Ewing 2001)

Large-scale econometric models and forecasting

Besides testing macroeconomic theories empirical macroeconomic analysis oftenseeks to forecast the future paths of economic aggregates Interest in forecast-ing stems not only from an obvious curiosity about the future path of aggregatevariables such as real output and prices but also from the fact that many ifnot all macroeconomic theories suggest that expectations of future events influ-ence current activity In this context forecasts can be used to proxy individualsrsquoexpectations of future events in tests of various aspects of macroeconomic models

One approach to making forecasts of future aggregate variables is to rely ontheoretical macroeconomics as a guide in the construction of large-scale econo-metric models Behavioral equations that are more detailed disaggregated versionsof equations (71)ndash(76) are estimated and coefficients are checked to make surethey agree with theory These models reflect attempts to produce large systems thatfaithfully represent the interrelationships in a complex national economy Givenpostulated paths of the exogenous variables the actual estimated equations arethen used to generate forecasts of the various aggregate variables such as outputand its components (eg consumption and investment) prices and interest rates

While large-scale econometric models as forecasting devices have their advo-cates (and many individuals reveal they have a positive value by willinglypaying for their forecasts) Granger and Newbold (1986 292ndash293) among oth-ers question the value of such a forecasting approach They note that ldquoteamsof macroeconomists have constructed forecasting models involving hundreds ofsimultaneous equations fitted to data that time series analysts would view as neitherplentiful nor of especially high qualityrdquo

An alternative to the large-scale macroeconomic models that is suggested areldquotime series modelsrdquo As summarized by Granger and Newbold (1986) ldquoin theiranxiety econometricians have failed to touch some very important basesrdquo thatinclude

bull the fact that there are many areas in which ldquoeconomic theory is not terriblywell developedrdquo

bull the fact that even where the theory is satisfactory it is ldquoalmost invariablyinsufficiently precise about dynamic specifications in the sense that it is clearthat one structure must be appropriaterdquo

bull the fact that even after appeal to economic theory there will be error termssince ldquono theory provides a completely accurate description of the behaviorof economic agents so that any postulated equation necessarily includes astochastic error termrdquo

Given these problems many macroeconomists suggest that the appropriate start-ing point to forecasting future macroeconomic variables is the use of time seriesmodels One result is that such time series modelling terms as AR ARMA andARIMA now abound in the macroeconomic literature It is thus useful to brieflyreview the nature of time series models However at the outset it should be madeclear that the discussion below is not complete but rather is provided simply asan introduction to some terms and concepts Proofs of various propositions andrigorous definitions of various properties of time series models can be found intime series texts such as Enders (2004) and Mills (1999)

Time series models

Let us take as our premise the idea that the actual observed time series of somevariable yt t = 0 1 T (eg the logarithm of economy-wide output for thelast 30 years) is the realization of some theoretical process which can be called aldquostochastic processrdquo As phrased by Harvey (1993) ldquoeach observation in a stochas-tic process is a random variable and the observations evolve in time according tocertain probabilistic laws Thus a stochastic process may be defined as a collectionof random variables which are ordered in timerdquo To forecast future values of yt one needs a model that defines the mechanisms by which the observations aregenerated

A distinguishing feature of a pure univariate time series model is that move-ments in yt are ldquoexplainedrdquo solely in terms of its own past or by its position inrelation to time13 That is time series models look for patterns in the past move-ments of a particular variable and use that information to predict future movementsof the variable In general a time series modelrsquos forecast of y based on knownvalues yT+1 is given by

yT+1 = E( yT+1|y0 yT )

As Pindyck and Rubinfeld (1991) suggest ldquoin a sense a time-series model is just asophisticated method of extrapolation yet it may often provide a very effec-tive tool for forecastingrdquo In this sense time series models are more along the linesof empirical analyses that ldquolet the data speak for themselvesrdquo rather than empiricalanalyses that (strictly speaking) ldquotest economic theoriesrdquo As Harvey (1993) statesldquoan essential feature of time series models is that they do not involve behaviouralrelationshipsrdquo Time series models reflect a ldquostatisticalrdquo approach to forecastingHowever the patterns in the data discovered by time series models do influencetheoretical discussions of the macroeconomy and tests of macroeconomic theo-ries concerning behavior have incorporated time series models14 An example of

how time series analysis has influenced theoretical macroeconomic discussions isgiven at the end of this section

Some properties of stochastic processes

There are several key properties of stochastic processes for time series that wecan introduce One is ldquostationarityrdquo For a stochastic process to be stationary thefollowing conditions must be satisfied for all t

E( yt) = micro

E[( yt minus micro)2] = σ 2y

E[( yt minus micro)( ytminusk minus micro)] = γk

Note that γ0 = σ 2y In words if a series is stationary the mean of the series is

invariant to time the variance of the series is invariant to time and the covarianceof the series is invariant to time As Pindyck and Rubinfeld (1991) note ldquoif astochastic process is stationary the probability distribution p( yt) is the same forall time t and its shape (or at least some of its properties) can be inferred by lookingat a histogram of the observations y1 yT that make up the observed seriesrdquo15

Also an estimate of the mean micro of the process can be obtained from the samplemean of the series

Tsumj=0

and an estimate of the variance σ 2y can be obtained from the sample variance16

σ 2y = 1

Tsumj=0

( yt minus y)2

As Granger and Newbold (1986 4) phrase it ldquoa stationarity assumption is equiv-alent to saying that the generating mechanism of the process is time-invariantso that neither the form nor the parameter values of the generation procedurechange through timerdquo The simplest example of a stationary stochastic process isa sequence of uncorrelated random variables with constant mean and variance

A second property of a stochastic process is the ldquoautocorrelation functionrdquo Theautocorrelation function provides us with a measure of how much correlation thereis (and by implication how much interdependency there is) between neighboringdata points in the series yt For stationary processes the autocorrelation with lagk is given by17

ρk = γkγ0

= E[( yt minus micro)( ytminusk minus micro)]σ 2y

For any stochastic process ρ0 = 1 If the stochastic process is simple ldquowhitenoiserdquo (ie yt = εt where εt is an independently distributed random variablewith zero mean and finite variance) then the autocorrelation function for thisprocess is given by

ρ0 = 1 ρk = 0 for k gt 018

A simple example of a stochastic process is the ldquorandom walkrdquo process inwhich each successive change in yt is drawn independently from a probabilitydistribution with zero mean19 Thus yt is determined by

yt = ytminus1 + εt (717)

where εt is a sequence of uncorrelated random variables (E(εtεs) = 0 for t = s)with mean zero (E(εt) = 0) and constant variance (E(ε2

t ) = σ 2e for all t) Recall

that a sequence εt of this kind is typically called ldquowhite noiserdquo If the stochasticprocess is a random walk the one-period-ahead forecast of yt+1 is simply yt

A simple extension of the random walk process is to incorporate a trend in theseries yt We then obtain the following stochastic process known as a random walkwith drift20

yt = ytminus1 + d + εt (718)

The one-period-ahead forecast of yt+1 is now yt + d By repeatedly substitutingfor past values of yt into (718) we obtain

yt =tminus1sumj=0

εtminusj + td + y0 (719)

For the random walk process without drift (d = 0) we see from (719) that thefirst requirement for stationarity namely that the mean be constant over time issatisfied if y0 is fixed21 That is E( yt) = E( y0) Nevertheless the process is notstationary since Var( yt) = tσ 2

e The random walk process tends to meander awayfrom its starting value but exhibits no particular trend in doing so22

Autoregressive processes

A process similar to the random walk that is stationary is called the ldquofirst-orderautoregressive processrdquo or AR(1)23

The AR(1) process is given by

yt = φytminus1 + (1 minus φ)micro + εt (720)

A necessary condition for stationarity is that |φ| lt 1 in which case E( yt) equalsthe constant term micro in (720)24

One possible example of an autoregressive process is the total numberunemployed each month (Granger and Newbold 1986) Let the total number unem-ployed in one month be yt This number might be thought to consist of a fixedproportion φ of those unemployed in the previous month (the others having foundemployment) plus a new group of workers seeking jobs If the new additionsare considered to form a white noise series with positive mean micro(1 minus φ) thenthe unemployment series is a first-order autoregressive process expressed byequation (720)

For convenience only it is often assumed that the process has a zero mean iemicro = 0 Note that we can always define a new variable yprime

t equiv yt minus micro such that yprimet

has a zero mean and is given by the AR(1) process

yprimet = φyprime

tminus1 + εt (721)

with |φ| lt 1 Setting micro = 0 thus simply means that the variable yt (egemployment real output etc) is measured in terms of deviations from its mean

By successive substitution for yt in (720) we obtain (assuming micro = 0)

yt =tminus1sumj=0

φjεtminusj + φty0 (722)

If the process is regarded as having started at some point in the remote past and|φ| lt 1 then we can write

yt =infinsum

φjεtminusj (723)

Given that εt is white noise we thus have that E( yt) = micro = 0 Assumingstationarity (|φ| lt 1) we know that the variance and covariances are constantRecall that εt is white noise such that E(εtεs) = 0 for s = t We thus have25

γ0 = σ 2y = E[( yt minus micro)2] = E

⎡⎢⎣⎡⎣ infinsum

φj(εtminusj)

⎤⎦

2⎤⎥⎦ = σ 2

e (1 minus φ2)

γ1 = E[( yt minus micro)( yt+1 minus micro)] = φσ 2e (1 minus φ2)

γ2 = E[( yt minus micro)( yt+2 minus micro)] = φ2σ 2e (1 minus φ2) etc

The autocorrelation function for AR(1) is thus particularly simple ndash it begins atρ0 = 1 and then declines geometrically ρk = φk Note that this process hasan infinite memory The current value of the process depends on all past valuesalthough the magnitude of this dependence declines with time

In general an autoregressive process of order p is generated by a weighted aver-age of past observations going back p periods together with a random disturbance

in the current period The AR( p) process is thus given by26

yt = φ1ytminus1 + φ2ytminus2 + φ3ytminus3 + middot middot middot + φpytminusp

+ (1 minus φ1 minus φ2 minus middot middot middot minus φp)micro + εt

= micro + εt +psum

φj( ytminusj minus micro)

with a necessary condition for stationarity being that φ1 + φ2 + middot middot middot + φp lt 127

The next section discusses necessary and sufficient conditions for stationarity

A brief digression on necessary and sufficient conditionsfor stationarity

To get some idea of what are necessary and sufficient conditions for stationaritylet us consider two specific cases AR(1) and AR(2) In the AR(1) case

yprimet minus φyprime

tminus1 = εt

where yprimet equiv yt minus micro Note that εt can be termed the ldquostochasticrdquo component of the

expression and the remainder the ldquodeterministicrdquo component28

Focusing on the deterministic component and assuming for convenience thatmicro = 0 we have a homogeneous first-order difference equation of the form29

yt minus φytminus1 = 0 (725)

To solve this equation let us try the general solution

yt = Abt (726)

which naturally implies ytminus1 = Abtminus130 The problem then is to find the valuesof A and b Substituting the trial solution into the above difference equation weobtain

Abt minus φAbtminus1 = 0

which by multiplying through by b1minustA can be rewritten as

b minus φ = 0 (727)

Equation (727) is called the ldquoauxiliaryrdquo or characteristic equation of (725) Thisequation provides us with the solution for b which is simply that b = φ Thissolution value is sometimes referred to as the ldquorootrdquo of the characteristic equation

Using (727) to substitute for b in (726) we thus have as the solution to thefirst-order difference equation (725) the expression

yt = Aφt (728)

The ldquoequilibriumrdquo solution for yt given a homogeneous difference equation suchas (725) is yt = 0 That is if yt equiv 0 then yt will not change over time Thisequilibrium solution is ldquostablerdquo if as t rarr infin yt rarr 0 That is deviations fromthe equilibrium yt will return toward that equilibrium value Obviously given oursolution (728) a necessary and sufficient condition for stability is that |φ| lt 1

In the context of an AR(1) process the notion of ldquostationarityrdquo corresponds tothe notion of stability for the first-order homogeneous difference equation (725)that is the deterministic component of yt for an AR(1) process The condition ofstationarity is thus that |φ| lt 1 The effect of a given shock will mitigate overtime if this stationarity condition is met A special nonstationary case is the ldquounitrootrdquo case of the random walk process in which φ = 1 In that case the effect ofa given shock will not dampen with the passage of time

Now let us consider the AR(2) case where

yprimet minus φ1yprime

tminus1 minus φ2yprimetminus2 = εt

in which yprimet equiv yt minus micro As before εt can be termed the ldquostochasticrdquo component of

the expression Assuming again for convenience that micro = 0 we can express thedeterministic component as a homogeneous second-order difference equation ofthe form

yt minus φ1ytminus1 minus φ2ytminus2 = 0 (729)

To solve this equation let us try yt = Abt which naturally implies ytminus1 = Abtminus1

and ytminus2 = Abtminus2 The problem then is to find the values of A and b Substitutingthe trial solution into the above difference equation we obtain

Abt minus φ1Abtminus1 minus φ2Abtminus2 = 0

b2 minus φ1b minus φ2 = 0 (730)

The quadratic equation (730) is called the ldquoauxiliaryrdquo or ldquocharacteristic equationrdquoof the second-order difference equation (729) The roots of this equation will bethe solutions of (730)31 The two ldquocharacteristicrdquo roots m1 and m2 may be foundin the usual way from the formula32

m1 m2 = [φ1 plusmn (φ21 + 4φ2)

12]2 (731)

each of which is acceptable in the solution Abt Note that for the quadraticequation (730) the roots m1 and m2 satisfy

(b minus m1)(b minus m2) = 0

where φ1 = m1 + m2 and φ2 = minusm1m2

Expression (731) provides us with two values for b and thus two potentialsolutions for yt

yt = A1mt1 and yt = A2mt

The general solution to a second-order difference equation combines these twoby taking the sum Thus the general solution of the second-order homogeneousdifference equation (729) is33

yt = A1mt1 + A2mt

2 (732)

Inspection of (732) suggests that stability for the linear second-order homo-geneous difference equation (729) requires that both roots of the characteristicequation have an absolute value less than one This reflects the fact that the gen-eral solution of the second-order homogeneous difference equation includes bothroots m1 and m2 The root with the higher absolute value is sometimes termed theldquodominant rootrdquo If both m1 and m2 are less than unity in absolute value yt willbe close to zero if t is large Equivalently if the dominant root is less than one inabsolute value convergence will occur

In the context of an AR(2) process the notion of ldquostationarityrdquo corresponds tothe notion of stability for the second-order homogeneous difference equation (729)that is the deterministic component of the process The condition of stationarityis thus that |m1| lt 1 and |m2 lt 1| That is for the AR(2) process stationarityrequires that the roots of the characteristic equation (730) are less than one inabsolute value ndash that is that they all lie inside the unit circle34

In terms of φ1 and φ2 the conditions for stationarity may thus be defined asfollows35

φ1 + φ2 lt 1 minusφ1 + φ2 lt 1 φ2 gt minus1

As you can see the prior necessary condition that φ1 + φ2 lt 1 is augmented withtwo other conditions

We have seen that stability for the second-order homogeneous differenceequation requires that the maximum of (|m1| |m2|) ndash that is the dominant root ndash beless than 1 To see how one obtains the three necessary conditions listed above forstabilitywe assume this is the case and determine the resulting restrictions placedon the coefficients φ1 and φ236 Recall that

m1 m2 = [φ1 plusmn (φ21 plusmn 4φ2)

Suppose φ21 + 4φ2 gt 0 so that the roots are real A maximum of (|m1| |m2|) less

than one means that

2 minus φ1 gt (φ21 + 4φ2)

12 gt minus2 minus φ1

2 minus φ1 gt minus(φ21 + 4φ2)

The sum of the roots is φ1 and since we are assuming each root is between minus1and 1 it must be the case that 2 gt φ1 gt minus2 or 0 gt minus2minusφ1 and 2minusφ1 gt 0 Thusthe second and third inequalities are always true and hence place no restrictionson φ1 and φ2

The first and fourth inequalities squared read

(2 minus φ1)2 gt φ2

1 + 4φ2 and φ21 + 4φ2 lt (minus2 minus φ1)

respectively These two expressions can be rewritten to obtain

φ1 + φ2 lt 1 and minus φ1 + φ2 lt 1

which are the first two conditions for stability The third condition for stability(φ2 gt minus1) is obtained from the case of complex conjugate roots37 The threenecessary conditions for the dominant root being less than one in absolute value arealso sufficient conditions This can be verified by showing using these inequalitiesthat it is possible to reverse the steps in the above calculations and arrive at themaximum of (|m1| |m2|) being less than one

In general a necessary and sufficient condition for stationarity of an AR( p)process is if the roots of the characteristic equation

b p minus φ1b pminus1 minus middot middot middot minus φp = 0 (733)

are less than one in absolute valueNote that for the AR(2) case there are three possible situations If φ2

1 +4φ2 gt 0the square root in (731) is a real number and m1 and m2 will be real and distinctIn this case the solution to (729) is given by

yt = A1mt1 + A2mt

where A1 and A2 are constants which depend on the starting values y0 and yminus1The second situation is that of repeated roots where φ2

1 + 4φ2 = 0 such thatm1 = m2 = m In this case the solution to (729) takes the form

yt = A3mt + A4tmt

If |m| lt 1 the damping force of mt will dominate both termsThe third situation for the AR(2) process is when φ2

1 + 4φ2 lt 0 in which casethe roots are a pair of complex conjugates (ie m1 m2 = h plusmn vi where h = φ12v = (4φ2 + φ2

1)122 and i is the imaginary number (minus1)12) Thus the solutionto (729) is given by

yt = A5(h + vi)t + A6(h minus vi)t

Appealing to De Moivrersquos theorem this expression can be transformed intotrigonometric terms The general form of the solution is

yt = pt(A7 cos λt + A8 sin λt)

where p is the modulus of the roots = (minusφ2)12) and λ satisfies the two conditions

cos λ = hp and sin λ = vp As in the other two cases if the absolute valueof the conjugate complex roots |h plusmn vi| lt 1 is less than one the process isstable The time path followed by yt in response to a shock is cyclical but theperiodic fluctuation will mitigate as time passes (ldquosinusoidal decayrdquo) if the processis stationary

Moving average processes

In a moving average process the process yt is described completely by a weightedsum of current and lagged random variables The simplest moving average processthe moving average process of order 1 or MA(1) takes the form

yt = micro + εt + θεtminus1 (734)

The term ldquomoving averagerdquo reflects the fact that a series so characterized will besmoother than the original white noise series εt In general such a moving averageprocess of order q MA(q) is written as38

yt = micro + εt + θ1εtminus1 + θ2εtminus2 + middot middot middot + θqεtminusq

= micro + εt +qsum

θjεtminusj (735)

As before if yt has nonzero mean we can focus with no loss of generality on thetransformed series yprime

t = yt minus micro which has zero meanOne possible example of a moving average process is economy-wide output

(Granger and Newbold 1986) Output yt could be in equilibrium (at mean micro)but is potentially moved from its equilibrium position each period by a seriesof unpredictable events such as periods of exceptional weather or strikes If thesystem is such that the effects of such events are not immediately assimilated butexert an influence on output for q periods then a moving average model can arise

Note that by repeatedly substituting for lagged valued of εt into an MA(1)process (equation (734)) one obtains39

yt =infinsum

(minusθ)jytminusj + εt (736)

If yt is not to depend on a shock to the system arising at some point in the remotepast θ must be less than one in absolute value Comparing (736) to (724) we

see that assuming what was previously denoted the stationarity condition but whatin this context is called the ldquoinvertibility conditionrdquo namely that |θ | lt 1 then anMA(1) process can be represented by an AR(infin) process40 The weights on pastvalues of yt for this AR(infin) process decline exponentially

In general if similar invertibility conditions are met then any finite-ordermoving average process has an equivalent autoregressive process of infinite orderLikewise we have already shown (see (723)) that if the AR(1) process is station-ary it is equivalent to a moving average process of infinite order MA(infin) In factfor any stationary autoregressive process of any order there exists an equivalentmoving average process of infinite order so that the autoregressive process isldquoinvertiblerdquo into a moving average process

Mixed autoregressive moving average processes

An obvious generalization of the above discussion is to combine an AR( p) process(ie (724)) and an MA(q) process (ie (735)) An autoregressive moving averageprocess of order ( p q) or ARMA( p q) can be written as41

yt = micro + εt +psum

φj( ytminusj minus micro) +qsum

θjεtminusj (737)

where as before εt is a zero-mean white noise For simplicity consider the casewhere micro = 0

Whether or not a mixed process is stationary depends solely on its autoregres-sive part If the ARMA process is stationary then there is an equivalent MA(qprime)process Similarly provided invertibility conditions hold there is an AR( pprime) pro-cess equivalent to the above ARMA( p q) process It thus follows that a stationaryARMA process can always be well approximated by a high-order MA process andthat if the process obeys the invertibility condition it can also be well approxi-mated by a high-order AR process However an ARMA process has the advantageof ldquoparsimonyrdquo in that the mixed model ARMA( p q) often can achieve as gooda fit as say an AR( pprime) but uses fewer parameters (ie p + q lt pprime)

As an example of an ARMA( p q) process we need only consider a case wherethe variable yt is the sum of a ldquotrue seriesrdquo that is AR( p) plus a white noiseobservation error Thus an ARMA( p q) series results42

Integrated processes

In series arising in economics the assumption of stationarity is often veryrestrictive That is often the characteristics of the underlying stochastic processgenerating a time series appear to change over time With economic data some-times a transformation (such as taking the logarithm of the variable) can result in astationary series In other cases we can obtain a stationary series by differencingone or more times We say that yt is a ldquohomogeneous nonstationary process of

order drdquo if

wt = dyt

is a stationary series Here denotes differencing that is

yt = yt minus ytminus1 2yt = yt minus ytminus1 = yt minus 2ytminus1 + ytminus2

The process yt is an ldquointegratedrdquo process if after a series yt has been differencedto produce a stationary series wt the series wt can be modelled as an ARMAprocess If wt = dyt and wt is an ARMA( p q) process then we say that yt is anldquointegrated autoregressive moving average process of order ( p d q)rdquo or simplyARIMA ( p d q)

An application of time series models

To gain a better understanding of how time series models are used in macroeco-nomic empirical work let us consider the time series for the logarithm of the levelof real output to be denoted by yt As we stated above theoretical macroeconomicmodels typically postulate shocks or ldquoimpulsesrdquo to the economy (eg shocks totechnology money supply and government policy) that in conjunction with aprediction of how various markets in the economy adjust to these shocks offersan explanation of the actual fluctuations in aggregate variables One importantquestion that can arise is whether such shocks are ldquotransitoryrdquo in that their effectsdo not persist or ldquopermanentrdquo where the economy moves to a new level of realoutput

The ldquotraditionalrdquo answer to the above question of transitory versus permanenteffects of shocks has been according to Campbell and Mankiw (1987a 111) thatfluctuations in real output ldquoprimarily reflect temporary deviations of productionfrom trendrdquo That is the traditional view has been that quarterly real output couldbe represented by

yt = Tt + St + Xt (738)

where Tt is a deterministic component representing trend St is a determinis-tic seasonal component and Xt is a stationary autoregressive process with nodeterministic component43

Trend and seasonal components were typically estimated in various fashionsand then eliminated The focus of empirical work had then been on examiningvariations in the detrended seasonally adjusted real output series For exampleBlanchard (1981) estimated the following second-order autoregressive process fordeviations of the log of seasonally adjusted real output from its estimated trendylowast

t and obtained the equation

ylowastt = 134ylowast

tminus1 minus 042ylowasttminus2 + εt (739)

Note that the necessary conditions for stationarity with respect to the series ylowastt

(namely φ1 + φ2 lt 1 minusφ1 + φ2 lt 1 and φ2 gt minus1) are metThe traditional view is embodied in (739) A shock to output increases for

a few quarters but the effect ultimately dies out In fact only 8 percent of ashock remains after 20 quarters Assuming an ARMA(22) process Blanchardand Fischer (1989 9) obtain a similar result

tminus1 minus 042ylowasttminus2 + εt minus 006εtminus1 + 025εtminus2 (740)

where by construction εt is that part of the deviation of current output from trendthat cannot be predicted from past output As Blanchard and Fischer note

A shock has an effect on GNP that increases initially and then decreases overtime After 10 quarters the effect is still 40 of the initial impact after 20quarters all but 3 of the effect has disappeared The view that reversiblecyclical fluctuations account for most of the short-term movements of realGNP and unemployment has been dominant for most of the last century

However as Granger and Newbold (1986 37) note ldquothe more modern view is thatas far as possible the trend seasonal and lsquoirregularrsquo components should be han-dled simultaneously in a single model aimed at depicting as faithfully as possiblethe behavior of a given time seriesrdquo Seasonality can be treated through a gener-alization of ARMA models44 More importantly for the question at hand trend isgenerally treated by differencing leading to the consideration of the ldquointegratedprocessesrdquo suggested above In fact Campbell and Mankiw (1987b) indicate thatthe answer to the question of the importance of temporary versus permanent shocksto output is biased if detrended data are used45 Campbell and Mankiw go on topoint out that examining the differences in the logarithm of real output does notprejudge the issue of whether shocks to the economy are transitory or permanentFor instance suppose that yt follows an IMA(11) process so that

yt minus ytminus1 = d + εt minus θεtminus1

Then a unit impulse in yt changes the forecast of yt+n by 1 minus θ regardless of nldquoHence depending on the value of θ news about current GNP could have a largeor small effect on onersquos forecast of GNP in ten yearsrdquo (Campbell and Mankiw1987b 860)46

Campbell and Mankiw consider ARMA( p q) processes for the difference in thelog of real output for p = 0 1 2 3 and for q = 0 1 2 347 Interestingly they finda high level of persistence in shocks One intriguing suggestion of Campbell andMankiw (1987b 868) is that ldquowhen we examine postwar annual data we cannotreject the hypothesis that the log of real GNP is a random walk with drift In thiscase the impulse response is unity at all horizonsrdquo Recall that the random walkprocess with drift (718) is an example of a nonstationary process that is first-order

homogeneous nonstationary To see this simply consider the series wt that resultsfrom differencing the random walk ndash that is the series

wt = yt minus ytminus1 = d + εt (741)

Since εt are assumed independent over time wt is clearly a stationary process Theterm d + εt is a white noise process In this example yt is an ARIMA(010) andyt is an ARMA(00) The impulse response to a shock is unity in such a case

As we will see later one way to interpret the above finding of persistence isto place weight on ldquoaggregate supplyrdquo shocks such as technological disturbancesrather than on ldquoaggregate demandrdquo shocks such as changes in the money supplyin explaining fluctuations in real output That is the finding gives credence tothe view of Nelson and Plosser (1982) among others that real (ie permanent)shocks dominate as a source of output fluctuations48

However as pointed out by Campbell and Mankiw (1987b 877) the Nelsonand Plosser conclusion is an extreme

one can attribute a major role to supply shocks without completely abandoninga role for demand shocks For example suppose that output Y (= log y) isthe sum of two components a supply-driven ldquotrendrdquo Y T and demand-driveldquocyclerdquo Y c that are uncorrelated at all leads and lags Suppose further thatY T is a first-order autoregressive process with parameter φ and that Y c issome stationary process If trend output is approximately a random walk sothat φ is small then the finding of great persistence implies that fluctuationsin the cycle are small relative to fluctuations in the trend If the change inthe trend is highly serially correlated (φ is large) however the finding ofpersistence is consistent with a substantial cyclical component

Campbell and Mankiw (1987b 877) go on to suggest that

a second way to interpret the finding of persistence is to abandon the

natural rate hypothesis Models of multiple equilibria might explain a long-lasting effect of aggregate demand shocks if shocks to aggregate demand canmove the economy between equilibria Shocks to aggregate demand couldhave permanent effects if technological innovation is affected by the businesscycle

Note that so far we have considered only a single or univariate time series Theconcepts involved however can be extended to multivariate series For instancea simple vector autoregression model (VAR) could take the form

yt = ytminus1 + εt

where now yt is considered an N times 1 vector reflecting N variables the randomdisturbance term εt is also an N times 1 vector and is an N times N matrix of parame-ters The disturbances are uncorrelated over time but may be contemporaneously

correlated As in the univariate case such a model is usually only fitted to vari-ables which are stationary (possibly obtained by logarithmic transformations orby taking first or second differences) As in the univariate case the objectivewould be to find a model that transforms a vector of time series into a whitenoise vector As Harvey (1993) notes ldquoalthough a model of (this) form is oftenused for forecasting in econometrics economic theory will typically place a priorirestrictions on elements of rdquo There are also questions concerning the correlationbetween innovations across different series of data for example the link betweeninnovations in money supply and output But given that our aim is to review basicmacroeconomic theory we will not pursue such topics

Conclusion

This chapter has emphasized the empirical nature of macroeconomics Inparticular time series analysis and econometric methods have been shown tobe useful tools for analyzing macroeconomic data Many of the policy conclu-sions that will be developed in the remainder of the book can be tested using thesemethods The results obtained from a thorough understanding of the time seriesproperties of important macroeconomic variables such as interest rates employ-ment numbers real output measures and the like together with the underlyingtheory may be used to provide policy recommendations Moreover more andmore businesses are relying on empirical macroeconomics when making operatingdecisions

Appendix translation of higher-order difference equationsinto lower order

Any higher-order difference equation can be interpreted in terms of an equivalentsystem of first-order difference equations To see this consider equation (729)

yt minus φ1ytminus1 minus φ2ytminus2 = 0

To facilitate matters we start by shifting the origin of this second-order homo-geneous difference equation from t = minus2 to t = minus1 such that we may rewrite(729) as

yt+1 minus φ1yt minus φ2ytminus1 = 0 (7A1)

Now let us introduce the new variable xt defined as

xt = ytminus1

which means that xt+1 = yt We may then express the second-order differenceequation (7A1) by means of two first-order simultaneous equations

yt+1 minus φ1yt minus φ2xt = 0

xt+1 minus yt = 0(7A2)

In matrix notation (7A2) becomes[yt+1xt+1

where A is the square nonsingular matrix defined as

A =[φ1 φ21 0

The ldquocharacteristic equation of a square matrixrdquo is defined by the determinant

|A minus λI | = 0

where I is the unit matrix[

1 00 1

]and λ is a scalar variable

Letting b = λ the ldquocharacteristic equation of matrix Ardquo

|A minus λI | =∣∣∣∣φ1 minus b φ2

1 minusb

∣∣∣∣ = b2 minus φ1b minus φ2 = 0

is identical to equation (730) which was termed the ldquocharacteristic equation of thesecond-order homogeneous difference equationrdquo The values for the λs (or bs) arethe roots of the characteristic equation One reason for restating the higher-orderdifference equation in matrix form is that necessary and sufficient conditions forstability can be expressed in terms of conditions on the matrix A

8 The neoclassical model

Introduction

This chapter turns our attention to one of the most popular and sometimescontroversial models used by macroeconomists the neoclassical model Themodel in its purest form is often used as a benchmark or starting point for addingldquorealismrdquo to the structure of the economy However one views the neoclassicalmodel there is no denying that it is a powerful tool for generating predictionsabout movements in economic aggregates As such this chapter works throughthe comparative static exercise of a change in the money supply This exerciseprovides a great deal of insight into how the monetary authority might influencethe economy or whether it can influence the economy at all A number of importantissues are raised for example money neutrality and money illusion Additionallythe chapter formally derives the aggregate supply curve in the neoclassical contextThe model has serious implications for the existence of a natural rate and also forthe formation of expectations

Comparative statics for the neoclassical model

We can use our previous method of analysis known as ldquocomparative staticrdquoanalysis to examine the effects of various shocks on the equilibrium level ofprices As the name suggests comparative static analysis is concerned with thecomparison of different equilibrium states associated with different sets of valuesof parameters and exogenous variables In the current context of the neoclassicalmodel the labor market can be isolated from the other markets (ie there is aldquoblock recursiverdquo character to the equilibrium solution) In such a context we candistinguish two types of exogenous variables

One type ldquosupply-siderdquo variables alter the level of output at any given pricelevel Such variables could include changes in technology and the existing capitalstock in the current version of the model or we could expand the analysis toinclude changes in the supply of other inputs (such as oil) changes in governmentpolicies that affect incentives to supply labor and changes in government policiesthat affect the incentives to invest and thus affect the productive capacity of theeconomy over time

Neoclassical model 117

The second type of exogenous variables ldquodemand-siderdquo variables do not alterthe current supply of output at prevailing prices but instead impact equilibriumprices and interest rates as determined in the output financial and money marketsBelow we consider the impact of demand-side ldquoshocksrdquo such as changes in initialmoney balances and the expected inflation rate

The macroeconomic approach to general equilibriumhow it can obscure

The neoclassical model can be viewed as a special case of a Walrasian generalequilibrium system distinguished by the existence of money1 As Clower (1965)notes

income magnitudes do not appear as independent variables in demand andsupply functions of the (Walrasian) general equilibrium model for incomesare defined in terms of quantities as well as prices and quantity variablesnever appear explicitly in the market excess demand functions of traditionaltheory To be sure income variables could be introduced by taking factorsupplies as given parameters but this would preclude the formulation of ageneral equilibrium model containing supply functions of all marketable factorservices

Referring to Chapter 9 of Patinkin (1965) Clower goes on to note that this point

was apparently overlooked by Patinkin when he formulated his ldquogeneral the-oryrdquo of macroeconomics It is instructive to notice that this chapter is notsupplemented by a mathematical appendix I do not mean to suggest thatauthors may not put such variables as they please into their models My pointis that such variables that can be shown to be fundamentally dependent onothers should not then be manipulated independently

What Clower is referring to is the fact that the neoclassical model with limitedperfect foresight effectively determines at time t the money wage for the labormarket and the (futures) price of output and interest rate Given perfect foresightindividual optimizing behavior will generate planned consumption and moneydemand functions at time t for period t that include the real wage rather thanincome That is since labor supply is a choice variable at time t labor income (theproduct of the real wage times labor supply) should not appear as an argument inhouseholdsrsquo demand and supply functions (Note that labor income along withdividend and interest payments equals output minus depreciation)

Formally we can discover how a ldquosmallrdquo change in one (or more) of the exoge-nous variables affects equilibrium values by totally differentiating the equilibriumconditions with respect to the prices to be determined and the exogenous variableand then solving for the implied change in equilibrium prices required to maintainequilibrium2 The resulting changes in equilibrium prices then imply changes in

118 Neoclassical model

the equilibrium levels of employment output consumption investment and realmoney holdings

However we choose instead the standard macroeconomic approach of arbitrar-ily separating the analysis into an analysis of the labor market and the resultingemployment and output (the ldquoaggregate supply equationrdquo) and then an analysisof the other markets and the detemination of the price level and the interest rate(the ldquoISrdquo and ldquoLMrdquo equations) While Clower is right in that this obscures thetraditional ldquogeneral equilibriumrdquo nature of the neoclassical model the approach isuseful in that it allows us to more easily extend the analysis to situations in whichprices are not market-clearing or to situations in which there is not limited perfectforesight

Given the assumption of perfect foresight on the part of both firms and workersconcerning the price level pt we have seen that the aggregate supply equation isindependent of the price level or the interest rate We know from the modifiedWalrasrsquo law that any two of the three excess demand conditions with respect tomoney financial assets and output can be used in the comparative static analysisUsing the equilibrium conditions with respect to output and money (the ldquoISrdquo andldquoLMrdquo equations) we thus have the three equations

yt = yt(K ) (81)

cdt (rt πe

t At Mpt yt) + I dnt(m

et + δ K) + δK + ψ(I d

nt) minus yt = 0 (82)

Ldt (rt πe

t At Mpt yt) minus Mpt = 0 (83)

to determine equilibrium output yt price level pt and interest rate rt Recallthat from the modified Walrasrsquo law we know that there is a fourth equation theequilibrium condition for the financial market that is implied by (82) and (83)This fourth equilibrium condition is

net Adt (rt πe

t At Mpt yt) minus net Ast (m

et + δ K) = 0 (84)

A change in the money supply the comparative statics

It is clear from the above that the neoclassical aggregate supply equation (81)determines output while the IS and LM equations (82) and (83) determine theprice level and interest rate given the equilibrium level of output Focusing on thelatter two equations and the determination of the price level and interest rate fora given level of output total differentiation with respect to the equilibrium prices( pt and rt) and the money supply change gives the following system of linearequations in matrix form3

[ minus(partcdpart(MP))Mp2

(1 minus partLdpart(Mp))Mp2partydpartrpartLdpartr

] [dpdr

]=[ minus(partcdpart(Mp))dMp(1 minus partLdpart(Mp))dMp

where partydpartr = partcdpartr + (1 + ψ prime)partI dpartm4 Solving the above linear-equationsystem for dp and dr using Cramerrsquos rule we obtain

dM= minus(partcdpart(Mp))(1p)partLdpartr minus (1 minus partLdpart(Mp))(1p)partydpartr

minus(partcdpart(Mp))(Mp2)partLdpartr minus (1 minus partLdpart(Mp))(Mp2)partydpartr

minus[(partcdpart(Mp))(Mp2)(partLdpart(Mp)minus1)minus(partcdpart(Mp))(Mp2)(partLdpart(Mp)minus1)](1p)

minus(partcdpart(Mp))(Mp2)partLdpartrminus(1minuspartLdpart(Mp))(Mp2)partydpartr

As you can see dpp = dMM and drdM = 0 That is a change in the moneysupply leads to the same proportional change in the price of the consumption goodand no change in the interest rate With regard to the latter result the interest ratedoes not change as the only thing that changes the interest rate is the rate of changein prices Since the price level adjusts there is no change in the interest rate and theLM curve does not change Note that from the labor market equilibrium conditionthat underlies the aggregate supply equation we know that the change in the pricelevel results in an equiproportionate change in the money wage The result is thatwe have ldquoneutrality of moneyrdquo In other words if individuals correctly anticipatethe effect of a change in the money supply on the price level as is the case inthe deterministic model under the assumption of limited perfect foresight thenmonetary changes have no ldquorealrdquo effects Real output the real wage the expectedreal rate of interest real consumption real investment and the real money supplyare all unaffected by the monetary change

The basic reason why changes in the money supply are ldquoneutralrdquo is the absenceof money illusion on the part of both firms and households In particular house-hold demand and supply functions indicate that if a change in money balancesis accompanied by an equiproportionate change in the price of output then thereis no change in any demands That is household demand and supply functionsare homogeneous of degree zero in money balances and prices This absence ofldquomoney illusionrdquo occurs assuming

bull perfect foresight at time t for price during period t so that a change in priceresults in no change in the real wage or employment (when coupled with thesimilar assumption of limited perfect foresight on the part of firms)

bull ldquoneutral distribution effectsrdquo such that the shift in wealth from prior creditorsto debtors that would accompany a rise in the price level leaves aggregatedemands unchanged

bull unit elastic expectations so that changes in the current price level can beviewed as leaving unaffected the expected rates of change in the pricelevel

As we will see later this money neutrality result of the neoclassical model formsthe basis of what has become known as the ldquopolicy ineffectiveness propositionrdquo(see McCallum 1979)

The ldquodynamicsrdquo of the system and the neutrality of money a review

As we discussed earlier the ldquostoryrdquo often told with respect to the dynamics of theabove situation is a ldquoloanable funds theoryrdquo of interest rate determination in whichthe interest rate moves to clear the financial market5 In this case the ldquotatonnementprocessrdquo or movement toward equilibrium involves

dpt = f ( ydt minus yt) f (0) = 0 df d( yd

t minus yt) gt 0

dpbt = fb(net Adt minus As

t ) fb(0) = 0 dfbd(net Adt minus net As

t ) gt 0

Note that we put ldquostoryrdquo in quotes since the analysis itself simply identifies variousequilibrium points with adjustments in prices to reach a different equilibrium pointgiven a shock essentially occurring without any passage of time Neverthelessconsider the following story with respect to a rise in the money supply

In the Patinkin analysis described earlier if the money supply were to doublefrom M to 2M the LL BB and CC curves would shift to the right so that the newequilibrium would occur at the original interest rate but at a price level double theoriginal one Such a once-and-for-all change in the initial money balances leavesthe money interest rate and real demands unchanged (the neutrality of money)

Superneutrality an informal review

Superneutrality of money occurs when changes in the rate of growth of the moneysupply leave the paths of capital and real output unaffected Although the aboveanalysis is static in nature it does provide some insight into the issue of thesuperneutrality of money To see how suppose each period the economy can bereplicated in every way That is the money supply equilibrium price level andinterest rate are identical each period If expectations of price changes are correctexpected inflation would equal zero For the economy to replicate itself it mustalso be the case that the initial capital stock is optimal such that the capital stockand thus output does not change over time With zero adjustment costs such asituation would imply a marginal product of capital equal to the expected real usercost of capital (me

t + δ where met equiv (rt minus πe

t )(1 + πet )) net investment demand

equal to zero each period and gross investment demand equal to δK Now let us compare this situation to an alternative sequence of temporary equi-

librium in which the economy is identical in every way except one the moneysupply is increased each period by a constant percentage from its level in the priorperiod Other things being equal the above analysis would suggest that one dif-ference across periods would be a rise in prices due to the positive growth in themoney supply Let us further assume that expectations of inflation adjust to reflect

what Sargent and Wallace (1975) would characterize as a new ldquosystematic moneysupply rulerdquo

Without fully developing the appropriate dynamic analysis it is easy to see thatone obvious adjustment of the static analysis given a growing money supply (asopposed to one that does not change across periods) is thus a higher expectedinflation rate (positive as opposed to zero) In fact the analysis of the static modelcan mimic to some extent the effects of a higher rate of growth in the money supplyby considering the impact of an increase in exogenous inflationary expectations

Following Sargent and Wallace (1975) among others assume that consump-tion demand depends on the expected real rate of interest r minus π not separatelyon its components (the money interest rate and the expected rate of inflation)6

Further assume that changes in expected inflation do not affect real moneydemand7 The comparative static results are then[ minus(partcdpart(MP))Mp2

(1 minus partLdpart(Mp))Mp2partydpart(r minus π)

partLdpartr

] [dpdr

]=[minus(partydpart(r minus π))dπ

where partydpart(r minusπ) = partcdpart(r minusπ)+ (1 +ψ prime)partI dpart(r minusπ) Solving the abovelinear equation system for dp and dr using Cramerrsquos rule we obtain

minus(partydpart(r minus π))partLdpartr

minus(partcdpart(Mp))(Mp2)partLdpartr minus (1 minus partLdpart(Mp))(Mp2)partydpart(r minus π)

minus(partLdpart(Mp))(Mp2)partydpart(r minus π)

As we have discussed before if there is no real balance effect with respect to con-sumption demand (partcdpart(Mp) = 0) or if real money demand does not respondto changes in the interest rate (partLdpartr = 0) then we can see from the above that

dr = minus(1 minus partLdpart(Mp))(Mp2)partydpart(r minus π)dπ

minus(1 minus partLdpart(Mp))(Mp2)partydpart(r minus π)= dπ

so that the change in the expected rate of inflation results in no change in theexpected real rate of interest (r minus π ) In this case money is superneutral in thata change in the growth of the money supply (although it alters inflation and thusgiven perfect foresight expected inflation) leaves the expected real rate of interestunchanged and thus does not affect investment and the future size of the capitalstock which depend inversely on the expected real interest rate

Recall that Sargent and Wallace (1975) have argued for superneutrality ofmoney In contrast Begg (1980) has noted that steady-state analyses of growthmodels with money like the analysis above do find that different rates of growthin the money supply have real effects (through changes in the expected real grossinterest rate)8 As the above analysis makes clear there are two conditions eitherof which is sufficient that will result in money being superneutral Begg (1980293) describes them as follows ldquoThe first condition is that the level of real moneybalances is not an argument in the consumption function and it is this conditionwhich distinguishes the rational expectations model of Sargent and Wallace fromthe analysis of growth models with money The second condition is that the demandfor money is independent of the nominal interest raterdquo (as well as π ) Otherwisea higher expected inflation will result in a fall in the expected real rate of interestand a higher price level

In the Patinkin framework with consumption demand and investment demanddepending on the expected real interest rate at a given price level and higherexpected inflation the CC curve must shift up vertically by the amount of theincrease in expected inflation so as to maintain the same expected real rate ofinterest But the LL curve does not shift with a change in expected inflationThus given a downward-sloping CC curve and an upward-sloping LL curve thenew equilibrium money interest rate does not rise by the extent of the increase inexpected inflation Superneutrality of money seems not to hold

In such a case the static analysis thus predicts a lower real stock of money eachperiod higher investment and a greater capital stock next period This suggests anew steady state in which the capital stock is greater In fact a complete dynamicanalysis confirms these predictions a higher steady-state capital stock is associatedwith an increase in the rate of growth of the money supply and consequent increasedexpected inflation

The role of the key assumptions of the neoclassical model

At this point it might be useful to review the role played by the two key assumptionsof the neoclassical model namely price flexibility and complete information onprices In most cases and this is no exception one can gain an understanding ofthe role of a particular assumption by exploring how the analysis would proceedif the assumption were not made Consider first the implications of dropping theneoclassical modelrsquos assumption that prices are perfectly flexible

Suppose that the demand for output decreases and firms cannot sell all theydesire at prevailing prices With flexible prices output prices will fall and thefalling price level restores output demand to its previous level However if outputprices are inflexible and do not adjust downward in response to a reduced demandfor output firms will respond by reducing production Reduced production willlead to a lower level of labor demand and thus a fall in employment Further labordemand will no longer depend upon the real wage but will be determined by whatcan be sold in the output market In short if output prices are inflexible then theoverall level of demand for goods and services is paramount in determining thelevel of employment

A second modification of the neoclassical model is to assume that output pricesare flexible but money wages are not If wages are ldquostickyrdquo relative to outputprices then changes in the price level will alter the real wage For instance inthe late 1970s high inflation rates led workers and employers in certain industriesto bargain for long-term contracts with a high rate of growth of the money wageThe high expected inflation did not materialize in the early 1980s The lower rateof inflation with no change in the rate of increase in wages meant a rise in thereal wage The resulting fall in the demand for labor led to lower employmentand output Given that wages are not perfectly flexible (eg ldquomultiperiod laborcontracts without complete indexationrdquo) output prices below that anticipated whenlabor contracts are signed lead to a fall in output and employment With inflexiblemoney wages the aggregate supply equation includes the price level of output asa determinant of output supply

Let us consider one more modification of the neoclassical model Suppose thatworkers have incomplete information on output prices and the real wage A firmdetermines its relevant real wage by dividing the money wage it pays its workersby the price it anticipates for the particular product its workers produce On theother hand workers must anticipate prices for a variety of different goods to bepurchased in order to determine their relevant real wage Thus firms may moreaccurately anticipate changes in prices and thus real wages than workers

Now suppose that there is an increase in output prices Given our currentassumption of incomplete information this will not only lead to an increase infirmsrsquo demand for labor and to higher wages as firms anticipate the fall in realwages but may also lead to an increase in the quantity of labor supplied for thefollowing reason Workers who have not anticipated the rise in output priceswill perceive the higher money wages as implying a rise in the real wage andwill increase their supply of labor accordingly As a result equilibrium employ-ment will rise with an increase in output prices Once again the aggregate supplyequation will incorporate the current price level as a potential determinant

The illusion model one modification of the neoclassical model

Our first departure from the neoclassical model is to introduce the potential forldquoimperfectrdquo foresight on the part of suppliers at time t concerning the price levelfor period t As a consequence the ldquonotionalrdquo or planned demands made at time tbased on anticipated prices for the period can differ from ldquoeffectiverdquo or realizeddemands andor supplies at the actual prevailing prices Further realized or effec-tive demands now depend on the quantity constraints experienced in other marketsas well as prices That is realized output now becomes a determinant of actualconsumption and money demand on the part of households

Real wage illusion the labor market and aggregate supply

As we have seen equilibrium employment is determined at the start of each periodin the labor market To formally show this let us start with the following statement

of equilibrium in the labor market in terms of a money wage wt and level ofemployment Nt such that

N st (wtpe

t ) minus Nt = 0

A critical aspect of the above is the fact that suppliers ndash in particular suppliers oflabor ndash may not correctly anticipate the price level that will exist with respect tooutput In particular we let pe

t denote suppliersrsquo expectation formed at the start ofperiod t of the price level for period t We assume this expectation is held withsubjective certainty so that we may express the real wage expected by supplierssimply by wtpe

t It is this anticipated real wage not the realized real wage wtpt that appears in the labor supply function

Firms on the other hand are presumed to correctly forecast the actual realwage9 Note that we retain the presumption that the money wage adjusts to clearthe labor market Similarly we implicitly have assumed that the price level adjuststo clear the output market such that firms are price-takers in the output marketand thus labor demand depends on the real wage rather than on a sales constraint

Initially let us assume that householdsrsquo anticipated level of prices is correctThat is we start with a money wage and level of employment consistent with theneoclassical model But we now assume that any change in the price level willnot be fully anticipated In particular we assume that

pet = g( pt) 1 gt gprime ge 0

The fact that gprime lt 1 implies imperfect foresight at time t on the part of householdsconcerning the price of output for period t If gprime gt 0 it indicates that householdsto some extent but not completely (gprime lt 1) anticipate changes in the equilibriumlevel of prices that will prevail for period t

Totally differentiating the above two equations representing equilibrium in thelabor market with respect to wt Nt pt and noting our prior assumption that pe

t = ptinitially one obtains[minus(partN d

t part(wtpt))pt(partN s

t part(wtpet ))pt

minus1minus1

] [dwtdNt

(partN dt part(wtpt))wtdpt( pt)

(partN st part(wtpe

t ))wtgprimedpt( pt)2

Applying Cramerrsquos rule gives

dpt= (partN d

t part(wtpt))(partN st part(wtpe

t ))(wt( pt)2)(gprime minus 1)

minus(partN dt part(wtpt))pt + (partN s

t part(wtpet ))pe

dpt= minuspartN d

t part(wtpt) + (partN st part(wtpe

t ))gprime

minuspartN dt part(wtpt) + partN s

t part(wtpet )

ptgt 0

Note that if gprime = 1 then we have the standard neoclassical result that dNtdpt = 0and dwtwt = dptpt so that a change in the price level results in no change in

either employment or the real wage However with gprime lt 1 we have that anincrease in the price level leads to a rise in employment and an increase in themoney wage less than proportional to the increase in the price level such thatthe real wage falls (ie dptpt gt dwtwt gt 0) Given the aggregate produc-tion function yt = f (Nt K) we thus have an ldquoaggregate supply functionrdquo ofthe form

yt = yt( pt minus pet K ) (81prime)

so that aggregate supply depends directly on the difference between the price levelfor period t and the price level anticipated by suppliers at time t

Consider the above findings with respect to the labor market and suppose thatthere is an increase in pt Given perfect foresight on the part of firms at time t thelabor demand curve shifts up vertically so that at the higher money wage associatedwith the same real wage demand would be the same However given 1 gt gprime ge 0the vertical shift upward in the supply curve is less than this with the result thatequilibrium employment rises as the money wage rises by proportionately lessthan the rise in prices

In undergraduate textbooks the fact that changes in the price of output can nowaffect real output is shown in ( pt yt) by an upward-sloping ldquoaggregate supplycurverdquo as in Figure 81 Recall that in the neoclassical model the aggregate supplycurve is vertical In either model such a curve summarizes the underlying eventsin the labor market

The above character of the money illusion model is sometimes said to reflect theldquonatural rate hypothesisrdquo The natural rate hypothesis posits that fully anticipatedincreases in prices have no effect on the rate of real economic activity ndash specificallyreal output employment and thus unemployment Thus we will refer to the abovemodel as a static version of a ldquonatural rate modelrdquo

Figure 81 Upward-sloping aggregate supply

Equilibrium aggregate supply and demand

An important feature of macroeconomic theories is that to a large extent they aredistinguished by their different treatment of labor markets What this means isthat the aggregate demand side is typical of macroeconomic models Recall thatthe aggregate demand side of macroeconomic models considers the equilibriumconditions of two of the remaining three markets in particular the output market(reflected by an ldquoISrdquo equation) and the money market (reflected by an ldquoLMrdquo orldquoportfoliordquo equation) Thus the equilibrium output price level and interest rate aregiven by equations (81prime) (82) and (83) Equation (81prime) is the aggregate supplyequation of a natural rate model (82) is the ldquoISrdquo equation depicting equilibriumbetween output demand and production and (83) is the portfolio or ldquoLMrdquo equationexpressing equilibrium with respect to the money market

At this point we will simplify equations (82) and (83) by removing the realbalance effect representing them as follows

cdt (rt πe

t At yt) + I dnt(m

t ) minus yt = 0 (82prime)

Ldt (rt πe

t At yt) minus Mpt = 0 (83prime)

In our model this implies that

partnet Adt part(Mpt) = 1

As we will see one justification for this form is if real money balances are notpart of household wealth which can be the case when we introduce depositoryinstitutions into the analysis In the meantime the above assumption makes theanalysis not only simpler but also more in line with traditional macroeconomicanalysis

Graphically the equilibrium price level and output can be shown using theaggregate demand and supply curves In Figure 82 the equilibrium output andprice level are thus given by plowast

t and ylowastt Looking at what underlies these curves

we can then infer the changes in money wages and employment (specifically froman analysis of the labor market that underlies the aggregate supply curve) as wellas changes in the interest rate and the components of investment and consumptioncomponents of output demand (specifically from an analysis of the output andmoney markets that underlie the aggregate demand curve)

Money supply change comparative statics for a naturalrate model

Collecting the aggregate supply IS and LM equations for the natural rate modelunder consideration we have (81prime) (82prime) and (83prime)

These three equations determine the equilibrium output the price level andthe interest rate Substituting (81prime) into (82prime) and (83prime) in order to focus on the

Aggregate supply

Aggregate demand

Figure 82 Macroeconomic equilibrium with upward-sloping supply

determination of the price level and interest rate and totally differentiating withrespect to the equilibrium prices ( pt and rt) and the money supply change givesus the following system of linear equations in matrix form10

[ minus(partcdparty minus 1)partyp(partLdparty)partypartp + Mp2

partydpartrpartLdpartr

] [dpdr

where partydpartr = partcdpartr + c1 + ψ prime)partI dpartm11 The term partypartp gt 0 reflectsthe direct effect of the price level on output as implied by the aggregate supplyequation (81prime)12 It is important to note that the equilibrium condition with respectto the labor market is incorporated into the above analysis in the form of thisaggregate supply equation

Solving the above linear equation system for dp and dr using Cramerrsquos rule weobtain

dM= pM

1 + (p2M )(partypartp)[(1 minus partcdparty)(partLdpartr)(partydpartr) + partLdparty]gt 0

dM= ((partcdparty minus 1)(partypartp)minus 1)(1p)

(partcdparty minus 1)partypartp(partLdpartr)minus((partLdparty)(partypartp)+ Mp2)(partydpartr)

As you can see we no longer have drdM = 0 Further letting x denote thedenominator for the expression for dp we have

Since x gt 1 1x lt 1 and we now have that

Thus the increase in the money supply leads to a less than proportionate increasein the price level so that the real money supply is greater13

Let us now consider the effect of the money supply shock on other variablesFrom our analysis of the labor market we know that

so that while the money wage rises the real wage falls We also know from thelabor market that the rise in the price level leads to higher employment and thusan increase in output From the demand functions we can derive the effects of thechange in the money supply on consumption and investment as equal to

dM= dcd

dM+ dcd

dMgt 0

dM= dId

dMgt 0

Graphically one can show the effect of the money supply shock in terms ofthe aggregate demand and aggregate supply curves Note that this exercise simplyinvolves a shift out of the aggregate demand curve

The natural rate hypothesis and expectation formation a preview

An important feature of the above analysis one already noted is that real activity ndashin particular employment output and unemployment ndash changes only to the extentthat price changes are not fully anticipated As we have just seen this ldquonatural ratehypothesisrdquo introduces the logical foundations for a monetary change to have realeffects

However note that monetary changes have real effects only to the extent that theresulting changes in the price level are not fully anticipated To understand whenthis might occur we first have to indicate why individuals may err in their formationof expectations concerning the price level The Lucas model suggests one way ofexplaining errors in forecasts such that suppliers only partially anticipate a changein the price level (ie 1 gt gprime ge 0) This model introduces the assumption ofrational expectations

Combining rational expectations with the natural rate hypothesis results in a verypowerful statement concerning monetary policy which has so far not been madeclear If the actions of monetary authorities are predictable under the presumptionof rational expectations individuals will correctly predict the consequences onprices The result in the context of a natural rate model is that such predictable

monetary changes will have no real effects In the deterministic world we are backto the neoclassical model The analysis above then refers only to an ldquounexpectedrdquoincrease in the money supply or to monetary ldquosurprisesrdquo Only such random shocksto the money supply will have real effects

Conclusion

A formal neoclassical model of the macroeconomy has been introduced and fullydeveloped The issues of money neutrality and money illusion have been discussedand it has been seen that money supply changes have no effect on real economicactivity when the assumptions of the neoclassical model hold However a numberof issues have been raised namely the existence of a natural rate and the potentialeffect of unanticipated money supply changes on economic activity

9 The ldquoKeynesian modelrdquo withfixed money wageModifying the neoclassical model

Introduction

The first modification of the neoclassical model is presented in this chapterTo begin we introduce the very realistic assumption that nominal wages are fixedat least for a period of time The ramifications of this change in the model aredeveloped in the context of the aggregate supply and demand model As with theneoclassical model we perform a comparative statics exercise in which the mone-tary authority changes the money supply and we trace out the effects of this actionon the economic aggregates in the model The model is then made slightly morecomplete and issues associated with sticky wages and the natural rate hypothesisare discussed We introduce the concept of rational expectations and the first ldquoover-lappingrdquo model and show that the Keynesian model has important implicationsfor the conduct of monetary policy

The ldquoKeynesian modelrdquo with fixed money wage modifyingthe neoclassical model

In the standard neoclassical model it is assumed that prices adjust in all marketsto equate demand and supply With respect to the labor market this implies a spotmarket at the start of each period in which one-period labor contracts are enteredinto and an associated one-period wage set Yet employment contracts are likelyto be multiperiod in the presence of hiring and training costs That is to minimizehiring and training costs firms seek long-term relationships with their employees

Firms promote long-term relationships with their employees by offering higherwages to their experienced workers As a consequence long-time employeesbecome attached or ldquoloyalrdquo to their employers since the wages they receive aregreater than those that other firms would offer them In essence employers aresharing the returns to their hiring and training investment with their workers inorder to reduce the number who quit A long-term attachment of workers to par-ticular firms could also stem from the high cost to workers of finding alternativeemployment as obtaining such employment means that workers must generallyinterview various employers visit employment agencies and spend valuable timesimply waiting for decisions on job applications to be made

Keynesian model 131

Given long-term employment contracts between firms and their workers wagesare typically specified for extended periods of time These long-term wage agree-ments are sometimes explicit as with many labor union contracts1 In other casesonly an implicit understanding exists on the wages that a firm will pay its employeesover some extended period of time If these contracts or understandings specifywages in money terms and if modifying these agreements is costly then thereexists an inherent inflexibility in money wages ndash that is there are ldquostickyrdquo wages2

This assumption of ldquostickyrdquo nominal wages is often viewed as the critical aspectof what has been termed the ldquoKeynesianrdquo macroeconomic model

If money wages are ldquostickyrdquo relative to prices then changes in the price levelwill alter the real wage For instance in the late 1970s high inflation rates ledworkers and employers in certain industries to bargain for long-term contractswith a high rate of growth of the money wage The high expected inflation did notmaterialize in the early 1980s The lower rate of inflation with no change in therate of increase in wages meant a rise in the real wage The resulting fall in thedemand for labor led to lower employment and output In this section we formallydevelop these results in the context of a static neoclassical macroeconomic modelwith the additional assumption of a fixed money wage The subsequent section thendevelops a linear rational expectations version of this model in which overlappingmultiperiod employment contracts introduce an element of nominal wage rigidity

Fixed money wage the labor market and aggregate supply

As we have seen in the competitive (spot) labor market of the neoclassical model(or in the Lucas-type macroeconomic model) the money wage and employmentare determined at the start of each period in the labor market If we accept theneoclassical modelrsquos assumption of limited perfect foresight on the part of bothlabor suppliers and firms we have equilibrium in the labor market in terms of amoney wage wt and level of employment Nt determined such that

N dt (wtpt K) minus Nt = 0 (91)

N st (wtpt) minus Nt = 0 (92)

In this case a change in the price level pt leads to an equiproportionate change inthe money wage wt and no change in employment Nt

We now seek to modify this analysis by assuming a fixed money wage wt = wfor period t As Sargent (1987a 21) states

the essential difference between the classical model and the Keynesian modelis the absence from the latter of the classical labor supply curve combined withthe labor market equilibrium condition Since there is one fewer equation inthe Keynesian model it can determine only six endogenous variables insteadof the seven determined in the classical model3 To close the Keynesianmodel the money wage is regarded as an exogenous variable one that atany point in time can be regarded as being given from outside the model

132 Keynesian model

perhaps from the past behavior of itself and other endogenous or exogenousvariables It bears emphasizing that the equation that we have deletedin moving from the classical to the Keynesian model [equation (92)] is acombination of a supply schedule (and) an equilibrium condition Notethat we continue to require that employment satisfy the labor demand schedule[equation (91)]

Sargent goes on to say that

we shall think of the labor supply schedule as being satisfied and helping todetermine the unemployment rate Usually the model is assumed to reachequilibrium in a position satisfying Nt lt N s

t so that there is an excess supplyof labor

Totally differentiating the labor demand condition (91) that determines the levelof employment with respect to the price level and employment we have

minus(partN dt (partwpt))(wp2

t )dpt minus dNt = 0 (93)

which can be rearranged to give

dNtdpt = minus(partN dt part(wpt))(wp2

t ) gt 0 (94)

where the sign reflects the presumption that partN dt part(wtpt) lt 0

In the simple case of no labor adjustment costs labor demand is defined bythe equality between the marginal product of labor and the real wage that ispartf (Nt K)partNt = wpt 4 Differentiating this implies that

[part2ftpartN 2t ]dNt = minus(wp2

t )dpt

or rearranging

dNtdpt = minus(wp2t )[part2ftpartN 2

t ] gt 0 (95)

given diminishing returns to the labor input (ie part2ftpartN 2t lt 0)

Combining the above analysis with the aggregate production function yt =f (Nt K) we thus have the ldquoaggregate supply equationrdquo

yt = yt( ptw K ) with partytpart( ptw) gt 0 (96)

Thus (as in a Lucas-type model) we have an aggregate supply that can dependdirectly on the price level for period t

The above findings can be understood with respect to the labor market Considera decrease in pt Given limited perfect foresight on the part of both firms andhouseholds at time t there is a downward (vertical) shift in labor demand so thatat the lower money wage wlowast

t associated with the same real wage demand would

Keynesian model 133

be the same Similarly the labor supply curve shifts down vertically so that atthis lower money wage wlowast

t labor supply would be the same as well Howevermultiperiod labor contracts fix the money wage at w so that the lower price level(and implied higher real wage) results in a fall in employment (which is nowdemand-determined) and an excess supply of labor

In the opposite case of a rise in the price level that can lead to an excessdemand in the labor market at the fixed money wage the presumption remainsthat employment is demand-determined This presumption reflects the view thatat least temporarily firms can direct workers with whom they have long-termemployment contracts to work overtime or extra shifts which the workers wouldotherwise not volunteer for

The above story provides a rationale for an upward-sloping ldquoaggregate sup-ply curverdquo Both contrast with the neoclassical model in which the aggregatesupply curve is vertical as a fall in the price level results in an equiproportionatefall in the money wage so that the real wage and employment remain unchangedIn the fixed wage model the underlying events in the labor market summarized bythe aggregate supply curve are the change in the real wage and thus labor demandand employment that accompany a price change when the money wage is fixedSuch an aggregate supply curve is upward-sloping

As we have noted before an important feature of macroeconomic theories is that toa large extent they are distinguished by their different treatment of labor marketsWhat this means is that the aggregate demand side is similar across macroeconomicmodels Recall that the aggregate demand side of macroeconomic models typicallyconsiders the equilibrium conditions of two of the remaining three markets inparticular the output market (reflected by an ldquoISrdquo equation) and the money market(reflected by an ldquoLMrdquo or ldquoportfoliordquo equation) Thus for the Keynesian modelwith fixed money wage the equilibrium output price level and interest rate aregiven by the following three equations

yt = yt( ptw K ) (96)

cdt (rt πe

t+1 At yt) + I dnt(m

Ldt (rt πe

t+1 At yt) minus Mpt = 0 (98)

Equation (96) is the aggregate supply equation of a Keynesian model with fixedmoney wage (97) is the ldquoISrdquo equation depicting equilibrium between outputdemand and production and equation (98) is the portfolio or ldquoLMrdquo equationexpressing equilibrium with respect to the money market Note that we havesimplified the IS and LM equations by removing the real balance effect for con-sumption demand and money demand5 Equations (97) and (98) can be combinedto eliminate the interest rate The resulting equation is referred to as the ldquoaggregatedemand equationrdquo

134 Keynesian model

The equilibrium price level and output ( plowastt and ylowast

t ) can be shown graphicallyusing aggregate demand and supply curves Looking at what underlies thesecurves we can then infer the change in employment (specifically from an analysisof the labor market that underlies the aggregate supply curve) as well as changes inthe interest rate and the investment and consumption components of output demand(specifically from an analysis of the output and money markets that underlie theaggregate demand curve)

A change in the money supply the comparative statics for theKeynesian model

The aggregate supply IS and LM equations (96)ndash(98) for the static Keynesianmodel under consideration determine the equilibrium output the price level andthe interest rate Substituting (96) into (97) and (98) in order to focus on thedetermination of the price level and interest rate and totally differentiating withrespect to the equilibrium prices ( pt and rt) and the money supply change givesus the following system of linear equations in matrix form6[ minus(partcdparty minus 1)partypartp

(partLdparty)(partypartp) + Mp2partydpartrpartLdpartr

] [dpdr

where partydpartr = partcdpartr + (1 + ψ prime)partI dpartm7 The term partypartp gt 0 reflects thedirect effect on the price level as implied by the aggregate supply equation (96)8

Note that the equilibrium condition with respect to the labor market is incorporatedinto the analysis in the form of this aggregate supply equation

dM= pM

dM= ((partcdparty minus 1)(partypartp) minus 1)p

(partcdparty minus 1)partypartp(partLdpartr) minus (partLdparty)(partypartp) + (Mp2)(partydpartr)

In contrast to the neoclassical model we no longer have drdM = 0 Furtherletting x denote the denominator for the expression for dp we have

Keynesian model 135

Thus the increase in the money supply leads to a less than proportionate increasein the price level so that the real money supply is greater

Consider now the effect of the money supply shock on other variables Fromour analysis of the labor market we know that w is fixed so that the increase inthe price level means a fall in the real wage and thus increased labor demandemployment and thus an increase in output From the demand functions for con-sumption and investment we can derive the effects of the change in the moneysupply on consumption and investment as equal to

dM= dcd

dM+ dcd

dMgt 0

dM= dId

dMgt 0

Note that the effect of the money supply shock is to increase the aggregate demandwhile not affecting aggregate supply

Sticky wages and the natural rate hypothesis

Expectations play an important role in the two modifications of the neoclassicalmodel In the modification with fixed wages the level at which negotiators fixfuture money wages depends on the expectation formed when wages were set con-cerning future prices The higher the expectation of future prices the higher thelevel of wages set in the labor agreements between workers and firms The pre-sumption is that workers and firms attempt to set future wages at their anticipatedmarket-clearing levels Associated with these anticipated market-clearing wagesis a particular real wage a natural rate of unemployment and a full employmentor natural rate of output

If price expectations turn out to be incorrect then output will vary from itsnatural rate For instance a shock that causes actual output prices to fall belowthose expected means that the money wage is fixed at a level that is too highfor full employment Consequently employment and output fall below the fullemployment level

In the typical Lucas-type model firms and workers set wages for the currentperiod based on incomplete information as well As we saw if suppliersrsquo expec-tations are incorrect then output will deviate from the full employment level Forexample a shock that causes actual output prices to fall below those expectedmeans lower employment and output as workers mistake lower money wages forlower real wages In fact higher real wages accompany the lower price level andthis is the source of the reduced demand for labor and employment

The two modifications of the neoclassical model have a second common ele-ment Both predict that a macroeconomic demand shock ultimately affects onlythe level of prices Even though money wages in the Keynesian model are fixed

136 Keynesian model

for the current period we know that money wages are not fixed forever Over timelabor agreements are renegotiated and money wages change to once again equatethe ldquoexpectedrdquo future demand for and supply of labor Over time in the absenceof further shocks the economy would thus tend to behave as neoclassical analysispredicts money wages and output prices would adjust to restore equilibrium tothe various markets in the economy

While the Keynesian model with fixed money wage admits the tendency foroutput to approach its natural level over time it does introduce a potential rolefor monetary policy to play in dampening fluctuations in output In so doing itchallenges the policy ineffectiveness view of Sargent and Wallace The best-knownexamples employing the Keynesian model to demonstrate the potential stabilizingpowers of monetary policy under rational expectations are the dual papers byFischer (1977) and Phelps and Taylor (1977)

A linear rational expectations version of the Keynesian model

Counting the above discussion we have so far considered three different modelsthat can be used to assess monetary policy One model is along the lines of theneoclassical model with limited perfect foresight Since this view of the economypredicts the neutrality of money a role for monetary policy either as an instigatorof output fluctuations or as an instrument to dampen output fluctuations is missingAs Mankiw (1987) suggests people who adopt this model ldquoview economic fluctu-ations through the lens of real business cycle theoryrdquo in which output fluctuationsare traced to ldquosupply-siderdquo disturbances

As Mankiw goes on to note however ldquothere are surely readers who believe thatmonetary policy has real short-run effects because of temporary misperceptions ornominal rigiditiesrdquo Mankiw is referring to individuals who adopt either the Lucas-type model or the ldquoKeynesianrdquo fixed money wage model9 Either one as we haveseen introduces a role for monetary policy as an instigator of output fluctuationsHowever these two models do differ as to whether monetary policy can be aninstrument to dampen output fluctuations in the context of rational expectations

A Lucas-type model built on ldquotemporary misperceptionsrdquo when coupled withrational expectations leaves little if any room for countercyclical monetary policyIn fact following the analysis of Sargent and Wallace it can be shown that whilerandom monetary shocks can impact output deterministic monetary policy basedon a set of policy rules is ineffective in counteracting fluctuations in output givenrational expectation10 Further attempts at discretionary monetary policy in thiscontext only result in a suboptimal (ldquotoo highrdquo) rate of inflation (Barro and Gordon1983) Thus in this model there remains a ldquostochasticrdquo neutrality of money

As the analysis in the previous section suggests however a ldquoKeynesian-typerdquomodel built on ldquonominal rigiditiesrdquo might introduce a role for monetary policyin stabilizing output even in the context of rational expectations The reasoningfor this is that wages (or as we will see later prices) can be set prior to thereceipt of information by the monetary authority that enters into the money supplyrule In this context as Phelps and Taylor (1977) state ldquoeven systematic and

Keynesian model 137

correctly anticipated policy can make a difference for the stability of output in arational expectations model with sticky prices and wagesrdquo11 Below we considerone example of such a model that counters the Sargent and Wallace ineffectivenessproposition a model proposed by Fischer that assumes ldquostickyrdquo wages

The supply equation with overlapping two-period labor contracts

The Lucas aggregate supply equation with adjustment costs can be expressed as

Yt = γ θ(Pt minus Etminus1Pt) + λYtminus1 (99)

where Yt = ln yt minus ln yn denotes difference between the logarithm of output forperiod t and the logarithm of the natural rate of output (which we have normalizedto equal zero) Pt = ln Pt is the logarithm of the price level for period t Etminus1Pt isthe expectation of the logarithm of the price level for period t using all informationavailable up to the end of period t minus 1 (at time t) and γ θ is a positive constant12

The supply of output as expressed by equation (99) satisfies the conditionthat employment equals labor demand (92) The fact that a higher price level Ptinduces firms to increase employment and thus output reflects the underlying lowerequilibrium real wage that accompanies the higher price level when suppliers donot anticipate the higher price level Thus we could express (99) in the form13

Yt = (Pt minus Wt + φ) + λYtminus1 (99prime)

where Wt is the logarithm of the equilibrium nominal wage for period t Theterm φ in (99prime) is defined such that if Etminus1Pt = Pt then the resulting log ofthe equilibrium real wage (ie ln(wtpt)) equals φ Ignoring the lagged outputterm in equation (99prime) this equilibrium real wage is the one associated with thenatural level of output and employment14 We will follow others and assume forconvenience that φ = 0 implying an equilibrium real wage with no surprisesequal to one

According to (99prime) if an increase in the price level is accompanied by a lessthan proportionate increase in the equilibrium money wage Pt minusWt rises (the realwage falls) and employment and output increase In the Lucas-type model such anevent occurs if suppliers do not forecast the price increase However Sargent andWallacersquos ineffectiveness proposition eliminates deterministic monetary policyrules as a source of such a price rise not matched by a similar rise in wages if(a) wages are set each period to equate labor demand and supply and (b) rationalexpectations are assumed In this case individuals and the monetary authoritiesare assumed to have a common set of information based on events up to the endof period t minus 1 (at time t) Individuals are also privy to the monetary policy ruleand they know the structure of the economy Thus they have knowledge of thedeterministic monetary policy to be followed during period t and its effect on theprice level for period t as predicted by the model Assuming flexible wages thispredicted effect of monetary policy on prices will be factored into the setting of themoney wage for period t As a consequence such deterministic monetary policy

138 Keynesian model

cannot change the real wage and thus leaves employment and output unaffectedas well

Obviously this chain of reasoning breaks down if money wages for period twere set prior to time t For example this policy ineffectiveness doctrine disap-pears if some wages are set at the end of period t minus 2 for period t In this casenew information that arrives during period t minus 1 can be incorporated by the mon-etary authorities into their money supply rule Even with rational expectationsthe implications of this cannot be used by individuals to adjust the money wagefor period t since by assumption the money wage is fixed Thus deterministicmonetary policy based on information revealed during period t minus 1 can alter thereal wage employment and output

To formally develop this potential stabilizing role of monetary policy in a moreldquoelegantrdquo fashion let us consider Fischerrsquos model The model disaggregates theeconomy into two sectors and assumes that the sectors alternate in setting multi-period employment contracts that fix nominal wages In particular ldquosuppose thatall labor contracts run for two periods and that the contract drawn up at the endof period t minus 2 specifies nominal wages for periods t minus 1 and t [Assume] thatcontracts are drawn up to maintain constancy of the real wagerdquo (Fischer 1977198) In other words

tminusiWt = EtminusiPt i = 1 2 (910)

where tminusiWt is the logarithm of the wage set at the end of period tminus i for period t15

The idea embodied in (910) that wages are set for more than one period iscritical to Fischerrsquos finding It essentially means that in any period half of thelabor contracts have fixed money wages16 Given that the wage is predeterminedfor each firm the aggregate supply equation is given by

Yt = 12 (Pt minus Etminus1Pt) + 1

2 (Pt minus Etminus2Pt) + ut (911)

where ut is a stochastic ldquorealrdquo disturbance or ldquosupply shockrdquo that impinges onproduction in each period17 Substituting (910) into (911) we can rewrite theaggregate supply equation as

2 (Pt minus Etminus2Pt) + ut (911prime)

A complete model except for specifying the source of expectations

Equation (911prime) provides us with one part of the standard macroeconomic modelthe ldquoaggregate supply equationrdquo To close the model we require LM and ISequations The explicit derivation of these is left to the next chapter but let usassume for the time being that the LM equation is given by

mt minus Pt = α1Yt minus α2 middot rt minus εt

Keynesian model 139

and the IS equation by

Yt = Xt minus β1(rt minus πet+1) + ut

Here mt = ln Mt mt = mt + εt such that the deterministic component mt is setby government authorities (ie the monetary authority) according to a monetaryrule Further rt minus πe

t+1 represents the expected real rate of interest Xt denotesa vector of exogenous variables that affect output demand εt and ut are randomterms associated with output demand and money supply respectively and assumedindependent (ie E(εtut) = 0) and well behaved

Combining the LM and IS equations to eliminate the interest rate rt we obtain

Yt = Xt + ut minus (β1α2)(minusmt minus εt + Pt minus α1Yt) + β1πet+1

which on rearranging becomes an ldquoaggregate demand equationrdquo of the form

Yt = [α2(α2 + α1β1)][Xt + ut + (β1α2)(mt + εt minus Pt) + β1πet+1] (912)

Note that if α2 = 0 (changes in the interest rate do not affect money demand) andα1 = 1 (the income elasticity of real money demand is one) then this simplifies towhat Fischer refers to as a ldquovelocity equationrdquo

Yt = mt minus Pt + vt (913)

where vt = εt is now to be interpreted as a money demand disturbance termaffecting the ldquovelocityrdquo of money18

To see why (913) is called a ldquovelocity equationrdquo note that the assumption ofmoney demand being independent of the interest rate allows us to capture therelationship between income and the price level summarized by the aggregatedemand equation by looking solely at the LM equation (ie neglecting the ISequation) In particular if we assume that real money demand can be expressedby the equation

Ldt = yt(exp(minusvt))

then equating real money demand to real money supply (Mtpt) gives us

yt(exp(minusvt)) = Mtpt (914)

Taking the logarithm of the equilibrium condition with respect to the money market(914) we obtain

ln yt minus vt = ln Mt minus ln pt

Given Pt = ln pt Yt = ln yt and mt = ln(Mt) this is simply equation (913)

140 Keynesian model

Note that equilibrium velocity is defined as the ratio of nominal output to themoney supply Thus rearranging (914) we have that

Equilibrium velocity equiv ptyt

Mt= exp(vt)

which explains the interpretation of vt as a ldquovelocityrdquo disturbance Fischer assumesthat vt has a zero mean so that expected velocity is one A vt above zero meansa decrease in real money demand relative to real output and thus an increase inequilibrium velocity As (914) makes clear for a given Mt a higher vt implies ahigher pt andor a higher yt to maintain equilibrium with respect to the demandfor and supply of money

As Fischer states (913) is ldquothe simplest way of taking demand consid-erations into accountrdquo In sum then the macroeconomics model considered byFischer is given by the aggregate supply equation (911prime) and the aggregate demandequation (913)

Combining (911prime) and (913) to eliminate Yt we have

12 (Pt minus Etminus1Pt) + 1

2 (Pt minus Etminus2Pt) + ut = mt minus Pt + vt

This can be solved to give the reduced-form equation for the price level Pt

Pt = 12

[(12 Etminus1Pt + 1

2 Etminus2Pt

)minus ut + mt + vt

] (915)

Combining (911prime) and (913) to eliminate Pt we have

Yt = 12 (mt minus Yt + vt minus Etminus1Pt) + 1

2 (mt minus Yt + vt minus Etminus2Pt) + ut

This can be solved for the reduced-form equation for output Yt

Yt = 12

[mt + vt + ut minus 1

2 Etminus1Pt minus 12 Etminus2Pt

] (916)

According to equations (915) and (916) an increase in the ldquorealrdquo disturbanceterm ut leads to higher equilibrium output and a reduced price level Intuitivelythis corresponds to a shift to the right in the ldquoaggregate supply curverdquo On the otherhand an increase in the ldquovelocityrdquo disturbance term vt corresponds to a shift to theright in the ldquoaggregate demand curverdquo and thus leads to a higher output and pricelevel given an upward-sloping aggregate supply curve Note that an increase in vtmeans a lower real money demand at each level of income The shift in the aggre-gate demand curve reflects the fact that a higher price level andor higher output isrequired to restore equilibrium in the money commodity and financial markets

If expectations can be taken as exogenous with respect to money supply changesthen (916) indicates that money supply changes can affect output But this wasalso the case for a Lucas-type model The next step is thus to see what happens

Keynesian model 141

when we assume rational expectations As will become clear even with rationalexpectations expectations formed at the end of period t minus 2 can be viewed asexogenous with respect to monetary changes planned for period t based on infor-mation obtained during period t minus 1 Thus monetary policy has the potential tooffset the persistent effect of disturbances that originate during period t minus 1

Introducing rational expectations

Let us now assume that individuals form their expectations Etminus1Pt and Etminus2Ptldquorationallyrdquo in that

Etminus2Pt = E(Pt |tminus2) (917)

indicating that EtminusiPt is the mathematical expectation of Pt conditional on theinformation set tminusi which is all information available at the end of period t minus ii = 1 2 Taking the expectation of (915) at the end of period t minus 2 we thus have

Etminus2Pt = Etminus2

[12 Etminus1Pt + 1

2 Etminus2Pt minus ut + vt + mt

] (919)

Note that Etminus2Etminus1Pt = Etminus2Pt Thus (919) becomes

Etminus2Pt = Etminus2minusut + vt + mt (920)

Taking the expectation of (915) at the end of period t minus1 (at time t) we then have

[12 Etminus1Pt + 1

] (921)

Substituting (920) into (921) gives

[12 Etminus1Pt + 1

2 Etminus2minusut + vt + mt minus ut + vt + mt

Rearranging

34 Etminus1Pt = 1

4 Etminus2minusut + vt + mt + 12 Etminus1minusut + vt + mt

Etminus1Pt = 13 Etminus2minusut + vt + mt + 2

3 Etminus1minusut + vt + mt (923)

which is Fischerrsquos (1977) equation (16)19

142 Keynesian model

Let the money supply be determined by the simple linear rule

mt = a1utminus1 + b1vtminus1

Since mt is a function only of information available up to the end of period t minus 1(at time t) Etminus1mt = mt Accordingly (923) can be written as

3 Etminus1minusut + vt + 23 mt (924)

Substituting (920) and (924) into the reduced-form equation for output (916)we obtain

Yt = 12 [ut + vt + mt] minus 1

[13 Etminus2minusut + vt + mt

+ 23 Etminus1minusut + vt + 2

]minus 1

4 [Etminus2minusut + vt + mt](925)

Equation (925) simplifies to

Yt = 13 (mt minus Etminus2mt) + 1

2 (ut + vt) + 16 Etminus1ut minus vt

+ 13 Etminus2ut minus vt

which is Fischerrsquos (1977) equation (18)As Fischer (1977 196) notes

disturbances aside this very simple macro model would be assumed in equi-librium to have the real wage set at its full employment level would imply theneutrality of money and would obviously have no role for monetary policyin affecting the level of output A potential role for monetary policy is createdby the presence of the disturbances ut and vt that are assumed to affect thelevel of output each period Each of the disturbances is assumed to follow afirst-order autoregressive scheme

ut = ρ1 middot utminus1 + εt where |ρ1| lt 1 (927)

vt = ρ2 middot vtminus1 + ηt where |ρ2| lt 1 (928)

where εt and ηt are mutually and serially uncorrelated stochastic terms withexpectation zero and finite variances σ 2

e and σ 2n respectively

Given equations (927) and (928) and the money supply rule

mt = a1utminus1 + b1vtminus1 (929)

Etminus2mt = a1ρ1utminus2 + b1ρ2vtminus2 (930)

Keynesian model 143

so that

mt minus Etminus2mt = a1utminus1 + b1vtminus1 minus [a1ρ1utminus2 + b1ρ2vtminus2]= a1εtminus1 + b1ηtminus1

According to (931)

the difference between the actual money stock in period t and that stockas predicted two periods earlier arises from the reactions of the monetaryauthority to the disturbances εtminus1 and ηtminus1 occurring in the interim It isprecisely these disturbances that cannot influence the nominal wage for thesecond period of wage contracts entered into at time t minus 2

(Fischer 1977 199)

Substituting (931) into (926)

Yt = 13 (a1εtminus1 + b1ηtminus1) + 1

+ 13 Etminus2ut minus vt (932)

From (927) and (928) we know that

ut + vt = (ρ1utminus1 + εt) + (ρ2vtminus1 + ηt)

= (ρ1utminus1 + εt) + (ρ22vtminus2 + ρ2ηtminus1 + ηt)

since by substitution vt = ρ22vtminus2 + ρ2ηtminus1 + ηt we also have that

Etminus1ut minus vt = ρ1utminus1 minus ρ2vtminus1 = ρ1utminus1 minus ρ22vtminus2 minus ρ2ηtminus1

Etminus2ut minus vt = ρ21utminus2 minus ρ2

2vtminus2

Thus we can rewrite (932) as

Yt = 12 (εtminus1 + ηtminus1) + 1

3 [εtminus1(a1 + 2ρ1) + ηtminus1(b1 + ρ2)] + ρ21utminus2

Fischer (1977 199) notes that

before we examine the variance of output as a function of the parameters a1and b1 it is worth explaining why the values of those parameters affect thebehavior of output even when the parameters are fully known The essential

144 Keynesian model

reason is that between the time the two-year contract is drawn up and the lastyear of operation of that contract there is time for the monetary authorityto react to new information about recent economic disturbances Given thenegotiated second-period nominal wage the way the monetary authority reactsto disturbances will affect the real wage for the second period of the contractand thus output

Optimal monetary policy rules the effectiveness of policy

As in our discussion of Sargent and Wallace let us presume that the goal of themonetary authority focuses solely on output In particular suppose that the mone-tary authority desires to set the money supply in order to minimize the fluctuationin the log of output around some desired level Then the objective can be expressedas to

min Etminus1(Yt minus Y lowast)2

Let us assume that Y lowast = Yn = 0 so that the objective becomes to

min Etminus1(Yt)2 (934)

From (933) we have that

(Yt)2 =

[12 (εt + ηt) + 1

3 [εtminus1(a1 + 2ρ1) + ηtminus1(b1 + ρ2)] + ρ21 utminus2

]times[

12 (εt + ηt) + 1

Note that E(εi) = E(ηi) = 0 E(ε2i ) = σ 2

e E(η2i ) = σ 2

n and that our independenceassumptions imply that E(εiηi) = 0 and for i = s E(εiεs) = 0 and E(εiηs) = 0Thus substituting (935) into (934) we have the following explicit form for theobjective

min Etminus1(Yt)2 = σ 2

[14 + 1

9 (a1 + 2ρ1)2]

+ (ρ21utminus2)

+ σ 2n

[14 + 1

9 (b1 + ρ2)2]

Given (936) the optimal monetary rule is to choose values for a1 and b1 suchthat

a1 = minus2ρ1 b1 = minusρ2 (937)

The above findings correspond to Fischerrsquos (1977) equation (23)21

Keynesian model 145

As Fischer (1977 200) states

to interpret the monetary rule examine [equation (933)] It can be seen therethat the level of output is affected by current disturbances (εt + ηt) that can-not be offset by monetary policy by disturbances (εtminus1 and ηtminus1) that haveoccurred since the signing of the older of the existing labor contracts andby a lagged real disturbance utminus2 The disturbances εtminus1 and ηtminus1 can bewholly offset by monetary policy and that is precisely what equation [(937)]indicates The utminus2 disturbance on the other hand was known when the olderlabor contract was drawn up and cannot be offset by monetary policy becauseit is taken into account in wage setting Note however that the stabilizationis achieved by affecting the real wage of those in the second year of laborcontracts and thus should not be expected to be available to attain arbitrarylevels of output ndash the use of too active a policy would lead to a change in thestructure of contracts

[A] more general interpretation of the monetary rule is to accommodatereal disturbances that tend to increase the price level and to counteract nominaldisturbances that tend to increase the price level

Fischer concludes by noting that

given a structure of contracts there is some room for maneuver by the mon-etary authorities ndash which is to say that their policies can though will notnecessarily be stabilizing

Conclusion

This chapter presented the sticky money wage or Keynesian model of the macro-economy We find that in contrast to the neoclassical model changes in the pricelevel affect real variables and the amount of labor employed in the economy Thusmoney has real effects The development of an upward-sloping aggregate supplycurve has dramatic implications for the conduct of monetary policy However it isshown that the expectations of agents in the economy also play an important rolein whether or not monetary policy is effective

10 The Lucas model

Introduction

As we have seen anticipated changes in prices have no impact on real vari-ables in the neoclassical model A key element of this model is the ldquoessentialpresumption that nominal output is determined on the aggregate demand sideof the economy with the division into real output and the price level largely depen-dent on the behavior of suppliers of labor and goodsrdquo (Lucas 1973) As such thismodel implies no link between price changes and real output

We have also seen how the natural rate model allows one to introduce a linkbetween unanticipated price changes and real output The seminal paper by Lucas(1973) formally develops a more complete model of the potential for ldquoshort-runsupply behavior (resulting) from suppliersrsquo lack of information on some of theprices relevant to their decisionsrdquo Lucasrsquos explanation of a tradeoff between unem-ployment and inflation ldquois that the positive association of price changes and outputarises because suppliers misinterpret general price movements for relative pricechangesrdquo

As with the ldquoillusion modelrdquo Lucas postulates ldquorational agents whose deci-sions depend on relative prices only placed in an economic setting where theycannot distinguish relative from general price movementsrdquo That is we retain thehypothesis that prices adjust to clear markets

Lucas adds to the simple (static) natural rate model so far discussed by explicitlymodeling the source of forecast errors In doing so he assumes that ldquoinferenceson these relevant unobserved prices are made optimally (or lsquorationallyrsquo) in lightof the stochastic nature of the economyrdquo Below we outline Lucasrsquos model1

The ldquoislandrdquo paradigm

Lucasrsquos model begins by disaggregating the economy into a number of what havebeen called ldquosectorsrdquo ldquomarketsrdquo or ldquoislandsrdquo As Lucas says ldquowe imagine sup-pliers as located in a large number of scattered competitive markets Demand forgoods in each period is distributed unevenly over markets leading to relative aswell as general price movementsrdquo In terms of our previous analysis one couldthink of each of the n sectors in the economy as inhabited by firms producing the

Lucas model 147

ith commodity (i = 1 n) Associated with each sector or ldquoislandrdquo is a set ofworkers and thus a labor market

For firms producing commodity i in period t the key relative price is the priceof their output relative to the wage paid in sector i or pitwit where pit is themoney price of commodity i (produced in sector i) and wit is the money wage forlabor in sector i It is assumed that individuals (firms and workers) in sector irsquoslabor market know the money wage and price of commodity i For labor suppliersin sector i however the key relative price is witpt where pt is the economy-wideprice level reflecting the fact that suppliers plan to use money wages to purchasea bundle of goods consisting of all n commodities2

It is this setup of ldquodispersed marketsrdquo and ldquoinformational discrepanciesrdquo thatLucas uses to generate a correlation between price changes and output ndash the famousLucas supply equation

The supply function for a particular sector

The Lucas model assumes a competitive labor market for sector or ldquoislandrdquo i suchthat the equilibrium level of employment and money wage equate market demandand supply With respect to the labor demand function let us start by assumingthe simple CobbndashDouglas production function such that the marginal product oflabor is given by

a(Nit)minusα where a gt 03

The profit-maximizing condition for the representative firm producingcommodity i is to equate the money wage to the marginal product of labor multi-plied by the money price of output The resulting optimal labor demand can thusbe defined by the equation4

wit = pita(N dit )

minusα

Taking the natural log of this equation and rearranging we have

ln N dit = 1

α[ln pit minus ln wit + ln a] (101)

With respect to labor supply let us assume for the moment that the real wage isknown Further let us assume that the labor supply function takes the followinglogarithmic form

ln N sit = 1

where β is a positive constantEmployment contracts entered into at time t in sector i specify the money

wage wit so that element of the real wage is known However if the price level is

148 Lucas model

unknown then the expected real wage based on information available at time t isequal to witEt(1pt) The associated expected logarithm of the real wage is then5

Et ln(witpt) = ln wit minus Et(ln pt)

Given the assumed log-linear labor supply function and ignoring the implicationsof uncertainty for labor supply we thus have

ln N sit =

)(ln wit minus Et(ln pt)) (102)

Equilibrium in the labor market for sector i entails a level of employment Nitand money wage wit such that the demand for labor equals the supply In loga-rithmic form and using the specific labor demand and supply functions given byequations (101) and (102) equilibrium requires that the log of the money wageln wit and the log of employment ln Nit be such that

ln Nit = 1

α(ln pit minus ln wit + ln a)

ln Nit = 1

β(ln wit minus Et(ln pt))

Substituting the first expression into the second to eliminate the logarithm of themoney wage we have

ln Nit = 1

α(ln pit + ln a) minus 1

α(β ln Nit + Et(ln pt))

which upon rearranging becomes

ln Nit = ln Nni + [1 + (α + β)][ln pit minus Et(ln pt)] (103)

where ln Nni = (ln a)(α + β)Equation (103) indicates that the logarithm of equilibrium employment in the ith

sector and thus the production of commodity i depends directly on the expectationof logarithm of the ratio of the price of commodity i(pit) to the general level ofprices pt The term Nni can be viewed as the ldquonormalrdquo level of employment Notethat we have abstracted from population growth and other factors that would resultin this ldquonormalrdquo level of employment varying across time

Given the assumption of a simple CobbndashDouglas production function of theform yit = (Nit)

1minusα(Ki)α we thus have

ln yit = ln yni + γ [ln pit minus Et(ln pt)] (104)

where γ = (1 minus α)(α + β) and ln yni = (1 minus α) ln Nni + α ln Ki The term yni isdenoted by Lucas as the ldquonormalrdquo level of output in the particular sector or market i

Lucas model 149

under consideration As Lucas (1973 327) states the ldquoquantity supplied in eachmarket will be viewed as the product of a normal (or secular) component commonto all markets and a cyclical component which varies from market to market

The cyclical component varies with perceived relative prices and with its ownlagged valuerdquo Note that for the moment we do not include the lagged value ofoutput in the above supply function One can justify the inclusion of such byassuming adjustment costs

The source of forecasting errors

According to (104) output of commodity i depends critically on suppliersrsquo fore-cast of the log of the general level of prices Et(ln pt) Now consider how sucha forecast may be obtained in a stochastic environment First it is assumed thatagents in sector i ndash in particular suppliers of labor involved in the production ofcommodity i ndash know commodity irsquos price pit However the exact extent to whichany change in the money price of commodity i reflects a change in the overalllevel of money prices as opposed to a change in commodity irsquos price relative toother prices is unknown It is this uncertainty that leads suppliers in sector i tomisinterpret a change in the general price level in terms of a change in a relativeprice

To be concrete suppose Etminus1(ln pt) incorporates all information available atthe end of period t minus 1 The logarithm of the actual price level will vary from thelogarithm of this expected price level to the extent that there are ldquosurprisesrdquo withrespect to the aggregate price level Letting ξt denote this ldquosurpriserdquo for period twe have

ln pt = Etminus1(ln pt) + ξt (105)

We assume that ξt which is that part of the price level that cannot be predictedfrom past data is a normally distributed random variable with zero expectationand variance σ 26

At the start of period t suppliers in market i receive one additional piece ofinformation the logarithm of the price of commodity i ln pit This signal isassumed to contain some information about the logarithm of the overall pricelevel in that

ln pit = ln pt + zit (106)

where zit is a normally distributed random variable with zero mean and varianceσ 2

z Thus using equation (105) to substitute for ln pt

ln pit = Etminus1(ln pt) + ξt + zit (107)

In words the logarithm of the nominal price for the sector ln pit is assumed toinform the supplier of the sum of the current ldquowhite noiserdquo innovations to the

150 Lucas model

relative price process in that sector (zit) and the innovations to aggregate demandand thus the economy-wide level of prices (ξt)7

We can then express the expectation of the logarithm of the price level at time tfor suppliers in sector i given the observed logarithm of the price of commodity iln pit by

Et(ln pt) equiv Eln pt |Etminus1(ln pt) ln pit (108)

Formally we have the joint distribution of two random variables f (ln pit ln pt)where one of them ln pit is known to take a particular value The problem is thebasic one of ldquobivariate regressionrdquo in that we have to determine the conditionalexpectation Eln pt | ln pit namely the ldquoaveragerdquo value of ln pt for the givenvalue of ln pit 8 As we shall see in the next section the resulting expression for theexpected general price level (in logs) given ln pit is observed can be expressedin linear form as

Et(ln pt) = Etminus1(ln pt) + (1 minus θ)(ln pit minus Etminus1(ln pt)) where 0 le θ le 1(109)

A digression on linear regression analysis

Let us assume a linear regression equation that links the observed logarithm of theprice of commodity i to the logarithm of the general level of prices of the form9

Etln pt | ln pit = a0 + a1(ln pit) (1010)

We can express the regression coefficients a0 and a1 in terms of some of the lowermoments of the joint distribution of ln pt and ln pit namely in terms of10

Eln pit = Etminus1(ln pt) (from (107))

Eln pt = Etminus1(ln pt) (from (105))11

Varln pit = σ 2 + σ 2z (from (107))

Cov(ln pit ln pt) = E(minusξt minus zit)(minusξt) = σ 2 + Ezit middot ξtIn general Ezit middot ξt = Cov(zit ξt) + EzitEξt However given that Ezit =Eξt = 0 and given the assumption that zit and ut are independent variables sothat Cov(zit ξt) = 0 we have12

Cov(ln pit ln pt) = σ 2

From (1010) we have that13

E(ln pt | ln pit) equivint

(ln pt)φ(ln pt | ln pit)d ln pt = a0 + a1(ln pit) (1011)

Lucas model 151

where φ(middot) is the conditional density function of ln pt given ln pit If we thenmultiply the expression on both sides of (1011) by the marginal density functionof ln pit denoted by g(ln pit) and integrate on ln pit we obtainintint

(ln pt)φ(ln pt | ln pit)g ln(pit)d ln ptd ln pit

a0g(ln pit)d ln pit +int

a1(ln pit)g(ln pit)d ln pit

Eln pt = a0 + a1Eln pit (1012)

since φ(ln pt | ln pit)g ln(pit) = f (ln pit ln pt) Had we multiplied the expressionon both sides of (1011) also by ln pit before integrating on ln pit we would haveobtainedintint

(ln pt)(ln pit)φ(ln pt | ln pit)g ln(pit)d ln ptd ln pit

a0(ln pit)g(ln pit)d ln pit +int

a1(ln pit)2g(ln pit)d ln pit

E(ln pit)(ln pt) = a0Eln pit + a1E(ln pit)2 (1013)

Solving (1012) and (1013) for a0 and a1 and making use of the fact that

E(ln pit)(ln pt) = Cov(ln pit ln pt) + E(ln pit)Eln ptand

E(ln pit)2 = Var(ln pit) + [Eln pit]2

we find that

a0 = Eln pt minus [Cov(ln pit ln pt)Eln pit](Var(ln pit))

a1 = (Cov(ln pit ln pt))(Var(ln pit))

Hence we can write equation (1010) as

Et(ln pt) equiv E(ln pt | ln pit)

= E(ln pt) + [Cov(ln pit ln pt)Var(ln pit)](ln pt minus Eln pit)Substituting in the above expressions for means variance and covariance we havethus derived (109) with

1 minus θ = σ 2(σ 2 + σ 2z )

152 Lucas model

Equation (109) indicates that agentsrsquo rational expectation of the current pricelevel is a ldquolinear least-squares projectionrdquo That is one could rewrite (109) as

Et(ln pt) = θEtminus1(ln pt) + (1 minus θ) ln pit (1014)

where θ = σ 2z (σ 2 + σ 2

z ) To see why this is called ldquolinear least-squaresrdquo notethat we could start with (1014) (the ldquolinearrdquo part of the projection) and thenpick θ to minimize the variance in this forecast or projection of ln pt (the ldquoleast-squaresrdquo part of the projection) In particular substituting in (105) for Etminus1(ln pt)

(ie Etminus1(ln pt) = ln pt minus ξt) and (106) for ln pit (ie ln pit = ln pt + zit)(1014) becomes

Et(ln pt) = ln pt minus θξt + (1 minus θ)zit (1014prime)

The problem of picking θ to minimize the variance of this projection can then beexpressed as14

Eln pt minus θξt + (1 minus θ)zit minus ln pt2 = θσ 2 + (1 minus θ)σ 2z

Taking the derivative of the above expression with respect to θ setting it equal tozero (ldquoleast squaresrdquo) and solving for θ we verify that θ = σ 2

z (σ 2 + σ 2z )

As Sargent (1987a 442) points out

the parameter θ is the fraction of the conditional variance in ln pit due torelative price variation The larger is this fraction the smaller is the weightplaced on ln pit in revising Etminus1(ln pt) to form Et(ln pt) This makes sensesince the larger is θ the more likely it is that a change in ln pit reflects arelative rather than a general price change15

Equation (1014) can be substituted into the supply function for commodity i(104) to obtain

ln yit = ln yni + γ θ(ln pit minus Etminus1(ln pt)) (1015)

where as before ln yni = (1 minus α)(ln Nni) + α(ln Ki) As noted above if weassumed adjustment costs then a lagged output term could be added to (1015)In this case we would have16

ln yni = ln yni + γ θ [ln pit minus Etminus1(ln pt)] + λ(ln yitminus1 minus ln yni) (1015prime)

If suppliers were able to observe the actual value of the price level so thatEt(ln pt) = ln pt then going back to (104) one could express the resulting ldquofullinformationrdquo output produced in sector i by

ln ylowastit = ln yni + γ [ln pit minus ln pt]

Lucas model 153

which given ln pit = ln pt + zit simply becomes

ln ylowastit = ln yni + γ zit

As you can see since the expectation of the random shock to relative prices zit iszero ln yni has the natural interpretation as the expected output of sector i givenfull information

The Lucas aggregate supply function

Equation (1015) is close to what is known as the ldquoLucas aggregate supplyfunctionrdquo Without adjustment costs the Lucas supply function takes the form

ln yt = ln yn + γ θ(ln pit minus Etminus1(ln pt)) (1016)

With adjustment costs the Lucas supply function takes the general form

ln yt = ln yn + γ θ(ln pit minus Etminus1(ln pt)) + λ(ln ytminus1 minus ln yn) (1016prime)

where yn denotes the natural rate of output17 For simplicity we have assumed thatthe natural rate of total output is constant across periods

The term ln yn can be interpreted either as the logarithm of output for theldquorepresentativerdquo sector or as the logarithm of total output across the n sectorsLet us assume the former interpretation The average level of real output can bedefined by

yt equiv 1

⎡⎣ nprod

pityit

⎤⎦

where [prodni=1 pityit]1n is the geometric mean of nominal output across the n markets

or sectors Taking logs we have the following definition for the logarithm ofaverage output

ln yt equiv minus ln pt + 1

nsumi=1

(ln pit + ln yit) (1017)

We will assume that the overall price level is constructed as a geometric mean ofindividual prices such that

pt equiv⎡⎣ nprod

⎤⎦

Taking logs

ln pt equiv 1

nsumi=1

ln pit

154 Lucas model

Substituting the above into (1017) we have the following definition for thelogarithm of average output

ln yt equiv 1

nsumi=1

ln yit (1018)

Substituting into (1018) the supply functions for the individual sectors as givenby (1015) we thus have18

ln yt equiv 1

nsumi=1

[ln yni + γ θ(ln pit minus Etminus1(ln pt))] (1019)

Recall that ln pit = ln pt + zit where zit is a normal random variable indepen-dently distributed across markets with a mean of zero and variance σ 2

z Substitutingthis into (1015) and rearranging we have

ln yt = 1

nsumi=1

[ln yni] + γ θ(ln pt minus Etminus1(ln pt)) + 1

nsumi=1

γ θzit (1020)

As the number of markets n approaches infinity from the law of large numberswe know that the sum of the zit divided by n approaches zero19 Thus for a largenumber of markets we may approximate (1020) by

ln yt = 1

⎡⎣ nsum

ln yni + γ θ(ln pt minus Etminus1(ln pt))

⎤⎦ (1021)

By definition the logarithm of the geometric average of ldquonormalrdquo output acrossmarkets ln yn is given by nminus1 sumn

i=1 ln yni Thus we can rewrite (1021) as (1016)in which ln yn is the logarithm of ldquonormalrdquo output that would occur if there were nosurprises with respect to the aggregate price level that is when ln pt = Etminus1(ln pt)Note that for simplicity we assume the natural rate is constant over time

Equation (1016) is the Lucas aggregate supply equation with the last termmissing As noted above if we include adjustment costs then we obtain (1016prime)indicating that the deviation of real output from its ldquonaturalrdquo level or trend isassociated with a deviation in the price level from that expected and past deviationsof output from the natural rate This last term makes output serially correlatedover time

The Lucas supply function and the Phillips curve

The Lucas supply function predicts a direct correlation between unanticipatedprice changes and output and thus a potential tradeoff between price changesand unemployment if one assumes that unemployment and output are inverselyrelated This potential inverse relationship between unemployment and inflation

Lucas model 155

is sometimes referred to as the ldquoPhillips curverdquo after AW Phillips who noted theempirical relationship between wage inflation and unemployment for the Britisheconomy for the 100 years up to 1957 (see Phillips 1958) Later depictions of thePhillips curve replaced the rate of change in wages with the inflation rate

To see this Phillips relationship more clearly rearrange the aggregate Lucassupply function without adjustment costs (1016) to obtain

ln pt = (ln yt minus ln yn)γ θ + Etminus1(ln pt)

Subtracting ln ptminus1 from both sides of this aggregate supply equation we have

ln(ptptminus1) = (ln yt minus ln yn)γ θ + Etminus1(ln(ptptminus1)) (1022)

Let the term πt denote the rate of inflation between periods t minus 1 and t20

πt equiv (pt minus ptminus1)ptminus1 = (ptptminus1) minus 1

It is common in macroeconomics to approximate the above rate of change inprices by the log of the ratio of the two prices If the ratio equals one then thelog equals zero which is the rate of inflation If the ratio is 1 + x and x is a smallproportion then the log of this ratio approximately equals the actual inflation rateFor instance if ptptminus1 = 105 so that inflation is 005 or 5 percent then the logof 105 is 00488 which approximates this 005 rate of inflation Thus we have

πt asymp ln(ptptminus1) = ln pt minus ln ptminus1

Using the above approximation for the rate of inflation we can rewrite(1022) as

πt = (ln yt minus ln yn)γ θ + Etminus1πt (1023)

which as Sargent (1987a 443) states

is in the form of a standard natural rate Phillips curve relating inflation (πt)

directly to output (ln yt) and to expected inflation (Etminus1πt) According to[(1023)] the Phillips curve shifts up in the (πt yt) plane by the exact amountof any increase in expected inflation This characteristic of equation [(1023)]is often taken as the hallmark of the natural unemployment rate hypothesisIt seems to offer an explanation for why the Phillips curve tradeoff worsenedas average inflation rates increased over the 1970s in many western countries

If we assumed that due to adjustment cost the lagged deviation in output fromthe natural level affects the current deviation as the Lucas aggregate supplyequation (1016prime) suggests then in terms of rates of change in prices we wouldhave

πt = (ln yt minus ln yn)γ θ + Etminus1πt minus (λγ θ)(ln ytminus1 minus ln yn) (1023prime)

156 Lucas model

We could instead express (1023) in terms of unemployment by assuming thereis a linear inverse relationship between deviations in output from the natural rateand deviations in the actual level of unemployment Ut from its natural rate Unsuch that

ln ytminus1 minus ln yn = minus(Ut minus Un)

where is a positive constant Substituting the above into (1023) we thus have

πt = minus(γ θ)(Ut minus Un) + Etminus1(πt) (1023primeprime)

indicating the inverse relationship between unanticipated price changes and theactual level of unemployment Rearranging (1023primeprime) we have that

Ut = Un minus (γ θ)[πt minus Etminus1(πt)] (1023primeprimeprime)

where γ θ gt 0 Equation (1023primeprimeprime) is the typical expression of the Phillips curvefound in the literature It indicates that deviations in the unemployment rate belowits natural level must be accompanied by deviations in the actual rate of inflationabove that expected It reflects the ldquonatural rate of unemployment hypothesisrdquo asoriginally coined by Friedman (1968 11)

There is always a temporary trade-off between inflation and unemploymentthere is no permanent trade-off The temporary trade-off comes not frominflation per se but from unanticipated inflation which generally means froma rising rate of inflation

Recall that as Barro and Gordon (1983 592) observed the term Etminus1(πt) in(1023primeprimeprime) is the

prior expectation of inflation for period t [which is] distinguished from theexpectation that is conditional on partial information about current pricesThis distinction arises in models (eg Lucas 1972 1973 Barro 1976) inwhich people operate in localized markets with incomplete information aboutcontemporaneous nominal aggregates In this setting the Phillips curve slopecoefficient (γ θ) turns out to depend on the relative variances for generaland market-specific shocks

Variability in prices and the tradeoff

As Lucas (1973 333) states

demand policies [can] tend to move inflation rates and output (relative totrend) in the same direction or alternatively unemployment and inflation inopposite directions The conventional Phillips curve account of this observedco-movement says that the terms of the tradeoff arise from the relatively stable

Lucas model 157

structural features of the economy and are thus independent of the nature ofthe aggregate demand policy pursued The alternative explanation of the sameobserved tradeoff is that the positive association of price changes and outputarises because suppliers misinterpret general price movements for relativeprice changes

Taking Lucasrsquos alternative viewpoint two aspects concerning the tradeoff aresuggested First as Lucas states ldquochanges in average inflation rates will notincrease average outputrdquo As we have seen if we compare the expected pricelevel for period t with the price level for the prior period the difference wouldincorporate individualsrsquo expectation of this average rate of inflation (along with anumber of other potentially relevant variables) Second ldquothe higher the variancein average prices the less lsquofavorablersquo will be the observed tradeoffrdquo We considerthis second point below by referring back to the simple Lucas supply functionwithout lagged output (1016)

Recall that the term ξt given by (105) denotes that part of the price level thatcannot be predicted from past data We have assumed that this ldquosurpriserdquo term ξtis a normally distributed random variable with zero expectation and variance σ 2Substituting this into (1016) we have that

ln yt minus ln yn = γ θξt (1024)

z ) and γ = (1 minus α)(α + β) Recall that θ is the weightattached to the expected price level prior to observing pit

Equation (1024) indicates that deviations in output from the natural level dependsolely on surprises In his statement concerning the variance of prices Lucas ispointing out that the impact of ldquosurprisesrdquo on output relative to its natural leveldepends on the ldquosloperdquo term λθ which is given by

λθ = 1 minus α

α + β

σ 2 + σ 2z

As Sargent (1987a 444) notes

a ldquofavorablerdquo tradeoff between output and unexpected inflation (that is a largevalue of γ θ ) will exist only when σ 2 is small relative to σ 2

z An attempt byauthorities to exploit the tradeoff between output and unexpected inflationmore fully by changing aggregate demand regimes might increase the vari-ance σ 2 relative to σ 2

z and thus change the slope γ θ This is yet anotherexample of how agentsrsquo optimal decision rules change in response to changesin the random processes governing the exogenous variables they base theirdecisions on

Sargentrsquos last point is another example of the ldquoLucas critiquerdquo in this contextwith respect to the validity of using past econometric estimates of a tradeoff inpredicting future tradeoffs

158 Lucas model

Note that although the tradeoff worsens with higher variability in prices theeffect of higher variability in prices on the variance of output about the natural rateis unclear In particular from (1016) we know that the variance in the differencebetween output and the natural rate is simply (γ θ)2σ 2 Given our definition ofγ θ the variance of the logarithm of output becomes

Var(ln yt) = γ 2(

σ 2 + σ 2z

Differentiating with respect to σ 2 we have

partVar(ln yt)

partσ 2 = γ 2(

σ 2 + σ 2z

minus 2γ 2σ 2 σ 2z

(σ 2 + σ 2z )

(σ 2 + σ 2z )2

γ 2σ 2z

σ 2 + σ 2z

)2 (1 minus 2σ 2

σ 2 + σ 2z

As the above expression indicates by itself an increase in the variation in theaverage price level (σ 2) will increase the variation in the logarithm of output for agiven ldquosloperdquo (γ θ) On the other hand as Lucas pointed out such an increase inthe variation in price level will result in a reduction in the effect of any given pricechange on output which by itself would decrease the variation in the logarithm ofoutput

Substituting (105) into the expanded Lucas supply function with lagged outputwe have the Lucas supply function of the form

ln yt minus ln yn = γ θξt + λ(ln ytminus1 minus ln yn)

where as before θ = σ 2z (σ 2 + σ 2

z ) Substituting for prior differences in outputfrom its natural level we thus have

ln yt minus ln yn = γ θ

infinsumi=0

λiξtminusi (1025)

which shows that the deviation of output from its natural rate depends on thecurrent and all previous values of the ldquoaggregate demand shockrdquo that affects theequilibrium price level

A complete model except for specifying thesource of expectations

Equation (1016) provides us with one part of the standard macroeconomic modelthe ldquoaggregate supply equationrdquo To simplify the analysis we will normalize outputso that the natural level of real output is equal to 1 We have already assumed that

Lucas model 159

the natural level of real output is constant over time These two assumptions allowus to write (1016) in the more compact form

where Yt denotes log of real output supply for period t (or equivalently the deviationin output from its natural level for period t) Pt denotes the log of the price leveland Etminus1Pt denotes the expectation of log of the price level What we now requireis a characterization of the aggregate demand side of the economy as typicallysummarized by the LM and IS equations

To obtain an explicit form for the portfolio or LM equation we start by assuminga real money demand function for the end of period t of the form

Ldt = yα1

t exp[minus(α2rt)] (1027)

where yt is real output α1 gt 0 and α2 gt 0 indicating that real money demand isdirectly related to real output but inversely related to the nominal interest rate21

Let Mt denote the nominal money supply at the end of period t (previously thishas been denoted by M ) and let mt denote the logarithm of this money supplyfor period t such that mt = ln Mt Further let us assume a logarithmic supply ofmoney function of the form

mt = mt + εt (1028)

Equation (1028) separates the logarithm of the money supply into two compo-nents a deterministic component mt set by government authorities according toa rule tying money supply changes to past variables and a random component εt which is assumed to be normally distributed with zero mean This random term isalso assumed to be serially independent (ie E(εtεs) = 0 for s = t)

The LM equation is simply the money market equilibrium condition equatingthe real supply of money to real money demand and thus is given by

Mtpt = Ldt (1029)

Taking logs of the equilibrium condition (1029) and substituting the logarithm ofthe money demand function (1027) and the money supply function (1028) wehave

mt minus Pt = α1Yt minus α2rt minus εt (1030)

where Pt = ln pt Equation (1030) is the standard log-linear form of the portfolioor LM equation22

To obtain an explicit form for the IS equation which is the equilibrium conditionin terms of equating output production to the demand for output we must postulatea specific form for output demand One common assumption is to include the

160 Lucas model

expected real rate of interest rt minus πet+1 as a determinant of output demand To do

so let

πet+1 equiv (pe

t+1 minus pt)pt = (pet+1pt) minus 1

As before we can approximate the expected rate of change in prices by the expecta-tion of the log of the ratio of the future to current price level (ie πe

t+1 asymp Pet+1minusPt

where Pet+1 is the expected log of the price level for period t+1) Thus the expected

real rate of interest becomes rt minus πet+1

Letting the term Xt denote a vector of exogenous variables that also affectsoutput demand we have in log-linear form the following equilibrium conditionfor the output market

Yt = Xt minus β1(rt minus πet+1) + ut (1031)

where ut is a serially independent stationary random process with mean zeroand finite variance equal to σ 2

u The random terms for output demand and moneysupply εt and ut respectively are assumed independent (ie E(εtut) = 0)

Summarizing we have a model consisting of the aggregate supply equation(1026) the LM equation (1030) and the IS equation (1031) which can be solvedfor the equilibrium output price and interest rate

In particular combining the LM and IS equations to eliminate the interest ratert we obtain

Yt = Xt + ut minus (β1α2) middot (minusmt minus εt + Pt minus α1Yt) + β1 middot πet+1

Yt = α2

α2 + α1β1

[Xt + ut + β1π

et+1 + β1

α2(mt + εt minus Pt)

] (1032)

Or in terms of the price level we have an ldquoaggregate demand equationrdquo of theform

Pt = mt + εt minus α2 + α1β1

β1Yt + α2

β1(Xt + ut + β1π

et+1) (1033)

Equations (1032) and (1033) indicate the inverse relationship between the pricelevel and output that is shown graphically by a downward-sloping aggregatedemand curve

With respect to (1033) note that the expected rate of inflation for the nextperiod πe

t+1 is viewed as a distinct entity Our prior assumption of unit elasticexpectations concerning the expected log of the future price level Pe

t+1 wouldimply that this term is in fact independent of changes in the current price levelNote also that if output were unchanged then an x percent change in the moneysupply would result in an x percent change in the price level This is the standardresult of the ldquoneoclassicalrdquo model23

Lucas model 161

Combining the above aggregate demand equation (1032) with the aggregatesupply equation to eliminate the output term Yt we have24

α2 + α1β1

[Xt + ut + β1π

et+1 + β1

= γ θ(Pt minus Etminus1Pt) + λYtminus1

Solving the above for the equilibrium price level one obtains

Pt = 1

J0 + β1

[β1(mt + εt) + α2(Xt + ut + β1π

(Etminus1Pt minus λYtminus1

)] (1034)

where J0 = γ θ(α2 + β1α1) Expression (1034) is sometimes called the reduced-form equation for the price level

Rearranging (1034) we have

Pt minus J0

J0 + β1Etminus1Pt

J0 + β1

[β1(mt + εt) + α2(Xt + ut + β1π

et+1) minus J0

λYtminus1

] (1034prime)

Let us assume perfect foresight meaning that Etminus1Pt = Pt Noting that1 minus J0(J0 + β1) = β1(J0 + β1) we can solve (1034prime) for the equilibriumprice level under this hypothesis of perfect foresight obtaining

Pt = mt + εtα2

β1(Xt + ut + β1π

et+1) minus J0

β1γ θYtminus1 (1034primeprime)

As equation (1034primeprime) makes clear under the presumption of limited perfectforesight the predictions are those of the neoclassical model

(a) a change in the money supply (in log form given by mt + εt) results in anequiproportionate change in the price level (in log form given by Pt)

(b) an increase in expected inflation πet+1 raises the price level

(c) a higher level of lagged output (Ytminus1) lowers the price level(d) an increase in output demand (Xt + ut) raises prices

Following a procedure similar to that used to derive (1034) if we combine theaggregate demand equation (1033) and aggregate supply equation to eliminatethe price level we have

Yt = λYtminus1 minus γ θEtminus1Pt

+ γ θ

mt + εt minus α2 + α1β1

β1Yt + α2

β1(Xt + ut + β1π

162 Lucas model

Solving for the equilibrium real output we have

Yt = β1

J0 + β1

[λYtminus1 + γ θ

[mt + εt minus Etminus1Pt + α2

β1(Xt + ut + β1π

(1035)

The above expression is sometimes call the reduced-form equation for real outputAccording to (1035) changes in the money supply (in logs given by mt = mt +εt)will affect real output to the extent that the impact of such changes on prices is notfully anticipated

Note that with perfect foresight we have Etminus1Pt = Pt In this case substituting(1034primeprime) into (1035) for Etminus1Pt we obtain

Yt = β1

J0 + β1

(λYtminus1 + J0

β1Ytminus1

)= λYtminus1 (1035primeprime)

Thus we have the standard neoclassical result that output in the current period isindependent of demand-side changes such as changes in expected inflation or themoney supply

One source of expectations autoregressive expectations

As Shiller (1978) has noted

one of the most difficult problems which confronts builders of macroeco-nomic models is the need to model the mechanism by which the public formsits expectations of future economic variables Many of the most importanttheoretical macroeconomic behavioral relations (eg the supply equationinvestment saving) depend critically on public expectations of future eco-nomic variables yet we often do not even have any data on what theseexpectations are

This and the next section suggest two approaches that have been taken to modelexpectations in particular the expected price level that enters into the aggre-gate supply equations These two approaches to expectation formation are thedistributed lag (or adaptive) scheme and the rational expectations scheme

To understand the ideas behind distributed lag schemes as the source of expec-tations we start by noting that the price level pt can be broken down into acombination of the price level for the previous period ptminus1 multiplied by theratio of the price level this period to last period

pt equiv ptminus1(ptptminus1)

Taking logs and recalling that ln(ptptminus1) is approximately equal to the rate ofinflation πt we thus have

Pt equiv ln pt = ln ptminus1 + πt

Lucas model 163

Taking expectations at time t assuming that at a minimum the price level for theprior period is known we have

Etminus1Pt equiv ln ptminus1 + Etminus1πt (1036)

Until the 1970s the approach to modeling the source of the expected rate of infla-tion embedded in (1036) was to assume individuals forecast the rate of inflationby looking at past inflation rates A common quantitative representation of thishypothesis originated by Fisher (1930) was to have individualsrsquo expectation ofthe inflation rate behave like a weighted average or ldquodistributed lagrdquo of recent pastinflation rates That is

Etminus1πt =qsum

ηiπtminusi (1037)

where the ηi are fixed numbers A typical idea behind this distributed lag approachto anticipated inflation was that individuals have ldquoadaptive expectationsrdquo whichmeant that individuals adjusted or ldquoadaptedrdquo their expectations of the rate of infla-tion in light of the actual forecast error made concerning the prior periodrsquos inflationrate Specifically adaptive expectations can be expressed as

Etminus1πt = Etminus2πtminus1 + δ(πtminus1 minus Etminus2πtminus1)

= δπtminus1 + (1 minus δ)Etminus1πtminus1(1038)

where 1 gt δ gt 0 Successive substitution allows us to rewrite (1038) as

Etminus1πt = δπtminus1 + (1 minus δ)δπtminus2 + (1 minus δ)2δπtminus3 + middot middot middotor

Etminus1πt =infinsum

δ(1 minus δ)iminus1πtminusi (1039)

As you can see (1039) is simply a specific form of equation (1037) in whichthe ηi place declining weight on past inflation rates the more distant they are andq = infin Given declining weights we can obtain a reasonable approximation of(1039) even if we truncate the distributed lag on past inflation after q periods aslong as q is reasonably large andor δ is reasonably large

Now let us place the above discussion not in terms of past rates of inflation butinstead in terms of past price levels Recalling the approximation

πtminusi asymp ln ptminusi minus ln ptminusiminus1

we can rewrite (1037) in the form

Etminus1πt =qsum

ηi(ln ptminusi minus ln ptminusiminus1) (1040)

164 Lucas model

Writing this out we have

Etminus1πt = η1(ln ptminus1 minus ln ptminus2) + η2(ln ptminus2 minus ln ptminus3)

+ η3(ln ptminus3 minus ln ptminus4) + middot middot middot + ηq(ln ptminusq minus ln ptminusqminus1)

Thus we may rewrite (1040) as

Etminus1πt = η1 ln ptminus1 + (η2 minus η1) ln ptminus2 + (η3 minus η2) ln ptminus3 + middot middot middot+ (ηq minus ηqminus1) ln ptminus4 minus ηq ln ptminusqminus1

(1041)

Combining (1041) with equation (1036) we obtain the following expression forthe expectation formed at time t concerning the log of the price level

Etminus1Pt = (1 + η1) ln ptminus1 + (η2 minus η1) ln ptminus2 + (η3 minus η2) ln ptminus3 + middot middot middot+ (ηq minus ηqminus1) ln ptminus4 minus ηq ln ptminusqminus1

Etminus1Pt =q+1sumi=1

vi ln ptminusi (1042)

Equation (1042) is what Sargent and Wallace (1975) refer to as ldquoautoregressiveexpectationsrdquo

A second source of expectations rational expectations

The papers by Lucas (1972 1973) and Sargent and Wallace (1975) suggestedthat in macroeconomic model building a different approach to specifying thesource of expectation is preferred As Cukierman (1986) summarizes this ldquorationalexpectationsrdquo approach to the modeling of inflationary expectations is

based on the maintained hypothesis that individuals know the structure of theeconomy and of governmentrsquos decision rule and that they use this structurein conjunction with the available information in order to form an optimalpredictor of future inflation [this approach] requires a precise specificationof the model of the economy as well as of the information sets of individualsEmpirical tests of this hypothesis are therefore joint tests of the validity of theexpectational hypothesis as well as of the postulated structure of the economyand of the particular assumptions made about the information possessed byindividuals

A ldquostructure of the economyrdquo was derived based on the Lucas aggregate sup-ply equation that included the assumption of market-clearing wages and pricesSuppose that individuals know this model and accept it as reflecting the structureof the economy As we saw above this model was solved to obtain (1034) the

Lucas model 165

reduced form for the equilibrium price level in period t We assume that at time tindividuals form their expectations Etminus1Pt ldquorationallyrdquo in that

indicating that Etminus1Pt is the mathematical expectation of Pt conditional on theinformation set tminus1 which is all information available at period t minus1 As Sargentnotes (1987a 440)

Lucas assumed that tminus1 included information on all lagged values of ln pitand lagged values of real output in all markets One could equally well con-ceive of less comprehensive definitions of tminus1 For now along with Lucaswe suppose that tminus1 includes a comprehensive list of variables includinglagged outputs and prices in all markets

At the end of period t minus 1 individuals of course know Etminus1Pt as well as Ytminus1and πe

t+1 Thus taking the expectation of (1034) and subtracting it from Pt wehave

Pt minus Etminus1Pt = β1

J0 + β1(mt + εt minus Etminus1mt) + α2

J0 + β1(Xt minus Etminus1Xt + ut)

(1044)

In Sargent and Wallace (1975 244) the deterministic part of the money supplymt is assumed to reflect a ldquolinear feedback rulerdquo of the form

mt = Gθlowastt (1045)

where ldquoθlowastt represents the set of current and past values of all of the endogenous and

exogenous variables in the system as of the end of period t minus 1 and G is a vectorof parameters conformable to θlowast

t rdquo A simple example of a monetary feedback rulewould be

mt = a0 + a1Ytminus1 (1046)

where a0 and a1 are positive constantsLet us assume that individualsrsquo information set tminus1 includes not only the

structure of the economy (as summarized by the above linear macroeconomicmodel) but also the money supply rule In the particular example of (1046) theyknow a0 a1 and Ytminus1 and thus mt Then the assumption of rational expectationsimplies that Etminus1mt = mt Furthermore since Xt represents the deterministicpart of the vector of exogenous variables affecting output demand we have thatEtminus1Xt = Xt Thus (1044) becomes

Pt minus Etminus1Pt = 1

J0 + β1(β1εt + α2ut) (1047)

166 Lucas model

Substituting (1047) into the aggregate supply equation one obtains

Yt = γ θ1

J0 + β1(β1εt + α2ut) + λYtminus1 (1048)

An important feature of (1048) is that a deviation in output from its natural levelwhich is represented by the term Yt different from zero given our assumption thatthe natural output level is normalized to equal one is determined only by pastdeviations and ldquosurprisesrdquo with respect to the money supply (the term εt) andoutput demand (the term ut) The ldquodeterministicrdquo or predictable component of anymoney supply change has no real effects Equation (1048) should look familiarGiven λ lt 1 it suggests that the times series for deviations of the logarithmof output from its natural rate is a stationary autoregressive process of order 1or AR(1)

The above analysis is an example of a ldquolinear rational expectations modelrdquoThe result that ldquopredictablerdquo monetary policy has no real effects reflects the twinassumptions of the natural rate hypothesis and rational expectations It should notbe surprising that deterministic monetary policy has no effect for the model ishomogeneous of degree 0 in mt Pt and Etminus1Pt This is a critical characteristic ofa ldquonatural rate modelrdquo

Conclusion

The main focus of this chapter has been on the development of the Lucas supplyfunction The model is often discussed in the context of the ldquoislandrdquo paradigmin which we specify the supply function for a particular sector in the economyThe role of forecasting errors is introduced and from that the Lucas aggregatesupply function is constructed and the relationship of this function to the Phillipscurve is discussed A number of other issues were discussed involving variabilityin prices and the corresponding economic tradeoff and then a complete modelwas introduced except for specifying the source of expectations Two sources ofexpectations were then described autoregressive expectations and rational expec-tations The implications of expectations were then discussed in their historicalcontext

11 Policy

Introduction

This chapter extends the earlier discussions about the actions of the monetaryauthority and how these actions affect the macroeconomy Perhaps the most inter-esting issue of monetary economics is addressed here that is the optimal roleof monetary policy The chapter highlights the differences in model results thatdepend on what type of expectations are assumed Particular attention is givento the Sargent and Wallace ldquoineffectiveness propositionsrdquo and the Phillips curveOther issues are then introduced including the ldquorule versus discretionrdquo debatetime inconsistency and the role of credibility and enforcement

Optimal monetary policy

In the 1970s the articles by Sargent and Wallace (1975 1976) and Lucas (19721973) altered the view of how one should assess the impact of monetary policy onthe economy and by implication what is optimal monetary policy The discussionstarts with the premise that monetary policy should be conducted according to arule or set of rules As Sargent and Wallace (1976 169) state

It is widely agreed that monetary policy should obey a rule that is a scheduleexpressing the setting of the monetary authorityrsquos instrument (eg the moneysupply) as a function of all the information it has received up through thecurrent moment Such a rule has the happy characteristic that in any givenset of circumstances the optimal setting for policy is unique If by remotechance the same circumstances should prevail at two different dates theappropriate settings for monetary policy would be identical1

The SargentndashWallace premise that monetary rules are preferred leads them toexplore the form of the optimal rule But as we will discover in going over thepaper by Barro and Gordon (1983) there is a question of whether monetary rulescan be enforced over time If not then what is typically left in these models isa ldquosecond bestrdquo solution involving the determination of optimal ldquodiscretionaryrdquo

168 Policy

policy Note that we use the term ldquosecond bestrdquo because enforceable rules tend todominate discretion in these models2

Accepting the premise that monetary policy can adopt enforceable rules stillleaves open the specification of the optimal set of rules The simplest rule sug-gested by Friedman (1959) would increase the money supply at a constant rateeach year perhaps 3 percent3 More complex rules known as ldquoreactive rulesrdquowould specify in advance how the growth of the money supply will change basedon new information on the state of the economy One such rule suggested is thatthe growth in the monetary base and thus the money supply automatically adjustwhenever the growth of nominal GNP deviates from its trend (McCallum 1985)Another reactive rule suggests that the government commit itself to holding theCPI to a preannounced target and adjust the monetary base and thus the moneysupply accordingly (Hall 1982)4

Below we begin our discussion of optimal monetary policy by reviewing theanalysis of optimal enforceable rules as suggested by the Sargent and Wallace(1975 1976) papers and reviewed in Sargent (1987a Chapter 17) This discussionis in the context of a natural rate model without rational expectations and then inthe context of a model which assumes rational expectations

Optimal monetary policy exogenous expectations

As we saw previously the reduced form for the log of the output in period t canbe expressed as

Yt = H0

[λYtminus1 + γ θ

[minusEtminus1Pt + mt + εt + α2

β1(Xt + ut + β1π

where H0 = β1(J0 +β1) and J0 = γ θ(α2 +β1α1) Recall that Yt is the differencein period t between the logarithm of output and the logarithm of the natural levelof output Normalizing so that the natural level of output equals one we canequivalently interpret Yt in equation (111) as total output As Sargent and Wallace(1976) state ldquoYt can be thought of as the unemployment rate or the deviation ofreal GNP from lsquopotentialrsquo GNP This equation should be thought of as the reducedform of a simple econometric modelrdquo

Recall that the log of the money supply in period t mt is the sum of adeterministic component mt and the random component εt with variance σ 2

To understand the impact of monetary changes on real GNP we must firstconsider how the expected log of the price level Etminus1Pt varies with changes inmonetary policy One approach in the spirit of ldquoautoregressive expectationsrdquo isto assume that the expected price level is independent of the current monetarypolicy This essentially means viewing the expectation of the log of the price level(Etminus1Pt) as exogenous6 Given this assumption we may rewrite the reduced-form

Policy 169

equation for output (111) as

Yt = a0 + a1Ytminus1 + a2mt + vt (112)

where vt = H0γ θ(α2β1)ut is a serially independent normally distributed randomvariable with variance σ 2

v and mean zero mt is the log of the money supply forperiod t where mt = mt + εt Ytminus1 is lagged output and a0 a1 and a2 areparameters

Suppose that the monetary authority desires to set the money supply in orderto minimize the fluctuation in the log of output around some desired level Let usassume that the log of this desired level denoted Y lowast is above the log of the naturallevel of output Yn7 Then the objective can be expressed as

We can break this expression into two terms in that the objective can beequivalently expressed as

min Etminus1(Yt minus Etminus1Yt)2 + (Etminus1Yt minus Y lowast)2 (113)

The second way of expressing the objective allows us to see the objective asminimizing the sum of two terms the variance of Yt conditional on informationup to the end of period t minus 1 and the ldquobias squaredrdquo around Y lowast The secondterm the bias squared around Y lowast is the reason for an ldquoactivistrdquo monetary policyEquation (113) indicates that the optimal monetary policy entails

1 minimizing the variance in the random component of the money supply Thisfollows since the first term in equation (113) the variance of Yt is given bya2

2σ2e +σ 2

v 8 If feasible complete elimination of the random component to themoney supply (ie a purely ldquodeterministicrdquo money supply) is optimal suchthat εt = 0 for all t and thus σ 2

e = 02 setting Etminus1Yt = Y lowast so as to make the second term in equation (113) equal

to zero From equation (112) this means a monetary policy such that

Etminus1(a0 + a1 middot Ytminus1 + a2 middot mt + vt) = Y lowast

Noting that Etminus1Yt = 0 and that Etminus1mt = mt we see that this deterministic partof the optimal monetary policy is defined by the equation

a0 + a1Ytminus1 + a2mt = Y lowast

which can be solved for the optimal deterministic monetary policy rule

mt = g0 minus g1Ytminus1 (114)

where g0 = (Y lowast minus a0)a2 and g1 = a1a2

170 Policy

An equivalent expression for the optimal monetary rule (114) is derived inSargent (1987a Chapter 17) under the presumption of exogenous expectations Inparticular Sargent uses equation (112) to substitute out for Ytminus1 in (114) so thatthe optimal (deterministic) monetary rule (114) becomes

mt = g0 minus g1[a0 + a1Ytminus2 + a2mtminus1 + vtminus1] (115)

Now substituting into (115) the expression for Ytminus2 suggested by equation (112)we have

mt = g0 minus g1[a0 + a2mtminus1 + vtminus1] minus g1a1[a0 + a1Ytminus3 + a2mtminus2 + vtminus2](116)

Continuing to successively substitute for Ytminusi i = 3 4 we have Sargentrsquosequivalent expression for the optimal (deterministic) policy rule as given by9

mt = g0 minus g1a0 minus⎛⎝g1a2

infinsumiminus1

aiminus11 mtminusi

⎞⎠ minus

infinsumi=1

aiminus11 + vtminusi

) (117)

Following the above optimal monetary policy (ie reducing any random compo-nent to the money supply to its minimum level and establishing the rule for thedeterministic component of the money supply as specified by (114) or (117)) wehave by construction that

Etminus1Yt = Y lowast and Yt = Y lowast + a2εt + vt

where vt = H0γ θ(α2β1)ut and the variance of the random component of themoney supply εt is set at its lowest feasible level Thus optimal monetary policyin essence sets output each period equal to Y lowast plus irreducible noise As Sargentand Wallace (1976 171) note

the application of the rule eliminates all serial correlation in output since thisis the way to minimize the variance in output The basic idea is that wherethe effects of shocks to a goal variable (like GNP) display a stable pattern ofpersistence (serial correlation) and hence are predictable the authority canimprove the behavior of the goal variable by inducing offsetting movementsin its instruments

Note that without the lag term for output g1 in (114) equals zero

Adaptive expectations and the accelerationist result

The well-known ldquoaccelerationist outcomerdquo concerning the path of inflation isimplied by the above analysis if expectations are adaptive and if the aim of mone-tary policy is to keep output above its natural level To see this let us go back to the

Policy 171

aggregate supply equation that underlies the reduced form for output To simplifylet us abstract from the stochastic elements in demand εt and ut (as well as anysupply-side disturbances) and also from adjustment costs (ie omit the laggedoutput term) Then the aggregate supply equation

Yt = γ θ [πt minus Etminus1πt] (118)

reflects the actual path that output will take10

To derive the accelerationist result assume that Y lowast gt Yn By normalizationYn = 0 so that to have Etminus1Yt = Y lowast gt Yn we must have Etminus1Yt gt 0Equation (118) suggests that to achieve an expected level of output greater thanits natural level the government must pursue a monetary policy that results inthe actual rate of inflation typically being above that expected In particular thedesire to keep output above its natural level means that monetary policy in period tresults in the actual inflation rate πt such that

πt minus Etminus1πt = Y lowastγ θ gt 0

If expectations are adaptive then we have that

Etminus1πt minus Etminus2πtminus1 = δ(πtminus1 minus Etminus2πtminus1)

The assumption of adaptive expectations coupled with Y lowast gt Yn = 0 thus impliesthat

Etπt+1 = Etminus1πt + δ(πt minus Etminus1πt) = Etminus1πt + δY lowastγ θ

In words the fact that individuals underestimate inflation this period (by theamount Y lowastγ θ ) leads them to adjust (ldquoadaptrdquo) their expectations of inflationupward (by the amount δY lowastγ θ ) The result is that to keep expected output atthe level Y lowast gt Yn next period means an increase in the inflation rate by theamount δY lowastγ θ each period As Blanchard and Fischer (1989 572) note

this is the famous accelerationist result derived by Friedman (1968) andPhelps (1968) using their Phillips curve together with the adaptive expec-tations assumption The explanation is simple if the government is trying tokeep output above the natural rate it has to produce inflation at a higher ratethan expected each period Since the expected inflation is a weighted averageof past inflation rates the actual rate must be increasing

Now let us assume instead that Y lowast = 0 To provide a role for an activist monetarypolicy let us reintroduce the lagged output term λYtminus1 into the right-hand sideof (118) Thus monetary policy can eliminate the effect of past deviations inoutput from the natural rate on current output In other words a policy aimed

172 Policy

at setting Etminus1Yt = Y lowast would imply altering inflation relative to expected onlywhen lagged output deviated from the natural level Rather than the prior result ofan ever-increasing inflation with Y lowast gt Yn with Y lowast = Yn we have that inflationsimply wanders11 For instance if logged lagged output Ytminus1 fell below the log ofthe natural level of output Yn = 0 then the difference Ytminus1 would be negativeOther things being equal this would imply a lower output in the current periodTo counteract this the government would pursue a monetary policy that results inthe actual rate of inflation being above that expected

The above discussion helps us understand the comment of Hall (1976) that

the benefits of inflation derive from the use of expansionary power to trickeconomic agents into behaving in socially preferable ways even thoughtheir behavior is not in their own interests The gap between actual andexpected inflation measures the extent of the trickery the optimal pol-icy is not nearly as expansionary when expectations adjust rapidly andmost of the effect of an inflationary policy is dissipated in costly anticipatedinflation

The above extract raises the following question Can the monetary authoritiessystematically trick the public in order to exploit the link between inflation andoutput For Sargent and Wallace and others the answer is no due to the existenceof rational expectations

Rational expectations and the SargentndashWallace ineffectivenessproposition

As Sargent (1987a) notes a critical aspect of the simple example of an optimalmonetary rule as given by (114) (or equivalently (117)) ldquois the implicit assumptionthat agentrsquos decision rules remain unchanged in the face of alternative stochas-tic processes for the control variable that different feedback rules implyrdquo WhatSargent means in this context is that the optimal monetary rule has been derivedunder the presumption that private agents do not take this rule into account informing their expectation of the price level Under this assumption one couldestimate the parameters of the reduced-form equation output (a0 a1 and a2 in(112)) independently of the feedback rule (114)

However Sargent and Wallace (1976) criticize this view In particular theyargue that ldquoin the reduced forms are embedded the responses of expectations tothe way policy is formed Changes in the way policy is made then ought not toleave the parameters of estimated reduced forms unchangedrdquo12 In other wordsrational individuals would clearly seek out and use information on how monetaryauthorities act as well as on the structure of the economy in forming expectationsof prices

Let us now consider the following version of the reduced-form equation foroutput (112) that explicitly includes the potential role of expected monetary policy

Policy 173

when individuals form expectations on prices13

Yt = a0 + a1Ytminus1 + a2(mt minus Etminus1) + vt (119)

For a given anticipated log of the money supply Etminus1mt we have as before theoptimal (deterministic) monetary rule of the form

so that

mt = g0 minus g1Ytminus1 + εt (1111)

where εt is the irreducible random element in the money supply determinationprocess Now assume that the public knows the monetary authoritiesrsquo feedbackrule Then our assumption of rational expectation (ie individuals use all availableinformation in forming expectations) implies that

Etminus1(mt) = g0 minus g1Ytminus1 (1112)

Combining (119) (1111) and (1112) the reduced form for output is now given by

Yt = a0 + a1Ytminus1 + a2εt + vt (1113)

so that the biased squared term in the objective of the monetary authorities(Etminus1Yt minus Y lowast)2 equals (a0 + a1Ytminus1 minus Y lowast)2

As is clear from (1113) there is no role for systematic monetary policy to affectreal output As Sargent (1987a 459) notes ldquothe bias squared is independent ofthe parameters of the money supply rulerdquo The optimal policy is then to makemonetary policy deterministic if feasible for then the variance of output (given bya2

2σ2e + σ 2

v ) is minimized by setting σ 2e = 0 Until we add an inflation objective

any deterministic rule will be equally as good for none will have any impact onoutput This is once again an example of the neutrality of money

As Sargent (1987a 458) notes ldquopolicy rules should be deterministic and involveno surprisesrdquo He goes on to argue that we

have therefore established the following stochastic neutrality theorem thatcharacterizes our model one deterministic feedback rule on the basis of theinformation set tminus1 which is common to the public and to the authority is asgood as any other deterministic feedback rule Via deterministic feedbackrules the monetary authority is powerless to combat the business cycle (theserial correlation in Yt)

Naturally if one abandons rational expectations or the natural rate hypothesis thenthis ldquostochastic neutralityrdquo result need not hold

174 Policy

The SargentndashWallace ineffectiveness proposition in thecontext of the Phillips curve

The above finding of the ldquoneutrality of moneyrdquo in the context of a stochasticlinear natural rate model with rational expectations is viewed by Sargent (1987a459) as

the antithesis of our earlier result rationalizing the activist Keynesian policyrules The reader is invited to verify that the truth of the neutrality theoremis not dependent on the particular information set assumed It will continueto hold for any specification of tminus1 so long as the public and the authorityshare the same information set

He concludes

The preceding results provide a [weak] defense for following rules with-out feedback Simple x-percent growth rules do as well as any deterministicfeedback rule and dominate rules with a stochastic component

Below we recast the SargentndashWallace ineffectiveness proposition in terms ofthe expectational Phillips curve This makes the discussion more in line with thenext sectionrsquos review of some implications of non-enforceable monetary rules Inaddition we add to the government goalrsquos an inflation objective In particular wemodify our analysis in the following four ways

1 we alter the objective to be in terms of unemployment rather than output2 we expand our objective function to include inflation3 we link inflation to unemployment via a modified Lucas supply equation4 we incorporate rational expectations

Our first task is to convert the objective of the government into unemploymentterms Before we assumed that the government simply sought to minimize thefluctuations in output about a particular level In particular if we let Zt denote thecost incurred in period t we assumed the objective was to

min Etminus1Zt where Zt equiv (Yt minus Y lowast)2 (1114)

We have previously assumed that the deviation of unemployment from its naturalrate is linearly related to the deviation of the log of output from the log of its naturallevel In particular we assumed

minus(Ut minus Un) = Yt

Policy 175

given the normalization of the natural level of output such that Yn equiv ln yn = 0Substituting the above into (1114) the problem facing the government policy-maker becomes

min Etminus1Zt where Zt = a(Ut minus kUn)2 a = 2 gt 0

k = 1 minus Y lowastUn (1115)

Note in (1115) that we assume k lt 1 which is equivalent to assuming thatthe log of optimal level of output Y lowast is greater than the log of the natural levelof output which by normalization has been set equal to zero As Blanchard andFischer (1989 596ndash597) suggest

The most plausible justification [for k lt 1] is the presence of distortions orimperfections that causes the natural rate of unemployment to be too high Thisjustification allows the loss function to be consistent with the single-periodutility function of private agents Another is that the governmentrsquos objectivefunction as shaped by the electoral process leads the government to seek toraise output above the natural level

Having converted the objective function into unemployment terms the next stepis to expand the objective function to include an inflation goal In particular let usassume that the cost in period t includes a term reflecting differences between theactual inflation rate πt and an optimal rate of inflation πlowast Assuming a simplequadratic form we have14

Zt = a(Ut minus kUn)2 + b(πt minus πlowast)2 (1116)

The problem of the government policy-maker is then

min Etminus1Zt where Zt equiv a(Ut minus kUn)2 + b(πt minus πlowast)2

As before we can decompose this objective to obtain the following equivalentexpression for the object of the government policy-maker

min a[Etminus1(Ut minus kUn)2 + (Etminus1Ut minus kUn)

2]+ b[Etminus1(πt minus πlowast)2 + (Etπt minus πlowast)2] (1117)

Our third task is to link inflation to unemployment Recall that if we ignore thelagged term with respect to output in the Lucas supply equation assume unem-ployment is linearly related to real output and approximate inflation by the log ofthe ratio of the price level this period to the price level last period then we canmanipulate the Lucas supply equation to obtain the expression

Ut = Un minus α(πt minus Etminus1πt) (1118)

176 Policy

where α = (γ θ) gt 015 Substituting (1118) into (1116) the governmentrsquosobjective becomes

min Etminus1a(1 minus k)Un minus α(πt minus Etminus1πt)2 + b middot (πt minus πlowast)2

Our fourth and final task is to introduce rational expectations As we saw earlierwe obtain the reduced form for the deviation of the price level from that expectedin period t

J0 + β0(β1εt + α2ut) (1119)

if (a) the government follows a particular rule in determining monetary policy(b) that rule is known to the public and (c) there exist rational expectations

An algebraic manipulation ndash simultaneously subtracting and adding the log ofthe price level for period t minus 1 to the left-hand side of equation (1119) ndash bringsus closer to having an expression that may be interpreted in terms of inflation

ln( ptptminus1) minus Etminus1(ln( ptptminus1) = 1

J0 + β1(β1εt + α2ut) (1120)

Using the log of the price ratio as an approximation for inflation we thus canapproximate (1120) as

πt minus Etminus1πt = 1

J0 + β1(β1εt + α2ut) (1121)

Substituting the above which reflects the rational expectations approach tomodeling expectations into the new government objective we have

min Etminus1

[(1 minus k)Un minus α

J0 + β1(β1εt + α2ut)

+ b(πt minus πlowast)2

min a[(1 minus k)Un]2 +(

J0 + β1

(β21σ 2

e + α22σ 2

+ Etminus1 b(πt minus πlowast)2 (1122)

since Etminus1εt = Etminus1ut = Etminus1εtut = 0It is clear from (1122) that with rational expectations monetary policy can play

no role in helping the government meet its objective concerning unemploymentIn other words in the context of the Lucas model the assumption of rationalexpectations means that the expected loss from deviations in unemployment fromits desired level and thus production from its desired level is independent ofthe deterministic monetary rule As a consequence the minimization problem asgiven by the first half of equation (1117) becomes simply one of specifying any

Policy 177

deterministic monetary policy rule and eliminating any random changes in themoney supply (ie setting σ 2

e = 0 if feasible)But the objective of the government now also includes an objective concerning

the rate of inflation so we need an expression for the equilibrium rate of changein prices πt that incorporates rational expectations To do so we start with thereduced form for the log of the price level obtained previously which is given by

Pt = 1

J0 + β1

[β1mt + εt + α2(Xt + ut + β1π

et+1) + J0

(1123)

Taking the difference between the reduced form for the log of the price level forperiod t and for period tminus1 we can obtain an expression for πt the rate of inflationbetween period t and t minus 1 of the form

πt = 1

J0 + β1[β1mt + α2(Xt minus Xtminus1 + ut minus utminus1) + α2β1(π

et+1 minus πw

+ J0(Etminus1Pt minus Etminus2Ptminus1) minus J0λ

γ θ(Ytminus1 minus Ytminus2)] (1124)

where πmt approximates the rate of change in the money supply (ie πmt =mt minus mtminus1 = ln(MtMtminus1)) Assuming rational expectations we have fromequation (1119) that

Etminus1Pt = Pt minus 1

and similarly

Etminus2Ptminus1 = Ptminus1 minus 1

J0 + β1(β1εtminus1 + α2utminus1)

so that

Etminus1Pt minus Etminus2Ptminus1 = πt + 1

J0 + β1[β1(εtminus1 minus εt) + α2(utminus1 minus ut)]

(1125)

Substituting this expression into equation (1124) one obtains

πtβ1

J0 + β1= 1

J0 + β1[β1πmt + α2(Xt minus Xtminus1 + ut minus utminus1)

+ α2β1 middot (πet+1 minus πe

t ) minus J0λ

178 Policy

where we use the fact that 1 minus J0(J0 + β1) = β1(J0 + β1) Solving for πt we have

πt = πmt + α2

β1(Xt minus Xtminus1 + ut minus utminus1)

+ α2(πet+1 minus πe

t ) minus J0λ

γ θ(Ytminus1 minus Ytminus2)

(1127)

which given πmt = mt minus mtminus1 = mt + εt minus (mtminus1 + εtminus1) can be rearranged toobtain

πt = πmt + εt minus εtminus1 + α2

β1(Xt minus Xtminus1 + ut minus utminus1) + α2(π

et+1 minus πe

minus J0λ

β1γ θ(Ytminus1 minus Ytminus2) (1128)

where πmt = mt minus mtminus1If we assume no change in exogenous (nonrandom) demand factors this period

compared to the last period (ie Xt = Xtminus1) no change in the expected future infla-tion between last period and this period (ie πe

t+1 = πet ) and ignore adjustment

costs (ie λ = 0) we can simplify equation (1128) to obtain

β1(ut minus utminus1) (1129)

The three assumptions we made to derive equation (1129) from equation (1128)largely limit any differences between period t and t minus 1 to differences in the sizeof the money supply In fact the two periods differ only by the deterministiccomponent of the money supply and by random factors where these randomfactors ndash the term ut minus utminus1 with respect to output demand and the term εt minus εtminus1with respect to the irreducible random component of the money supply ndash havemean zero In other words all potential changes in aggregate demand or productionexcept money supply changes that would lead to different expected price levels inthe two periods have been removed

Substituting equation (1129) into (1122) we thus obtain the following completegovernment objective function under rational expectations

[[(1 minus k)Un]2 +

J0 + β1

(β21σ 2

e + α22σ 2

+ Etminus1b[πmt + εt minus εtminus1 + α2

β1(ut minus utminus1) minus πlowast]2

from which we see that constant monetary growth will not achieve a constant rateof inflation unless we neglect the lagged disturbance terms16 To obtain the resultof an optimal monetary rule in the form of a constant rate of growth in the moneysupply we must further simplify and neglect the lagged disturbance terms If we

Policy 179

for the moment ignore the lagged stochastic terms εtminus1 and utminus1 we have theobjective

[[(1 minus k)Un]2 +

J0 + β1

(β21σ 2

e + α22σ 2

+ b(πmt minus πlowast)2 + σ 2e +

As before to minimize the loss requires that one reduces the random variation inthe money supply to zero (if feasible) so that σ 2

e = 0 That is the optimal monetaryrule is ldquodeterministicrdquo Further given an inflation objective and no reason otherthan monetary policy for prices to change the obvious optimal rule is simply toset the determinant rate of change in the money supply equal to the desired rate ofinflation That is the optimal policy is

πmt = πlowast

Note that this rule assumes no shocks to the economy If there were a steady rateof increase in output (and thus the real demand for output) then that would raisethe optimal constant rate of change in the money supply to achieve a given rate ofchange in prices Thus Friedmanrsquos ldquo3 percentrdquo rule for monetary growth presumeda 3 percent growth in real output so as to be consistent with a zero rate of inflation

Rules versus discretion monetary policy and timeinconsistency

The SargentndashWallace ineffectiveness proposition has been used by many to supportarguments for the government not adopting activist policy rules to offset fluctua-tions ndash particularly downturns ndash in an economyrsquos real output reflecting demand-sidedisturbances The reason as we have seen is that such policies have no real effectsonce one assumes rational expectations although the attempt can lead to higherinflation

Yet as Barro and Gordon (1983) point out

empirical studies indicate the presence of countercyclical monetary policyat least for the post-World War II United States ndash rises in the unemploymentrate appear to generate subsequent expansions in monetary growth Within thenatural rate framework it is difficult to reconcile this countercyclical behaviorwith rationality of the policymaker

As Barro and Gordon go on to say ldquoa principal object of our analysis is to achievethis reconciliationrdquo That is rather than saying what policy rules governmentshould follow Barro and Gordon want to explain why government acts the way itdoes ndash thus the term ldquopositive theoryrdquo in the title of their paper

180 Policy

The BarrondashGordon approach combines a number of topics that we haveconsidered before First they utilize a natural rate model like the Lucas modelSecond they assume rational expectations And third and most interestingly theyprovide us with a nice example of the phenomenon of ldquotime inconsistencyrdquo in thecontext of optimal monetary policy when there exists the potential Phillips curvetradeoff

Contrasting the BarrondashGordon and SargentndashWallace policyenvironments

In the SargentndashWallace view of the optimal monetary policy choice it is assumedthat the policy-maker makes a once-and-for-all decision with respect to the partic-ular monetary policy rule (reactive or not) Under certain assumptions as we haveseen this leads to the natural conclusion that optimal monetary policy entails thesimple rule of a constant rate of growth in the money supply so that the resultingaverage rate of inflation equals the desired level If the desired level were zerothen inflation would be set equal to zero through appropriate monetary policy

Barro and Gordon (1983 598) have questioned this result on the basis that ldquotheremay be no mechanism in place to constrain the policymaker to stick to the rule astime evolvesrdquo The result is that the policy-maker decides each period the optimalmonetary policy to follow In other words ldquothough the objective function anddecision rules of private agents are identicalrdquo Barro and Gordon obtain differentresults from Sargent and Wallace because ldquothe problems differ in the opportunitysets of the policymakerrdquo Below we illustrate the exact nature of this differenceby first showing how Barro and Gordonrsquos setup provides an example of the ldquotimeinconsistency problemrdquo not present in the Sargent and Wallace problem and thenderive the equilibrium for an economy characterized by the policy-makers whoare allowed each period to pick a potentially new optimal monetary policy

Time inconsistency an example in the context of optimalmonetary policy

In an important paper on the ldquotime inconsistencyrdquo problem of optimal policyKydland and Prescott (1977) point out situations in which the optimal policiesdecided at time t would be changed at time t + 117 In the context of monetarypolicy as Kydland and Prescott note

the reason that such policies are suboptimal is not due to myopia The effect of(monetary policy) upon the entire future is taken into consideration Rather thesuboptimality arises because there is no mechanism to induce future policy-makers to take into consideration the effect of their policy via the expectationsmechanism upon current decisions of agents

Let us see what this means by way of a concrete example

Policy 181

A simple example of ldquotime inconsistencyrdquo following Kydland and Prescott isconstructed below To simplify the discussion somewhat for the moment we ignoreuncertainty and restrict our attention to two periods In this setting the govern-ment through monetary policy can determine each period the actual inflation raterather than the expected rate of inflation For periods 0 and 1 the choice variablesfacing the monetary policy-maker are thus π0 the actual inflation in period 0 andπ1 the actual inflation in period 1 The inflation choice impacts the unemploymentrate in that U0 and U1 the unemployment rates in periods 0 and 1 are given bythe Phillips curve formulation (1118)

U0 = Un minus α(π0 minus Eminus1π0)

U1 = Un minus α(π1 minus E0π1)

where the state variables Eminus1π0 and E0π1 denote the expected rate of inflation inperiods 0 and 1 respectively In periods 0 and 1 the objective for periods 0 and 1is to minimize the simple quadratic forms

Z0 = a(U0 minus kUn)2 + b(π0 minus πlowast)2

where the constants a and b are positive It is assumed that k lt 1 to capture theidea that distortions exist in the economy that an activist policy can address18

Substituting in the ldquoPhillips curverdquo relations in period 0 the present value of theobjective function is

Z = Z0 + β middot Z1 = a[(1 minus k)Un minus α(π0 minus Eminus1π0)]2 + b[π0 minus πlowast]2

+ βa[(1 minus k)Un minus α(π1 minus E0π1)]2 + βb[π1 minus πlowast]2

where β is the constant discount factorLet us presume that the expectations of inflation the ldquostaterdquo variables are

exogenous and equal to the desired level each period (ie Eminus1π0 = E0π1 = πlowast)In period 0 the optimal inflation rates for periods 0 and 1 (as determined bymonetary policy) are then

partZpartπ0 = minus2aαq0 + 2b(π0 minus πlowast) = 0

partZpartπ1 = β[minus2aαq1 + 2b(π1 minus πlowast)] = 0

where qi = (1 minus k)Un minus α(πi minus πlowast) i = 0 1 In period 1 the objective is

Z1 = a[(1 minus k)Un minus α(π1 minus E0π1)]2 + b[π0 minus πlowast]2

and the optimal solution given π0 and E0π1 = πlowast is thus

partZ1partπ1 = minus2aαq1 + 2b(π1 minus πlowast) = 0

182 Policy

Comparing this first-order condition to partZ0partπ1 it is clear that in the case ofexogenous expectations the optimal solution is time-consistent Further it impliesa rate of inflation greater than the expected (optimal) rate of πlowast each period toachieve an unemployment rate below the natural This follows given that one ofthe objectives is to approach a level of unemployment equal to kUn with k lt 1

What happens however if individualsrsquo expectations of inflation for period 1 isa forecast that correctly anticipates the optimal future policy decision That is wehave ldquorational expectationsrdquo such that in a deterministic world

E0π1 = h(0) = π1

where 0 is the information set at the end of period zeroWhat the optimal inflation policy is now depends on whether policy-makers

can ldquocommitrdquo to future policy actions Let us start by assuming they can placeconstraints on future policy We know that in period 0 the optimal solution for π1is then given by

βpartZ1

partπ1+ partZ

partE0π1

dh(middot)dπ1

= β[2b(π1 minus πlowast)] = 0

Since dh(middot)partπ1 = 1 the optimal planned (at time 0) rate of inflation for period 1is equal to the desired level πlowast This is the SargentndashWallace ineffectivenessproposition

However once period 1 occurs E0π1 is set (let us assume it equals πlowast) If thepolicy-maker was not then constrained by the prior specification of a monetarypolicy to achieve a rate of inflation equal to πlowast they would choose π1 such that

which implies π1 gt πlowast The ldquotime inconsistencyrdquo arises as you can see becauseparth(middot)partπ1 = 0 As Barro and Gordon state ldquothe term lsquotime inconsistencyrsquo refersto the policymakerrsquos incentives to deviate from the rule when private agents expectit to be followedrdquo

Equilibrium when monetary rules are not enforceable

As Barro and Gordon note in the time inconsistency problem ldquoconstraints onfuture policy actions are infeasible by assumptionrdquo In contrast in the SargentndashWallace view

rules are enforceable so that the policymaker can commit the course of futurepolicy (and thus of expectations) In the former case the time-inconsistentsolution is not an equilibrium given the problem facing the policymaker Inthe latter case the incentives to deviate from the rule are irrelevant sincecommitments are assumed to be binding

(Barro and Gordon 1983 599)

Policy 183

As the above extract suggests in the ldquotime inconsistency caserdquo we have not yetcharacterized a rational expectationsrsquo equilibrium since in our example individu-alsrsquo expectations of inflation for period 1 were incorrect According to Barro andGordon there are three features of an equilibrium The first is ldquoa decision rulefor private agents which determines their actions as a function of their currentinformationrdquo These actions of private agents based on current information aresummarized by the Phillips curve

Ut = Un + ζt minus α middot (πt minus Etminus1πt) (1130)

where ζt a random variable with zero mean and variance σ 2z has been added

to denote a real shock that affects the natural unemployment rate for the currentperiod only19

The second feature of an equilibrium is ldquoa policy rule which specifies the behav-ior of policy instruments as a function of the policymakerrsquos current informationsetrdquo This policy rule is given by the choice of inflation rates πt+i i = 0 1 2 by the monetary authorities with the following objective

min Et

⎡⎣ infinsum

β iZt+i|It

⎤⎦

where Zt+i equiv a(Ut+i minus kUn)2 + b(πt+i minus πlowast)2 a gt 0 b gt 0 and 0 le k le 1

The term It denotes the initial state of information and 1 gt β gt 0 is the constantdiscount factor (β = 1(1+rreal) where rreal is the exogenous real rate of interest)For simplicity we will assume that the optimal level of inflation is zero so thatπlowast = 0

The third feature is an expectations function which determines the expectationsof private agents as a function of their current information Assuming that ldquothepublic understands the nature of the policymakerrsquos optimization problem in eachperiodrdquo then Etminus1πt = πt where πt is the optimal choice of inflation by thepolicy-maker for period t

Combining the information contained in our discussion of the first and secondfeatures of equilibrium the policy-makerrsquos optimal choice of πt minimizes

EtZt = Et[a(Ut minus kUn)2 + b(πt)

2]= Et[a((1 minus k)Un minus α(πt minus Etminus1πt))

2 + b(πt)2]

Given that Eξt = 0 the expression to be minimized by the policy-maker can bewritten as

EtZt = [a((1 minus k)Un minus α(πt minus Etminus1πt))2 + b(πt)

2]A critical point to note is that each period the policy-maker inherits Etminus1πt andtakes that expected rate of inflation as given in the above optimal choice of πt

184 Policy

The first-order condition for period t is thus

partEtZtpartπt = 2a((1 minus k)Un minus α(πt minus Etminus1πt))(minusα) + 2b(πt) = 0

which can be simplified and rearranged to obtain the following expression for theoptimal inflation rate

πt = aα

b[(1 minus k)Un minus α(πt minus Etminus1πt)] (1131)

From the third feature of equilibrium we know that the public realizes the problemfaced by the policy-maker in terms of choosing the optimal rate of inflation forperiod t as defined by (1131) and thus Etminus1πt = πt so that the second term dropsout and we have

πt = aα

b(1 minus k)Un = Etminus1πt gt πlowast = 0 (1132)

as long as k lt 1 We thus have that expectations are rational and individualsoptimize subject to these expectations Since Etminus1πt = πt we have that EtUt = UnThus ldquothe equilibrium solution delivers the same unemployment rate and a higherrate of inflation at each daterdquo than is the case in the SargentndashWallace problem wherethere is a ldquorules-type equilibriumrdquo Given the optimal rate of inflation πlowast = 0a rules-type equilibrium with rational expectations would have the actual rate ofinflation equal to zero

As Barro and Gordon (1983 608) conclude

under a discretionary regime the policymaker performs optimally subject toan assumed inability to commit future actions The framework assumes ratio-nality within the given institutional mode Excessive inflation apparentlyunrewarding countercyclical policy responses and reactions of monetarygrowth and inflation to other exogenous influences can be viewed as prod-ucts of rational calculation under a regime where long-term commitments areprecluded

The model stresses the importance of monetary institutions which deter-mine the underlying rules of the game A purely discretionary environmentcontrasts with regimes such as a gold standard or paper-money constitutionin which monetary growth and inflation are determined via choices amongalternative rules (the SargentWallace approach) The rule of law or equivalentcommitments about future governmental behavior are important for inflationjust as they are for other areas that are influenced by possibly shifting publicpolicies

An alternative to the ldquorule of lawrdquo is reputation As Blanchard and Fischer(1989 599) state

reputation is the most interesting and persuasive explanation of how gov-ernments avoid dynamic inconsistency Governments know that they can do

Policy 185

better than the shortsighted solution over the long run They hope by actingconsistently over long periods to build a reputation that will cause the privatesector to believe their announcements The key to the answer [to the ques-tion of whether reputation can sustain the optimal policy] is the specificationof private sector expectations of how the public reacts to broken promises

Conclusion

This chapter has presented an overview of many of the major themes and issuesfaced in macroeconomics in terms of the conduct of monetary policy The roles ofexpectations the ldquoineffectiveness propositionrdquo the modified Phillips curve rulesversus discretion time inconsistency credibility and enforcement are all dealt within this chapter Perhaps the major contribution of this chapter is that it highlightsmany of the issues and problems that real-world monetary authorities face whendeciding what course of action to take

12 Open economy

Introduction

The field of international economics can be roughly categorized as concernedwith either the real side or the finance side of international issues The ldquorealrdquoside focuses on such basic questions as why trade occurs between countries whatdetermines the terms of trade and how government policies such as tariffs do orquotas affect trade The ldquofinancerdquo side makes explicit the fact that countries differin currencies in order to focus on such questions as what determines exchangerates and how macroeconomic shocks in one country (eg a change in the supplyof money) affect its economy and the economies of the countries with which ittrades

In the discussion below we examine questions more like those considered by theldquofinancerdquo side Namely we extend simple macroeconomic analysis to an ldquoopenrdquoeconomy that is an economy that incorporates a foreign sector This analysisdiffers from traditional macroeconomic analysis of a ldquoclosedrdquo economy in thefollowing respects

1 There is trade of composite commodities and financial assets between twocountries For the moment we assume not only that the two countries producedifferentiated output but also that the financial assets issued by the firms andgovernment of one country are not perfect substitutes for the financial assetsissued by firms and government of the second country

2 The two economies are isolated in that individuals in each country can onlypurchase or sell labor services in their own labor markets

3 The two economies are differentiated in that each has its own media ofexchange This means that an individual in the domestic economy whoseeks to purchase foreign goods must exchange domestic money for foreignmoney1 Similarly an individual in the foreign country must exchange hisforeign money for the domestic money to purchase the domestic goods Suchexchanges take place in the foreign exchange market at the prevailing ldquoforeignexchange raterdquo or the price of one currency in terms of the second currencyWe will let et denote the exchange rate for domestic money For instanceif the domestic country is the USA and the foreign country is Japan then

Open economy 187

et on December 1 1989 was 1434 yen in that 1434 yen = 1 dollar Thisimplies that 1et the price of dollars in terms of yen was 0006973 dollarson December 1 1989

4 There is distinct government policy (fiscal and monetary) in each country inthat each countryrsquos government determines spending taxation and monetarypolicy for its economy

To understand how macroeconomic analysis is altered in the above ldquoopen econ-omyrdquo setting it is instructive to first consider the nature of the constraints faced bythe various participants in an open economy We then examine how the behaviorof these participants is affected by changes in such variables as exchange ratesWith that background we are ready to consider some simple examples of openmacroeconomic analysis such as the effect of a change in the money supply in thecontext of an open economy neoclassical model

Open economy participants constraints and Walrasrsquo law

To begin our task of modeling an open economy recall that it is typical ofmacroeconomic models to simplify the economy by grouping markets into broadcategories In a closed economy there were three important markets the outputfinancial and labor markets One could add to this the money ldquomarketrdquo sinceequilibrium required that the demand for money equaled supply In an open econ-omy we add another market the foreign exchange market where the currencyof one country is traded for that of the other country In a closed economy theparticipants in the various markets in the economy could be placed in one of fivecategories households firms government (fiscal side) the central bank and pri-vate depository institutions In an open economy we add one more participantforeigners

As we have seen a common theme of macroeconomic models is their emphasison the interdependencies among markets Macroeconomics recognizes that eventsin one market imply changes in other markets as well This ldquogeneral equilibriumrdquoapproach contrasts with the ldquopartial equilibriumrdquo approach of microeconomicswhich is less concerned with how changes in one market affect all other marketsTo fully understand the links across markets it is useful to specify the ldquofinancingconstraintsrdquo faced by the participants in the various markets Our discussion ofopen economy macroeconomics thus begins by introducing these financing con-straints for firms households government (fiscal side) the central bank privatedepository institutions and foreigners We then sum these constraints to obtain amodified Walrasrsquo law

The financing constraints in an open economy

We start our discussion of the constraints faced by the participants in an openeconomy by considering the financing constraint faced by the new participant for-eigners Foreigners purchase domestic output and financial assets Let X d

t denote

188 Open economy

foreignersrsquo demand for domestic output (exports) and let net Adft denote foreignersrsquo

desired real change in their holdings of domestic financial assetsTo purchase domestic output and financial assets foreigners must first acquire

domestic money That is foreigners must finance these purchases either fromincome generated from their previously acquired holdings of domestic financialassets or by supplying foreign currency in exchange for the domestic currencyin the foreign exchange markets In particular we have the following foreignerfinancing constraint

X dt + net Ad

ft minus αf (zBf pt + dt) minus FCst etpt = 0 (121)

Note that for simplicity we limit foreignersrsquo holdings of domestic financial assetsto private financial assets (ie we do not have them holding government bonds)Further we assume foreignersrsquo portfolio of holdings of domestic financial assetsis identical to2 that of domestic households and that they own αf 1 gt αf ge 0 ofthe total value of financial assets issued by domestic firms Thus αf (zBf pt + dt)

is the real income foreigners gain from their holdings of domestic financial assets3

In equation (121) the term FCst etpt denotes foreignersrsquo real supply of foreign

currency in the foreign exchange market FCst is foreignersrsquo supply in units of the

foreign currency in period t Multiplying by 1et the price of the foreign currencyin terms of the domestic currency puts this in terms of the domestic currencyDividing by the price level pt then puts it in real terms (ie in terms of the domesticcomposite commodity) Embedded in the desired change in foreignersrsquo holdingsof domestic financial assets are changes in the desired holdings by foreign centralbanks (ie changes in foreign central banks ldquointernational reservesrdquo)

In addition to the above new constraint that accompanies the introduction ofa new participant to the economy foreigners we have constraints faced by thedomestic households the central bank private depository institutions government(fiscal side) and firms In the case of households and the central bank we modifythe constraints introduced in previous chapters to incorporate the exchanges ofcommodities and financial assets with foreigners In particular for householdsthe budget constraint in an open economy can be expressed by

bdt + cd

t + zdt + (M d

t minus M )pt + net Adht + net AFd

ht minus [yt minus αf (zBf pt + dt)

+ α(zBff pft + pftdft)etpt minus δK minus Tnt] = 0 (122)

where the new term zdt denotes real imports net Ad

ht denotes householdsrsquo desiredchange in their real holdings of foreign assets and α(zBff pft + pftdft) denotes theincome (in terms of the foreign currency) gained from holding the proportion aof the financial assets issued by foreign firms4 This income (in foreign currency)times the price of foreign currency in terms of domestic currency (1et) gives thedomestic currency value of income from foreign asset holdings Dividing by theprice level pt puts the income in terms of the composite commodity (ie in realterms)

Open economy 189

Total consumption of commodities in period t is now the sum of cdt purchase

of output and zdt imports of commodities produced abroad5 Similarly the total

desired change in financial assets is the sum of net Adht the desired change in hold-

ings of domestic financial assets and net AFdht the desired change in holdings of

foreign financial assets Note that in incorporating the firm distribution constraintinto the household budget constraint we have taken into account the fact that notall domestic output yt is income to domestic households since foreigners own theshare αf of domestic firms On the other hand households have an additionalsource of income from the ownership of foreign financial assets

For the central bank the (stock) financing constraint equates the sum of thechange in real (domestic) financial asset holdings and international assets to thereal change in the monetary base or

net Adct + net AFd

ct minus (MBst minus MB)pt = 0 (123)

where net Adct = pbt(Bd

gct minus Bgc)pt MB = R + C and MBst = Rs

t + Cst 6 We

have added the term net AFdct to denote the real demand for additional international

(foreign currency denominated) assets by the central bank In particular

net AFdct = pfbt(B

dfct minus Bfc)etpt

The quantity pfbt(Bdfct minusBfc) is the change in the amount of foreign assets demanded

by the central bank in terms of the foreign currency pfbt denotes the price of foreignbonds in terms of foreign currency and the numbers of such bonds demanded andinitially held by the domestic central bank are denoted respectively by Bd

fct and

Bfc Multiplying this quantity by the price of foreign currency in domestic currencyterms (1et) gives the domestic currency value of foreign assets demanded by thecentral bank Then dividing by the price level pt puts the net demand for interna-tional assets by the central bank net AFd

ct in terms of the composite commodity(ie in real terms)

For private depository institutions the (stock) financing constraint indicates thatprivate depository institutions can be viewed as financing additions to reserves andto financial assets holdings by creating deposits or

net Atpb + (Rd

t minus R)pt minus (Dst minus D)pt = 0 (124)

where net reserves demanded or initially held are denoted by Rdt and R respec-

tively and checkable deposits supplied or initially outstanding are denoted byDs

t and D respectively For simplicity we assume that all private demands forforeign financial assets as well as for foreign commodities are captured in thehousehold budget constraint To the extent that private banks purchase foreignfinancial assets they can be viewed as acting as financial intermediaries forhouseholds

190 Open economy

For the domestic government (fiscal side) the financing constraint is

gdct minus T lowast

nt minus net Asgt = 0 (125)

where T lowastnt = Tt minustrt minusz(Bgh+Bgp)pt tr denotes transfer payments and net As

gt =Pbt(Bs

gt minusBg)pt Equation (125) incorporates the ldquoflowrdquo financing constraint forthe central bank In doing so it is assumed that the central bank claims on realresources just exhaust its interest payments on government debt holdings plus anyincome associated with central bank holdings of foreign assets Finally for firmsthe financing constraint on capital purchases is given by

Idnt + ψ(I d

nt) minus net Asft = 0 (126)

Note that we assume for simplicity that neither firms nor the government purchaseforeign commodities

Walrasrsquo law and the balance of payments

We now sum the constraints faced by the six participants in an open economy(foreigners households the central bank private banks the government andfirms) as given by equations (121)ndash(126) In doing so assume equilibrium withrespect to the demand and supply of bank reserves (ie the Rs

t part of MBst in the

central bank constraint equals Rdt in the private banks constraint) and note that

the initial monetary base MB equals R + C that the money supply is defined byM s

t = Dst + Cs

t and that the initial money supply is given by M = D + C Weobtain

[Bdt + Cd

t + X dt + gd

ct + δK + I dnt + ψ(I d

nt) minus yt] + [M dt pt minus Mpt]

+ [net Adht + net Ad

ft + net Adct + net Ad

pt minus net Asgt minus net As

ft]+ [zdlowast

t + net AFdht + net AFd

ct minus FCst etpt] = 0 (127)

where zdlowastt + net AFd

ht + net AFdct minus FCs

t etpt denotes householdsrsquo imports notfinanced out of income generated from foreign asset holdings Equation (127) isan example of the modified Walrasrsquo law for an open economy7 As we discussbelow (127) can be viewed as stating that the sum of excess demands in fourmarkets must equal zero

The first term in (127) reflects excess demand in the output market where thedemand now includes foreignersrsquo demand for domestic output (exports) Note thatby adding and subtracting import demand we could express the excess demandfor output in the form

bdt + cdlowast

t + (xdt minus zd

t ) + gdct + δK + I d

nt + ψ(I dnt) minus yt (128)

where the term cdlowastt denotes householdsrsquo total consumption in terms of both domes-

tic and foreign output (ie cdlowastt = cd

t + zdt ) Excess demand in the output market

Open economy 191

is often expressed this way with the consumption term denoting total householdconsumption such that the ldquonetrdquo export demand term (xd

t minus zdt ) appears

Setting the excess demand for output term in (127) to zero gives us the ISequation in an open economy The second term in (127) reflects the excess demandfor money Note that we assume that only domestic households desire to hold thedomestic money (foreigners seek the domestic money in the foreign exchangemarket not to hold but as a means to purchase the domestic output or financialassets) Setting this second term in (127) to zero gives us the standard LM equation

The third term in (127) reflects excess demand in the financial market In goingto an open economy we add to the demand side of the financial market a demandfor domestic financial assets by foreigners (which could include private foreignagents as well as foreign central banks) In addition note that householdsrsquo demandfor domestic financial assets (net Ad

ht) no longer reflects householdsrsquo total demandfor financial assets since they now have the option of purchasing foreign financialassets (net AFd

t )The fourth and final term in (127) reflects excess real demand for foreign

currency in the foreign exchange market8 As before we can view output ( yt)or the price of output ( pt) as determined in the output market and the domesticinterest rate (rt) as determined in the financial market Now we can view the foreignexchange rate (et) as determined in the foreign exchange market The demand forforeign currency reflects zdlowast

t the demand associated with householdsrsquo purchases offoreign commodities that could not be financed from the foreign currency earningsof their holdings of foreign financial assets net AFd

ht the demand for foreigncurrency associated with householdsrsquo purchases of additional foreign financialassets and net AFd

ct the demand associated with the central bankrsquos desired changein international reserves Note that the real demand for the foreign currency couldinstead be stated as the real supply of the domestic currency in the foreign exchangemarket

FCst etpt denotes the real supply of foreign currency or real demand for the

domestic currency in the foreign exchange market which from (121) reflectsforeignersrsquo purchases of financial assets and exports net of those financed throughdomestic currency earnings on financial assets held by foreigners Foreignersrsquopurchases of financial assets include both foreign private (ie foreign householdsrsquo)and foreign public (ie foreign central bank) purchases

The balance of payments accounts

Let us assume for now that the domestic economy discussed above is the USA TheUS Department of Commerce actually measures the various sources of the demandfor and supply of dollars in the foreign exchange markets cited above These data ofinternational transactions are presented as the US balance of payments accountsTable 121 summarizes its major components Transactions are categorized aseither sources of the real demand for or the real supply of dollars in the foreignexchange market for the US dollar9 Equivalently they reflect the real supply ofor real demand for foreign currency in the foreign exchange market

192 Open economy

Table 121 The US balance of payments accounts (in ldquorealrdquo terms)

Real demand for dollars Real supply of dollars

1 US exports of goods and services xt US imports of goods and services zt2 Transfers (interest and dividends to US

holders of foreign financial assetsgovernment grants and gifts)

Transfers (interest and dividends toforeign holders of US financial assetsgovernment grants and gifts)

α(zBff pft + pftdft)etpt αf (zBf pt + dt)

3 Foreign purchases of US financialassets (capital inflow) net Ad

US purchases of foreign financial assets(capital outflow) net AFd

h + net AFdct

The net demand for dollars associated with the first component of the balanceof payments accounts (exports minus imports) is called the balance of trade Ifthe balance of trade is negative as it has been recently for the USA then the USAis said to experience a balance of trade deficit If it is positive as was the case for106 consecutive years from the end of the Civil War to 1971 as well as throughmost of the 1970s then a balance of trade surplus is said to exist

The third component of the demand for and supply of dollars in the balance ofpayments accounts is the capital account This capital account can be divided intoa private part and a public part On the demand side the private part measures thereal dollars demanded by foreigners other than foreign central banks to financepurchases of US financial assets On the supply side the private part measuresthe real dollars supplied by US households to buy foreign financial assets Thesecurrency exchanges are called private international capital flows Private inter-national capital flows associated with the demand for dollars are referred to asUS private international capital inflows since they reflect the inflow of foreigncurrency due to private foreignersrsquo purchases of US financial assets Private inter-national capital flows associated with the supply of dollars are referred to as USprivate international capital outflows since they reflect the outflow of dollars dueto US householdsrsquo purchases of foreign financial assets

Summing the net demand (demand minus supply) for dollars associated withthe first two components plus the private international capital flows and adjustingfor measurement errors (the discrepancy term) we obtain what is called the USbalance of payments10 In years in which the balance of payments is negative it isreferred to as a balance of payments deficit On the other hand a positive balanceof payments is called a balance of payments surplus

When there is a surplus or deficit in the balance of payments accounts thenequality between the real demand for and real supply of dollars is brought about byan offsetting deficit or surplus on what is known as ldquothe official reserve transactionbalancerdquo The official reserve transaction balance reflects the intervention into theforeign exchange market by the US central bank (the Fed) andor by foreign centralbanks This is the ldquopublicrdquo part of international capital flows Whenever there existsa balance of payments deficit in the USA then (on net) central banks demand USdollars in the foreign exchange markets If this were solely the US central bank

Open economy 193

intervening this means that net AFdct wasis negative (the Fed wasis a net supplier

of dollars in the foreign exchange market) and the US central bank would havelostlose international reserves

Behavior in an open economy

With an understanding of the constraints faced by the various participants in anopen economy we now consider the behavior of these participants in particularthe determinants of imports (zd

t ) exports (xdt ) private international capital inflows

(a part of net Adft) and private international capital outflows (net AFd

ht) In the firstpart of this section we examine how a change in the foreign exchange rate for thedomestic currency (to be concrete the US dollar) can alter the relative price offoreign goods and thus lead to a change in the division of household consumptionbetween purchases of domestic goods and purchases of foreign goods11 We alsoexamine other factors that influence US imports of goods and services and thusthe real demand for foreign currency (real supply of US dollars) in the foreignexchange markets

In the second part of this section we consider the factors that influence howhouseholds divide their real accumulation of financial assets between US stocksand bonds and the financial assets of foreign countries As we saw above house-hold purchases of foreign financial assets constitute capital outflows since inorder to make these purchases households must demand foreign currency (supplydollars) in the foreign exchange market We will see how differences in foreignand domestic interest rates and the expected appreciation or depreciation of theUS dollar determine the relative returns on foreign and domestic financial assetsThese relative returns in turn affect householdsrsquo portfolio choices between foreignand domestic financial assets and the real demand for foreign currency (real supplyof dollars) in the foreign exchange market

In the third and fourth parts of this section we turn to the other side of theforeign exchange market to examine determinants of foreignersrsquo purchases of USgoods and services and of US financial assets In particular we consider how suchfactors as exchange rates affect foreignersrsquo demand for US goods (US exports)and how such factors as relative interest rates and the expected rate of changein the exchange rate affect foreignersrsquo demand for US financial assets We finishthis section by illustrating graphically the behavior discussed in terms of the realdemand for and supply of the domestic currency (the US dollar)

Householdsrsquo demand for imports

When deciding whether to purchase foreign or domestic goods households look attheir relative prices To be concrete consider two countries The domestic countryis the USA and the foreign country is Japan The relative price of Japanese goods isthen the real quantity of US goods that must be sacrificed to purchase the foreigngood For example if the price of a Japanese car is $6000 and the price of aUS computer is $1500 then the relative price of a Japanese car in terms of US

194 Open economy

computers is 4 computers If the relative price of Japanese cars rises the USAwill import fewer Japanese cars The relative price of foreign goods sometimesreferred to as the terms of trade is an important determinant of the quantity ofimports The relative prices of foreign goods depend on the dollar prices of USgoods the prices of foreign goods in their own currency and foreign exchangerates (the price of one currency in terms of a second currency) The simple examplethat follows illustrates this point Suppose that the price of a US computer is $1500and that in Japan the price of a Japanese car is 600000 yen The third ldquopricerdquo thatwe need to know in order to compute the relative price of a Japanese car in terms ofUS computers is the foreign exchange rate in particular the price of a yen in termsof dollars Suppose that it takes et yen to buy one dollar in the foreign exchangemarkets Then it takes 1et dollars to buy one yen Returning to our example ofJapanese cars and US computers if et = 100 yen per dollar and the Japanese carhad a yen price of 600000 then its dollar price would be 6000

In general the calculation of the relative price of Japanese goods is thus

Relative price of Japanese goods (in terms of US goods)

= Yen price of Japanese goods ( pft)

times Price of yen in terms of dollars (1et)Dollar price of US goods ( pt)

According to this expression a rise in the yen price of Japanese goods ( pft)

raises their relative price Similarly a fall in the dollar price of US commodities( pt) raises the relative price of Japanese goods Finally a rise in the price of yen interms of dollars (1et) also increases the relative price of Japanese goods12 Sinceall three changes mean a higher relative price of Japanese goods and thus anincrease in the cost to US buyers of Japanese goods in terms of US goods forgoneall three changes reduce US imports of Japanese goods The above relative priceexpression for a basket of foreign goods is sometimes termed the real exchangerate for foreign goods ndash that is

Real exchange rates (foreign goods)

= Dollar price of foreign goods

Dollar price of US goods= pft(1et)

pt= pft

While a rise in the relative price of foreign goods will lead to a reduction in thequantity of foreign goods bought we have to be careful not to infer from this thatUS import demand will necessarily be inversely related to the relative or real priceof imports The reason for this is that our measure of US real import demand is interms of the US good and services not in terms of the foreign good An examplewill highlight this distinction

Open economy 195

Suppose that the dollar depreciates (et falls) With the implied appreciation offoreign currency (1et rises) the price of the foreign good in terms of US goodsbecomes greater as our expression for the real exchange rate for foreign goods( pftpt et) indicates With a higher price fewer foreign goods will be purchased ndashthis is clear This by itself would suggest a fall in the value of imports into theUSA But the higher price also means that each foreign good purchased will costmore in terms of US goods that must be sacrificed This by itself would suggest arise in the value of US imports (measured in terms of US goods that must be paidto obtain the imports) The net impact of a change in the relative price of foreigngoods on US imports measured in terms of US goods depends on which of thesetwo effects is stronger

It is typically assumed that over time the effect of a change in the relative pricesof imports on the quantity of imports purchased dominates so that the value ofimports in terms of US goods will fall with a rise in the relative price of importsThis is what we will assume13 Formally this condition requires that the priceelasticity of demand for foreign goods be greater than one That is a 1 percentincrease in the price of foreign goods must cause a greater than 1 percent reductionin the amount of foreign goods that US households demand14

Besides the relative prices of imports real disposable income affects US importdemand An increase in disposable income can lead to a rise in household con-sumption demand In an open economy with foreign trade a rise in consumptiondemand means an increase in purchases not only of domestically produced goodsbut also of foreign goods Thus householdsrsquo import demand is directly related totheir disposable income To summarize household real import demand zd

t dependsinversely on the relative price of foreign goods pftetpt and directly on disposableincome yt minus δK minus Tnt

zdt = zd

t ( pftetpt yt minus δK minus Tnt ) (129)

Capital outflows householdsrsquo demand for foreign financial assets

When deciding whether to purchase US or foreign financial assets householdscompare domestic and foreign rates of return The nominal rate of return on USfinancial assets is simply the money interest rate rt The comparable nominal rateof return on foreign financial assets is not so simple to identify To explain howto compute this return which we will denote rlowast

t let us suppose that a householdlends one dollar in the foreign financial market

If the price of a dollar is et units of the foreign currency say yen then in termsof the foreign currency the household lends et yen If foreign financial assets offerthe interest rate rft then one period from now the household will have et(1 + rft)

yen At that time the household can convert these yen holdings back to dollars atthe exchange rate then existing At the time the money is lent this future exchangerate may be uncertain15 Let householdsrsquo expectation of this future exchange ratebe denoted by ee

t+116 Then the household expects to convert its et(1 + rft) yennext year into et(1 + rft)ee

t+1 dollars Subtracting the one dollar with which the

196 Open economy

household started the rate of return to lending in foreign financial markets rlowast isgiven by

rlowastt = (et(1 + rft)ee

t+1) minus 1 = [et(1 + rft) minus eet+1]ee

We can simplify the above equation by noting that the expected future dollarexchange rate ee

t+1 yen per dollar equals the current exchange rate of et yen times(1 + θe

t+1) where θet+1 is the expected rate of change in the price of a dollar in

terms of yen Substituting the expression et(1+θet+1) for the expected future dollar

exchange rate eet+1 into the above expression for rlowast

t we have

rlowastt = [et(1 + rft) minus et(1 + θe

t+1)]et(1 + θet+1) = (rft minus θe

t+1)(1 + θet+1)

Since the expected rate of appreciation of a dollar is typically small we canapproximate the above expression by

rlowastt = rft minus θe

t+1 (1210)

Equation (1210) has a straightforward interpretation The return to lending in theforeign financial market equals the difference between the foreign interest rateand the expected rate of change in the price of the dollar The return to lendingin foreign financial markets increases with a higher foreign interest rate rft anddecreases with a higher expected rate of increase in the price of the dollar (θe

t+1)The higher the expected rate of increase in the price of the dollar the lower theexpected return to lending in foreign financial markets since for a given numberof dollars sold for foreign currency at the start of the period fewer dollars can bebought back at the end of the period

When households choose between purchasing domestic and foreign financialassets they compare the domestic interest rate rt with the rate of return to lendingin the foreign financial markets rlowast

t We can thus express household real demandfor additional foreign financial assets as

net AFdht = net AFd

ht(rt rft minus θet+1 ) (1211)

In (1211) an increase in the US interest rate or a fall in the rate of return onforeign financial assets implies a reduction in householdsrsquo real demand for foreignfinancial assets The three dots in (1211) reflect other factors that have been leftunspecified For instance changes in the political stability of foreign governmentsare one unspecified factor that would likely impact on US householdsrsquo demandfor foreign financial assets Equation (1211) suggests that we should add anotherfacet to our previous discussion on householdsrsquo demand for US financial assetsnet Ad

ht In addition to such factors as real income taxes the US money interestrate and the expected rate of inflation household demand for US financial assetsdepends on the expected return to lending abroad rft minus θe

Open economy 197

Foreignersrsquo demand for exports

Just as US demand for foreign goods depends on relative prices so too isforeignersrsquo demand for US goods based on the relative prices of those goodsThe relative prices of US goods to foreigners measure what foreigners have togive up of their own goods in order to purchase US goods As we have seenthese relative prices depend on the money prices of US goods the money pricesof foreign goods and the exchange rate Considering ldquocompositerdquo goods for eachcountry the relative price of US goods to foreigners or the real exchange rate forforeign goods is given by

Real exchange rates (US goods)

= Dollar price of US goods

Dollar price of foreign goods= pt

pft(1et)= ptet

Note that the real exchange rate for US goods is simply the inverse of the previouslyobtained real exchange rate for foreign goods The price of a dollar in terms ofan index of foreign currencies rose by 70 percent in 1984 and 1985 The resultingrise in the relative prices of US goods contributed significantly to a reduction inforeign demand for US goods and the large US trade deficit of the mid-1980sSimilarly the dramatic fall in the price of a dollar in the subsequent period fromlate 1985 to 1988 led to an increase in US exports Thus we have

xdt = xd

t ( ptetpft ) (1212)

Equation (1212) indicates that export demand falls with a rise in the relative priceof US goods to foreigners

We know from our previous discussion that an increase in US disposable incomeleads to a rise in household purchases of both domestically produced output andforeign goods and services By the same token an increase in foreignersrsquo dispos-able income leads to a rise in their purchases of US goods Thus among the itemsmissing in (1212) that determine foreignersrsquo real export demand xd

t is foreigndisposable income

Capital inflows foreignersrsquo demand for financial assets

In 1960 purchases of US financial assets by foreigners were approximately one-half the amount of purchases of foreign financial assets by US citizens In the USfinancial markets foreign purchases of new US financial assets were less than5 percent of household and depository institution purchases Twenty-five yearslater foreigners were purchasing four times as many US financial assets than theUSA was purchasing abroad In the US financial market close to 30 percent ofnew US financial assets were being purchased by foreigners This dramatic changein capital inflows to the USA is one indication of the growing importance to theUS economy of international trade not only in goods but also in financial assets

198 Open economy

As with US households we will assume that foreigners decide to purchase eitherUS assets or financial assets of their own country by comparing the rates of returnon the two types of financial assets For foreigners the nominal rate of return ondomestic assets is the money interest rate in their own country (rft) The expectedrate of return to foreigners on US financial assets equals the US interest rate plusthe expected change in the price of the dollar in the foreign exchange market (iert + θe

t+1)Not surprisingly the expected return to foreigners lending in US financial mar-

kets increases when the US interest rate rt increases Not as obvious is that thereturn also increases with an increase in the expected rate of change in the price ofthe dollar θe

t+1 This is because foreigners lending in US financial markets converttheir currency to dollars to make the loans When the loans are repaid they thenconvert dollars back to their own currency If the dollar is anticipated to appreciateduring the course of the year then part of their expected return to lending in theUSA is the increase in the value of the dollars (in terms of their own currency)

Summarizing we can express the foreign demand for US financial assetsnet Ad

net Adft = net Ad

ft(rft rt θet+1 ) (1213)

Equation (1213) indicates that foreignersrsquo demand for US financial assetsincreases if the US interest rate (rt) rises or the expected rate of change in theprice of the dollar (θe

t+1) increases or the foreign interest rate (rft) falls

The foreign exchange market the real demand and supply of dollars

The change in the price of a dollar (in terms of a second currency) affects the realquantity of dollars supplied and demanded in the foreign exchange market Letus start with the price of a dollar set at the equilibrium level of (et)0 let us say100 yen If the price of a dollar now falls to (et)1 say 50 yen then this depreciationof the dollar (appreciation of the yen) leads to a reduction in the real quantity ofdollars supplied from Q0 to Q1

A fall in the price of a dollar from 100 to 50 yen means a rise in the dollar priceof a yen from 001 to 002 dollars or from 1 cent to 2 cents Even though there isno increase in the yen price of Japanese goods the dollar price of Japanese goodsrises For example a 600000 yen Japanese car that formerly cost 6000 dollars(600000 times 001) now costs 12000 dollars (600000 times 002) If the dollar prices ofUS goods have not changed then the relative or real prices of Japanese cars haverisen In our example this means that US households must give up an increasedamount of US goods to obtain one more Japanese car

The depreciation of the dollar and resulting higher relative price for Japanesegoods leads US households to reduce the quantity of Japanese goods demandedHowever as we discussed above the fact that the quantity of Japanese goodspurchased falls does not necessarily mean that the quantity of dollars suppliedin the foreign exchange market also falls There are two countervailing forces

Open economy 199

at work here While the purchase of fewer Japanese goods would reduce thequantity of dollars supplied the fact that each Japanese good has a higher pricewould increase the quantity of dollars supplied By assuming that the first effectoutweighs the second effect we conclude that a depreciation of the dollar causesthe real quantity of dollars supplied in the foreign exchange market to fall Thusthere is an upward-sloping supply of dollars curve17

Naturally behind the supply of dollars in the foreign exchange market is notonly US real import demand but also the real demand by households and the UScentral bank for additional foreign financial assets (net AFd

ht + net AFdct) As we

saw above this sum of the US import demand and demand for foreign financialassets can be interpreted as representing not only a supply of dollars but also ademand for foreign currency18

The above discussion also highlights the effect of a change in the price ofa dollar (in terms of a second currency) on the quantity of dollars demandedin the foreign exchange market The depreciation of the dollar (or appreciationof the yen) from (et)0 to (et)1 leads to an increase in the quantity of dol-lars demanded US in real terms The fall in the price increases the quantity of dollarsdemanded because it lowers the relative price of US goods to foreigners A fall inthe price of a dollar from 100 yen to 50 yen causes a rise in the dollar price of theyen from 1 cent to 2 cents Even though there has been no increase in the dollarprice of US goods the yen price of US goods falls and the Japanese increase theirdemand for US goods As a consequence the real quantity of dollars demandedin the foreign exchange market increases

Naturally behind the demand for dollars in the foreign exchange market isnot only foreignersrsquo export demand but also their demand for US financial assets(net AFd

ht) As we have seen this sum of the export demand and foreignersrsquo demandfor US financial assets can be interpreted as representing not only a demand fordollars but also a supply of foreign currency

Simple examples of open economy (static) macroeconomicanalysis

The statement of Walrasrsquo law for an open economy indicates that for static macro-economic analysis of an open economy we need look at only three of the fourexcess demand conditions reflecting the output financial foreign exchange andmoney markets Standard practice is to look at the output money and foreignexchange markets In the neoclassical model these three equations would besolved to obtain the equilibrium price level interest rate and exchange rate ( pt rt and et respectively) In the BarrondashGrossman disequilibrium analysis thesethree equilibrium conditions could be solved to obtain the equilibrium outputinterest rate and exchange rate ( yt rt and et respectively) In the Lucas modelor the Keynesian fixed money wage model these three equilibrium conditionsalong with the appropriate aggregate supply equation could be solved for theequilibrium price level output interest rate and exchange rate ( pt yt rt and et respectively)

200 Open economy

With flexible exchange rates (ie exchange rates determined without theintervention of central banks) the basic change in the macroeconomic analysisis to recognize that the net export component of output demand (seeequation (128)) is sensitive to interest rate changes which implies the IS curve(in (interest rate output) space) is flatter The reason why a higher US interest ratereduces net export demand is that a higher interest rate increases capital inflows(net Ad

ft) and reduces capital outflows (net AFdht) The first change increases the

demand for dollars in the foreign exchange market while the second reduces thesupply of dollars in the foreign exchange market The result is an appreciationof the dollar that reduces exports and increases imports Note that with flexibleexchange rates a change in either the price level or income does not affect netexport demand

Money supply changes in the neoclassical model purchasingpower parity

In general when considering two countries the country with the lower inflationrate will tend to have an exchange rate that is appreciating at a rate approximatelyequal to the difference in inflation rates between the two countries This patternof changes in foreign exchange rates is sometimes said to reflect the purchasingpower parity condition The condition of purchasing power parity means that thepurchasing power of each countryrsquos currency is the same whether the currency isused to purchase domestic goods or foreign goods Purchasing power parity existsfor a monetary shock in the neoclassical model since a money supply change willlead to changes not only in domestic prices but also in foreign exchange rates suchthat relative prices remain constant

Interest-rate parity

There are of course a number of variations to the above analysis For instancewe could assume that domestic and foreign financial assets are perfect substitutesand that the domestic country is sufficiently small in its interactions with foreignfinancial markets that it takes the foreign interest rate rft as a given In this caseof ldquointerest rate parityrdquo since the return to lending at home rt must equal that oflending abroad rlowast

t = rft minus θet+1 we have that

rft = rt + θet+1 (a constant) (1214)

One use of (1214) is in the well-known Dornbusch model (see Dornbusch 1976)Assume that output and the price of output are fixed initially Assume θe

t+1 = 0initially such that rt = rft Now consider an increase in the money supply Tomaintain equilibrium with respect to the demand and supply of money the interestrate rt must decrease According to (1214) the resulting increase in internationalcapital outflows and reduction in international capital inflows must reduce theexchange rate such that it is expected to appreciate at a rate equal to the difference

Open economy 201

between xdt and the new lower rt This is Dornbuschrsquos famous ldquoexchange rate

overshootingrdquo That is the analysis implies a fall in the exchange rate below whatit will ultimately be after income or prices adjust

A second use of (1214) is to combine it with the assumption that exchangerates are fixed (through appropriate central bank intervention in the foreignexchange markets) An example taking this approach is the MundellndashFlemingmodel (Fleming 1962 Mundell 1968)

Conclusion

This chapter has brought together many of the issues associated with analyzinga macroeconomy that participates in the open or global economy In keepingwith the basic model framework of this book the household is now viewed ashaving a demand for imported products as well as domestically produced goodsand services Additionally firms are allowed to sell to agents in foreign coun-tries This flow of goods and services across borders logically leads to a flow offunds Moreover this interconnectedness among trading partners implies that theactions of one country in particular those of the monetary authority may influenceconditions in the other country

1 Introduction

1 Empirically an economyrsquos total output is measured by real gross domestic product (orindustrial production) employment is estimated from company records or householdsurveys estimates of unemployment are compiled from household surveys or statisticson recipients of unemployment benefits and price indexes are computed to measurechanges in the overall level of prices

2 Leon Walras first derived the result in his book Elements drsquoEconomie Politique Purefirst published in 1874ndash77 (see Walras 1954)

3 An endogenous variable is one that is determined by the model An exogenous variableis one that is taken as given by the model

4 An element that distinguishes both static and dynamic macroeconomic analysis fromArrowndashDebreu analysis is the incorporation of money The exceptions to this in macro-economic analysis are real business cycle theories which for the most part are purelyldquorealrdquo in the sense that money is not an intrinsic part of the analysis

5 Note that an alternative to exogenous expectations is to specify an ldquoexpectation func-tionrdquo If this function relates expectations in certain specific ways to all currentinformation such that the expectation function reflects ldquorational expectationsrdquo thenstatic analysis is converted to dynamic analysis

6 In dynamic models this would be termed ldquoperfect foresightrdquo in a deterministic settingand ldquorational expectationsrdquo in a stochastic setting

7 One could say that the effect of such a variable is implicit in the exogenous level ofexpected future prices of the static analysis

8 A similar example of an incomplete listing of effects is a change in current governmentpolicies A change in current government actions may require changes in governmentactions in subsequent periods that will impact future markets Yet the construction ofstatic analysis does not require such effects to be spelled out

9 Forward markets are markets in which agreements are made that specify the prices atwhich goods will be exchanged in the future Goods are thus in essence indexed by thedate of trade

10 This is the ArrowndashDebreu contingent-claim interpretation of a competition equilibriummodel (see Arrow 1964 Debreu 1959)

11 In some cases one can attain the stationary state under the less restrictive requirementthat certain exogenous variables simply grow at a steady rate over time

12 While classical economists did not fully articulate their model Patinkin (1965) is widelycited as providing a comprehensive review and formalization of many of the key ideasunderlying the classical economistsrsquo views The result may be denoted the prototypeof the neoclassical (static) model Dynamic neoclassical macroeconomic models arebroadly based on neoclassical growth models with the addition of shocks of variouskinds Among the classic works developing neoclassical growth models are Solow(1956) Cass (1965) Koopmans (1965) and Sidrauski (1967a 1967b)

Notes 203

13 Two of the most widely known proponents of new classical economics are Robert Lucasand Thomas Sargent It has been suggested by some however that this line of analysisis less the extension of classical analysis than it is the antithesis of Keynesian analysisas discussed below See Niehans (1987) for this interpretation

14 This is changing as indicated by Howitt (1985) Shapiro and Stiglitz (1984) Weitzman(1985) and by the papers cited in the May 1988 American Economic Review

15 Howitt goes on to say that there is the ldquoreciprocal Keynesian question of how exactlythe economic system manages to overcome all the obvious coordination problems thatstand in the way of attaining the state of equilibrium common to new classical economicsmodelsrdquo

2 Walrasian economy

1 A brief description of the theory is given in Patinkin (1965 note B)2 In reality of course there is no auctioneer In fact some think that Walras would have

been the first to question such a description of the economy as realistically capturingthe true dynamic process by which the economy reaches the equilibrium described bysupply and demand curves See Walker (1987) who points out that Walras promoted aldquodisequilibrium-production model of tatonnementrdquo as more representative of Walrasrsquothought than the tatonnement model with an ldquoauctioneerrdquo

3 We use the letter T to denote the number of commodities because later we distinguishthe T commodities according to time of availability (ie from period 1 to period T )

4 One could replace the assumption of known prices by the assumption that individualsform expectations at time t about prices at time t and that such expectations are correctThis has been referred to as a situation in which expectations satisfy the assumption ofldquoweak consistencyrdquo

5 With zero transactions costs equality of purchase and sale prices could be viewed asforced by arbitrage conditions

6 Walras was one of the first to note this point Note that πjj = 17 See Debreu (1959 Chapter 2) for a discussion of such accounting prices8 A distinctive feature of macroeconomics is to alter traditional ArrowndashDebreu general

equilibrium analysis in such a way that we can reinterpret accounting prices as moneyprices and determine the level of money prices

9 Others characterize transactions costs in similar fashion For instance Alchian and Dem-setz (1972) cite the costs of ldquoformingrdquo ldquonegotiatingrdquo and ldquoenforcing contractsrdquo whileDahlman (1979) writes of ldquosearch and information costsrdquo ldquobargaining and decisioncostsrdquo and ldquopolicing and enforcingrdquo costs

10 See Varian (1992) for a discussion on these points By ldquowell behavedrdquo we mean thatpartuapartcai gt 0 and ua is strictly quasi-concave

11 In this case the Lagrangian is written ignoring the non-negativity constraints onconsumption of commodity i i = 1 T Below we provide an equivalent character-ization of the constrained maximization problem that incorporates these constraints inthe Lagrangian

12 A function y = f (x1 xn xn+1 xm) is said to be homogeneous of degree k inthe arguments x1 through xn if

f (λx1 λxn xn+1 xm) = λk f (x1 xn xn+1 xm)

1 With zero ldquoadjustment costsrdquo there could exist a market for capital at time t such thatthe capital employed during the initial period Kt is conceptually distinct from capitalinherited K In this case however equilibrium in the capital market at time t wouldthen imply that Kt = K

204 Notes

2 Note that output is a ldquoflow variablerdquo That is for a period i of length h the outputproduced is given by hyi The term yi is thus the ldquorate of outputrdquo over period i Sinceoutput is a flow variable at a point in time the rate of production is not defined (Asyou can see for any rate of output the limit of production as the length of the periodgoes to zero is zero)

3 In general for a period i of length h labor services are denoted by hNi such that totalwage payments at the end of the period are wihNi Like the labor input one shouldthink of the capital input in flow terms with the stock of capital determining the rateor flow of capital services

4 For simplicity of notation these expressions presume perfect foresight5 In general for a period i of length h the nominal interest rate from the end of that period

to the end of the next period is given by hri = hzpbi + (pbi+h minus pbi)pbi 6 For simplicity of notation these expressions assume perfect foresight with respect to

future prices of equity shares In addition the expressions presume future dividends areknown

7 To minimize notational clutter we have chosen to not denote such anticipations withthe expectation operator We do presume that agents (firms and households) have com-mon expectations (assumed to be held with subjective certainty) concerning plans withrespect to the issuing of equity shares

8 For instance if ri lt rei for period i there would be zero demand for bonds Theresulting excess supply of bonds would lead to a fall in the price of bonds and thus arise in the return on bonds until equality across rates of return held

9 We assume for the moment perfect foresight at time t with respect to prices at the endof the period (at time t + 1)

10 As in the prior models including money balances in the utility function reflectshow money can save (leisure) time required to make exchanges within a period Forsimplicity we limit money holdings to agents labelled ldquohouseholdsrdquo

11 For the representative household holdings of bonds issued by other households mustbe zero so that there is no real indebtedness effect with respect to the bonds exchangedamong representative households

12 In general if we assume there are ni firms producing commodity i and that there are mdifferent commodities then

yt =msum

⎡⎣ nisum

( pip)fij(Nijt Kijt)

⎤⎦

The above is a stylized view of how the empirical counterpart to total output real grossdomestic product is actually computed by the Commerce Department

13 Note that it is assumed that the capital stock K generates a fixed rate of capital servicesThat is we do not consider variation in the ldquoutilization of capitalrdquo If we did so thenvariation in the services flowing from the capital stock could be considered an additionalchoice variable with capital utilization presumably affecting the extent of depreciationin the capital stock over time

14 Fama and Miller (1972) cite conditions under which the ldquoowners of the firmrdquo iethe holders of the S equity shares will direct the managers of the firm at time t tomaximize Vt

15 In continuous time

petminus1 =int infin

t[psdsSs]eminusr(sminust)ds

where the interest rate r in the above expression is formally rs

Notes 205

16 Note that Stminus1 equiv S Assuming ri = rt i = t + 1 we have

pe = [1(1 + r)]⎡⎣ptdtS +

infinsumk=1

[pt+k dt+kSt+kminus1](1 + rt)k

⎤⎦

17 If the change is negative net revenues available for distribution as dividends would bereduced

18 We will explain more fully below the nature of these ldquoadjustment costsrdquo19 As suggested earlier if the utilization of capital were viewed as a choice variable then

it would be natural to have δ directly related to utilization of capital during the period20 Later we will say more about investment and the nature of adjustment costs21 Note that this discussion assumes for simplicity zero adjustment costs22 We follow Sargent and implicitly assume adjustment costs are related to net not gross

investment Net investment measures the change in the capital stock As we will seelater the result is a slightly different expression for Tobinrsquos Q than found elsewherewhen adjustment costs depend on gross investment We could also expand the natureof ψ so that adjustment costs depend on the size of the capital stock as is done in suchpapers as Lucas (1967) Uzawa (1969) and Gould (1968)

23 In the National Income and Product Accounts of the USA that report various measuresof the activity in the economy depreciation is measured by what is called the ldquocapitalconsumption allowancerdquo

24 The presence of such markets reflects the existence of ldquoperfect capital marketsrdquo in themacroeconomics literature Remember that with respect to the choice of investment forthe individual firm zero adjustment costs imply a potential discrete jump in the capitalstock at a point in time so that investment for the individual firm may not be defined

25 LrsquoHospitalrsquos rule states that if a is a number if f (x) and g(x) are differentiable and g(x)does not equal zero for all x on some interval 0 lt |x minusa| lt ε if the limit of f (x) equalszero as x approaches a and if the limit of g(x) is zero as x approaches zero then whenthe limit of the ratio f (x)gprime(x) as x approaches a exists or is infinite it equals the limitof f (x)g(x) as x approaches a

26 In continuous-time or discrete-time models such adjustment costs mean that the ldquocapitalmarketrdquo at time t is eliminated

27 Note that since bonds and equity shares are perfect substitutes the optimization problemwill not provide a breakdown into the optimal number of bonds versus equity shares

28 Recall that we are holding the stock of equity shares outstanding constant29 Note that we ignore the potential choice of capital at time t Kt and associated choice

of bonds setting Kt = K This in fact would be the case with capital adjustment costs30 If we evaluated returns from the start of a period each of the return functions would be

multiplied by 1R31 The equivalence of bond equity share and retained earnings financing of changes in

the capital stock can be shown32 This expression reflects the envelope theorem In particular we have that

dW (Kt+1)

dInt+1

dKt+1= dW (Kt+1)

dKt+1= 0

33 If the price of capital differed from the price of output but both prices were expected tochange at the same rate so that the relative price of capital was assumed to be constantover time then the expected real user or rental cost of capital would be (pkp)(mt + δ)where pkp denotes the relative price of capital

206 Notes

34 Formally at time t the inherited debt-to-equity ratio is given by pbBpeS35 Naturally there are other factors not discussed36 Note that during period t when the length of the period between planned purchases of

capital (at time t) and final installation (at time t + 1) is 1 then net investment Int isgiven by Int = Kt+1 minus K and adjustment costs are given by ψ(Int)

37 During period t+1 net investment is defined by Int+1 = Kt+2 minusKt+1 and adjustmentcosts are given by ψ(Int+1)

38 We can see from the general nature of the investment demand functions that if theproduction function were not separable then the assumption of a constant real wageover time as well as a constant expected real rate of return over time would obtain thisresult

39 See Sargent (1987a 11) for a similar expression in continuous time Note that thetwo expressions would be exact if we take the limit as the length of the period goesto zero The equality between ψ prime

t+h and ψ primet+2h that typically would be an approx-

imation in discrete time holds exactly in the limit In addition the definition ofthe real interest rate for a period of length h 1 + hm = (1 + hr)(1 + hπ) orm = (r minus π)(1 + hπ) indicates that in the limit (as h goes to zero) m = r minus π If we had assumed that adjustment costs were based on gross not net investmentthen the fraction on the left-hand side of (312) would include the term minusδψ in thedenominator

40 The Q theory of investment demand was suggested by Tobin (1969) Sargent is oneauthor who expresses investment demand in this way

41 In fact empirical measures of Tobinrsquos average Q have been constructed although suchmeasures are more complex than those discussed here since they must incorporateinfluences of the tax system (such as investment tax credits accelerated depreciationallowances and the like) on the optimal choice of the capital stock

42 In a two-sector model Tobinrsquos average Q is given by pV p1K where pV is the nominalvalue of the firm and p1 denotes the price of capital which differs from p the price ofoutput

43 In our case there is a single argument of the adjustment function ndash net investment Ifone follows Hayashi (1982) and assumes as he does that adjustment costs depend ongross investment then one must add the assumption that there is a linear homogeneousadjustment function such that ψ(δk) = ψ primeδk over the appropriate range Hayashirsquosadjustment cost function (like others) includes the stock of capital as well

44 Eulerrsquos theorem (or law) is that if the function f (middot) is a differentiable functionhomogeneous of degree 1 (linear homogeneous) with f Rn rarr R then

f (x) equivnsum

[partf (x)partxi]xi

Replacing the real wage with the marginal product of labor reflects the assumption thatthe firm is a price-taker in both the output and labor markets

45 See for example Azariadis (1976) A second reason is that new workers and previouslyemployed workers may not be considered perfect substitutes (eg see Oi 1962)

46 Taylor (1972) and Hall (1980) are among those who have examined thisphenomenon

1 Time consistency means in this context that households will follow through on priorplans as the starting date advances Strotz (1955ndash56) discussed this point in terms of autility maximizing problem

Notes 207

2 An example of nonseparable preferences is a ldquohabit persistence modelrdquo in which utilityis given by

infinsumi=t

βiminustu(ci ciminus1 )

so that consumption at time t depends on prior consumption Alternatively one couldhave utility given by

infinsumi=t

βiminustu(ci 1 minus Ni 1 minus Niminus1)

such that past work becomes a pertinent state variable for the current period Kydlandand Prescott (1982) is one important exception to the macroeconomic literature inassuming nonseparable preferences

3 This is sometimes referred to as an ldquoend-of-periodrdquo equilibrium specification in themarket for assets Alternative asset specifications for discrete-time analysis have beendiscussed by among others Foley (1975) As Edi Karni (1978) pointed out Patinkinrsquosmodel is an end-of-period model

4 For simplicity we ignore the financial asset markets at time t If we assumed portfolioadjustment costs then it would be the case that at time t desired bond and moneyholdings would be B and S respectively

5 That is (partW (xt+1)partcdt+1)(partcd

t+1partxt+1) = 0 since partW (xt+1)partcdt+1 = 0 similarly

the indirect effects of a change in xt+1 on partW (xt+1) through its impact on optimallabor supply and real money balance holdings are zero

6 For completeness note that by substituting equation (43prime) into (45) we could have theequivalent expression

partW (xt+1)partxt+1 = β[partut+1part(Mt+1pt+1)]Rt+1[Rt+1 minus Rmt+1]7 These conditions appear in various forms throughout the macroeconomics literature

see for instance Mankiw et al (1985) or Barro and King (1984)8 The resulting demand for leisure function is termed a ldquoHicksian or compensated demand

functionrdquo as it is constructed by varying the price of leisure (the real wage) and incomeso as to keep the individual at a fixed level of utility

9 The resulting demand function for leisure that incorporates both the income and sub-stitution effects of changes in the real wage is an example of a ldquoMarshallianrdquo demandfunction

10 Borjas and Heckman (1978) See also Pencavel (1985) for a survey of estimates Forevidence on the effect of wage increases on the labor supply of working women seeNakamura and Nakamura (1981) and Robinson and Tomes (1985)

11 Note that we continue to assume nonsatiation such that partutpartct gt 012 The second statement presumes sufficient dispersion in preferences so that at each real

wage there are some individuals who are just indifferent between a zero and positivelabor supply Thus any rise in the real wage will increase labor force participation

13 This point was first formally developed in the classic paper by Lucas and Rapping(1970) The role of ldquomarket clearingrdquo in macroeconomic analysis will be clearer laterwhen we consider alternative characterizations of the labor market

14 To be exact this result assumes that utility is separable and concave in leisure That isit is assumed that part2upartcpartN = part2upartNpart(Mp) = 0 and part2upartN 2 lt 0

15 Alogoskoufis (1987) provides a good review of the empirical analysis in this area

208 Notes

16 Fisherian ndash or in Patinkinrsquos (1965) terminology ldquoFisherinerdquo ndash analysis takes itsname from the classic ldquotime-preferencerdquo analysis of Fisher see in particular Fisher(1930)

17 To assure this one could let duadcai gt 0 with limcairarr0(duadcai) rarr infin andlimcairarrinfin(duadcai) rarr 0

18 Equivalently one could assume partuapart(Maip) equiv 0 for all i Given the nonnegativityconstraint on Maipi i = t t + T and the presumption of a positive nominalinterest rate ri i = t t +T minus1 the optimal solution would be M d

aipi = 0 for all iThat is bonds will dominate money as an asset and only bonds will be held if assetholdings are positive The reason is simple ndash the unique attribute of money holdingsas a way to reduce ldquotransaction costsrdquo has not been introduced by providing a ldquoutilityyieldrdquo to holding money

19 Note that the equality sign here rather than the ge sign reflects the fact that the first-order conditions imply that λi gt 0 so that the condition λipartLpartλi = 0 must be met bypartLpartλi = 0

20 Note that we assume a time-invariant one-period utility function The notation uai

reflects the fact that utility of agent a in period i depends on consumption inperiod i cai

21 In general for a period of length h we have that 1 + hm = (1 + hri)(1 + hπi)Solving for hmi and then dividing through by h we have that mi = (ri minusπi)(1+hπi)Thus in the limit as the length of each period goes to zero mi = ri minus πi Thus incontinuous-time analysis the real rate of interest is exactly defined by ri minus πi

22 In the discussion to follow we maintain the assumption of a time-invariant utilityfunction such that ua

i (cai) = ua(cai) for i = t t + T 23 At the point where cat = cat+1 the slope of the indifference curve is minus1β Assuming

cat = cat+1 and no initial assets or debt (ie zBat = 0) if 1β = Rt then the resultof the same consumption in each period would imply the individual would be neither alender nor a borrower

24 As Modigliani (1996) has stated ldquothe consumption and saving decisions of householdsat each point of time reflect a more or less conscious attempt at achieving the preferreddistribution of consumption over the life cycle subject to the constraint imposed bythe resources accruing to the household over the lifetimerdquo Modigliani summarizes hiscontribution to the analysis of consumption behavior in his Nobel lecture of December1985 (see Modigliani 1986)

25 Examples of such discussions are Diamond and Hausman (1984) and Hurd (1987)However what happens in the aggregate does mask different behavior among subgroupsof the populations For instance Burbidge and Robb (1985) find for Canadian data thatwhile an inverted U-shaped profile exists for the ldquoaveragerdquo Canadian household ldquowhitecollarrdquo households do appear to continue to accumulate wealth years after both husbandand wife have left the labor force

26 If individual a were the representative agent then consumption smoothing would implythat the aggregate endowment of the consumption good is identical across periods ieci = ci+1 i = t t + T minus 1

27 Note that in the case of multiperiod bonds agent arsquos future income could includepayments derived from the initial holdings of assets Since the present value of suchfuture ldquoincomerdquo is incorporated in the current value of the assets operationally futureincome cat i = t+1 t+T is defined as income other than derived from initial assetholdings In a production context the source of such income would be compensationfor labor services sold

28 In an economy with production this transitory component of current income can reflectsuch events as a temporary layoff a short-run opportunity to work overtime or atemporary tax rebate

29 This discussion ignores the effects of uncertainty on optimal consumption plans

Notes 209

30 Note that the equality sign here rather than the ge sign reflects the fact that the first-orderconditions imply that λi gt 0 so that the condition λipartLpartλi = 0 must be met bypartLpartλi = 0

31 Note that with one-period bonds individual a has a zero initial endowment of bondsthat continue into the future at time t

32 Other papers on this topic include Hansen and Singleton (1983) and Mankiw et al(1985)

33 An example of a compositional change that could affect aggregate consumption butwould not be accounted for in the analysis to follow is a change in the proportionof retired individuals in the economy Thus on this ground at least Hallrsquos empiricalfindings cannot be taken as the last word on consumption behavior at the level of theindividual

5 Summarizing the behavior and constraints of firms and households

1 Caballero and Engel (1999) provide an empirical study of investment dynamics in thecontext of manufacturing firms

2 Weber (1998) provides empirical evidence on the link between the financial marketsand consumption spending

3 We refer to this perfect foresight as ldquolimited perfect foresightrdquo since it concerns only thecurrent period This focus on current markets alone typical of static analysis impliesldquoexpectations functions of the agentsrdquo with respect to prices in subsequent periodsthat do not reflect the underlying analysis of markets beyond the current period Thusbeyond the current period expectations do not have the property of perfect foresight (orrational expectations in a nondeterministic setting)

4 In an open economy ndash that is one which admits a foreign sector ndash we would add afourth market the market for foreign exchange

5 Note that in a fully monetized economy money enters on one side of every exchange ndashpurchase or sale ndash in these three markets

6 Note that money holdings and money demand arise only for ldquohouseholdsrdquo To the extentfirms do hold money and make choices with respect to the size of such holdings wepresume their behavior would be similar to that of households and so lump firms withhouseholds with respect to such activity Thus the behavior of the firm is restricted tolabor demand output supply investment demand and financial asset supply

7 Recall that the expected real user cost of capital is mt + δ where δ is the rate ofdepreciation of capital and mt is the expected real rate of interest (ie mt equiv (rt minusπe

t )(1+πet ) where rt is the money interest rate and πe

t is the expected rate of inflationbetween periods t and t + 1)

8 As previously for simplicity we continue to assume that expectations of future pricesare held with subjective certainty

9 If part2f partKpartN = 0 then the real wage would not enter as an argument in the capitaldemand function nor would the existing capital stock affect labor demand

10 That is we can express a demand function in a form similar to one that would beobtained if the analysis were to consider only two periods

11 Note that we do allow the expected wage inflation between period t and t + 1 πewt

to differ from subsequent wage inflation so that the expected real wage next periodwe

t+1pet+1 equiv wt(1 + πe

wt)pt(1 + πet ) can differ from the current real wage This

introduces the possibility of intertemporal substitution of labor supply in response to achange in the current real wage

12 Assuming less than unit elastic expectations with respect to future income streams wouldassure from the standard Fisherian problem that the marginal propensity to consumewould be less than one

13 Recall that πet equiv ( pe

t+1 minus pt)pt

210 Notes

14 Recall that firms are presumed not to hold money balances so there are no real balanceeffects to concern us with respect to firmsrsquo demands or supplies

15 As Lucas and Rapping (1970) point out introducing other expectation assumptionscan retain the ldquointertemporal substitution hypothesisrdquo in the context of changes in thecurrent price level but in doing so money illusion is introduced In their own wordsthe

labor-supply equation is not homogeneous in current wages and current prices(such that) there is ldquomoney illusionrdquo in the supply of labor ldquomoney illusionrdquoresults not from a myopic concentration on money values but from our assumptionthat the suppliers of labor are adaptive on the level of prices expecting a returnto normal price levels regardless of current prices and from the empirical factthat the nominal interest rate does not change in proportion to the actual rate ofinflation With these expectations it is to a supplierrsquos advantage to increase hiscurrent supply of labor and his current money savings when prices rise

(Lucas and Rapping 1970 268ndash269)

Lucas and Rapping are particularly looking at the effect of a change in the current pricelevel on the expected real rate of interest Their assumption of ldquoadaptiverdquo expectationsimplies that an increase in pt results in a fall in πe

t equiv (pet+1 minus pt)pt a rise in the

expected real rate of interest and thus a rise in labor supply16 That is future technology capital demand labor demand the real wage the rate of

depreciation and future real interest rates are all unchanged by such a change in pricesand the money supply

6 The simple neoclassical macroeconomic model (without governmentor depository institutions)

1 In particular it is assumed that the money wage rate adjusts to continuously maintainequality between the demand for and supply of labor the price of output adjusts tomaintain equality between the demand for and supply of output and the price of financialassets (and thus the interest rate) adjusts to maintain equality between the demand for andsupply of financial assets Patinkin (1965) provides one of the first complete accountingsof this model

2 That is we would expect prices in the various markets to eventually adjust to eliminateany possible excess demands or supplies in the economy We would also expect agentsultimately to correctly anticipate the price level The neoclassical model can be modifiedto explain the workings of the economy in the face of incomplete information and priceinflexibility

3 As before the ldquolaws of motionrdquo dictating how prices change to reach equilibrium aregiven by Walrasrsquo excess demand hypothesis and we maintain the assumption that noexchange occurs until an equilibrium is reached (the recontracting assumption)

4 Alternatively one could assume that expectations at time t concerning these futurevariables are constant

5 Note that Patinkin has firms as well as households managing a portfolio of financialassets and money balances which is why he includes the demand function for labor inthe above statement In our analysis this statement applies to the labor supply functionalone

6 This last sentence anticipates the intertemporal substitution hypothesis7 We ignore the potential effect of changes in the interest rate on labor supply and thus

employment8 This reflects the assumption that households and firms share common expectations

concerning the price level (in fact for both pet = pt)

Notes 211

9 Reasons such as these for changes in output form the basis of much of the currentanalysis in the literature with respect to ldquoreal business cyclesrdquo

10 This is perhaps too extreme a statement To the extent that a higher price level isanticipated the resulting lower real money balances could lead to an increase in laborsupply at any given real wage and consequently increased employment and output Alsothere is a potential effect of changes in the price level on aggregate labor supply throughthe impact of such changes on the expected real rate of interest if unit elastic expecta-tions concerning future prices are not assumed Recall that we follow macroeconomictradition and abstract from these possibilities

11 In a recent study Ewing et al (2002) develop a model of the equilibrium unemploymentrate and examine how it responds to unanticipated changes in real output

12 Fairlie and Kletzer (1998) discuss the issues revolving around job displacement13 Unemployment may also result if prices in the economy do not adjust quickly enough to

ensure that all markets (particularly the labor market) are continuously in equilibriumUnemployment associated with labor market disequilibrium is sometimes referred toas ldquoinvoluntaryrdquo unemployment We analyze such unemployment later

14 See for instance the paper by Evans (1989) that examines the relationship betweenoutput and unemployment in the United States

15 The ldquoISrdquo equation is sometimes referred to as the aggregate demand equation indicatingthat it reflects equality between total or aggregate demand for output and productionNote that equilibrium in the output market is being described in terms of demand equalto what is produced ylowast

t The aggregate supply equation indicates what will be produced16 This is the case only for this simple aggregate model without government17 Such an assumption removes the anticipated real wage next period as an argument

in these demand functions Recall that earlier ldquostaticrdquo assumptions concerning futureinterest rates and rates of inflation have already simplified the form of these functions

18 However ldquorealrdquo or ldquosupplyrdquo shocks such as the above-mentioned changes in technologycapital stock supply of other inputs such as oil or in labor supply at prevailing realwages can affect real output

19 This analysis should be familiar since we performed a similar analysis for the economywithout production

20 Note that unit elastic expectations imply that partπet partpt = 0

21 Note that without a real balance effect the CC curve would be horizontal22 The lower interest rate abstracts from a real balance effect in the output market so that

the CC curve is horizontal

7 Empirical macroeconomics traditional approaches and time series models

1 To reduce notational clutter we suppress time subscripts All variables are period tvariables

2 Note that in Sargent (1987a 20) equation (71) is replaced by the ldquorepresentativerdquofirmrsquos first-order condition for the optimal use of the labor input given a competitivelabor market that is the condition that the real wage equals the marginal product oflabor wp = Fn(n K) Note that if the production function F(n k) is separable in thelabor and capital inputs such that f (n K) = v(n)+u(K) and v(n) = (1g)(fnminusn22)then equation (71) is identical to Sargentrsquos equation since fn(n K) = (1g)(f minus n)

3 Unless otherwise noted all parameters in this model such as f g h and j in equations(71) and (72) are assumed to be positive

4 In this context ldquolump-sumrdquo taxes are taxes independent of income Equation (73) isan example of the ldquoconsumption functionrdquo

5 Endogenous variables are variables whose values are determined by the analysis6 Sargent adds equations (73) and (74) to the system (712) in order to determine

consumption and investment demand as well

212 Notes

7 To derive wlowast start with the equilibrium condition nd = ns Substituting the first twoequations of 712 the equilibrium money wage satisfies

f minus g middot (wlowastp) = h + j middot (wlowastp)

Then one can simply solve this equation for the equilibrium money wage wlowast We canthen obtain the equilibrium employment level nlowast by substituting the expression for wlowastinto either of the first two equations of 712

8 This is a property of ldquoclassicalrdquo macroeconomic models in that monetary changesthat alter the price level do not affect real variables such as the real wage output andemployment

9 The IS equation indicates combinations of the interest rate r and output y at which ifthe output were produced and that interest rate prevailed output demand would equaloutput produced The LM equation indicates combinations of r and y that will equatethe demand for and supply of money

10 For the model under consideration the ldquoaggregate demand curverdquo (a plot in ( p y) spaceof the aggregate demand equation) slopes downward and the ldquoaggregate supply curverdquo(a plot in ( p y) space of the aggregate supply curve) is vertical The intersection of theaggregate demand and supply curves graphically determines the equilibrium output andprice level

11 See Altonji and Siow (1987) Ewing and Payne (1998) examined the relation-ship between the personal savings rate and consumer sentiment in the context of aconsumption model

12 Note that Taylor assumes that certain demands for instance consumption demand maydepend on past as well as current values of output and the money supply so that thereduced-form expression estimated includes lagged values of income and the real moneysupply

13 Time series analysis can be viewed as primarily the art of specifying the most likelystochastic process that could have generated an observed time series

14 That is forecasts generated by time series models have been used to proxy individualsrsquoexpectations of future events in tests of various theoretical macroeconomic models

15 A histogram is a plot of the frequency distribution of a set of observations16 If the process is also ldquoergodicrdquo these statistics give consistent estimates of the mean and

variance Ergodicity basically requires that observations sufficiently far apart should bealmost uncorrelated Then by averaging a series through time one is continually addingnew and useful information to the average For a rigorous explanation of this conceptsee Hannan (1970 201)

17 The variable γk k = 0 1 2 is termed the autocovariance function18 Or more generally k = 0 Note that ρk = ρminusk 19 Some have suggested that stock market prices follow a random walk See Campbell

et al (1997)20 Note that if yt is taken to be the logarithm of real output then the trend reflects a constant

rate of growth of output equal to d in the absence of shocks21 This assumes y0 is the initial value of the function yt 22 A random walk is an example of a class of nonstationary processes known as ldquointe-

gratedrdquo processes that can be made stationary by the application of a time-invariantldquofilterrdquo As defined by Granger and Newbold (1986) ldquoif a series wt is formed by a linearcombination of terms of a series yt so that wt = summ

j=minuss cjytminusj then wt is called a

lsquofilteredrsquo version of yt If only past terms of yt are involved so that wt = summj=0 cjytminusj

then wt might be called a one-sided or backward-looking filterrdquo23 This is a special case of a class of stochastic processes known as Markov processes24 Recall that for the random walk process φ = 1 in which case the process was not

stationary as the variance of the process becomes larger and larger with time

Notes 213

25 To obtain this result note that

⎡⎣ infinsum

(φ2)jε2tminusj

⎤⎦ =

infinsumj=0

(φ2)jσ 2e = σ 2

e (1 minus φ2)

26 Often the ldquolag operatorrdquo L or equivalently the ldquobackward shift operatorrdquo B is usedto express this equation Lτ yt (or Bτ yt) = ytminusτ τ = 1 2 3 There are associatedpolynomials in the lag operator such that

d(L) = d0 + d1L + d2L2 + d3L3 + middot middot middot + dpLp

Letting

φ(L) = 1 minus φ1L minus φ2L2 minus φ3L3 minus middot middot middot minus φpLp

we can thus express an AR( p) process for yt as φ(L)yt = εt 27 If yt is an AR( p) process it may be described in the following way ldquoan appropriate

finite backward-looking filter applied to yt will produce a white noise seriesrdquo (Grangerand Newbold 1986 32)

28 If there were a single shock to yt at time 0 (ie εt = 0 for all t gt 0) then the deterministiccomponent would signify the deviation of the time path from its equilibrium level Aswe will see in this case stationarity would imply ldquostability of equilibriumrdquo in that ytwould converge to its equilibrium value over time

29 A linear difference equation of order s is of the form

yt =ssum

ajytminusj + c

30 Successive substitution (the ldquoiterative method of solutionrdquo) reveals this essential natureof the solution In particular substituting for past values of yt in (725) results inyt = φty0

31 The appendix to this chapter shows how one can reinterpret higher-order differenceequations as a system of first-order difference equations

32 Note that in general a quadratic equation of the form

ax2 + bx + c = 0

can be solved using the quadratic formula

x1 x2 = [minusb plusmn (b2 minus 4ac)12]2a

In our case a = 1 b = minus(m1 + m2) = minusφ1 and c = m1m2 = minusφ233 This assumes m1 = m2The solution for repeated roots is discussed briefly below34 A variable x is said to be inside the unit circle if x lt |1| An alternative way of

expressing this condition is in terms of the associated polynomial 1 minusφ1L minusφ2L2 = 0This polynomial equation is similar to (730) except that b is replaced by 1L and thewhole equation is multiplied through by L2 The stationarity condition is that the rootsof this polynomial equation should lie outside the unit circle

35 Recall that φ1 = m1 + m2 and φ2 = minusm1m236 The following discussion follows Goldberg (1958 171ndash172)

214 Notes

37 As discussed below in this case the two roots are m1 = h + vi and m2 = h minus vi whereh = φ12 v = (4φ2 + φ2

1)122 and i is the imaginary number (minus1)12 The product

of these two roots is (φ21 minus 4φ2 minus φ2

1)4 = minusφ2 which is the square of the modulus

of the roots We are assuming (minusφ2)12 lt 1 so we have the condition that φ2 lt 1 orφ2 gt minus1

38 Note that if yt is an MA process then it is a ldquobackward looking filterrdquo applied to a whitenoise process As before we can use a lag operator L (or backward shift operator B) toexpress an MA(q) process for yt as yt = micro + θ(L)εt where the polynomial in the lagoperator θ(L) is given by

θ(L) = 1 + θ1L + θ2L2 + middot middot middot + θqLq

39 Recall that as discussed above we assume for simplicity zero mean for yt that ismicro = 0

40 For a more detailed description of invertibility see Granger and Newbold (1986) orBox and Jenkins (1970)

41 Note that ARMA( p 0) equiv AR( p) and ARMA(0 q) equiv MA(q) Using the lag operatorthe ARMA model (in the case of a zero mean) can be simply expressed as φ(L)Yt =θ(L)εt

42 In general if yt is ARMA( p1 q1) and xt is ARMA( p2 q2) the sum zt = yt + xt isARMA( p3 q3) where p3 le p1 + p2 and q3 le max(p1 + q2 p2 + q1) A proof of thisis found in Granger and Morris (1976)

43 A time sequence T (t) is called ldquodeterministicrdquo if there exists a function of past andpresent values gt = g(T (t minus j)) j = 0 1 such that E[(Tt+1 minus gt)

2] = 0 If thefunction gt is a linear function of Ttminusj j ge 0 then Tt is called ldquolinear deterministicrdquo

44 For instance for a stationary series of quarterly data one could postulate a simplefourth-order seasonal AR process of the form

yt = φ4ytminus4 + εt

This is a special case of AR(4) with φ1 = φ2 = φ3 = 0 The model could be extendedto include both AR and MA terms at other seasonal lags for instance the followingARMA(21)

yt = φ4ytminus4 + φ8ytminus8 + θ4εtminus4 + εt

To add other than seasonal components one could simply fill in the gaps (eg addterms such as φ1ytminus1 θ1εtminus1 and the like to the above) Other options are discussedby Harvey (1993) and others

45 Campbell and Mankiw argue that even if the log of output followed a random walk withdrift indicating that the effect of any shock persists indefinitely into the future estimatesusing the detrended series would be biased and erroneously conclude otherwise

46 Note that if the log of real output is an ARMA( p q) process then the differencedprocess will be an ARMA( p q + 1) process This means that to allow for stationaritywith respect to the level of real output requires at least one moving average process forthe differenced series

47 Equivalently for the logarithm of real output they are considering ARIMA(p 1 q)processes for p = 0 1 2 3 and for q = 0 1 2 3

48 Such a finding is often termed as supportive of real business cycle theories

Notes 215

1 The presence of money reflects the introduction of ldquoimperfectrdquo or costly informationA medium of exchange can arise to minimize costs incurred by participants in theeconomy when there exists imperfect information on potential exchange partners

2 In so doing we assume that the new equilibrium like the initial one exists Further wedisregard the process of adjustment of the variables to the new equilibrium Alterna-tively we could introduce ldquolaws of motionrdquo for the equilibrium values (eg the excessdemand hypothesis for price changes) and examine whether the equilibriums are stable

3 For notational simplicity we let Ld = Ldt cd = cd

t Id = Idnt p = pt r = rt

yd = cdt + Id

nt + δK + ψ(Idt ) and M = M

4 For simplicity we let the expected real rate of interest component of the expected realuser cost of capital ((rt minus πe

t )(1 + πet )) be approximated by rt minus πe

t as representedby r minus π

5 One alternative is for the interest rate to adjust to equate the demand for and supply ofmoney

6 That is we assume that consumption demand is a function of the expected gross realrate of interest Rt not its components (rt and πe

t ) This would be the case if one couldview the consumption decision from the point of view of the pure ldquoFisherian problemrdquoin essence separating the allocation of consumption across time decisions from theldquoportfolio problemrdquo Note that for notational ease we not only let rt denote the moneyinterest rate rt but also let π denote the expected rate of inflation πe

t 7 This would be the case if our focus was solely on the portfolio choice problem8 Examples of growth models with money include Tobin (1965) and Sidrauski (1967b) As

Begg (1980 293) notes ldquoin a steady state any expectations generating mechanism willyield correct predictions thus the steady state analysis of growth models with moneymay be viewed as a special case of the rational expectations model with systematicmonetary policyrdquo

9 This asymmetry in information can reflect an aggregation across labor markets in whicheach firm determines labor demand based on its correct anticipation of the price of theparticular commodity it produces while suppliers of labor determine labor supply basedon their potentially incorrect anticipation of the overall level of prices reflecting theidea that suppliers are concerned with the purchasing power of wages in terms ofcommodities not restricted to the particular commodity that they produce

10 As before for notational simplicity we let Ld = Ldt cd = cd

t p = pt r = rt y = yt and M = M

11 Recall that for simplicity we let the expected real rate of interest component of theexpected real user cost of capital (rt minus πe

t )(1 + πet ) be approximated by rt minus πe

t asrepresented by r minus π

12 Note that partypartp = 0 for the neoclassical model given limited perfect foresight13 This is obvious from the graph of aggregate demand and supply curves if one compares

the vertical aggregate supply curve of the neoclassical model dpp = dMM with theupward-sloping aggregate supply curve of the model with real wage illusion Note thatthe shift in the aggregate demand curve for a given change in the money supply isidentical in either case

9 The ldquoKeynesian modelrdquo with fixed money wage modifyingthe neoclassical model

1 Typically a union contract runs for three years Often however there are provisions thatpermit parts of the agreement to be renegotiated at specific times during the three-yearcontract period

2 As a general rule labor agreements are specified in money terms An exception to thisis the cost of living agreements (COLAs) as part of union wage contracts in the United

216 Notes

States which became popular during the 1960s and 1970s COLAs adjust money wagesautomatically to changes in prices typically using changes in the Consumer Price Indexto measure price changes However the percentage of all workers who have contractswith COLAs is fairly small as less than 20 percent of the total labor force is covered bycollective bargaining agreements Further not all COLA clauses offer full protectionagainst general price increases as wages may rise by only some fraction of the increaseof the CPI Considering these qualifications for the time being we simplify by assumingthat all labor agreements are specified in money terms

3 These seven variables for the classical model are made up of three ldquopricesrdquo along withfour variables implied from the behavioral equations The three ldquopricesrdquo are moneywage price level and interest rate determined by the equilibrium conditions for the labormarket and two of the three other markets (output financial andor money markets)The variable implied from the behavioral equations are employment (labor demandfunction) output (production function) consumption (consumption demand function)and investment (investment demand function)

4 Note that this analysis is inconsistent with the idea that long-term employment contractsthat fix the money wage for several periods arise due to adjustment costs with respectto the labor input

5 As discussed before one justification for this form is if real money balances are not partof household wealth which can be the case when we introduce depository institutionsinto the analysis

t p = pt Idnt = Id r = rt

y = yt and M = M 7 Recall that for simplicity we let the expected real rate of interest component of the

expected real user cost of capital (rt minus πet )(1 + πe

t ) be approximated by rt minus πet as

represented by r minus π 8 Note that partypartp = 0 for the neoclassical model given limited perfect foresight9 The Lucas model was introduced in Chapter 8 and is covered in more depth in

Chapter 1010 An exception to this statement occurs if monetary authorities can react to period t

disturbances that is if monetary authoritiesrsquo information set includes the values of therandom shocks in period t which are not known to private agents

11 Phelps and Taylor (1977) go on to state that they ldquodo not pretend to have a rigorousunderstanding of [why prices andor wages are set in advance] In the ancient andhonorable tradition of Keynesians past we take it for granted that there are disadvantagesfrom too-frequent or too-precipitate revisions of price lists and wage schedulesrdquo

12 Note that θ is affected by the variability of relative price shocks in relation to generalprice shocks

13 We assume for simplicity that the coefficient on Pt minus Wt + φ is one14 Note that if Pt minus Wt + φ = 0 then equation (99prime) is a first-order linear difference

equation of the form YtminusλYtminus1 = 0 with solution Yt = c0λt In the limit as t approachesinfinity Yt equals zero Recall that the logarithm of the natural level of output is zero

15 Recall our assumption that the scale factor in the determination of the real wage is equalto zero for convenience

16 As Fischer shows if wages were set only for the current period t that is tminusiWt = φ +EtminusiPt then his results would be like those obtained by the Sargent and Wallace modelwith rational expectations for similar reasons This can be seen clearly on substituting(910) into (99prime) in which case one obtains the standard Lucas-like aggregate supplyequation

17 Later it will be assumed that ut and utminus1 are correlated so that information obtainedduring period t minus 1 will help predict variations in output for period t Like the laggedoutput term in the Lucas equation this introduction of serial correlation in output isnecessary (but not sufficient) if monetary policy rules that dictate the money supply

Notes 217

for period t based on information obtained up to the end of period t minus 1 are to serve astabilizing role

18 Note that in Fischerrsquos paper minusvt rather than vt is used in (913) Our vt is more in linewith other models

19 With the one exception noted before that Fischer has minusvt replacing vt 20 Again note that Fischer has minusηt and minusρ2 in the above equation since our vt is his minusvt 21 Fischer has b1 = minusρ2 since our vt is his minusvt

10 The Lucas model

1 The discussion follows that found in Lucas (1973) and Sargent (1987a 438ndash446)2 Or as Sargent (1987a 483) states ldquoan employee cares about his prospective wage

measured not in terms of own-market goods but in terms of an economy-wide averagebundle of goods the assumption is that the labor supplier works in one market butshops in many other marketsrdquo

3 The CobbndashDouglas production function would be given by yit = (Nit)1minusα(Ki)

α whereKi denotes the inherited capital stock for sector or ldquoislandrdquo i Nit is the employment oflabor for sector i and yit is the production of commodity i by sector i during period t Inthis case the marginal product of labor is given by partyitpartNit = (1 minus α)(Nit)

minusα(Ki)α

which implies that a = (1 minus α)(Ki)α

4 Solving this expression for the demand for labor and differentiating with respect to thereal wage relevant for a firm producing commodity i we have

partN dit part(witpit) = (1 minus α)minus1α(minus1α)(witpit)

(minus1minusα)α lt 0

Note that our particular form for the labor demand function implies a constant elasticityIn particular the elasticity of demand for labor is given by

partN dit

part(witpit)

witpit

= minus 1

5 Up to this point we have assumed that expectations are held with ldquosubjective certaintyrdquoso that Et(1pt) = 1Etpt and the expected real wage can be represented as witEtpt However we now assume that individuals view the price level as a random variableIn this setting the expression for the expected real wage is Et(witpt) or witEt(1pt)which is not the same as witEtpt In fact the relationship between witEt(1pt) andwitEtpt is shown by Jensenrsquos inequality witEt(1pt) ge wit(1Etpt) This inequalityreflects the fact that the function f ( pt) = 1pt is convex rather than linear in pt Forinstance if pt = p0 with probability t and p1 with probability 1 minus t then we haveEt(1pt) = tf ( p0) + (1 minus t)f ( p1) gt f (tp0 + (1 minus t)p1) = 1Etpt where 1 gt t gt 0On the other hand ln(1pt) is linear in ln pt so the log of the real wage is linear in thelog of the price level Thus we express the labor supply function in logarithmic formand consider the expectation of the log of the price level

6 The term ξt is assumed to be serially independent which means that E(ξtξs) = 0 fort = s

7 We assume that zit and ξt are statistically independent8 In problems involving more than two random variables ndash that is a ldquomultivariate regres-

sionrdquo ndash we are correspondingly concerned with the term E(z|x y) the expected valueof z for given values of x and y and so on

9 The discussion that follows is standard statistical theory Note that if the joint distributionof the two variables is a bivariate normal density function then the regression of ln pton ln pit is linear

218 Notes

10 In general if x1 x2 xn are random variables a1 a2 an are constants andq = a1x1 + a2x2 + middot middot middot + anxn then

E(q) =nsum

aiE(xi)

Var(q) =nsum

a2i Var(xi) + 2

sumiltj

aiajCov(xixj)

wheresum

iltj means that the summation extends over all values of i and j from 1 to nfor which i lt j This expression is derived from the definition of the variance

Var(q) equiv E([q minus E(q)]2) = E

⎛⎜⎝⎡⎣ nsum

ai(xi minus microi)

⎤⎦

2⎞⎟⎠

which can be expanded by means of the multinomial theorem according to which forexample (a+b+ c +d)2 = a2 +b2 + c2 +d2 +2ab+2ac +2ad +2bc +2bd +2cdNote that

Cov(xixj) = E[(xi minus microi)(xj minus microj)]

If the xi i = 1 n are independent then

Var(q) =nsum

a2i middot Var(xi)

11 Note that these first two expectations are taken without information on ln pit 12 While it is true that if two random variables are independent they are also uncorre-

lated the converse does not necessarily hold That is two random variables that areuncorrelated are not necessarily independent However two random variables havingthe bivariate normal distribution are independent if and only if they are uncorrelated

13 In general for any two random variables x1 and x2 with joint density function f (x1 x2)the marginal density of x2 g(x2) is obtained by integrating out from minusinfin to +infinthe other variable Thus g(x2) = int

f (x1 x2)dx1 Similarly we can obtain the marginaldensity of x1 by integrating out x2 The conditional probability density function of therandom variable x1 given that the random variable x2 takes on the value x2 is definedby φ(x1|x2) = f (x1 x2)g(x2) assuming that g(x2) does not equal zero

14 To obtain the following expression we use the fact that the random variables ξt and zitare independent with zero means and variances σ 2 and σ 2

z respectively15 Note that this ldquosignal extractionrdquo problem appears in a number of different contexts

such as in statistical theories of discrimination in labor economics (Aigner and Cain1977 Lundberg and Startz 1983) and in the industrial organization literature

16 If there were a trend in the natural level of output then ln ynit would replace the firstln yni and in the lagged term ln ynitminus1 would replace the second ln yni

17 The last term in (1016prime) indicates the deviation of output in the prior period from itsldquonormalrdquo level It is presumed that λ lt 1

18 Recall that this supply function assumes no adjustment costs and thus does not have alagged output term in it

Notes 219

19 In general if z1 z2 zn are independent random variables having the same distribu-tion with mean micro and variance σ 2

z and if z = (z1 + z2 + + zn)n then E(z) = micro

and Var(z) = σ 2z n Note that as n goes to infinity the variance of z goes to zero

20 Recall that we have let π(πtminus1) denote ( pt minus ptminus1)ptminus1 This was done so thatthe terms making up the expected real interest rate rt minus πe

t would have the sametime subscript πe

t = ( pet+1 minus pt)pt In the discussions to follow however we shift

time subscripts so that the previously denoted π = π and the previously denoted πet

becomes πet+1

21 Note that the natural log of real output is ln( yt) = Yt 22 Sometimes the money demand function is simplified by assuming α2 = 0 so that there

are no effects of interest rate changes on real money demand23 This follows given mt = mt +εt so that dPtdmt = 1 Recall that Pt and mt are logs of

the price level and the money supply respectively so that dPt = d(ln pt) = (1pt)dptand dmt = d(ln Mt) = (1Mt)dMt

24 Note that we are somewhat imprecise in our statement that we are eliminating the termfor output from the equation Recall that the exact interpretation of Yt would be as thedifference between the logarithm of output and the logarithm of the natural level ofoutput Equivalently we may call Yt the log of the ratio of output to its natural level

11 Policy

1 The alternative to a rule is called ldquopurely discretionary monetary policyrdquo in whichmoney supply changes are made purely at the discretion of the government dependingon its current reading of the economy and current set of objectives

2 This view is not universally accepted For instance according to the Nobel prize winnerJames Tobin (1985) the government should be free to do as it sees fit and he sees manyreasons ldquofor the Fedrsquos reluctance to tie its own hands as much as lsquorulesrsquo advocates wishrdquo Sargent and Wallace (1976) identify as ldquoKeynesiansrdquo individuals who believegovernment monetary policy should attempt to ldquolean against the windrdquo in an effort toattenuate the business cycle In the view of Sargent and Wallace and others the monetaryauthority has no scope to conduct countercyclical policy Not surprisingly those whodisagree with this view suggest alternative models of the economy such as those withldquostickyrdquo prices more favorable to their alternative views

3 Those who argue for a rule such as this are sometimes called ldquomonetaristsrdquo Someadvocates of a constant growth rate in the money supply would restrict the length oftime during which a particular rate of growth was fixed For instance William Pooleformer member of the Presidentrsquos Council of Economic Advisers (1982ndash1985) andnow with the Fed suggests that monetary rules should be adopted but that the ruleshould be ldquosubject to change at any time upon presentation of a convincing case withsupporting evidencerdquo (Poole 1985)

4 Hallrsquos suggestion echoes Simonsrsquo (1936) proposal for ldquoa monetary rule of maintainingthe constancy of some price indexrdquo Wayne Angell a 1986 appointee to the Board ofGovernors of the Federal Reserve (the monetary authorities for the USA) has arguedfor a monetary policy that will stabilize a price index constructed from a basket of basiccommodities (perhaps including gold)

5 Recall also that we assume the random variables εt and ut (the random variable asso-ciated with output demand) are independent with zero mean and respective variancesσ 2

e and σ 2u

6 An example of such ldquoexogenousrdquo price expectations would be the autoregressiveexpectations

7 Recall that the natural level of output was normalized to equal one so that Yn equiv ln yn = 0Thus if the optimal level of output is the natural level then the objective would be tominimize Etminus1(Yt)

220 Notes

8 Recall that we are assuming that εt and vt are independent random variables From(112)

Etminus1Yt = a0 + a1Ytminus1 + a2mt

so that Yt minus Etminus1Yt = a2εt + vt Squaring this expression and noting that E(εt) = 0E(vt) = 0 and given independence E(εt middot vt) = 0 we obtain

Etminus1(Yt minus Etminus1Yt)2 = a2

2σ 2e + σ 2

9 See Sargentrsquos (1987a 453) equation (13) for the corresponding expression Recall thatg0 = (Y lowast minus a0)a2

10 Note that ln yn = 0 due to normalization11 To tie down the inflation rate we would have to add an inflation objective to the goals

of the government12 Note that this is another example of the Lucas critique in which econometric estimates

of a specific set of parameters based on past policies cannot be used to project the impactof new different monetary rules to be followed in the future for with these new policiesthe parameters will change

13 This equation is derived from combining the price error expression with the aggregatesupply equation given earlier

14 The assumptions leading up to (1116) have been chosen so that (1116) matches thecriterion found in Barro and Gordon (1983) We will contrast the results of that paperwith our findings here shortly

15 The form of this equation mirrors that in Barro and Gordon16 Consider what would happen if we did not neglect the lagged disturbance terms For

instance let us say one of the disturbance terms (εtminus1 or utminus1) is positive rather thanzero while the other is held equal to zero According to the reduced-form equation forthe price level for period t minus 1 the result would be a higher price level in period t minus 1with the positive lagged disturbance term If the rate of growth in the money supplybetween period t minus 1 and t was the same in both cases then the inflation rate wouldbe higher in the case when both lagged disturbance terms are zero Thus to maintain aconstant inflation rate a lower money supply growth is implied if lagged disturbanceterms are zero instead of positive

17 Kydland and Prescott were awarded the 2004 Nobel prize in economics for their work18 This form of the objective function introduced above is discussed in Barro and Gordon

(1983) Without the assumption of k lt 1 a zero average rate of inflation would beoptimal regardless of the nature of expectation formation

19 For simplicity we assume there is no persistence in real shocks Thus we assume λ = 0in the original Barro and Gordon model

12 Open economy

1 For discussion purposes the terms ldquodomesticrdquo and ldquoforeignrdquo are used with respect tothe domestic perspective

2 For simplicity we assume foreigners do not desire to hold domestic money3 We assume for simplicity that foreigners are not taxed by the domestic government4 That is the domestic households own the remaining share of bonds issued by foreign

firms Bff and α of the equity share issues by foreign firms The total dividends (interms of the foreign currency) paid by foreign firms in period t are denoted by pftdftwhere pft is the price level of the foreign country and dft are real dividends of foreignfirms

Notes 221

5 One might also include bdt as purchase of output by private depository institutions is

included in the household budget constraint This reflects the replacement of dividendspaid by private depository institutions to their shareholders (households) by the differ-ence between banksrsquo interest income and their purchases of the composite commodityThis definition of dividends for private banks has been termed the ldquoflowrdquo constraint forprivate depository institutions

6 Recall that MB denotes the monetary base R denotes reserves and C denotes currencyin the hands of the nonbank public

7 Recall that we use the term ldquomodifiedrdquo as we have substituted the firm distribution con-straint into the household budget constraint for labor income We thus have effectivelysuppressed the labor market from the markets under consideration This modified lawis useful in understanding how the aggregate demand equation is derived

8 By ldquoreal demandrdquo we mean in units of the domestic commodity9 The term ldquorealrdquo means in terms of the domestic composite commodity

10 In measuring the exports imports and international capital flows a number of itemsare often missed For instance the clandestine transfer of funds from the Philippines toUS bank accounts would generate a demand for dollars On the other hand secretiveimports of heroin from Turkey result in a supply of dollars in international marketsThe net of such unmeasured transactions are lumped under the heading of ldquostatisticaldiscrepancyrdquo in the balance of payments accounts We have omitted this item fromTable 121

11 For simplicity we assume that the exchange rate affects only the division of totalconsumption between imports and purchases of the domestic output total consumptionis assumed to be unaffected by such exchange rate changes

12 Note that an increase in the price of the yen or an ldquoappreciationrdquo of the yen means afall in the price of a dollar in terms of yen or a ldquodepreciationrdquo of the dollar

13 If the price elasticity with respect to imports was less than one in short run a rise in therelative price of imports due to a depreciation of the dollar could in fact lead to a risein the value of imports as well This short-run phenomenon when applied to the path ofnet exports over time is referred to as the ldquoJ-curve effectrdquo

14 Actually for much of the analysis to follow we need only assume the weaker ldquoMarshallndashLernerrdquo condition that the sum of the price elasticity of demand for imports and theprice elasticity of demand for exports exceeds one This assures that a price of the dollarbelow its equilibrium level will be associated with an excess demand for dollars in theforeign exchange markets while a price of the dollar above its equilibrium level willbe associated with an excess supply of dollars

15 Given future markets for foreign currency this is not always the case16 We assume that such expectation is held with subjective certainty17 There are several reasons why in the short run the supply curve may not be upward-

sloping First it often takes time for US purchasers to adjust purchases in light of achange in relative prices Second the prices of domestic goods that are close substitutesto the imported goods can rise significantly in the short run as domestic producers hitshort-run production constraints The third complication that has the effect of makingthe supply of dollars curve less likely to be upward-sloping is that foreign producersat least in the short run often adjust the foreign currency prices of goods they exportto partially offset the impact of exchange rate changes on the prices of their goodsin foreign markets For instance Knetter (1987) found that with a depreciation of thedollar (appreciation of the West German Mark) West German exporters often reducedthe Mark price of their exports so as to minimize the rise in the dollar price of Germangoods that would result from the appreciation of the Mark For the time being weabstract from such short-run considerations although this is not to lessen the importanceof this phenomenon as the experience during the 1985ndash1987 period indicates With adepreciation of the dollar the dollar value of imports grew as the USA had to supply

222 Notes

more dollars for each unit of imported goods and there was initially little reduction inthe quantity of goods imported

18 It is important to remember that while we talk of households as being the only privatedemanders of foreign goods and financial assets this is purely a simplifying deviceFirms also demand foreign goods and private depository institutions demand foreignfinancial assets The analysis would be more complex if we explicitly recognized thesedemands but our conclusions would be unchanged since we can subsume in householdactions the actions of firms and depository institutions in foreign markets

References

Aigner DJ and Cain GC (1977) Statistical theories of discrimination in labor marketsIndustrial and Labor Relations Review 30(2) 175ndash187

Alchian A and Demsetz H (1972) Production information costs and economicorganization American Economic Review 62 777ndash795

Alogoskoufis GS (1987) On intertemporal substitution and aggregate labor supplyJournal of Political Economy 95 938ndash960

Altonji J and Siow A (1987) Testing the response of consumption to income changeswith (noisy) panel data Quarterly Journal of Economics 102 293ndash328

Arrow K (1964) The role of securities in the optimal allocation of risk-bearing Review ofEconomic Studies 31 (April) 91ndash96

Arrow K and Hahn FH (1971) General Competitive Analysis San Francisco CAHolden-Day

Azariadis C (1976) On the incidence of unemployment Review of Economic Studies 43115ndash125

Barro RJ (1976) Rational expectations and the role of monetary policy Journal ofMonetary Economics 2 1ndash32

Barro RJ and Gordon D (1983) A positive theory of monetary policy in a natural ratemodel Journal of Political Economy 91 589ndash610

Barro RJ and Grossman HI (1971) A general disequilibrium model of income andemployment American Economic Review 61 82ndash93

Barro RJ and King RG (1984) Time-separable preferences and intertemporal substitu-tions models of business cycles Quarterly Journal of Economics 99 817ndash839

Begg DKH (1980) Rational expectations and the non-neutrality of systematic monetarypolicy Review of Economic Studies 47 293ndash303

Blanchard OJ (1981) What is left of the multiplier accelerator American EconomicReview 71 150ndash154

Blanchard OJ and Fischer S (1989) Lectures on Macroeconomics Cambridge MA MITPress

Borjas GJ and Heckman JJ (1978) Labor supply estimates for public policy evaluationNBER Working Paper No W0299 November

Box GEP and Jenkins GM (1970) Time Series Analysis San Francisco CAHolden-Day

Burbidge JB and Robb AL (1985) Evidence on wealthndashage profiles in Canadian cross-section data Canadian Journal of Economics 18 854ndash875

Caballero R and Engel E (1999) Explaining investment dynamics in US manufacturingA generalized (S s) approach Econometrica 67 783ndash826

224 References

Campbell JY and Mankiw NG (1987a) Permanent and transitory components inmacroeconomic fluctuations American Economic Review 77 111ndash117

Campbell JY and Mankiw NG (1987b) Are output fluctuations transitory QuarterlyJournal of Economics 102 857ndash880

Campbell JY Lo AW and MacKinlay AC (1997) The Econometrics of FinancialMarkets Princeton NJ Princeton University Press

Cass D (1965) Optimal growth in an aggregate model of capital accumulation Review ofEconomic Studies 32 233ndash240

Clower RW (1965) The Keynesian Counter-revolution A theoretical appraisal InFH Hahn and FPR Brechling (eds) The Theory of Interest Rates London Macmillan

Coase RH (1960) The problem of social cost Journal of Law and Economics 31ndash44

Cukierman A (1986) Measuring inflationary expectations Journal of Monetary Eco-nomics 17 315ndash324

Dahlman C (1979) The problem of externality Journal of Law and Economics 22141ndash162

Debreu G (1959) The Theory of Value New York WileyDiamond PA and Hausman JA (1984) Individual retirement and savings behaviour

Journal of Public Economics 23 81ndash114Dornbusch R (1976) Expectations and exchange rate dynamics Journal of Political

Economy 84 1161ndash1176Enders W (2004) Applied Econometric Time Series 2nd edn Hoboken NJ WileyEvans G (1989) Output and unemployment dynamics in the United States 1950ndash1985

Journal of Applied Econometrics 4 213ndash237Ewing BT (2001) Cross-effects of fundamental state variables Journal of

Macroeconomics 23 633ndash645Ewing BT and Payne JE (1998) The long-run relation between the personal savings rate

and consumer sentiment Financial Counseling and Planning 9(1) 89ndash96Ewing BT Levernier W and Malik F (2002) Differential effects of output shocks on

unemployment rates by race and gender Southern Economic Journal 68 584ndash599Fama EF and Miller MH (1972) The Theory of Finance New York Holt Rinehart and

WinstonFischer S (1977) Long-term contracts rational expectations and the optimal money supply

rule Journal of Political Economy 85 191ndash205Fisher I (1930) The Theory of Interest New York MacmillanFleming MJ (1962) Domestic financial policies under fixed and under floating exchange

rates IMF Staff Papers 9 369ndash379Foley KD (1975) On two specifications of asset equilibrium in macroeconomic models

Journal of Political Economy 83 303ndash324Friedman M (1959) A Program for Monetary Stability New York Fordham University

PressFriedman M (1968) The role of monetary policy American Economic Review 58 1ndash17Goldberg S (1958) Difference Equations New York WileyGould JP (1968) Adjustment cost in the theory of investment of the firm Review of

Economic Studies 35 47ndash56Grandmont JM (1977) Temporary general equilibrium theory Econometrica 45

535ndash572Granger CWJ and Morris M (1976) Time series modelling and interpretation Journal

of the Royal Statistical Society Series A 139 246ndash257

References 225

Granger CWJ and Newbold P (1986) Forecasting Economic Time Series 2nd ednOrlando FL Academic Press

Hall R (1976) The Phillips curve and macroeconomic policy In K Brunner andAH Meltzer (eds) The Phillips Curve and Labor Markets Carnegie-RochesterConference Series on Public Policy Amsterdam North-Holland

Hall R (1980) Employment fluctuation and wage rigidity In GL Perry (ed) BrookingsPapers on Economic Activity pp 91ndash123 Washington DC Brookings Institution

Hall RE (1982) Explorations in the Gold Standard and related policies for stabilizingthe dollar In RE Hall (ed) Inflation Causes and Effects Chicago IL University ofChicago Press

Hall RE (1988) Intertemporal substitution in consumption Journal of Political Economy96 339ndash357

Hannan EJ (1970) Multiple Time Series New York WileyHansen B (1970) A Survey of General Equilibrium Systems New York McGraw-HillHansen LP and Singleton KJ (1983) Stochastic consumption risk aversion and the

temporal behavior of asset returns Journal of Political Economy 91 249ndash265Harvey AC (1993) Time Series Models Cambridge MA MIT PressHayashi F (1982) Tobinrsquos marginal q and average q A neoclassical interpretation

Econometrica 50(1) 213ndash224Hicks J (1939) Value and Capital Oxford Clarendon PressHowitt P (1985) Transaction costs in the theory of unemployment American Economic

Review 75 88ndash100Howitt P (1986) Conversations with economists A review essay Journal of Monetary

Economics 18 103ndash118Hurd M (1987) Savings of the elderly and desired bequests American Economic Review

77 298ndash312Karni E (1978) Period analysis and continuous analysis in Patinkinrsquos macroeconomic

model Journal of Economic Theory 17 134ndash140Kester WC (1986) Capital ownership structure A comparison of the United States and

Japanese manufacturing corporations Financial Management 15(1) 5ndash16Keynes JM (1936) General Theory of Employment Interest and Money London

MacmillanKletzer LG and Fairlie R (1998) Jobs lost jobs regained An analysis of blackwhite

differences in job displacement in the 1980s Industrial Relations 37 460ndash477Knetter M (1987) Export prices and exchange rates Theory and evidence Working paper

Stanford University NovemberKoopmans T (1965) On the concept of optimal economic growth In Proceedings ndash

Study Week on the Econometric Approach to Development Planning Chicago ILRand-McNally

Kydland FE and Prescott EC (1977) Rules rather than discretion The inconsistency ofoptimal plans Journal of Political Economy 85 473ndash491

Kydland FE and Prescott EC (1982) Time to build and aggregate fluctuationsEconometrica 50 1345ndash1370

Leonard JS (1988) In the wrong place at the wrong time The extent of frictional andstructural unemployment NBER Working Paper No 1979

Lucas RE (1967) Adjustment costs and the theory of supply Journal of Political Economy75 321ndash334

Lucas RE (1972) Expectations and the neutrality of money Journal of Economic Theory4 103ndash124

226 References

Lucas RE (1973) Some international evidence on outputndashinflation tradeoffs AmericanEconomic Review 63 326ndash334

Lucas R Jr (1981) Methods and problems in business cycle theory In Studies in Business-Cycle Theory Cambridge MA MIT Press First published in Journal of Money Creditand Banking 12 November 1980

Lucas RE and Rapping LA (1970) Real wages employment and inflation InES Phelps (ed) Microeconomic Foundations of Employment and Inflation TheoryNew York WW Norton

Lundberg S and Startz R (1983) Private discrimination and social intervention incompetitive labor markets American Economic Review 73(3) 340ndash347

McCallum B (1979) The current state of the policy-ineffectiveness debate AmericanEconomic Review 69 240ndash245

McCallum B (1985) On consequences and criticisms of monetary targeting Journal ofMoney Credit and Banking 17 570ndash597

Mankiw NG (1987) The optimal collection of seigniorage Theory and evidence Journalof Monetary Economics 20 327ndash341

Mankiw NG Rotemberg JJ and Summers LH (1985) Intertemporal substitution inmacroeconomics Quarterly Journal of Economics 100 225ndash251

Marx K (1976) Capital Vol 1 A Critique of Political Economy HarmondsworthPenguin

Mills TC (1999) The Econometric Modelling of Financial Time Series 2nd ednCambridge Cambridge University Press

Modigliani F (1966) Life cycle hypothesis of saving the demand for wealth and the supplyof capital Social Research 33 160ndash217

Modigliani F (1986) Life cycle individual thrift and the wealth of nations AmericanEconomic Review 76 297ndash313

Mundell RA (1968) International Economics New York MacmillanNakamura A and Nakamura M (1981) A comparison of the labor force behavior of

married women in the US and Canada with special attention to the impact of incometaxes Econometrica 49 451ndash489

Nelson CR and Plosser CI (1982) Trends and random walks in macroeconomictime series Some evidence and implications Journal of Monetary Economics 10139ndash162

Niehans J (1987) Classical monetary theory new and old Journal of Money Credit andBanking 19 409ndash424

Oi WY (1962) Labor as a quasi-fixed factor Journal of Political Economy 70 538ndash555Patinkin D (1965) Money Interest and Prices New York Harper amp RowPencavel J (1985) Labor supply of men A survey In Orley Ashenfelter (ed) Handbook

of Labor Economics Amsterdam North-HollandPhelps ES (1968) Money-wage dynamics and labor market equilibrium Journal of

Political Economy 76 678ndash711Phelps ES and Taylor JB (1977) Stabilizing powers of monetary policy under rational

expectations Journal of Political Economy 85 163ndash190Phillips AW (1958) The relationship between unemployment and the rate of change in

money wage rates in the United Kingdom 1861ndash1957 Economica 25 283ndash299Pindyck RS and Rubinfeld DL (1991) Econometric Models and Economic Forecasts

3rd edn New York McGraw-HillPoole W (1985) Comment on ldquoOn consequences and criticisms of monetary targetingrdquo

Journal of Money Credit and Banking 17 602ndash605

References 227

Radford RA (1945) The economic organization of a prisoner of war camp Economica12 189ndash201

Robinson C and Tomes N (1985) More on the labour supply of Canadian womenCanadian Journal of Economics 18 156ndash163

Sargent T (1987a) Macroeconomic Theory 2nd edn Boston MA Academic PressSargent T (1987b) Dynamic Macroeconomic Analysis Cambridge MA Harvard

University PressSargent TJ and Wallace N (1975) ldquoRationalrdquo expectations the optimal monetary

instrument and the optimal money supply rule Journal of Political Economy 83241ndash254

Sargent TJ and Wallace N (1976) Rational expectations and the theory of economicpolicy Journal of Monetary Economics 2 169ndash183

Shapiro C and Stiglitz JE (1984) Equilibrium unemployment as a worker disciplinedevice American Economic Review 74 433ndash444

Shiller RJ (1978) Rational expectations and the dynamic structure of macroeconomicmodels Journal of Monetary Economics 4 1ndash44

Sidrauski M (1967a) Rational choice and patterns of growth in a monetary economyAmerican Economic Review 57 534ndash544

Sidrauski M (1967b) Inflation and economic growth Journal of Political Economy 75797ndash810

Simons HC (1936) Rules versus authorities in monetary policy Journal of PoliticalEconomy 44 1ndash30

Sims C (1972) Money income and causality American Economic Review 62 540ndash552Solow R (1956) A contribution to the theory of economic growth Quarterly Journal of

Economics 70 65ndash94Strotz RH (1955ndash1956) Myopia and inconsistency in dynamic utility maximization

Review of Economic Studies 23 165ndash180Stuart CE (1981) Swedish tax rates labor supply and tax revenues Journal of Political

Economy 89 1020ndash1038Taylor J (1972) The behaviour of unemployment and unfilled vacancies Great Britain

1958ndash71 An alternative view Economic Journal 82 1352ndash1365Taylor JB (1979) Estimation and control of a macroeconomic model with rational

expectations Econometrica 47 1267ndash1286Tobin J (1965) Money and economic growth Econometrica 33 671ndash684Tobin J (1969) A general equilibrium approach to monetary theory Journal of Money

Credit and Banking 1(1) 15ndash29Tobin J (1985) Comment on ldquoOn consequences and criticisms of monetary targetingrdquo or

Monetary targeting Dead at last Journal of Money Credit and Banking 17 605ndash610Uzawa H (1969) Time preference and the Penrose effect in a two-class model of economic

growth Journal of Political Economy 77 628ndash652Varian H (1992) Microeconomic Analysis 3rd edn New York WW NortonWalker DA (1987) Walrasrsquo theories of tatonnement Journal of Political Economy 95

758ndash774Walras L (1954) Elements of Pure Economics trans W Jaffeacute London George Allen amp

UnwinWeber CE (1998) Consumption spending and the paper-bill spread Theory and evidence

Economic Inquiry 36 575ndash589Weitzman M (1985) The simple macroeconomics of profit sharing American Economic

Review 75 937ndash953

accelerationist outcome 170ndash2aggregate demand 7 86 90ndash1 94ndash5 99

Keynesian model 133ndash4 135 139 140Lucas model 160ndash1 neoclassical model126 127 128 146 shocks 158 seealso demand

aggregate supply 82 86ndash90 91 94ndash5 98113 Keynesian model 132 133ndash4 135138 140 145 Lucas model 137153ndash4 155 158 160ndash1 164 166monetary policy 170ndash1 neoclassicalmodel 118 123 125ndash6 127 128 133see also supply

aggregation issues 5ndash6 8 14ndash15Alchian A 203n9Alogoskoufis GS 51Angell Wayne 219n4Arrow-Debreu theory 2 202n4 n10

203n8assets 51 52 53 portfolio decision

39ndash40 57ndash9 tangible 25ndash6 see alsofinancial assets

autocorrelation function 102ndash3autoregressive expectations 162ndash4 168autoregressive processes 103ndash5 110 111

113 166

balance of payments 190ndash3bankruptcy 32 33Barro RJ 156 167 179ndash80 182ndash4Begg DKH 122 215n8behavioral hypotheses 96 98 99 100Bellman equation firms 29 30 31 34

35 households 43 44 45 48ndash9Blanchard OJ 111 112 171 175 184ndash5bonds 19ndash20 23 27ndash9 32ndash3 financial

market equilibrium 84ndash5 firmfinancing constraint 24ndash6 64 Fisherian

problem 52 foreign 189 households39 40 43ndash4 72 204n11 money illusion59 portfolio choice 57 58 59 93 realvalue 77 temporary equilibrium 80 81

budget constraint 7 16 household 43 4455 65ndash6 67 70ndash1 73ndash4 open economy188 189

Burbidge JB 208n25business cycle 5 6 49 50 173 real

business cycle theory 79 136 202n4technological innovation 113

Caballero R 209n1Campbell JY 111 112 113 214n45capital 23 24 31 68 69ndash70 cost of

31ndash2 36 69 70 94 120 209n7 firmfinancing constraint 24 25 26 64international flows 192 193 195ndash6197ndash8 200 Tobinrsquos Q 36 see alsocapital stock

capital account 192capital adjustment costs 21 26ndash7 34ndash7

69ndash70capital markets 27 203n1 205n24capital stock 3 18ndash19 20ndash1 23 25 27ndash9

optimal investment 30 31 69 70retained earnings financing 29ndash30 32superneutrality of money 120 121 122Tobinrsquos Q 35 36

central banks 187 188 189 190191 192ndash3

classical economics 4 5 79 212n8Clower RW 117 118Coase RH 11CobbndashDouglas production function 147

148 217n3commodities 8 9ndash10 14 16 18ndash19 20

individual experiments 11 13ndash14

Index 229

Lucas model 146ndash7 market equilibrium92 open economy 188ndash9

comparative static analysis 116ndash23consumer behavior 11 12ndash13consumption 20 39 40ndash6 47ndash8 67

70ndash4 aggregate 55 61 demand 97 99195 215n6 financial asset demand 94Fisherian problem 51ndash5 individualexperiments 11 12ndash13 intertemporalsubstitution hypothesis 51 60ndash3Keynesian model 134 135 laborparticipation 49 life-cycle hypothesis50 55ndash6 57 money supply shocks 128neoclassical model 117ndash18 123 126neutrality of money 119 open economy190ndash1 193 permanent incomehypothesis 56ndash7 portfolio choice 5758 59 real balance effect 77 92superneutrality of money 121 122

continuous-time analysis 4cost of capital 31ndash2 36 69 70 94 120

209n7cost of living agreements (COLAs) 215n2costs bankruptcy 33 capital adjustment

21 26ndash7 34ndash7 69ndash70 dividends 23ndash4firm financing constraint 24

Cramerrsquos rule 87 119 121 127 134Cukierman A 164currency depreciation 193 195 198ndash9

221n17

Dahlman C 203n9De Moivrersquos theorem 109Debreu G 8debt 32 60debt-to-equity ratio 32ndash4decision-making behavior 5ndash6 38 51 see

also portfolio decisiondemand excess 13ndash14 15ndash16 67 118

133 190ndash1 199 individual demandfunctions 13 14 labor marketequilibrium 83 Lucas model 146156ndash7 market demand functions 1415 money 86 neoclassical model82ndash3 119 122 123 new classicaleconomics 5 price elasticity of 195real balance effect 60 77 Sayrsquos law 716 temporary equilibrium 80Walrasian model 7 15 see alsoaggregate demand

Demsetz H 203n9depository institutions 126 187 188 189

221n5 222n18

depreciation 26 31 74 205n23deterministic feedback rules 172 173 174deterministic models 6disequilibrium 78 199distributed lag schemes 162ndash4dividends depository institutions 221n5

firms 20 22 23ndash4 29 30 32 34ndash5future 76ndash7 households 43 44 66

Dornbusch model 200ndash1dynamic analysis 2 3ndash4 6 79 122dynamic programming 28ndash9

econometric models 96 100ndash1employment 1 19 128 148 full

employment model 88 142 Keynesianmodel 130ndash1 132ndash3 134 135 137neoclassical model 82 83 84 91 122123ndash5 126 see also labor labor marketlabor supply

Engel E 209n1equilibrium 1 5 7 15 199 aggregate

demand 90ndash1 94ndash5 99 aggregatesupply 86ndash8 94ndash5 employment123ndash5 financial markets 80ndash1 84ndash6illusion model 79 Keynesian model79 133ndash4 137 139ndash40 labor market83ndash4 88 90 119 123ndash5 127 131 148loanable funds theory 120 Lucas model159 160 162 monetary policy 182ndash4money market 85ndash6 92ndash3 126 159multiple equilibria models 113 naturalrate of unemployment 89 90neoclassical model 78 79 117ndash18 126non- market-clearing model 79 partial187 Patinkin analysis 91ndash4 stationarity106 stochastic models 6 temporary 278 80ndash3 86 120 velocity 139ndash40 seealso general equilibrium

equity shares 18 19ndash20 21ndash2 23 27ndash932ndash3 firm financing constraint 24ndash664 households 40 43 72 real value77 temporary equilibrium 80 81

ergodicity 212n16Eulerrsquos theorem 206n44Ewing BT 211n11 212n11exchange rate 186ndash7 191 193ndash6 197

199 200ndash1expectations 41 59 84 119 202n5

adaptive 163 170ndash2 210n15 aggregatedemand 90ndash1 autoregressive 162ndash4168 inflation 62 69 71 77 120ndash2163 164 interest rate 60 61 62 6971 75ndash7 Keynesian model 135 136ndash8

230 Index

expectations (Continued)140 141ndash4 145 labor marketequilibrium 83 Lucas model 157 160optimal monetary policy 168ndash70 180182 183ndash4 output 68 rational 3128ndash9 136ndash8 141ndash4 164ndash6 172ndash4176ndash8 182ndash4 suppliers 124 135 weakconsistency 203n4 see also forecasting

exports 190ndash1 193 197 200

financial assets 77 85 93ndash4 118 firms19ndash20 25 26 28 29 68ndash9 70households 40 42 43ndash4 66 71 72ndash4labor market equilibrium 83 openeconomy 186 187ndash8 189 191 192193 195ndash6 197ndash9 temporaryequilibrium 81ndash2 see also bondsequity shares

financial market 1 36 67 96ndash7equilibrium 80ndash1 84ndash6 93 94 95 118open economy 191 195ndash6 197 199

firm distribution constraint 23ndash4 28 4165 67 74 76ndash7 189

firm financing constraint 23 24ndash6 28ndash964 67 68ndash70 84ndash5

firms 18ndash38 64ndash5 68ndash70 130 135aggregate supply 87 capital adjustmentcosts 26ndash7 34ndash7 dividends 23ndash4investment demand 97 laboradjustment costs 37ndash8 money illusion75 119 neoclassical model 79objectives 21ndash3 open economy 187188 189 190 222n18 optimalinvestment 30ndash2 real wage illusion124 125 retained earnings financing29ndash30 34

Fischer S 112 136 137 138ndash45 171175 184ndash5

Fisher I 79 163 208n16Fisherian problem 39 46 51ndash5 209n12

215n6forecasting 96 100ndash1 101ndash14 128

149ndash50 see also expectationsforeign exchange market 186ndash7 188 191

193ndash5 198ndash9 200foreigners 187ndash90 191 197ndash8Friedman Milton 56 156 168 171 179futures market 3 7 8 10 65 66

general equilibrium 1 2 11 95 97 187neoclassical model 117 118 prices 710 15ndash16 Walrasian model 8 see alsoequilibrium

GNP see gross national productGordon D 156 167 179ndash80 182ndash4government 187 190Grandmont JM 78Granger CWJ 100 102 112 212n22gross national product (GNP) 112 168

habit persistence model 207n2Hall Robert 60ndash1 62 172Hansen B 8Harvey AC 101 114Hayashi F 36 206n43Hicks John 2 8households 18 19 39ndash63 65ndash6 70ndash4

aggregate supply 87 bonds and equityshares 20 24 consumption demand97 dividends 23 24 Fisherian problem39 46 51ndash5 imperfect foresight 124intertemporal substitution hypothesis49ndash51 60ndash3 labor supply problem46ndash51 life-cycle hypothesis 55ndash6money illusion 75ndash6 77 119neoclassical model 79 open economy187 188 189 191 193ndash6 222n18permanent income hypothesis 56ndash7portfolio decision 39ndash40 46 57ndash9price changes 21

Howitt P 5 203n15

imperfect foresight 123 124imports 193ndash5 199 200income 59 74 97 208n27 foreign goods

195 197 life-cycle hypothesis 55ndash657 neoclassical model 117 permanentincome hypothesis 56ndash7 see also wages

income effects 47ndash8 49individual experiments 11ndash14 21 39inflation 53 59 72 196 200

expectations 62 69 71 77 120ndash2 163164 Lucas model 146 154ndash6 157160 161 monetary policy 136 170ndash2174ndash9 180 181ndash2 183ndash4superneutrality of money 121 122wages 123 131 see also prices

integrated processes 110ndash11 112interest rate 1 4 20 31 36 59

consumption demand 97 equilibrium91ndash4 equity shares 22 23 expectations60 61 62 69 71 75ndash7 financialmarket equilibrium 84 85 94Fisherian problem 53 54 foreignfinancial assets 195 196 198household problem 43 45 46 47

Index 231

intertemporal substitution hypothesis49 50 51 63 72 Keynesian model133 134 labor demand 83 loanablefunds theory 120 Lucas model159ndash60 neoclassical model 118 119126ndash7 neutrality of money 119 parity200ndash1 superneutrality of money 121ndash2temporary equilibrium 81 82

intertemporal substitution hypothesis (ISH)49ndash51 60ndash3 72 73 75ndash6

invertibility condition 110investment 3 29 84 85 128 capital

adjustment costs 26ndash7 35 36 demand32 68ndash70 75 90 94 97 99 120206n40 Keynesian model 134 135neoclassical model 118 126 neutralityof money 119 optimal 30ndash2superneutrality of money 121 122supply-side variables 116

IS equation 79 90ndash1 98ndash9 118 126 191export demand 200 Keynesian model133 134 138ndash9 Lucas model 159 160

ISH see intertemporal substitutionhypothesis

lsquoislandrsquo paradigm 146ndash9

J-curve effect 221n13

Karni Edi 207n3Keynes John Maynard 4Keynesian model 5 79 88 130ndash45 199Knetter M 221n17Kydland FE 180ndash1 207n2 220n17

labor 19 21 27 42 66 adjustment costs37ndash8 demand 68 76 97 132ndash3 147215n9 217n4 Keynesian models 5marginal product of 30 optimalinvestment 31 participation 48ndash9temporary equilibrium 80 see alsoemployment labor supply wages

labor market 1 18 37 65 96ndash7 186aggregate supply 86ndash8 94 95 119intertemporal substitution hypothesis49ndash51 Keynesian model 131ndash3 134Lucas model 147 148 natural rate ofunemployment 89 90 neoclassicalmodel 83ndash4 118 124ndash5 127 Walrasrsquolaw 66ndash7 82 see also employment

labor reserve hypothesis 37ndash8labor supply 39 40 41 45 46ndash9 215n9

intertemporal substitution hypothesis49ndash51 72 73 75ndash6 210n15 Keynesian

model 132 133 Lucas model 147ndash8neoclassical model 79 83ndash4 117 123124 perfect foresight 66 70 71 realbalance effect 77 supply-side variables116 tax rates 99 see also employmentlabor

leisure 40ndash1 42 44 46ndash8 49 50 72life-cycle hypothesis 50 55ndash6 57linear regression analysis 150ndash3LM equation 79 90ndash1 98ndash9 118 126

191 Keynesian model 133 134 138ndash9Lucas model 159 160

loanable funds theory 120Lucas RE 6 50ndash1 146 149 156ndash7

164 167 203n13 210n15Lucas model 88 128 135ndash6 140 146ndash66

176 199Lucas supply function 79 137 153ndash6

158 174 175

Mankiw NG 111 112 113 136 214n45market clearing 4ndash5 6 7 15 50 79

commodity market 92 financial market93 Lucas model 146 164 wages 135

market experiments 11 14ndash16markets 1 2 20 78 187 aggregate

demand 86 90ndash1 capital 27 203n1205n24 foreign exchange 186ndash7 188191 193ndash5 198ndash9 200 futures 3 7 810 65 66 Lucas model 146 149 154sequential 67 see also financial marketlabor market money market

Marx Karl 79microeconomics 5 6 78 88 187mixed autoregressive moving average

processes 110Modigliani Franco 55ndash6 208n24ModiglianindashMiller theorem 32monetary policy 166 167ndash85 187

activist 169 171 174 discretionary136 167ndash8 184 219n1 Keynesianmodel 136ndash8 141 142 144ndash5 policyineffectiveness proposition 120 136137ndash8 172ndash9 182

money classical analysis 4 comparativestatic analysis 118 demand 71ndash4 7797 121 122 139ndash40 159 financialmarket equilibrium 84 householdproblem 41 42 43ndash4 45 46 66neoclassical model 117ndash18 neutralityof 119ndash20 136 142 173 174portfolio decision 39 57 58 59 72superneutrality of 120ndash2

232 Index

money (Continued)temporary equilibrium 80 82transaction costs 11 utility yield of 42see also money market money supply

money illusion 59 75ndash6 77 88 119 125210n15

money market 7 16 67 96ndash9 199aggregate demand 90 94 95equilibrium 85ndash6 92ndash3 126 159Keynesian model 133 134

money supply 3 76 100 119 168 190Keynesian model 134ndash5 142 Lucasmodel 159ndash60 161 162 monetarypolicy 144 169 170 173 178ndash9 180money market equilibrium 86 91neoclassical model 116 118ndash20 126ndash9purchasing power parity 200superneutrality of money 120ndash2 seealso money

moving average processes 109ndash10multiple equilibria models 113MundellndashFleming model 201

natural rate hypothesis 125 128ndash9 135166

Nelson CR 113neoclassical model 4ndash5 116ndash29 146 199

202n12 Keynesian model comparison133 136 Lucas model comparison 160162 modification of 130ndash1 purchasingpower parity 200 simple 78ndash95

new classical economics 4ndash5 6 7Newbold P 100 102 112 212n22numeraire 7 8 9ndash10

official reserve transaction balance 192open economy 186ndash201output 1 18 19 20 66ndash7 202n1

aggregate demand 86 90 94 95 139aggregate supply 86 87ndash8 classicalanalysis 4 comparative static analysis118 Dornbusch model 200 equilibrium80 82 91ndash2 94 126 expectations 6875 firm distribution constraint 65Keynesian model 5 133ndash4 135 137ndash8143ndash4 145 labor market equilibrium83 84 Lucas model 79 146 147148ndash9 153ndash5 157ndash62 165ndash6 monetarypolicy 144 168 170ndash2 173 174ndash5179 money supply shocks 128 movingaverage processes 109 neoclassicalmodel 117ndash18 122 123 125 126neutrality of money 119 open economy

187ndash8 189 190ndash1 199 simpletheoretical model 96ndash9 supply-sidevariables 116 136 time series model111ndash14

overlapping generations models 6

Patinkin D 8 11 120 202n12 critiqueof 117 equilibrium 83 84ndash5 91ndash4firms 210n5 labor supply 47

Payne JE 212n11perfect competition 7 11perfect foresight 3 87 firms 65 68ndash70

75 households 41 43 56ndash7 66 70ndash475 labor market equilibrium 83 125Lucas model 161 162 neoclassicalmodel 78 79 84 117 118 119 131Walrasrsquo law 66 67 82

perfect substitution 20 22 29 32period (discrete) analysis 4permanent income hypothesis 56ndash7Petty William 79Phelps ES 136ndash7 171 216n11Phillips AW 155Phillips curve 154ndash5 156ndash7 171 174ndash9

180 181 183Pindyck RS 101 102Plosser CI 113policy see monetary policypolicy ineffectiveness proposition 120

136 137ndash8 172ndash9 182Poole William 219n3portfolio decision 39ndash40 46 51 57ndash9 72

93 193preferences 11 40Prescott EC 180ndash1 207n2 220n17price elasticity of demand 195prices 1 2 3 19 68 accounting 7 10

13 15 aggregate demand 90 91 9495 aggregate supply 86 98autoregressive expectations 162ndash4classical analysis 4 comparative staticanalysis 116 demand-side variables117 equity shares 20 22 forecastingerrors 149ndash50 foreign goods 193ndash5197 future 41 59 75 76 householdbudget constraint 66 74 individualexperiments 11 Keynesian model 133134 135 137 145 labor marketequilibrium 83 84 Lucas model 146147 149 153 154 156ndash8 159ndash64market experiments 11 microeconomicfoundations 6 monetary policy 172ndash3money market equilibrium 86 natural

Index 233

rate hypothesis 125 natural rate ofunemployment 88ndash9 neoclassicalmodel 78ndash9 117ndash20 122ndash3 124ndash5126ndash8 130 136 146 new classicaleconomics 5 7 perfect foresight 71purchasing power parity 200 realindebtedness effect 21 relative 7 89ndash10 11 13 15ndash16 sequential markets67 superneutrality of money 120tatonnement process 15 temporaryequilibrium 80 82 Walrasian model 79ndash10 15 see also inflation

private international capital flows 192193 195ndash6 197ndash8 200

production 18ndash19productivity of labor 37purchasing power parity 200

Radford RA 8lsquorandom walkrsquo process 103 106 112ndash13Rapping LA 50ndash1 210n15rational expectations 3 128ndash9 136ndash8

141ndash4 164ndash6 172ndash4 176ndash8 182ndash4 seealso expectations

real balance effect 60 77 86 92ndash3 94121

real indebtedness effect 60real user cost of capital 31ndash2 36 69 70

94 120 209n7recontracting assumption 15reduced-form expressions 96 98 99ndash100

Lucas model 161 162 monetary policy168ndash9 172ndash3 177

reputation 184ndash5retained earnings financing 29ndash30 32 34risk 41Robb AL 208n25Rubinfeld DL 101 102

Sargent T 8 96 131ndash2 211n2 n6autoregressive expectations 164employees 217n2 investment demand206n40 Keynesians 219n2 linearregression analysis 152 monetarypolicy 136ndash7 167 168 170 172ndash4180 182 money supply 165 newclassical economics 203n13outputinflation tradeoff 157 Phillipscurve 155 superneutrality of money121 122

Say Jean-Baptiste 7Sayrsquos law 8 16seasonality 111 112

shareholders 25ndash6shares see equity sharesShiller RJ 162shocks 1 6 153 158 186 demand-side

117 monetary policy 170 moneysupply 128 129 135 Phillips curve156 time-series models 111ndash13 wages135ndash6

Simons HC 219n4Sims Christopher 100spot markets 2 37 130 131spot prices 10static analysis 2ndash4 47 50 79 116ndash23stationarity 102 103 104 105ndash9 110ndash11

113 114stationary analysis 2 3ndash4stochastic processes 6 41 101 102ndash3

110 172 173Stuart CE 99substitution effects 47ndash8 49 see also

intertemporal substitution hypothesissupply labor market equilibrium 83

Lucas supply function 79 137 153ndash6158 174 175 neoclassical model 119new classical economics 5 real balanceeffect 77 Sayrsquos law 7 16 temporaryequilibrium 80 86 Walrasian model 715 see also aggregate supply laborsupply money supply

tatonnement process 7 15 120tax issues 3 32 33 99Taylor JB 100 136ndash7 212n12 216n11technology 18ndash19 21terms of trade 194lsquotime inconsistency problemrsquo 180ndash2time series models 96 100 101ndash14Tobin James 206n40 219n2Tobinrsquos Q 27 35ndash7transaction costs 5 7 10ndash11 20

42 208n18

uncertainty 6 41unemployment 1 50ndash1 104 125

frictional 89 involuntary 211n13Lucas model 146 154ndash6 monetarypolicy 174ndash5 176 179 181ndash2 183184 natural rate of 88ndash90 135 156174ndash5

utility 13 17 40ndash1 42 46 48 61ndash2

velocity equations 139ndash40

234 Index

wages 18 19 23 24 32 36 aggregatedemand 90 91 aggregate supply 86ndash7anticipated 69 70 71 75ndash6 77 124household problem 42 43 44 47ndash866 intertemporal substitution hypothesis50 51 72 73 Keynesian model 130ndash3135ndash6 137ndash8 142ndash4 145 labordemand 68 97 labor marketequilibrium 83 84 labor participation48ndash9 Lucas model 147ndash8 moneysupply shocks 128 neoclassical model117 122ndash3 126 neutrality ofmoney 119 real wage illusion 123ndash5sticky 123 131 135ndash6 137temporary equilibrium 80 82 see alsoincome

Wallace N autoregressive expectations164 Keynesians 219n2 monetarypolicy 136ndash7 167 168 170 172 180182 money supply 165 superneutralityof money 121 122

Walrasrsquo law 1 8 14 16 85 91 balance ofpayments 190ndash1 excess demand 118financial assets 93 labor market 66ndash7market equilibrium 97 99 openeconomy 187 199 perfect foresight82 sequential markets 67

Walras Leon 1 16 202n2 203n2Walrasian models 7ndash11 15 18ndash20wealth 55 59 60 83Weber CE 209n2working hours 47ndash8

Book Cover

Half-Title

Series-Title

Copyright

Contents

1 Introduction

2 Walrasian economy

6 The simple neoclassical macroeconomic model (without government or depository institutions)

7 Empirical macroeconomics Traditional approaches and time series models

9 The Keynesian model with fixed money wage Modifying the neoclassical model

10 The Lucas model

11 Policy

12 Open economy

References

At each point in time individuals in an economy are making choices with respectto the acquisition sale andor use of a variety of different goods Such activitycan be summarized by aggregate variables such as an economyrsquos total productionof various goods and services the aggregate level of unemployment the generallevel of interest rates and the overall level of prices Macroeconomics is the studyof movements in such economy-wide variables as output employment and prices

The focus of this book is on developing simple theoretical models that provideinsight into the reasons for fluctuations in such aggregate variables These modelsexplore how shocks or ldquoimpulsesrdquo to the economy impact individualsrsquo behaviorin specific markets and the resulting implications in terms of changes in aggregatevariables

Understanding Macroeconomic Theory will provide the reader with anin-depth understanding of standard theoretical models Walrasian Keynesian andneoclassical It is written in a concise accessible style and will be an indispensabletool for all students who wish to gain firm grounding in the complexities ofmacroeconomic theories

John M Barron is the Loeb Professor of Economics in the Krannert School ofManagement at Purdue University

Bradley T Ewing is the Jerry S Rawls Endowed Professor in OperationsManagement in the Rawls College of Business at Texas Tech University

Gerald J Lynch is Professor of Economics in the Krannert School of Managementat Purdue University

HB1725B3753 2006339ndashdc22 2005026372

Contents

1 Introduction 1

11 Policy 167

12 Open economy 186

1 Introduction

2 Introduction

Introduction 3

4 Introduction

Introduction 5

6 Introduction

2 Walrasian economy

Introduction

8 Walrasian economy

Walrasian economy 9

msuma=1

cai = ci gt 0

The numeraire

(π1T πTminus1T )

(π1 πTminus1)

Tsumi=1

πi cai minusTsum

πicai ge 0

wheresumT

maxCa1CaT

ua(ca1 caT )

subject to

Tsumi=1

πi cai minusTsum

maxca1caT λ

πi cai minusTsum

πicai

partcaile 0 i = 1 T

cai ge 0 i = 1 T

partλge 0

λpartL

partλ= 0

λ ge 0

πi cai minusTsum

πicai

partcai= 0 i = 1 T

partλge 0

λpartL

partλ= 0

microipartL

λ ge 0

microi ge 0 i = 1 T

(π1 πTminus1

Tsumi=1

πi cai

) i = 1 T

partλ=

Tsumi=1

πi cai minusTsum

πicdai = 0

sumTi=1 πi(cd

Market experiments

(π1 πTminus1

Tsumi=1

πi ci1 Tsum

)equiv

msuma=1

cdai(middot)

zi equivmsum

(zai) equivmsum

(cdai minus cai)

Aggregation issues

(π1 πTminus1

Tsumi=1

πi ci

Tsumi=1

πicai equivmsum

πi cai

dπi = dpi

pT= f (zi)

pT i = 1 T minus 1

clowastai = cd

(πlowast

1 πlowasttminus1

Tsumi=1

πi cai

) i = 1 T a = 1 m (21)

rArr clowasti equiv

msuma=1

clowastai =

msuma=1

cdai equiv cd

cdi = ci i = 1 T

Tsumi=1

πicdai minus

Tsumi=1

πi cai = 0

msuma=1

πi cai minusTsum

πicdai

summaminus1 cd

ai and ci forsumm

Tsumi=1

πizi = 0

Conclusion

Introduction

yt = f (Nt K)

yi = f (Ni Ki)

1 + r = (z + pbt)pb

1 + rt = (z + pbt+1)pbt

pb = 1

1 + r(z + pbt) = 1

infinsumi=1

(1 + rt)i

(z + z

yt = f (Nt K)

Vt = peS

pe = [1(1 + r)]⎡⎣ptdtS +

infinsumk=1

(1 + rt+jminus1)

⎤⎦

infinsumk=1

(1 + rt+jminus1)

⎤⎦

pt equiv p(1 + π)

⎛⎝ kprod

(1 + rt+jminus1)

⎞⎠ k = 1 2 3

Vt = (SR)

⎡⎣dtS +

infinsumk=1

Rt+jminus1

⎤⎦

minus wages

it = limhrarr0

h= lim

hrarr0

Kt+h minus K + hδK

h= Kt + δKt

it = limhrarr0

Kt+h minus K

limhrarr0 dhdh= K

ψ(0) = 0

Vt = (SR)

⎡⎣dtS +

infinsumk=1

Rt+jminus1

⎤⎦

max(SR)

⎡⎣dtS +

infinsumk=1

Rt+jminus1

⎤⎦

yt = f (Nt K)

yt+1 = f (Nt+1 Kt+1)

Kt+1 = Int + K

dt + W (Kt+1)

Kt+1 = Int + K

yt = f (Nt K)

partftpartNt

minus wt

pt= 0 (31)

t = dψ(Int)

dInt (32)

Kt+2 = Int+1 + Kt+1

yt+1 = f (Nt+1 Kt+1)

Thus we have32

dW (Kt+1)

dKt+1= 1

(partft+1

partKt+1 minus δ

)+ dW (Kt+2)

dKt+2 (33)

minus 1

Rt+ dW (Kt+2)

dKt+2= 0 (34)

dW (Kt+1)

dKt+1= 1

(partft+1

partKt+1 minus δ

Rt (35)

minus1 + 1

(partft+1

partKt+1 minus δ

Rt= 0 (36)

partft+1

= mt + δ

N dt = N d(wtpt K)

I dnt = I d

where I dnt = Kd

dt + W (Kt+1)

Kt+1 = Int + K

yt = f (Nt K)

partftpartNt

minus wt

pt= 0 (31)

t = dψ(Int)

dInt (32)

dt+1Rt + W (Kt+2)

Kt+2 = Int+1 + Kt+1

yt+1 = f (Nt+1 Kt+1)

partft+1

partNt+1minus wt+1

pt+1= 0 (38)

t+1 = dψ(Int+1)

dInt+1 (39)

dW (Kt+1)

dKt+1= 1

(partft+1

partKt+1 minus δ

)+ dW (Kt+2)

dKt+2 (310)

(partft+1

partKt+1 minus δd

)+ 1 + ψ prime

Rt= 0 (311)

partft+1

t minus ψ primet+1

Idnt = I d

nt(Q minus 1)dId

d(Q minus 1)gt 0

V =infinsum

Conclusion

Introduction

infinsumi=t

minus [ct+1 + Mt+1pt+1 + At+1] = 0

W (xt) = maxct Nt

Mtpt xt+1

partutpartct= wt

pt (44)

partutpartct

partutpart(Mtpt)

partutpartct= (Rt)

partutpart(Mtpt)

partutpartct

Mt+1pt+1xt+2

partutpartct

partutpart(Mtpt)

wt+1Rt

pt (48)

Leisure

partuatpartcat= wt

pt+ micron

partuatpartcat

Fisherian analysis

maxcat cat+T

xat+1xat+T+1

t+Tsumi=t

β iminustua(cai)

subject to

xat+T+1 ge 0

i = 2 T + 1

microT ge 0

condition

β iminustduai dcai

= λiRi

duai dcai

βduai+1dcai+1

1 + mi equiv Ri = (1 + ri)(1 + πi)

mi = ri minus πi

βRi = 1

ai gt cdai+1

t+1dcat+1)dcat+1

dcat+1

dcat= minus dua

t dcat

βduat+1dcat+1

minus1cat+1

Rearranging we have

duat dcat

βduat+1dcat+1

cdat =

t+Tsumi=t

(caiRiminust)

t+Tsumi=t

(caiRiminust)

cdat = cp

Portfolio choice

maxcat cat+T

t+Tsumi=t

subject to

xat+T+1 ge 0

microT ge 0

λi = λi+1Ri+1

we obtain

Rearranging gives

duai d(Maipi)

βduai+1dcai+1

= Ri minus Rmi

duai d(Maipi)

βduai+1dcai+1

1 minus πi

cdat = cd

M datpt = M d

⎡⎣ jprod

⎤⎦

minus1

(caj+1)

at + M datpt

atpt 31

t+Tsumi=tminus1

ctminus1 = ct then

tminus1 lt 0

dudctminus1

dudct= Rtminus1

which simplifies to

Then (412) becomes

Conclusion

Introduction

I dnt + ψ(I d

net Ast equiv As

t minus At

Ast equiv [

pbtBst + petS

Idnt equiv Kd

t+1 minus K

yst = f (N d

t K) (53)

cdt + (M d

t minus (wtpt)eN s

net Adt equiv Ad

t minus At

Adt equiv

dt + petS

cdt + (M d

nt) minus yst

]+[net Ad

t minus net Ast

]+ (wtpt)

t minus N st

]+ [M d

N st = N d

nt) minus ylowastt

]+[net Ad

t minus net Ast

]+ [M d

t minus M ]pt = 0

N st = N d

t (wtpt K)

yst = yd

t (wtpt K)

t part(wtpt) lt 0

Kdt+1 = Kd

t+1(met + δ we

t+1pet+1) (58)

partKdt+1part(me

t+1pet+1) lt 09

I dnt = I d

nt(met + δ we

t+1pet+1 K)

net Ast = net As

t (met + δ we

t+1pet+1 K)

Kdt+1 = Kd

t+1(met + δ me

t+1 + δ wet+1pe

t+1 wet+2pe

t+2 ) (59)

Kdt+1 = Kd

t+1(met + δ we

t+1pet+1 we

t+2pet+2 )

wet+2 equiv we

we have that wei pe

i = wet+1pe

Kdt+1 = Kd

t+1(met + δ we

t+1pet+1) (510)

partKdt+1part(me

t + δ) lt 0

partKdt+1part(we

ntpart(wet+1pe

t+1) lt 0

partnet Ast part(me

t+1pet+1) lt 0

Idt equiv Kd

N st = N s

t (wtpet we

t πet+1 At Mpe

+ zBpt)

cdt = cd

t (wtpet we

t πet+1 At Mpe

+ zBpt)

Ldt = Ld

t (wtpet we

t πet+1 At Mpe

+ zBpt)

cdt + (M d

t minus (wtpet )N

N st = N s

t (wtpt wet+1pe

t+1 xt)

cdt = cd

t (wtpt wet+1pe

t+1 xt)

Ldt = Ld

t (wtpt wet+1pe

t+1 xt)

where Ldt equiv M d

N st = N s

t (wtpt wet+1pe

t+1 rt πet xt)

cdt = cd

t (wtpt wet+1pe

t+1 rt πet xt)

Ldt = Ld

t (wtpt wet+1pe

t+1 rt πet xt)

t part(wet+1pe

partN st partπe

t part(wet+1pe

partcdt partπe

t part(wet+1pe

net Adt = (wtpt)N

t part(wet+1pe

t+1) lt 0

partnet Adt partπe

t partMpt) gt 0

net Adt + (M d

t minus M )pt = 0

t part(wtpt) lt 0

cdt + (M d

cdt = cd

t (wet+1pe

Ldt = Ld

t (wet+1pe

partcdt partylowast

t gt 0

partLdt partylowast

t gt 0

Since N dt and ys

t+1pet+1)N

dt+1 minus δKd

t+1 Kdt+1)minus (we

t+1pet+1)N

dt+1 minusδKd

At = [ f (N dt+1 Kd

t+1) minus (wet+1pe

t+1]met

pbt =infinsum

z(1 + rt)i = zrt

Introduction

t clowastt Blowast

t ylowastt Ilowast

t Blowastt Slowast

t ) suchthat

N st = N d

yst = cd

Bst = Bd

Sst = Sd

Mpt = M dt pt

I dnt = Kd

t+1 minus K

yst = f (N d

N st = N d

yst = cd

Ast = Ad

Mpt = M dt pt

Ast equiv [ pbtB

st + petS

st ]pt

Adt equiv [ pbtB

dt + petS

dt ]pt

N st = N d

yst = cd

net Ast = net Ad

Mpt = M dt pt

net Ast equiv As

t minus At

net Adt equiv Ad

t minus At

[cdt + I d

t ]+ (wtpt)[N d

t pt minus Mpt = 0

N st minus N d

[cdt + I d

t (wtpt)

pbt = 1rt

nt + ψ(I dbt)

t part(wtpt))(1pt)

minus1minus1

] [dwtdNt

]=[(partN d

ylowastt = ylowast

t (K ) (61)

N demand

N supply

Output

Aggregate supply

cdt + I d

t = 0 (62)

Idnt + δK + ψ(I d

cdt = cd

Idnt = I d

nt(met + δ K)

Ldt = Mpt (63)

Ldt = Ld

ntpartrt]drt = 0

t = 0= [partcd

t partrt] gt 0

net Adt = net As

net Ast = net As

t (met + δ K)

net Adt = net Ad

t partrt]drt = 0

t = 0= [partnet Ad

Output

Aggregate supply

Aggregate demand

Conclusion

Introduction

ns = h + j(wp) (72)

ys = f (n K) (75)

ns minus n = 0 (77)

nd minus n = 0 (78)

ys minus y = 0 (79)

f (n K) minus y = 0

nlowast = [1( j + g)][ jf + gh] (714)

Time series models

yT+1 = E( yT+1|y0 yT )

E( yt) = micro

Tsumj=0

σ 2y = 1

Tsumj=0

( yt minus y)2

ρk = γkγ0

ρ0 = 1 ρk = 0 for k gt 018

yt =tminus1sumj=0

yprimet = φyprime

tminus1 + εt (721)

yt =tminus1sumj=0

yt =infinsum

φjεtminusj (723)

φj(εtminusj)

⎤⎦

2⎤⎥⎦ = σ 2

e (1 minus φ2)

= micro + εt +psum

tminus1 = εt

yt = Abt (726)

b minus φ = 0 (727)

yt = Aφt (728)

m1 m2 = [φ1 plusmn (φ21 + 4φ2)

12]2 (731)

yt = A1mt1 + A2mt

2 (732)

than one means that

yt = A1mt1 + A2mt

yt = A3mt + A4tmt

= micro + εt +qsum

θjεtminusj (735)

yt =infinsum

θjεtminusj (737)

order drdquo if

wt = dyt

yt = Tt + St + Xt (738)

yt = ytminus1 + εt

Conclusion

xt = ytminus1

A =[φ1 φ21 0

|A minus λI | = 0

1 00 1

1 minusb

Introduction

yt = yt(K ) (81)

cdt (rt πe

Ldt (rt πe

net Adt (rt πe

et + δ K) = 0 (84)

] [dpdr

t minus yt) gt 0

t ) gt 0

partLdpartr

] [dpdr

N st (wtpe

t ) minus Nt = 0

t part(wtpet ))pt

minus1minus1

] [dwtdNt

(partN st part(wtpe

dpt= (partN d

t part(wtpet ))pe

dpt= minuspartN d

t ))gprime

t part(wtpet )

ptgt 0

cdt (rt πe

t At yt) + I dnt(m

Ldt (rt πe

Aggregate supply

Aggregate demand

] [dpdr

dM= pM

dM= dcd

dM+ dcd

dMgt 0

dM= dId

dMgt 0

Conclusion

Introduction

Keynesian model 131

132 Keynesian model

t ) gt 0 (94)

t )dpt

or rearranging

t ] gt 0 (95)

Keynesian model 133

yt = yt( ptw K ) (96)

cdt (rt πe

Ldt (rt πe

134 Keynesian model

] [dpdr

dM= pM

Keynesian model 135

dM= dcd

dM+ dcd

dMgt 0

dM= dId

dMgt 0

136 Keynesian model

Keynesian model 137

138 Keynesian model

Keynesian model 139

140 Keynesian model

Mt= exp(vt)

Pt = 12

[(12 Etminus1Pt + 1

2 Etminus2Pt

)minus ut + mt + vt

] (915)

Yt = 12

] (916)

Keynesian model 141

[12 Etminus1Pt + 1

] (919)

[12 Etminus1Pt + 1

] (921)

[12 Etminus1Pt + 1

Rearranging

34 Etminus1Pt = 1

142 Keynesian model

]minus 1

Keynesian model 143

so that

According to (931)

(Fischer 1977 199)

2vtminus2

144 Keynesian model

(Yt)2 =

[12 (εt + ηt) + 1

]times[

12 (εt + ηt) + 1

e E(η2i ) = σ 2

[14 + 1

9 (a1 + 2ρ1)2]

+ (ρ21utminus2)

+ σ 2n

[14 + 1

9 (b1 + ρ2)2]

Keynesian model 145

Conclusion

10 The Lucas model

Introduction

Lucas model 147

wit = pita(N dit )

minusα

ln N dit = 1

ln N sit = 1

148 Lucas model

ln N sit =

ln Nit = 1

Lucas model 149

150 Lucas model

Lucas model 151

we find that

1 minus θ = σ 2(σ 2 + σ 2z )

152 Lucas model

z (σ 2 + σ 2z )

Lucas model 153

yt equiv 1

⎡⎣ nprod

pityit

⎤⎦

nsumi=1

⎤⎦

Taking logs

ln pt equiv 1

nsumi=1

ln pit

154 Lucas model

ln yt equiv 1

nsumi=1

ln yit (1018)

ln yt equiv 1

nsumi=1

ln yt = 1

nsumi=1

γ θzit (1020)

ln yt = 1

⎡⎣ nsum

⎤⎦ (1021)

Lucas model 155

156 Lucas model

Lucas model 157

λθ = 1 minus α

α + β

σ 2 + σ 2z

158 Lucas model

Var(ln yt) = γ 2(

σ 2 + σ 2z

partVar(ln yt)

partσ 2 = γ 2(

σ 2 + σ 2z

(σ 2 + σ 2z )

(σ 2 + σ 2z )2

γ 2σ 2z

σ 2 + σ 2z

)2 (1 minus 2σ 2

σ 2 + σ 2z

infinsumi=0

λiξtminusi (1025)

Lucas model 159

Ldt = yα1

mt = mt + εt (1028)

Mtpt = Ldt (1029)

160 Lucas model

so let

πet+1 equiv (pe

Yt = α2

α2 + α1β1

[Xt + ut + β1π

et+1 + β1

] (1032)

β1Yt + α2

β1(Xt + ut + β1π

et+1) (1033)

Lucas model 161

α2 + α1β1

[Xt + ut + β1π

et+1 + β1

Pt = 1

J0 + β1

[β1(mt + εt) + α2(Xt + ut + β1π

)] (1034)

Pt minus J0

J0 + β1Etminus1Pt

J0 + β1

[β1(mt + εt) + α2(Xt + ut + β1π

et+1) minus J0

λYtminus1

] (1034prime)

Pt = mt + εtα2

β1(Xt + ut + β1π

et+1) minus J0

+ γ θ

β1Yt + α2

β1(Xt + ut + β1π

162 Lucas model

Yt = β1

J0 + β1

[λYtminus1 + γ θ

β1(Xt + ut + β1π

(1035)

Yt = β1

J0 + β1

(λYtminus1 + J0

β1Ytminus1

Lucas model 163

Etminus1πt =qsum

ηiπtminusi (1037)

Etminus1πt =qsum

164 Lucas model

(1041)

Lucas model 165

(1044)

J0 + β1(β1εt + α2ut) (1047)

166 Lucas model

Yt = γ θ1

Conclusion

11 Policy

Introduction

168 Policy

Yt = H0

[λYtminus1 + γ θ

β1(Xt + ut + β1π

Policy 169

2σ2e +σ 2

170 Policy

infinsumiminus1

aiminus11 mtminusi

⎞⎠ minus

infinsumi=1

) (117)

Policy 171

172 Policy

Policy 173

so that

2σ2e + σ 2

174 Policy

He concludes

Policy 175

176 Policy

J0 + β0(β1εt + α2ut) (1119)

J0 + β1(β1εt + α2ut) (1120)

J0 + β1(β1εt + α2ut) (1121)

min Etminus1

J0 + β1

(β21σ 2

e + α22σ 2

Policy 177

Pt = 1

J0 + β1

et+1) + J0

(1123)

πt = 1

et+1 minus πw

and similarly

so that

(1125)

πtβ1

J0 + β1= 1

t ) minus J0λ

178 Policy

πt = πmt + α2

t ) minus J0λ

(1127)

et+1 minus πe

minus J0λ

[[(1 minus k)Un]2 +

J0 + β1

(β21σ 2

e + α22σ 2

Policy 179

[[(1 minus k)Un]2 +

J0 + β1

(β21σ 2

e + α22σ 2

πmt = πlowast

180 Policy

Policy 181

182 Policy

E0π1 = h(0) = π1

βpartZ1

partπ1+ partZ

partE0π1

dh(middot)dπ1

Policy 183

min Et

⎡⎣ infinsum

β iZt+i|It

⎤⎦

2 + b(πt)2]

184 Policy

πt = aα

Policy 185

Conclusion

12 Open economy

Introduction

Open economy 187

t denote

188 Open economy

X dt + net Ad

bdt + cd

t + zdt + (M d

Open economy 189

net Adct + net AFd

t + Cst 6 We

net AFdct = pfbt(B

dfct minus Bfc)etpt

fct and

net Atpb + (Rd

190 Open economy

gdct minus T lowast

gt =Pbt(Bs

Idnt + ψ(I d

t = Dst + Cs

[Bdt + Cd

t + X dt + gd

ft]+ [zdlowast

bdt + cdlowast

t + (xdt minus zd

Open economy 191

192 Open economy

h + net AFdct

Open economy 193

194 Open economy

pt= pft

Open economy 195

zdt = zd

196 Open economy

t we have

t+1)(1 + θet+1)

t+1 (1210)

net AFdht = net AFd

Open economy 197

pft(1et)= ptet

xdt = xd

198 Open economy

net Adft = net Ad

Open economy 199

200 Open economy

Open economy 201

Conclusion

1 Introduction

Notes 203

2 Walrasian economy

204 Notes

yt =msum

⎡⎣ nisum

⎤⎦

Notes 205

pe = [1(1 + r)]⎡⎣ptdtS +

infinsumk=1

⎤⎦

dW (Kt+1)

dInt+1

dKt+1= dW (Kt+1)

dKt+1= 0

206 Notes

f (x) equivnsum

[partf (x)partxi]xi

Notes 207

infinsumi=t

208 Notes

Notes 209

t+1 minus pt)pt

210 Notes

Notes 211

212 Notes

Notes 213

⎡⎣ infinsum

(φ2)jε2tminusj

⎤⎦ =

infinsumj=0

(φ2)jσ 2e = σ 2

e (1 minus φ2)

Letting

yt =ssum

ajytminusj + c

ax2 + bx + c = 0

214 Notes

Notes 215

yd = cdt + Id

216 Notes

Notes 217

10 The Lucas model

minusα(Ki)α

partN dit

part(witpit)

witpit

= minus 1

218 Notes

E(q) =nsum

aiE(xi)

Var(q) =nsum

a2i Var(xi) + 2

sumiltj

aiajCov(xixj)

wheresum

ai(xi minus microi)

⎤⎦

2⎞⎟⎠

Var(q) =nsum

a2i middot Var(xi)

Notes 219

becomes πet+1

11 Policy

e and σ 2u

220 Notes

2σ 2e + σ 2

12 Open economy

Notes 221

222 Notes

References

224 References

References 225

226 References

References 227

113 166

Index 229

221n17

221n5 222n18

230 Index

Howitt P 5 203n15

Index 231

158 174 175

232 Index

Index 233

42 208n18

234 Index

Book Cover

Half-Title

Series-Title

Copyright

Contents

1 Introduction

2 Walrasian economy

10 The Lucas model

11 Policy

12 Open economy

References

HB1725B3753 2006339ndashdc22 2005026372

Contents

1 Introduction 1

11 Policy 167

12 Open economy 186

1 Introduction

2 Introduction

Introduction 3

4 Introduction

Introduction 5

6 Introduction

2 Walrasian economy

Introduction

8 Walrasian economy

Walrasian economy 9

msuma=1

cai = ci gt 0

The numeraire

(π1T πTminus1T )

(π1 πTminus1)

Tsumi=1

πi cai minusTsum

πicai ge 0

wheresumT

maxCa1CaT

ua(ca1 caT )

subject to

Tsumi=1

πi cai minusTsum

maxca1caT λ

πi cai minusTsum

πicai

partcaile 0 i = 1 T

cai ge 0 i = 1 T

partλge 0

λpartL

partλ= 0

λ ge 0

πi cai minusTsum

πicai

partcai= 0 i = 1 T

partλge 0

λpartL

partλ= 0

microipartL

λ ge 0

microi ge 0 i = 1 T

(π1 πTminus1

Tsumi=1

πi cai

) i = 1 T

partλ=

Tsumi=1

πi cai minusTsum

πicdai = 0

sumTi=1 πi(cd

Market experiments

(π1 πTminus1

Tsumi=1

πi ci1 Tsum

)equiv

msuma=1

cdai(middot)

zi equivmsum

(zai) equivmsum

(cdai minus cai)

Aggregation issues

(π1 πTminus1

Tsumi=1

πi ci

Tsumi=1

πicai equivmsum

πi cai

dπi = dpi

pT= f (zi)

pT i = 1 T minus 1

clowastai = cd

(πlowast

1 πlowasttminus1

Tsumi=1

πi cai

) i = 1 T a = 1 m (21)

rArr clowasti equiv

msuma=1

clowastai =

msuma=1

cdai equiv cd

cdi = ci i = 1 T

Tsumi=1

πicdai minus

Tsumi=1

πi cai = 0

msuma=1

πi cai minusTsum

πicdai

summaminus1 cd

ai and ci forsumm

Tsumi=1

πizi = 0

Conclusion

Introduction

yt = f (Nt K)

yi = f (Ni Ki)

1 + r = (z + pbt)pb

1 + rt = (z + pbt+1)pbt

pb = 1

1 + r(z + pbt) = 1

infinsumi=1

(1 + rt)i

(z + z

yt = f (Nt K)

Vt = peS

pe = [1(1 + r)]⎡⎣ptdtS +

infinsumk=1

(1 + rt+jminus1)

⎤⎦

infinsumk=1

(1 + rt+jminus1)

⎤⎦

pt equiv p(1 + π)

⎛⎝ kprod

(1 + rt+jminus1)

⎞⎠ k = 1 2 3

Vt = (SR)

⎡⎣dtS +

infinsumk=1

Rt+jminus1

⎤⎦

minus wages

it = limhrarr0

h= lim

hrarr0

Kt+h minus K + hδK

h= Kt + δKt

it = limhrarr0

Kt+h minus K

limhrarr0 dhdh= K

ψ(0) = 0

Vt = (SR)

⎡⎣dtS +

infinsumk=1

Rt+jminus1

⎤⎦

max(SR)

⎡⎣dtS +

infinsumk=1

Rt+jminus1

⎤⎦

yt = f (Nt K)

yt+1 = f (Nt+1 Kt+1)

Kt+1 = Int + K

dt + W (Kt+1)

Kt+1 = Int + K

yt = f (Nt K)

partftpartNt

minus wt

pt= 0 (31)

t = dψ(Int)

dInt (32)

Kt+2 = Int+1 + Kt+1

yt+1 = f (Nt+1 Kt+1)

Thus we have32

dW (Kt+1)

dKt+1= 1

(partft+1

partKt+1 minus δ

)+ dW (Kt+2)

dKt+2 (33)

minus 1

Rt+ dW (Kt+2)

dKt+2= 0 (34)

dW (Kt+1)

dKt+1= 1

(partft+1

partKt+1 minus δ

Rt (35)

minus1 + 1

(partft+1

partKt+1 minus δ

Rt= 0 (36)

partft+1

= mt + δ

N dt = N d(wtpt K)

I dnt = I d

where I dnt = Kd

dt + W (Kt+1)

Kt+1 = Int + K

yt = f (Nt K)

partftpartNt

minus wt

pt= 0 (31)

t = dψ(Int)

dInt (32)

dt+1Rt + W (Kt+2)

Kt+2 = Int+1 + Kt+1

yt+1 = f (Nt+1 Kt+1)

partft+1

partNt+1minus wt+1

pt+1= 0 (38)

t+1 = dψ(Int+1)

dInt+1 (39)

dW (Kt+1)

dKt+1= 1

(partft+1

partKt+1 minus δ

)+ dW (Kt+2)

dKt+2 (310)

(partft+1

partKt+1 minus δd

)+ 1 + ψ prime

Rt= 0 (311)

partft+1

t minus ψ primet+1

Idnt = I d

nt(Q minus 1)dId

d(Q minus 1)gt 0

V =infinsum

Conclusion

Introduction

infinsumi=t

minus [ct+1 + Mt+1pt+1 + At+1] = 0

W (xt) = maxct Nt

Mtpt xt+1

partutpartct= wt

pt (44)

partutpartct

partutpart(Mtpt)

partutpartct= (Rt)

partutpart(Mtpt)

partutpartct

Mt+1pt+1xt+2

partutpartct

partutpart(Mtpt)

wt+1Rt

pt (48)

Leisure

partuatpartcat= wt

pt+ micron

partuatpartcat

Fisherian analysis

maxcat cat+T

xat+1xat+T+1

t+Tsumi=t

β iminustua(cai)

subject to

xat+T+1 ge 0

i = 2 T + 1

microT ge 0

condition

β iminustduai dcai

= λiRi

duai dcai

βduai+1dcai+1

1 + mi equiv Ri = (1 + ri)(1 + πi)

mi = ri minus πi

βRi = 1

ai gt cdai+1

t+1dcat+1)dcat+1

dcat+1

dcat= minus dua

t dcat

βduat+1dcat+1

minus1cat+1

Rearranging we have

duat dcat

βduat+1dcat+1

cdat =

t+Tsumi=t

(caiRiminust)

t+Tsumi=t

(caiRiminust)

cdat = cp

Portfolio choice

maxcat cat+T

t+Tsumi=t

subject to

xat+T+1 ge 0

microT ge 0

λi = λi+1Ri+1

we obtain

Rearranging gives

duai d(Maipi)

βduai+1dcai+1

= Ri minus Rmi

duai d(Maipi)

βduai+1dcai+1

1 minus πi

cdat = cd

M datpt = M d

⎡⎣ jprod

⎤⎦

minus1

(caj+1)

at + M datpt

atpt 31

t+Tsumi=tminus1

ctminus1 = ct then

tminus1 lt 0

dudctminus1

dudct= Rtminus1

which simplifies to

Then (412) becomes

Conclusion

Introduction

I dnt + ψ(I d

net Ast equiv As

t minus At

Ast equiv [

pbtBst + petS

Idnt equiv Kd

t+1 minus K

yst = f (N d

t K) (53)

cdt + (M d

t minus (wtpt)eN s

net Adt equiv Ad

t minus At

Adt equiv

dt + petS

cdt + (M d

nt) minus yst

]+[net Ad

t minus net Ast

]+ (wtpt)

t minus N st

]+ [M d

N st = N d

nt) minus ylowastt

]+[net Ad

t minus net Ast

]+ [M d

t minus M ]pt = 0

N st = N d

t (wtpt K)

yst = yd

t (wtpt K)

t part(wtpt) lt 0

Kdt+1 = Kd

t+1(met + δ we

t+1pet+1) (58)

partKdt+1part(me

t+1pet+1) lt 09

I dnt = I d

nt(met + δ we

t+1pet+1 K)

net Ast = net As

t (met + δ we

t+1pet+1 K)

Kdt+1 = Kd

t+1(met + δ me

t+1 + δ wet+1pe

t+1 wet+2pe

t+2 ) (59)

Kdt+1 = Kd

t+1(met + δ we

t+1pet+1 we

t+2pet+2 )

wet+2 equiv we

we have that wei pe

i = wet+1pe

Kdt+1 = Kd

t+1(met + δ we

t+1pet+1) (510)

partKdt+1part(me

t + δ) lt 0

partKdt+1part(we

ntpart(wet+1pe

t+1) lt 0

partnet Ast part(me

t+1pet+1) lt 0

Idt equiv Kd

N st = N s

t (wtpet we

t πet+1 At Mpe

+ zBpt)

cdt = cd

t (wtpet we

t πet+1 At Mpe

+ zBpt)

Ldt = Ld

t (wtpet we

t πet+1 At Mpe

+ zBpt)

cdt + (M d

t minus (wtpet )N

N st = N s

t (wtpt wet+1pe

t+1 xt)

cdt = cd

t (wtpt wet+1pe

t+1 xt)

Ldt = Ld

t (wtpt wet+1pe

t+1 xt)

where Ldt equiv M d

N st = N s

t (wtpt wet+1pe

t+1 rt πet xt)

cdt = cd

t (wtpt wet+1pe

t+1 rt πet xt)

Ldt = Ld

t (wtpt wet+1pe

t+1 rt πet xt)

t part(wet+1pe

partN st partπe

t part(wet+1pe

partcdt partπe

t part(wet+1pe

net Adt = (wtpt)N

t part(wet+1pe

t+1) lt 0

partnet Adt partπe

t partMpt) gt 0

net Adt + (M d

t minus M )pt = 0

t part(wtpt) lt 0

cdt + (M d

cdt = cd

t (wet+1pe

Ldt = Ld

t (wet+1pe

partcdt partylowast

t gt 0

partLdt partylowast

t gt 0

Since N dt and ys

t+1pet+1)N

dt+1 minus δKd

t+1 Kdt+1)minus (we

t+1pet+1)N

dt+1 minusδKd

At = [ f (N dt+1 Kd

t+1) minus (wet+1pe

t+1]met

pbt =infinsum

z(1 + rt)i = zrt

Introduction

t clowastt Blowast

t ylowastt Ilowast

t Blowastt Slowast

t ) suchthat

N st = N d

yst = cd

Bst = Bd

Sst = Sd

Mpt = M dt pt

I dnt = Kd

t+1 minus K

yst = f (N d

N st = N d

yst = cd

Ast = Ad

Mpt = M dt pt

Ast equiv [ pbtB

st + petS

st ]pt

Adt equiv [ pbtB

dt + petS

dt ]pt

N st = N d

yst = cd

net Ast = net Ad

Mpt = M dt pt

net Ast equiv As

t minus At

net Adt equiv Ad

t minus At

[cdt + I d

t ]+ (wtpt)[N d

t pt minus Mpt = 0

N st minus N d

[cdt + I d

t (wtpt)

pbt = 1rt

nt + ψ(I dbt)

t part(wtpt))(1pt)

minus1minus1

] [dwtdNt

]=[(partN d

ylowastt = ylowast

t (K ) (61)

N demand

N supply

Output

Aggregate supply

cdt + I d

t = 0 (62)

Idnt + δK + ψ(I d

cdt = cd

Idnt = I d

nt(met + δ K)

Ldt = Mpt (63)

Ldt = Ld

ntpartrt]drt = 0

t = 0= [partcd

t partrt] gt 0

net Adt = net As

net Ast = net As

t (met + δ K)

net Adt = net Ad

t partrt]drt = 0

t = 0= [partnet Ad

Output

Aggregate supply

Aggregate demand

Conclusion

Introduction

ns = h + j(wp) (72)

ys = f (n K) (75)

ns minus n = 0 (77)

nd minus n = 0 (78)

ys minus y = 0 (79)

f (n K) minus y = 0

nlowast = [1( j + g)][ jf + gh] (714)

Time series models

yT+1 = E( yT+1|y0 yT )

E( yt) = micro

Tsumj=0

σ 2y = 1

Tsumj=0

( yt minus y)2

ρk = γkγ0

ρ0 = 1 ρk = 0 for k gt 018

yt =tminus1sumj=0

yprimet = φyprime

tminus1 + εt (721)

yt =tminus1sumj=0

yt =infinsum

φjεtminusj (723)

φj(εtminusj)

⎤⎦

2⎤⎥⎦ = σ 2

e (1 minus φ2)

= micro + εt +psum

tminus1 = εt

yt = Abt (726)

b minus φ = 0 (727)

yt = Aφt (728)

m1 m2 = [φ1 plusmn (φ21 + 4φ2)

12]2 (731)

yt = A1mt1 + A2mt

2 (732)

than one means that

yt = A1mt1 + A2mt

yt = A3mt + A4tmt

= micro + εt +qsum

θjεtminusj (735)

yt =infinsum

θjεtminusj (737)

order drdquo if

wt = dyt

yt = Tt + St + Xt (738)

yt = ytminus1 + εt

Conclusion

xt = ytminus1

A =[φ1 φ21 0

|A minus λI | = 0

1 00 1

1 minusb

Introduction

yt = yt(K ) (81)

cdt (rt πe

Ldt (rt πe

net Adt (rt πe

et + δ K) = 0 (84)

] [dpdr

t minus yt) gt 0

t ) gt 0

partLdpartr

] [dpdr

N st (wtpe

t ) minus Nt = 0

t part(wtpet ))pt

minus1minus1

] [dwtdNt

(partN st part(wtpe

dpt= (partN d

t part(wtpet ))pe

dpt= minuspartN d

t ))gprime

t part(wtpet )

ptgt 0

cdt (rt πe

t At yt) + I dnt(m

Ldt (rt πe

Aggregate supply

Aggregate demand

] [dpdr

dM= pM

dM= dcd

dM+ dcd

dMgt 0

dM= dId

dMgt 0

Conclusion

Introduction

Keynesian model 131

132 Keynesian model

t ) gt 0 (94)

t )dpt

or rearranging

t ] gt 0 (95)

Keynesian model 133

yt = yt( ptw K ) (96)

cdt (rt πe

Ldt (rt πe

134 Keynesian model

] [dpdr

dM= pM

Keynesian model 135

dM= dcd

dM+ dcd

dMgt 0

dM= dId

dMgt 0

136 Keynesian model

Keynesian model 137

138 Keynesian model

Keynesian model 139

140 Keynesian model

Mt= exp(vt)

Pt = 12

[(12 Etminus1Pt + 1

2 Etminus2Pt

)minus ut + mt + vt

] (915)

Yt = 12

] (916)

Keynesian model 141

[12 Etminus1Pt + 1

] (919)

[12 Etminus1Pt + 1

] (921)

[12 Etminus1Pt + 1

Rearranging

34 Etminus1Pt = 1

142 Keynesian model

]minus 1

Keynesian model 143

so that

According to (931)

(Fischer 1977 199)

2vtminus2

144 Keynesian model

(Yt)2 =

[12 (εt + ηt) + 1

]times[

12 (εt + ηt) + 1

e E(η2i ) = σ 2

[14 + 1

9 (a1 + 2ρ1)2]

+ (ρ21utminus2)

+ σ 2n

[14 + 1

9 (b1 + ρ2)2]

Keynesian model 145

Conclusion

10 The Lucas model

Introduction

Lucas model 147

wit = pita(N dit )

minusα

ln N dit = 1

ln N sit = 1

148 Lucas model

ln N sit =

ln Nit = 1

Lucas model 149

150 Lucas model

Lucas model 151

we find that

1 minus θ = σ 2(σ 2 + σ 2z )

152 Lucas model

z (σ 2 + σ 2z )

Lucas model 153

yt equiv 1

⎡⎣ nprod

pityit

⎤⎦

nsumi=1

⎤⎦

Taking logs

ln pt equiv 1

nsumi=1

ln pit

154 Lucas model

ln yt equiv 1

nsumi=1

ln yit (1018)

ln yt equiv 1

nsumi=1

ln yt = 1

nsumi=1

γ θzit (1020)

ln yt = 1

⎡⎣ nsum

⎤⎦ (1021)

Lucas model 155

156 Lucas model

Lucas model 157

λθ = 1 minus α

α + β

σ 2 + σ 2z

158 Lucas model

Var(ln yt) = γ 2(

σ 2 + σ 2z

partVar(ln yt)

partσ 2 = γ 2(

σ 2 + σ 2z

(σ 2 + σ 2z )

(σ 2 + σ 2z )2

γ 2σ 2z

σ 2 + σ 2z

)2 (1 minus 2σ 2

σ 2 + σ 2z

infinsumi=0

λiξtminusi (1025)

Lucas model 159

Ldt = yα1

mt = mt + εt (1028)

Mtpt = Ldt (1029)

160 Lucas model

so let

πet+1 equiv (pe

Yt = α2

α2 + α1β1

[Xt + ut + β1π

et+1 + β1

] (1032)

β1Yt + α2

β1(Xt + ut + β1π

et+1) (1033)

Lucas model 161

α2 + α1β1

[Xt + ut + β1π

et+1 + β1

Pt = 1

J0 + β1

[β1(mt + εt) + α2(Xt + ut + β1π

)] (1034)

Pt minus J0

J0 + β1Etminus1Pt

J0 + β1

[β1(mt + εt) + α2(Xt + ut + β1π

et+1) minus J0

λYtminus1

] (1034prime)

Pt = mt + εtα2

β1(Xt + ut + β1π

et+1) minus J0

+ γ θ

β1Yt + α2

β1(Xt + ut + β1π

162 Lucas model

Yt = β1

J0 + β1

[λYtminus1 + γ θ

β1(Xt + ut + β1π

(1035)

Yt = β1

J0 + β1

(λYtminus1 + J0

β1Ytminus1

Lucas model 163

Etminus1πt =qsum

ηiπtminusi (1037)

Etminus1πt =qsum

164 Lucas model

(1041)

Lucas model 165

(1044)

J0 + β1(β1εt + α2ut) (1047)

166 Lucas model

Yt = γ θ1

Conclusion

11 Policy

Introduction

168 Policy

Yt = H0

[λYtminus1 + γ θ

β1(Xt + ut + β1π

Policy 169

2σ2e +σ 2

170 Policy

infinsumiminus1

aiminus11 mtminusi

⎞⎠ minus

infinsumi=1

) (117)

Policy 171

172 Policy

Policy 173

so that

2σ2e + σ 2

174 Policy

He concludes

Policy 175

176 Policy

J0 + β0(β1εt + α2ut) (1119)

J0 + β1(β1εt + α2ut) (1120)

J0 + β1(β1εt + α2ut) (1121)

min Etminus1

J0 + β1

(β21σ 2

e + α22σ 2

Policy 177

Pt = 1

J0 + β1

et+1) + J0

(1123)

πt = 1

et+1 minus πw

and similarly

so that

(1125)

πtβ1

J0 + β1= 1

t ) minus J0λ

178 Policy

πt = πmt + α2

t ) minus J0λ

(1127)

et+1 minus πe

minus J0λ

[[(1 minus k)Un]2 +

J0 + β1

(β21σ 2

e + α22σ 2

Policy 179

[[(1 minus k)Un]2 +

J0 + β1

(β21σ 2

e + α22σ 2

πmt = πlowast

180 Policy

Policy 181

182 Policy

E0π1 = h(0) = π1

βpartZ1

partπ1+ partZ

partE0π1

dh(middot)dπ1

Policy 183

min Et

⎡⎣ infinsum

β iZt+i|It

⎤⎦

2 + b(πt)2]

184 Policy

πt = aα

Policy 185

Conclusion

12 Open economy

Introduction

Open economy 187

t denote

188 Open economy

X dt + net Ad

bdt + cd

t + zdt + (M d

Open economy 189

net Adct + net AFd

t + Cst 6 We

net AFdct = pfbt(B

dfct minus Bfc)etpt

fct and

net Atpb + (Rd

190 Open economy

gdct minus T lowast

gt =Pbt(Bs

Idnt + ψ(I d

t = Dst + Cs

[Bdt + Cd

t + X dt + gd

ft]+ [zdlowast

bdt + cdlowast

t + (xdt minus zd

Open economy 191

192 Open economy

h + net AFdct

Open economy 193

194 Open economy

pt= pft

Open economy 195

zdt = zd

196 Open economy

t we have

t+1)(1 + θet+1)

t+1 (1210)

net AFdht = net AFd

Open economy 197

pft(1et)= ptet

xdt = xd

198 Open economy

net Adft = net Ad

Open economy 199

200 Open economy

Open economy 201

Conclusion

1 Introduction

Notes 203

2 Walrasian economy

204 Notes

yt =msum

⎡⎣ nisum

⎤⎦

Notes 205

pe = [1(1 + r)]⎡⎣ptdtS +

infinsumk=1

⎤⎦

dW (Kt+1)

dInt+1

dKt+1= dW (Kt+1)

dKt+1= 0

206 Notes

f (x) equivnsum

[partf (x)partxi]xi

Notes 207

infinsumi=t

208 Notes

Notes 209

t+1 minus pt)pt

210 Notes

Notes 211

212 Notes

Notes 213

⎡⎣ infinsum

(φ2)jε2tminusj

⎤⎦ =

infinsumj=0

(φ2)jσ 2e = σ 2

e (1 minus φ2)

Letting

yt =ssum

ajytminusj + c

ax2 + bx + c = 0

214 Notes

Notes 215

yd = cdt + Id

216 Notes

Notes 217

10 The Lucas model

minusα(Ki)α

partN dit

part(witpit)

witpit

= minus 1

218 Notes

E(q) =nsum

aiE(xi)

Var(q) =nsum

a2i Var(xi) + 2

sumiltj

aiajCov(xixj)

wheresum

ai(xi minus microi)

⎤⎦

2⎞⎟⎠

Var(q) =nsum

a2i middot Var(xi)

Notes 219

becomes πet+1

11 Policy

e and σ 2u

220 Notes

2σ 2e + σ 2

12 Open economy

Notes 221

222 Notes

References

224 References

References 225

226 References

References 227

113 166

Index 229

221n17

221n5 222n18

230 Index

Howitt P 5 203n15

Index 231

158 174 175

232 Index

Index 233

42 208n18

234 Index

Book Cover

Half-Title

Series-Title

Copyright

Contents

1 Introduction

2 Walrasian economy

10 The Lucas model

11 Policy

12 Open economy

References

HB1725B3753 2006339ndashdc22 2005026372

Contents

1 Introduction 1

11 Policy 167

12 Open economy 186

1 Introduction

2 Introduction

Introduction 3

4 Introduction

Introduction 5

6 Introduction

2 Walrasian economy

Introduction

8 Walrasian economy

Walrasian economy 9

msuma=1

cai = ci gt 0

The numeraire

(π1T πTminus1T )

(π1 πTminus1)

Tsumi=1

πi cai minusTsum

πicai ge 0

wheresumT

maxCa1CaT

ua(ca1 caT )

subject to

Tsumi=1

πi cai minusTsum

maxca1caT λ

πi cai minusTsum

πicai

partcaile 0 i = 1 T

cai ge 0 i = 1 T

partλge 0

λpartL

partλ= 0

λ ge 0

πi cai minusTsum

πicai

partcai= 0 i = 1 T

partλge 0

λpartL

partλ= 0

microipartL

λ ge 0

microi ge 0 i = 1 T

(π1 πTminus1

Tsumi=1

πi cai

) i = 1 T

partλ=

Tsumi=1

πi cai minusTsum

πicdai = 0

sumTi=1 πi(cd

Market experiments

(π1 πTminus1

Tsumi=1

πi ci1 Tsum

)equiv

msuma=1

cdai(middot)

zi equivmsum

(zai) equivmsum

(cdai minus cai)

Aggregation issues

(π1 πTminus1

Tsumi=1

πi ci

Tsumi=1

πicai equivmsum

πi cai

dπi = dpi

pT= f (zi)

pT i = 1 T minus 1

clowastai = cd

(πlowast

1 πlowasttminus1

Tsumi=1

πi cai

) i = 1 T a = 1 m (21)

rArr clowasti equiv

msuma=1

clowastai =

msuma=1

cdai equiv cd

cdi = ci i = 1 T

Tsumi=1

πicdai minus

Tsumi=1

πi cai = 0

msuma=1

πi cai minusTsum

πicdai

summaminus1 cd

ai and ci forsumm

Tsumi=1

πizi = 0

Conclusion

Introduction

yt = f (Nt K)

yi = f (Ni Ki)

1 + r = (z + pbt)pb

1 + rt = (z + pbt+1)pbt

pb = 1

1 + r(z + pbt) = 1

infinsumi=1

(1 + rt)i

(z + z

yt = f (Nt K)

Vt = peS

pe = [1(1 + r)]⎡⎣ptdtS +

infinsumk=1

(1 + rt+jminus1)

⎤⎦

infinsumk=1

(1 + rt+jminus1)

⎤⎦

pt equiv p(1 + π)

⎛⎝ kprod

(1 + rt+jminus1)

⎞⎠ k = 1 2 3

Vt = (SR)

⎡⎣dtS +

infinsumk=1

Rt+jminus1

⎤⎦

minus wages

it = limhrarr0

h= lim

hrarr0

Kt+h minus K + hδK

h= Kt + δKt

it = limhrarr0

Kt+h minus K

limhrarr0 dhdh= K

ψ(0) = 0

Vt = (SR)

⎡⎣dtS +

infinsumk=1

Rt+jminus1

⎤⎦

max(SR)

⎡⎣dtS +

infinsumk=1

Rt+jminus1

⎤⎦

yt = f (Nt K)

yt+1 = f (Nt+1 Kt+1)

Kt+1 = Int + K

dt + W (Kt+1)

Kt+1 = Int + K

yt = f (Nt K)

partftpartNt

minus wt

pt= 0 (31)

t = dψ(Int)

dInt (32)

Kt+2 = Int+1 + Kt+1

yt+1 = f (Nt+1 Kt+1)

Thus we have32

dW (Kt+1)

dKt+1= 1

(partft+1

partKt+1 minus δ

)+ dW (Kt+2)

dKt+2 (33)

minus 1

Rt+ dW (Kt+2)

dKt+2= 0 (34)

dW (Kt+1)

dKt+1= 1

(partft+1

partKt+1 minus δ

Rt (35)

minus1 + 1

(partft+1

partKt+1 minus δ

Rt= 0 (36)

partft+1

= mt + δ

N dt = N d(wtpt K)

I dnt = I d

where I dnt = Kd

dt + W (Kt+1)

Kt+1 = Int + K

yt = f (Nt K)

partftpartNt

minus wt

pt= 0 (31)

t = dψ(Int)

dInt (32)

dt+1Rt + W (Kt+2)

Kt+2 = Int+1 + Kt+1

yt+1 = f (Nt+1 Kt+1)

partft+1

partNt+1minus wt+1

pt+1= 0 (38)

t+1 = dψ(Int+1)

dInt+1 (39)

dW (Kt+1)

dKt+1= 1

(partft+1

partKt+1 minus δ

)+ dW (Kt+2)

dKt+2 (310)

(partft+1

partKt+1 minus δd

)+ 1 + ψ prime

Rt= 0 (311)

partft+1

t minus ψ primet+1

Idnt = I d

nt(Q minus 1)dId

d(Q minus 1)gt 0

V =infinsum

Conclusion

Introduction

infinsumi=t

minus [ct+1 + Mt+1pt+1 + At+1] = 0

W (xt) = maxct Nt

Mtpt xt+1

partutpartct= wt

pt (44)

partutpartct

partutpart(Mtpt)

partutpartct= (Rt)

partutpart(Mtpt)

partutpartct

Mt+1pt+1xt+2

partutpartct

partutpart(Mtpt)

wt+1Rt

pt (48)

Leisure

partuatpartcat= wt

pt+ micron

partuatpartcat

Fisherian analysis

maxcat cat+T

xat+1xat+T+1

t+Tsumi=t

β iminustua(cai)

subject to

xat+T+1 ge 0

i = 2 T + 1

microT ge 0

condition

β iminustduai dcai

= λiRi

duai dcai

βduai+1dcai+1

1 + mi equiv Ri = (1 + ri)(1 + πi)

mi = ri minus πi

βRi = 1

ai gt cdai+1

t+1dcat+1)dcat+1

dcat+1

dcat= minus dua

t dcat

βduat+1dcat+1

minus1cat+1

Rearranging we have

duat dcat

βduat+1dcat+1

cdat =

t+Tsumi=t

(caiRiminust)

t+Tsumi=t

(caiRiminust)

cdat = cp

Portfolio choice

maxcat cat+T

t+Tsumi=t

subject to

xat+T+1 ge 0

microT ge 0

λi = λi+1Ri+1

we obtain

Rearranging gives

duai d(Maipi)

βduai+1dcai+1

= Ri minus Rmi

duai d(Maipi)

βduai+1dcai+1

1 minus πi

cdat = cd

M datpt = M d

⎡⎣ jprod

⎤⎦

minus1

(caj+1)

at + M datpt

atpt 31

t+Tsumi=tminus1

ctminus1 = ct then

tminus1 lt 0

dudctminus1

dudct= Rtminus1

which simplifies to

Then (412) becomes

Conclusion

Introduction

I dnt + ψ(I d

net Ast equiv As

t minus At

Ast equiv [

pbtBst + petS

Idnt equiv Kd

t+1 minus K

yst = f (N d

t K) (53)

cdt + (M d

t minus (wtpt)eN s

net Adt equiv Ad

t minus At

Adt equiv

dt + petS

cdt + (M d

nt) minus yst

]+[net Ad

t minus net Ast

]+ (wtpt)

t minus N st

]+ [M d

N st = N d

nt) minus ylowastt

]+[net Ad

t minus net Ast

]+ [M d

t minus M ]pt = 0

N st = N d

t (wtpt K)

yst = yd

t (wtpt K)

t part(wtpt) lt 0

Kdt+1 = Kd

t+1(met + δ we

t+1pet+1) (58)

partKdt+1part(me

t+1pet+1) lt 09

I dnt = I d

nt(met + δ we

t+1pet+1 K)

net Ast = net As

t (met + δ we

t+1pet+1 K)

Kdt+1 = Kd

t+1(met + δ me

t+1 + δ wet+1pe

t+1 wet+2pe

t+2 ) (59)

Kdt+1 = Kd

t+1(met + δ we

t+1pet+1 we

t+2pet+2 )

wet+2 equiv we

we have that wei pe

i = wet+1pe

Kdt+1 = Kd

t+1(met + δ we

t+1pet+1) (510)

partKdt+1part(me

t + δ) lt 0

partKdt+1part(we

ntpart(wet+1pe

t+1) lt 0

partnet Ast part(me

t+1pet+1) lt 0

Idt equiv Kd

N st = N s

t (wtpet we

t πet+1 At Mpe

+ zBpt)

cdt = cd

t (wtpet we

t πet+1 At Mpe

+ zBpt)

Ldt = Ld

t (wtpet we

t πet+1 At Mpe

+ zBpt)

cdt + (M d

t minus (wtpet )N

N st = N s

t (wtpt wet+1pe

t+1 xt)

cdt = cd

t (wtpt wet+1pe

t+1 xt)

Ldt = Ld

t (wtpt wet+1pe

t+1 xt)

where Ldt equiv M d

N st = N s

t (wtpt wet+1pe

t+1 rt πet xt)

cdt = cd

t (wtpt wet+1pe

t+1 rt πet xt)

Ldt = Ld

t (wtpt wet+1pe

t+1 rt πet xt)

t part(wet+1pe

partN st partπe

t part(wet+1pe

partcdt partπe

t part(wet+1pe

net Adt = (wtpt)N

t part(wet+1pe

t+1) lt 0

partnet Adt partπe

t partMpt) gt 0

net Adt + (M d

t minus M )pt = 0

t part(wtpt) lt 0

cdt + (M d

cdt = cd

t (wet+1pe

Ldt = Ld

t (wet+1pe

partcdt partylowast

t gt 0

partLdt partylowast

t gt 0

Since N dt and ys

t+1pet+1)N

dt+1 minus δKd

t+1 Kdt+1)minus (we

t+1pet+1)N

dt+1 minusδKd

At = [ f (N dt+1 Kd

t+1) minus (wet+1pe

t+1]met

pbt =infinsum

z(1 + rt)i = zrt

Introduction

t clowastt Blowast

t ylowastt Ilowast

t Blowastt Slowast

t ) suchthat

N st = N d

yst = cd

Bst = Bd

Sst = Sd

Mpt = M dt pt

I dnt = Kd

t+1 minus K

yst = f (N d

N st = N d

yst = cd

Ast = Ad

Mpt = M dt pt

Ast equiv [ pbtB

st + petS

st ]pt

Adt equiv [ pbtB

dt + petS

dt ]pt

N st = N d

yst = cd

net Ast = net Ad

Mpt = M dt pt

net Ast equiv As

t minus At

net Adt equiv Ad

t minus At

[cdt + I d

t ]+ (wtpt)[N d

t pt minus Mpt = 0

N st minus N d

[cdt + I d

t (wtpt)

pbt = 1rt

nt + ψ(I dbt)

t part(wtpt))(1pt)

minus1minus1

] [dwtdNt

]=[(partN d

ylowastt = ylowast

t (K ) (61)

N demand

N supply

Output

Aggregate supply

cdt + I d

t = 0 (62)

Idnt + δK + ψ(I d

cdt = cd

Idnt = I d

nt(met + δ K)

Ldt = Mpt (63)

Ldt = Ld

ntpartrt]drt = 0

t = 0= [partcd

t partrt] gt 0

net Adt = net As

net Ast = net As

t (met + δ K)

net Adt = net Ad

t partrt]drt = 0

t = 0= [partnet Ad

Output

Aggregate supply

Aggregate demand

Conclusion

Introduction

ns = h + j(wp) (72)

ys = f (n K) (75)

ns minus n = 0 (77)

nd minus n = 0 (78)

ys minus y = 0 (79)

f (n K) minus y = 0

nlowast = [1( j + g)][ jf + gh] (714)

Time series models

yT+1 = E( yT+1|y0 yT )

E( yt) = micro

Tsumj=0

σ 2y = 1

Tsumj=0

( yt minus y)2

ρk = γkγ0

ρ0 = 1 ρk = 0 for k gt 018

yt =tminus1sumj=0

yprimet = φyprime

tminus1 + εt (721)

yt =tminus1sumj=0

yt =infinsum

φjεtminusj (723)

φj(εtminusj)

⎤⎦

2⎤⎥⎦ = σ 2

e (1 minus φ2)

= micro + εt +psum

tminus1 = εt

yt = Abt (726)

b minus φ = 0 (727)

yt = Aφt (728)

m1 m2 = [φ1 plusmn (φ21 + 4φ2)

12]2 (731)

yt = A1mt1 + A2mt

2 (732)

than one means that

yt = A1mt1 + A2mt

yt = A3mt + A4tmt

= micro + εt +qsum

θjεtminusj (735)

yt =infinsum

θjεtminusj (737)

order drdquo if

wt = dyt

yt = Tt + St + Xt (738)

yt = ytminus1 + εt

Conclusion

xt = ytminus1

A =[φ1 φ21 0

|A minus λI | = 0

1 00 1

1 minusb

Introduction

yt = yt(K ) (81)

cdt (rt πe

Ldt (rt πe

net Adt (rt πe

et + δ K) = 0 (84)

] [dpdr

t minus yt) gt 0

t ) gt 0

partLdpartr

] [dpdr

N st (wtpe

t ) minus Nt = 0

t part(wtpet ))pt

minus1minus1

] [dwtdNt

(partN st part(wtpe

dpt= (partN d

t part(wtpet ))pe

dpt= minuspartN d

t ))gprime

t part(wtpet )

ptgt 0

cdt (rt πe

t At yt) + I dnt(m

Ldt (rt πe

Aggregate supply

Aggregate demand

] [dpdr

dM= pM

dM= dcd

dM+ dcd

dMgt 0

dM= dId

dMgt 0

Conclusion

Introduction

Keynesian model 131

132 Keynesian model

t ) gt 0 (94)

t )dpt

or rearranging

t ] gt 0 (95)

Keynesian model 133

yt = yt( ptw K ) (96)

cdt (rt πe

Ldt (rt πe

134 Keynesian model

] [dpdr

dM= pM

Keynesian model 135

dM= dcd

dM+ dcd

dMgt 0

dM= dId

dMgt 0

136 Keynesian model

Keynesian model 137

138 Keynesian model

Keynesian model 139

140 Keynesian model

Mt= exp(vt)

Pt = 12

[(12 Etminus1Pt + 1

2 Etminus2Pt

)minus ut + mt + vt

] (915)

Yt = 12

] (916)

Keynesian model 141

[12 Etminus1Pt + 1

] (919)

[12 Etminus1Pt + 1

] (921)

[12 Etminus1Pt + 1

Rearranging

34 Etminus1Pt = 1

142 Keynesian model

]minus 1

Keynesian model 143

so that

According to (931)

(Fischer 1977 199)

2vtminus2

144 Keynesian model

(Yt)2 =

[12 (εt + ηt) + 1

]times[

12 (εt + ηt) + 1

e E(η2i ) = σ 2

[14 + 1

9 (a1 + 2ρ1)2]

+ (ρ21utminus2)

+ σ 2n

[14 + 1

9 (b1 + ρ2)2]

Keynesian model 145

Conclusion

10 The Lucas model

Introduction

Lucas model 147

wit = pita(N dit )

minusα

ln N dit = 1

ln N sit = 1

148 Lucas model

ln N sit =

ln Nit = 1

Lucas model 149

150 Lucas model

Lucas model 151

we find that

1 minus θ = σ 2(σ 2 + σ 2z )

152 Lucas model

z (σ 2 + σ 2z )

Lucas model 153

yt equiv 1

⎡⎣ nprod

pityit

⎤⎦

nsumi=1

⎤⎦

Taking logs

ln pt equiv 1

nsumi=1

ln pit

154 Lucas model

ln yt equiv 1

nsumi=1

ln yit (1018)

ln yt equiv 1

nsumi=1

ln yt = 1

nsumi=1

γ θzit (1020)

ln yt = 1

⎡⎣ nsum

⎤⎦ (1021)

Lucas model 155

156 Lucas model

Lucas model 157

λθ = 1 minus α

α + β

σ 2 + σ 2z

158 Lucas model

Var(ln yt) = γ 2(

σ 2 + σ 2z

partVar(ln yt)

partσ 2 = γ 2(

σ 2 + σ 2z

(σ 2 + σ 2z )

(σ 2 + σ 2z )2

γ 2σ 2z

σ 2 + σ 2z

)2 (1 minus 2σ 2

σ 2 + σ 2z

infinsumi=0

λiξtminusi (1025)

Lucas model 159

Ldt = yα1

mt = mt + εt (1028)

Mtpt = Ldt (1029)

160 Lucas model

so let

πet+1 equiv (pe

Yt = α2

α2 + α1β1

[Xt + ut + β1π

et+1 + β1

] (1032)

β1Yt + α2

β1(Xt + ut + β1π

et+1) (1033)

Lucas model 161

α2 + α1β1

[Xt + ut + β1π

et+1 + β1

Pt = 1

J0 + β1

[β1(mt + εt) + α2(Xt + ut + β1π

)] (1034)

Pt minus J0

J0 + β1Etminus1Pt

J0 + β1

[β1(mt + εt) + α2(Xt + ut + β1π

et+1) minus J0

λYtminus1

] (1034prime)

Pt = mt + εtα2

β1(Xt + ut + β1π

et+1) minus J0

+ γ θ

β1Yt + α2

β1(Xt + ut + β1π

162 Lucas model

Yt = β1

J0 + β1

[λYtminus1 + γ θ

β1(Xt + ut + β1π

(1035)

Yt = β1

J0 + β1

(λYtminus1 + J0

β1Ytminus1

Lucas model 163

Etminus1πt =qsum

ηiπtminusi (1037)

Etminus1πt =qsum

164 Lucas model

(1041)

Lucas model 165

(1044)

J0 + β1(β1εt + α2ut) (1047)

166 Lucas model

Yt = γ θ1

Conclusion

11 Policy

Introduction

168 Policy

Yt = H0

[λYtminus1 + γ θ

β1(Xt + ut + β1π

Policy 169

2σ2e +σ 2

170 Policy

infinsumiminus1

aiminus11 mtminusi

⎞⎠ minus

infinsumi=1

) (117)

Policy 171

172 Policy

Policy 173

so that

2σ2e + σ 2

174 Policy

He concludes

Policy 175

176 Policy

J0 + β0(β1εt + α2ut) (1119)

J0 + β1(β1εt + α2ut) (1120)

J0 + β1(β1εt + α2ut) (1121)

min Etminus1

J0 + β1

(β21σ 2

e + α22σ 2

Policy 177

Pt = 1

J0 + β1

et+1) + J0

(1123)

πt = 1

et+1 minus πw

and similarly

so that

(1125)

πtβ1

J0 + β1= 1

t ) minus J0λ

178 Policy

πt = πmt + α2

t ) minus J0λ

(1127)

et+1 minus πe

minus J0λ

[[(1 minus k)Un]2 +

J0 + β1

(β21σ 2

e + α22σ 2

Policy 179

[[(1 minus k)Un]2 +

J0 + β1

(β21σ 2

e + α22σ 2

πmt = πlowast

180 Policy

Policy 181

182 Policy

E0π1 = h(0) = π1

βpartZ1

partπ1+ partZ

partE0π1

dh(middot)dπ1

Policy 183

min Et

⎡⎣ infinsum

β iZt+i|It

⎤⎦

2 + b(πt)2]

184 Policy

πt = aα

Policy 185

Conclusion

12 Open economy

Introduction

Open economy 187

t denote

188 Open economy

X dt + net Ad

bdt + cd

t + zdt + (M d

Open economy 189

net Adct + net AFd

t + Cst 6 We

net AFdct = pfbt(B

dfct minus Bfc)etpt

fct and

net Atpb + (Rd

190 Open economy

gdct minus T lowast

gt =Pbt(Bs

Idnt + ψ(I d

t = Dst + Cs

[Bdt + Cd

t + X dt + gd

ft]+ [zdlowast

bdt + cdlowast

t + (xdt minus zd

Open economy 191

192 Open economy

h + net AFdct

Open economy 193

194 Open economy

pt= pft

Open economy 195

zdt = zd

196 Open economy

t we have

t+1)(1 + θet+1)

t+1 (1210)

net AFdht = net AFd

Open economy 197

pft(1et)= ptet

xdt = xd

198 Open economy

net Adft = net Ad

Open economy 199

200 Open economy

Open economy 201

Conclusion

1 Introduction

Notes 203

2 Walrasian economy

204 Notes

yt =msum

⎡⎣ nisum

⎤⎦

Notes 205

pe = [1(1 + r)]⎡⎣ptdtS +

infinsumk=1

⎤⎦

dW (Kt+1)

dInt+1

dKt+1= dW (Kt+1)

dKt+1= 0

206 Notes

f (x) equivnsum

[partf (x)partxi]xi

Notes 207

infinsumi=t

208 Notes

Notes 209

t+1 minus pt)pt

210 Notes

Notes 211

212 Notes

Notes 213

⎡⎣ infinsum

(φ2)jε2tminusj

⎤⎦ =

infinsumj=0

(φ2)jσ 2e = σ 2

e (1 minus φ2)

Letting

yt =ssum

ajytminusj + c

ax2 + bx + c = 0

214 Notes

Notes 215

yd = cdt + Id

216 Notes

Notes 217

10 The Lucas model

minusα(Ki)α

partN dit

part(witpit)

witpit

= minus 1

218 Notes

E(q) =nsum

aiE(xi)

Var(q) =nsum

a2i Var(xi) + 2

sumiltj

aiajCov(xixj)

wheresum

ai(xi minus microi)

⎤⎦

2⎞⎟⎠

Var(q) =nsum

a2i middot Var(xi)

Notes 219

becomes πet+1

11 Policy

e and σ 2u

220 Notes

2σ 2e + σ 2

12 Open economy

Notes 221

222 Notes

References

224 References

References 225

226 References

References 227

113 166

Index 229

221n17

221n5 222n18

230 Index

Howitt P 5 203n15

Index 231

158 174 175

232 Index

Index 233

42 208n18

234 Index

Book Cover

Half-Title

Series-Title

Copyright

Contents

1 Introduction

2 Walrasian economy

10 The Lucas model

11 Policy

12 Open economy

References

Understanding Macroeconomic Theory

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