Dirk Krueger - Macroeconomics Theory

7/31/2019 Dirk Krueger - Macroeconomics Theory

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Macroeconomic TheoryDirk Krueger1

Department of EconomicsUniversity of Pennsylvania

August 2007

1I am grateful to my teachers in Minnesota, V.V Chari, Timothy Kehoe and EdwardPrescott, my colleagues at Stanford, Robert Hall, Beatrix Paal and Tom Sargent,my co-authors Juan Carlos Conesa, Jesus Fernandez-Villaverde and Fabrizio Perri aswell as Victor Rios-Rull for helping me to learn modern macroeconomic theory. Allremaining errors are mine alone.


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Contents

1 Overview and Summary 1

2 A Simple Dynamic Economy 52.1 General Principles for Specifying a Model . . . . . . . . . . . . . 52.2 An Example Economy . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Denition of Competitive Equilibrium . . . . . . . . . . . 72.2.2 Solving for the Equilibrium . . . . . . . . . . . . . . . . . 82.2.3 Pareto Optimality and the First Welfare Theorem . . . . 112.2.4 Negishis (1960) Method to Compute Equilibria . . . . . . 132.2.5 Sequential Markets Equilibrium . . . . . . . . . . . . . . . 18

2.3 Appendix: Some Facts about Utility Functions . . . . . . . . . . 23

3 The Neoclassical Growth Model in Discrete Time 273.1 Setup of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Optimal Growth: Pareto Optimal Allocations . . . . . . . . . . . 28

3.2.1 Social Planner Problem in Sequential Formulation . . . . 293.2.2 Recursive Formulation of Social Planner Problem . . . . . 313.2.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.4 The Euler Equation Approach and Transversality Condi-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3 Competitive Equilibrium Growth . . . . . . . . . . . . . . . . . . 48

3.3.1 Denition of Competitive Equilibrium . . . . . . . . . . . 493.3.2 Characterization of the Competitive Equilibrium and the

Welfare Theorems . . . . . . . . . . . . . . . . . . . . . . 513.3.3 Sequential Markets Equilibrium . . . . . . . . . . . . . . . 553.3.4 Recursive Competitive Equilibrium . . . . . . . . . . . . . 56

4 Mathematical Preliminaries 574.1 Complete Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . 584.2 Convergence of Sequences . . . . . . . . . . . . . . . . . . . . . . 594.3 The Contraction Mapping Theorem . . . . . . . . . . . . . . . . 634.4 The Theorem of the Maximum . . . . . . . . . . . . . . . . . . . 69

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iv CONTENTS

5 Dynamic Programming 715.1 The Principle of Optimality . . . . . . . . . . . . . . . . . . . . . 715.2 Dynamic Programming with Bounded Returns . . . . . . . . . . 78

6 Models with Uncertainty 816.1 Basic Representation of Uncertainty . . . . . . . . . . . . . . . . 816.2 Denitions of Equilibrium . . . . . . . . . . . . . . . . . . . . . . 83

6.2.1 Arrow-Debreu Market Structure . . . . . . . . . . . . . . 836.2.2 Sequential Markets Market Structure . . . . . . . . . . . . 856.2.3 Equivalence between Market Structures . . . . . . . . . . 86

6.3 Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.4 Stochastic Neoclassical Growth Model . . . . . . . . . . . . . . . 88

7 The Two Welfare Theorems 917.1 What is an Economy? . . . . . . . . . . . . . . . . . . . . . . . . 917.2 Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947.3 Denition of Competitive Equilibrium . . . . . . . . . . . . . . . 967.4 The Neoclassical Growth Model in Arrow-Debreu Language . . . 977.5 A Pure Exchange Economy in Arrow-Debreu Language . . . . . 997.6 The First Welfare Theorem . . . . . . . . . . . . . . . . . . . . . 1017.7 The Second Welfare Theorem . . . . . . . . . . . . . . . . . . . . 1027.8 Type Identical Allocations . . . . . . . . . . . . . . . . . . . . . . 110

8 The Overlapping Generations Model 1118.1 A Simple Pure Exchange Overlapping Generations Model . . . . 112

8.1.1 Basic Setup of the Model . . . . . . . . . . . . . . . . . . 1138.1.2 Analysis of the Model Using Oer Curves . . . . . . . . . 1188.1.3 Inecient Equilibria . . . . . . . . . . . . . . . . . . . . . 1258.1.4 Positive Valuation of Outside Money . . . . . . . . . . . . 1298.1.5 Productive Outside Assets . . . . . . . . . . . . . . . . . . 1328.1.6 Endogenous Cycles . . . . . . . . . . . . . . . . . . . . . . 1348.1.7 Social Security and Population Growth . . . . . . . . . . 136

8.2 The Ricardian Equivalence Hypothesis . . . . . . . . . . . . . . . 1418.2.1 Innite Lifetime Horizon and Borrowing Constraints . . . 1428.2.2 Finite Horizon and Operative Bequest Motives . . . . . . 151

8.3 Overlapping Generations Models with Production . . . . . . . . . 1568.3.1 Basic Setup of the Model . . . . . . . . . . . . . . . . . . 1568.3.2 Competitive Equilibrium . . . . . . . . . . . . . . . . . . 1578.3.3 Optimality of Allocations . . . . . . . . . . . . . . . . . . 1648.3.4 The Long-Run Eects of Government Debt . . . . . . . . 168

9 Continuous Time Growth Theory 1739.1 Stylized Growth and Development Facts . . . . . . . . . . . . . . 173

9.1.1 Kaldors Growth Facts . . . . . . . . . . . . . . . . . . . . 1749.1.2 Development Facts from the Summers-Heston Data Set . 174

9.2 The Solow Model and its Empirical Evaluation . . . . . . . . . . 179


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CONTENTS v

9.2.1 The Model and its Implications . . . . . . . . . . . . . . . 1829.2.2 Empirical Evaluation of the Model . . . . . . . . . . . . . 184

9.3 The Ramsey-Cass-Koopmans Model . . . . . . . . . . . . . . . . 1959.3.1 Mathematical Preliminaries: Pontryagins Maximum Prin-

ciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1959.3.2 Setup of the Model . . . . . . . . . . . . . . . . . . . . . . 1959.3.3 Social Planners Problem . . . . . . . . . . . . . . . . . . . 1979.3.4 Decentralization . . . . . . . . . . . . . . . . . . . . . . . 206

9.4 Endogenous Growth Models . . . . . . . . . . . . . . . . . . . . . 2119.4.1 The BasicAK -Model . . . . . . . . . . . . . . . . . . . . 2119.4.2 Models with Externalities . . . . . . . . . . . . . . . . . . 2159.4.3 Models of Technological Progress Based on Monopolistic

Competition: Variant of Romer (1990) . . . . . . . . . . . 228

10 Bewley Models 24110.1 Some Stylized Facts about the Income and Wealth Distribution

in the U.S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24210.1.1 Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . 24210.1.2 Main Stylized Facts . . . . . . . . . . . . . . . . . . . . . 243

10.2 The Classic Income Fluctuation Problem . . . . . . . . . . . . . 24910.2.1 Deterministic Income . . . . . . . . . . . . . . . . . . . . 25010.2.2 Stochastic Income and Borrowing Limits . . . . . . . . . . 258

10.3 Aggregation: Distributions as State Variables . . . . . . . . . . . 26210.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26210.3.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . 269

11 Fiscal Policy 27311.1 Positive Fiscal Policy . . . . . . . . . . . . . . . . . . . . . . . . . 27311.2 Normative Fiscal Policy . . . . . . . . . . . . . . . . . . . . . . . 273

11.2.1 Optimal Policy with Commitment . . . . . . . . . . . . . 27311.2.2 The Time Consistency Problem and Optimal Fiscal Policy

without Commitment . . . . . . . . . . . . . . . . . . . . 273

12 Political Economy and Macroeconomics 275

13 References 277


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Chapter 1

Overview and Summary

After a quick warm-up for dynamic general equilibrium models in the rst partof the course we will discuss the two workhorses of modern macroeconomics, theneoclassical growth model with innitely lived consumers and the OverlappingGenerations (OLG) model. This rst part will focus on techniques rather thanissues; one rst has to learn a language before composing poems.

I will rst present a simple dynamic pure exchange economy with two in-nitely lived consumers engaging in intertemporal trade. In this model theconnection between competitive equilibria and Pareto optimal equilibria can beeasily demonstrated. Furthermore it will be demonstrated how this connec-tion can exploited to compute equilibria by solving a particular social plannersproblem, an approach developed rst by Negishi (1960) and discussed nicely byKehoe (1989).

This model with then enriched by production (and simplied by droppingone of the two agents), to give rise to the neoclassical growth model. Thismodel will rst be presented in discrete time to discuss discrete-time dynamicprogramming techniques; both theoretical as well as computational in nature.The main reference will be Stokey et al., chapters 2-4. As a rst economicapplication the model will be enriched by technology shocks to develop theReal Business Cycle (RBC) theory of business cycles. Cooley and Prescott(1995) are a good reference for this application. In order to formulate thestochastic neoclassical growth model notation for dealing with uncertainty willbe developed.

This discussion will motivate the two welfare theorems, which will then bepresented for quite general economies in which the commodity space may beinnite-dimensional. We will draw on Stokey et al., chapter 15s discussion of

Debreu (1954).The next two topics are logical extensions of the preceding material. We willrst discuss the OLG model, due to Samuelson (1958) and Diamond (1965).The rst main focus in this module will be the theoretical results that distinguishthe OLG model from the standard Arrow-Debreu model of general equilibrium:in the OLG model equilibria may not be Pareto optimal, at money may have

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2 CHAPTER 1. OVERVIEW AND SUMMARY

positive value, for a given economy there may be a continuum of equilibria(and the core of the economy may be empty). All this could not happen inthe standard Arrow-Debreu model. References that explain these dierences indetail include Geanakoplos (1989) and Kehoe (1989). Our discussion of theseissues will largely consist of examples. One reason to develop the OLG modelwas the uncomfortable assumption of innitely lived agents in the standardneoclassical growth model. Barro (1974) demonstrated under which conditions(operative bequest motives) an OLG economy will be equivalent to an economywith innitely lived consumers. One main contribution of Barro was to providea formal justication for the assumption of innite lives. As we will see thismethodological contribution has profound consequences for the macroeconomiceects of government debt, reviving the Ricardian Equivalence proposition. Asa prelude we will briey discuss Diamonds (1965) analysis of government debtin an OLG model.

In the next module we will discuss the neoclassical growth model in con-tinuous time to develop continuous time optimization techniques. After havinglearned the technique we will review the main developments in growth the-ory and see how the various growth models fare when being contrasted withthe main empirical ndings from the Summers-Heston panel data set. We willbriey discuss the Solow model and its empirical implications (using the arti-cle by Mankiw et al. (1992) and Romer, chapter 2), then continue with theRamsey model (Intriligator, chapter 14 and 16, Blanchard and Fischer, chapter2). In this model growth comes about by introducing exogenous technologicalprogress. We will then review the main contributions of endogenous growth the-ory, rst by discussing the early models based on externalities (Romer (1986),Lucas (1988)), then models that explicitly try to model technological progress(Romer (1990).

All the models discussed up to this point usually assumed that individualsare identical within each generation (or that markets are complete), so thatwithout loss of generality we could assume a single representative consumer(within each generation). This obviously makes life easy, but abstracts from alot of interesting questions involving distributional aspects of government policy.In the next section we will discuss a model that is capable of addressing theseissues. There is a continuum of individuals. Individuals are ex-ante identical(have the same stochastic income process), but receive dierent income realiza-tions ex post. These income shocks are assumed to be uninsurable (we thereforedepart from the Arrow-Debreu world), but people are allowed to self-insure byborrowing and lending at a risk-free rate, subject to a borrowing limit. Deaton(1991) discusses the optimal consumption-saving decision of a single individualin this environment and Aiyagari (1994) incorporates Deatons analysis into a

full-blown dynamic general equilibrium model. The state variable for this econ-omy turns out to be a cross-sectional distribution of wealth across individuals.This feature makes the model interesting as distributional aspects of all kindsof government policies can be analyzed, but it also makes the state space verybig. A cross-sectional distribution as state variable requires new concepts (de-veloped in measure theory) for dening and new computational techniques for


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3

computing equilibria. The early papers therefore restricted attention to steadystate equilibria (in which the cross-sectional wealth distribution remained con-stant). Very recently techniques have been developed to handle economies withdistributions as state variables that feature aggregate shocks, so that the cross-sectional wealth distribution itself varies over time. Krusell and Smith (1998)is the key reference. Applications of their techniques to interesting policy ques-tions could be very rewarding in the future. If time permits I will discuss suchan application due to Heathcote (1999).

For the next two topics we will likely not have time; and thus the corre-sponding lecture notes are work in progress. So far we have not consideredhow government policies aect equilibrium allocations and prices. In the nextmodules this question is taken up. First we discuss scal policy and we startwith positive questions: how does the governments decision to nance a givenstream of expenditures (debt vs. taxes) aect macroeconomic aggregates (Barro(1974), Ohanian (1997))?; how does government spending aect output (Baxterand King (1993))? In this discussion government policy is taken as exogenouslygiven. The next question is of normative nature: how should a benevolent gov-ernment carry out scal policy? The answer to this question depends cruciallyon the assumption of whether the government can commit to its policy. A gov-ernment that can commit to its future policies solves a classical Ramsey problem(not to be confused with the Ramsey model); the main results on optimal scalpolicy are reviewed in Chari and Kehoe (1999). Kydland and Prescott (1977)pointed out the dilemma a government faces if it cannot commit to its policy-this is the famous time consistency problem. How a benevolent governmentthat cannot commit should carry out scal policy is still very much an open

question. Klein and Rios-Rull (1999) have made substantial progress in an-swering this question. Note that we throughout our discussion assume that thegovernment acts in the best interest of its citizens. What happens if policies areinstead chosen by votes of selsh individuals is discussed in the last part of thecourse.

As discussed before we assumed so far that government policies were eitherxed exogenously or set by a benevolent government (that can or cant commit).Now we relax this assumption and discuss political-economic equilibria in whichpeople not only act rationally with respect to their economic decisions, but alsorationally with respect to their voting decisions that determine macroeconomicpolicy. Obviously we rst had to discuss models with heterogeneous agents since

with homogeneous agents there is no political conict and hence no interestingdierences between the Ramsey problem and a political-economic equilibrium.This area of research is not very far developed and we will only present twoexamples (Krusell et al. (1997), Alesina and Rodrik (1994)) that deal with thequestion of capital taxation in a dynamic general equilibrium model in whichthe capital tax rate is decided upon by repeated voting.


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4 CHAPTER 1. OVERVIEW AND SUMMARY


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Chapter 2

A Simple DynamicEconomy

2.1 General Principles for Specifying a ModelAn economic model consists of dierent types of entities that take decisionssubject to constraints. When writing down a model it is therefore crucial toclearly state what the agents of the model are, which decisions they take, whatconstraints they have and what information they possess when making theirdecisions. Typically a model has (at most) three types of decision-makers

1. Households: We have to specify householdspreferences over commodi-

ties and their endowments of these commodities. Households are as-sumed to maximize their preferences, subject to a constraint set thatspecies which combination of commodities a household can choose from.This set usually depends on the initial endowments and on market prices.

2. Firms: We have to specify thetechnology available to rms, describ-ing how commodities (inputs) can be transformed into other commodities(outputs). Firms are assumed to maximize (expected) prots, subject totheir production plans being technologically feasible.

3. Government: We have to specify whatpolicy instruments (taxes, moneysupply etc.) the government controls. When discussing government policyfrom a positive point of view we will take government polices as given

(of course requiring the government budget constraint(s) to be satised),when discussing government policy from a normative point of view wewill endow the government, as households and rms, with an objectivefunction. The government will then maximize this objective function bychoosing policy, subject to the policies satisfying the government budgetconstraint(s)).

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6 CHAPTER 2. A SIMPLE DYNAMIC ECONOMY

In addition to specifying preferences, endowments, technology and policy, wehave to specify whatinformation agents possess when making decisions. Thiswill become clearer once we discuss models with uncertainty. Finally we haveto be precise about how agents interact with each other. Most of economicsfocuses on market interaction between agents; this will be also the case in thiscourse. Therefore we have to specify ourequilibrium concept, by makingassumptions about how agents perceive their power to aect market prices.In this course we will focus on competitive equilibria, by assuming that allagents in the model (apart from possibly the government) take market pricesas given and beyond their control when making their decisions. An alternativeassumption would be to allow for market power of rms or households, whichinduces strategic interactions between agents in the model. Equilibria involvingstrategic interaction have to be analyzed using methods from modern gametheory, which you will be taught in the second quarter of the micro sequence.

To summarize, a description of any model in this course should always con-tain the specication of the elements in bold letters: what commodities aretraded, preferences over and endowments of these commodities, technology, gov-ernment policies, the information structure and the equilibrium concept.

2.2 An Example EconomyTime is discrete and indexed byt = 0 ; 1; 2; : : : There are 2 individuals that liveforever in this pure exchange economy. There are no rms or any government inthis economy. In each period the two agents trade a nonstorable consumptiongood. Hence there are (countably) innite number of commodities, namelyconsumption in periodst = 0 ; 1; 2; : : :

Denition 1 An allocation is a sequence (c1

; c2

) = f (c1t ; c

2t )g

1t =0 of consump-tion in each period for each individual.

Individuals have preferences over consumption allocations that can be rep-resented by the utility function

u(ci ) =1

Xt =0 t ln(cit ) (2.1)with 2 (0; 1):

This utility function satises some assumptions that we will often require inthis course. These are further discussed in the appendix to this chapter. Notethat both agents are assumed to have the same time discount factor:

Agents have deterministic endowment streamsei = f eit g1t =0 of the consump-tion goods given by

e1t =20

if t is evenif t is odd

e2t =02

if t is evenif t is odd


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2.2. AN EXAMPLE ECONOMY 7

There is no uncertainty in this model and both agents know their endowmentpattern perfectly in advance. All information is public, i.e. all agents knoweverything. At period0; before endowments are received and consumption takesplace, the two agents meet at a central market place and trade all commodities,i.e. trade consumption for all future dates. Letpt denote the price, in period0;of one unit of consumption to be delivered in periodt; in terms of an abstractunit of account. We will see later that prices are only determined up to aconstant, so we can always normalize the price of one commodity to1 and makeit the numeraire. Both agents are assumed to behave competitively in thatthey take the sequence of pricesf pt g1t =0 as given and beyond their control whenmaking their consumption decisions.

After trade has occurred agents possess pieces of paper (one may call themcontracts) stating

in period 212 I, agent1; will deliver 0.25 units of the consumptiongood to agent2 (and will eat the remaining 1.75 units)

in period 2525 I, agent1; will receive one unit of the consumptiongood from agent2 (and eat it).

and so forth. In all future periods the only thing that happens is that agentsmeet (at the market place again) and deliveries of the consumption goods theyagreed upon in period0 takes place. Again, all trade takes place in period0and agents are committed in future periods to what they have agreed upon inperiod 0: There is perfect enforcement of these contracts signed in period0:1

2.2.1 Denition of Competitive EquilibriumGiven a sequence of pricesf pt g1t =0 households solve the following optimizationproblem

maxf c it g

1t =0

1

Xt =0 t ln(cit )s.t.

1

Xt =0 pt cit1

Xt =0 pt eitcit 0 for allt

Note that the budget constraint can be rewritten as1

Xt =0 pt (eit cit ) 0

1A market structure in which agents trade only at period0 will be called an Arrow-Debreumarket structure. We will show below that this market structure is equivalent to a marketstructure in which trade in consumption and a particular asset takes place in each period, amarket structure that we will call sequential markets.


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The quantity eit cit is the net trade of consumption of agenti for periodt whichmay be positive or negative.

For arbitrary prices f pt g1t =0 it may be the case that total consumption inthe economy desired by both agents,c1t + c2t at these prices does not equal totalendowmentse1t + e2t 2: We will call equilibrium a situation in which pricesare right in the sense that they induce agents to choose consumption so thattotal consumption equals total endowment in each period. More precisely, wehave the following denition

Denition 2 A (competitive) Arrow-Debreu equilibrium are prices f ^ pt g1t =0 and allocations (f cit g1t =0 ) i =1 ;2 such that

1. Given f ^ pt g1t =0 ; for i = 1 ; 2; f cit g1t =0 solves

maxf c it g

1t =0

1

Xt =0 t ln(cit ) (2.2)

s.t.1

Xt =0 ^ pt cit1

Xt =0 ^ pt eit (2.3)cit 0 for all t (2.4)

2.c1t + c2t = e1t + e2t for all t (2.5)

The elements of an equilibrium are allocations and prices. Note that wedo not allow free disposal of goods, as the market clearing condition is statedas an equality.2 Also note the^s in the appropriate places: the consumptionallocation has to satisfy the budget constraint(2:3) only at equilibrium pricesand it is the equilibrium consumption allocation that satises the goods marketclearing condition(2:5): Since in this course we will usually talk about com-petitive equilibria, we will henceforth take the adjective competitive as beingunderstood.

2.2.2 Solving for the Equilibrium

For arbitrary prices f pt g1t =0 lets rst solve the consumer problem. Attachthe Lagrange multiplier i to the budget constraint. The rst order necessary

2Dierent people have dierent tastes as to whether one should allow free disposal or not.Personally I think that if one wishes to allow free disposal, one should specify this as part of technology (i.e. introduce a rm that has available a technology that uses positive inputs toproduce zero output; obviously for such a rm to be operative in equilibrium it has to be thecase that the price of the inputs are non-positive -think about goods that are actually badssuch as pollution).


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conditions forcit and cit +1 are then

tcit

= i pt (2.6)

t +1

cit +1= i pt +1 (2.7)

and hence pt +1 cit +1 = p t c

it for allt (2.8)

for i = 1 ; 2:Equations (2:8); together with the budget constraint can be solved for the

optimal sequence of consumption of householdi as a function of the innitesequence of prices (and of the endowments, of course)

cit = c

it (f pt g

1t =0 )

In order to solve for the equilibrium pricesf pt g1t =0 one then uses the goodsmarket clearing conditions(2:5)

c1t (f pt g1t =0 ) + c2t (f pt g1t =0 ) = e

1t + e2t for allt

This is a system of innite equations (for eacht one) in an innite numberof unknownsf pt g1t =0 which is in general hard to solve. Below we will discussNegishis method that often proves helpful in solving for equilibria by reducingthe number of equations and unknowns to a smaller number.

For our particular simple example economy, however, we can solve for theequilibrium directly. Sum(2:8) across agents to obtain

pt +1 c1t +1 + c2t +1 = p t (c1t + c2t )

Using the goods market clearing condition we nd that

pt +1 e1t +1 + e2t +1 = p t (e

1t + e

2t )

and hence pt +1 = p t

and therefore equilibrium prices are of the form

pt = t p0

Without loss of generality we can setp0 = 1 ; i.e. make consumption at period0 the numeraire.3 Then equilibrium prices have to satisfy

^ pt = t

3Note that multiplying all prices by > 0 does not change the budget constraints of agents,so that if pricesf pt g1t =0 and allocations(f cit g1t =0 ) i 2 1 ; 2 are an AD equilibrium, so are pricesf p t g1t =0 and allocations(f cit g1t =0 ) i =1 ;2


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so that, since < 1, the period0 price for periodt consumption is lower than theperiod 0 price for period0 consumption. This fact just reects the impatienceof both agents.

Using (2:8) we have that cit +1 = cit = ci0 for all t , i.e. consumption isconstant across time for both agents. This reects the agents desire to smoothconsumption over time, a consequence of the strict concavity of the period utilityfunction. Now observe that the budget constraint of both agents will hold withequality since agents period utility function is strictly increasing. The left handside of the budget constraint becomes

1

Xt =0 ^ pt cit = ci01

Xt =0 t = ci0

1

for i = 1 ; 2:

The two agents dier only along one dimension: agent 1 is rich rst, which,given that prices are declining over time, is an advantage. For agent1 the righthand side of the budget constraint becomes

1

Xt =0 ^ pt e1t = 21

Xt =0 2t = 21 2and for agent2 it becomes

1

Xt =0 ^ pt e2t = 2 1

Xt =0 2t = 2 1 2The equilibrium allocation is then given by

c1t = c10 = (1 )2

1 2=

21 +

> 1

c2t = c20 = (1 )2

1 2=

2 1 +

< 1

which obviously satises

c1t + c2t = 2 = e

1t + e

2t for allt

Therefore the mere fact that the rst agent is rich rst makes her consumemore in every period. Note that there is substantial trade going on; in eacheven period the rst agent delivers2 21+ = 21+ to the second agent and inall odd periods the second agent delivers2 21+ to the rst agent. Also notethat this trade is mutually benecial, because without trade both agents receivelifetime utility

u(eit ) = 1


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whereas with trade they obtain

u(c1) =1

Xt =0 t ln 21 + = ln2

1+

1 > 0

u(c2) =1

Xt =0 t ln 2 1 + = ln2

1+

1 < 0

In the next section we will show that not only are both agents better o inthe competitive equilibrium than by just eating their endowment, but that, ina sense to be made precise, the equilibrium consumption allocation is sociallyoptimal.

2.2.3 Pareto Optimality and the First Welfare Theorem

In this section we will demonstrate that for this economy a competitive equi-librium is socially optimal. To do this we rst have to dene what sociallyoptimal means. Our notion of optimality will be Pareto eciency (also some-times referred to as Pareto optimality). Loosely speaking, an allocation is Paretoecient if it is feasible and if there is no other feasible allocation that makes nohousehold worse o and at least one household strictly better o. Let us nowmake this precise.

Denition 3 An allocation f (c1t ; c2t )g1t =0 is feasible if

1.cit 0 for all t; for i = 1 ; 2

2.c1t + c2t = e1t + e2t for all t

Feasibility requires that consumption is nonnegative and satises the re-source constraint for all periodst = 0 ; 1; : : :

Denition 4 An allocation f (c1t ; c2t )g1t =0 is Pareto ecient if it is feasible and if there is no other feasible allocation f (~c1t ; ~c2t )g1t =0 such that

u(~ci ) u(ci ) for both i = 1 ; 2u(~ci ) > u (ci ) for at least one i = 1 ; 2

Note that Pareto eciency has nothing to do with fairness in any sense: anallocation in which agent1 consumes everything in every period and agent2starves is Pareto ecient, since we can only make agent2 better o by making

agent 1 worse o.We now prove that every competitive equilibrium allocation for the economydescribed above is Pareto ecient. Note that we have solved for one equilibriumabove; this does not rule out that there is more than one equilibrium. One can,in fact, show that for this economy the competitive equilibrium is unique, butwe will not pursue this here.


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Proposition 5 Let (f cit g1t =0 ) i =1 ;2 be a competitive equilibrium allocation. Then (f cit g1t =0 ) i =1 ;2 is Pareto ecient.

Proof. The proof will be by contradiction; we will assume that(f cit g1t =0 )i =1 ;2is not Pareto ecient and derive a contradiction to this assumption.

So suppose that(f cit g1t =0 )i =1 ;2 is not Pareto ecient. Then by the denitionof Pareto eciency there exists another feasible allocation(f ~cit g1t =0 ) i =1 ;2 suchthat

u(~ci ) u(ci ) for both i = 1 ; 2u(~ci ) > u (ci ) for at least onei = 1 ; 2

Without loss of generality assume that the strict inequality holds fori = 1 :Step 1: Show that

1

Xt =0 ^ pt ~c

1t >

1

Xt =0 ^ pt c

1t

where f ^ pt g1t =0 are the equilibrium prices associated with(f cit g1t =0 ) i =1 ;2 : If not,i.e. if

1

Xt =0 ^ pt ~c1t1

Xt =0 ^ pt c1tthen for agent1 the ~-allocation is better (rememberu(~c1) > u (c1) is assumed)and not more expensive, which cannot be the case sincef c1t g1t =0 is part of a competitive equilibrium, i.e. maximizes agent1s utility given equilibriumprices. Hence

1

Xt =0

^ pt ~c1t >1

Xt =0

^ pt c1t (2.9)

Step 2: Show that1

Xt =0 ^ pt ~c2t1

Xt =0 ^ pt c2tIf not, then

1

Xt =0 ^ pt ~c2t 0 such that

1

Xt =0

^ pt ~c2t + 1

Xt =0

^ pt c2t

Remember that we normalized p0 = 1 : Now dene a new allocation for agent2;by

c2t = ~c2t for allt 1

c20 = ~c20 + for t = 0


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Obviously1

Xt =0 ^ pt c

2t =

1

Xt =0 ^ pt ~c

2t +

1

Xt =0 ^ pt c

2t

andu(c2) > u (~c2) u(c2)

which cant be the case sincef c2t g1t =0 is part of a competitive equilibrium, i.e.maximizes agent2s utility given equilibrium prices. Hence

1

Xt =0 ^ pt ~c2t1

Xt =0 ^ pt c2t (2.10)Step 3: Now sum equations(2:9) and (2:10) to obtain

1

Xt =0 ^ pt (~c

1t + ~c

2t ) >

1

Xt =0 ^ pt (c

1t + c

2t )

But since both allocations are feasible (the allocation(f cit g1t =0 )i =1 ;2 because it isan equilibrium allocation, the allocation(f ~cit g1t =0 )i =1 ;2 by assumption) we havethat

~c1t + ~c2t = e1t + e2t = c1t + c2t for alltand thus

1

Xt =0 ^ pt (e1t + e2t ) >1

Xt =0 ^ pt (e1t + e2t );our desired contradiction.

2.2.4 Negishis (1960) Method to Compute EquilibriaIn the example economy considered in this section it was straightforward tocompute the competitive equilibrium by hand. This is usually not the case fordynamic general equilibrium models. Now we describe a method to computeequilibria for economies in which the welfare theorem(s) hold. The main idea isto compute Pareto-optimal allocations by solving an appropriate social plannersproblem. This social planner problem is a simple optimization problem whichdoes not involve any prices (still innite-dimensional, though) and hence mucheasier to tackle in general than a full-blown equilibrium analysis which consistsof several optimization problems (one for each consumer) plus market clearingand involves allocationsand prices. If the rst welfare theorem holds then weknow that competitive equilibrium allocations are Pareto optimal; by solving

for all Pareto optimal allocations we have then solved for all potential equilib-rium allocations. Negishis method provides an algorithm to compute all Paretooptimal allocations and to isolate those who are in fact competitive equilibriumallocations.

We will repeatedly apply this trick in this course: solve a simple socialplanners problem and use the welfare theorems to argue that we have solved


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for the allocations of competitive equilibria. Then nd equilibrium prices thatsupport these allocations. The news is even better: usually we can read o the prices as Lagrange multipliers from the appropriate constraints of the socialplanners problem. In later parts of the course we will discuss economies in whichthe welfare theorems do not hold. We will see that these economies are muchharder to analyze exactly because there is no simple optimization problem thatcompletely characterizes the (set of) equilibria of these economies.

Consider the following social planners problem

maxf (c1t ;c 2t )g

1t =0

u (c1) + (1 )u(c2) (2.11)

= maxf (c1t ;c 2t )g

1t =0

1

Xt =0 t ln(c1t ) + (1 )ln( c2t )s.t.

cit 0 for alli; all tc1t + c2t = e1t + e2t 2 for allt

for a Pareto weight 2 [0; 1]: The social planner maximizes the weighted sum of utilities of the two agents, subject to the allocation being feasible. The weightindicates how important agent1s utility is to the planner, relative to agent2sutility. Note that the solution to this problem depends on the Pareto weights,i.e. the optimal consumption choices are functions of

f (c1t ; c2t )g1t =0 = f (c1t ( ); c

2t ( ))g1t =0

We have the following

Proposition 6 An allocation f (c1t ; c2t )g1t =0 is Pareto ecient if and only if it solves the social planners problem (2:11) for some 2 [0; 1]

Proof. Omitted (but a good exercise)This proposition states that we can characterize the set of all Pareto e-

cient allocations by varying between 0 and 1 and solving the social plannersproblem for all s. As we will demonstrate, by choosing a particular; the asso-ciated ecient allocation for that turns out to be the competitive equilibriumallocation.

Now let us solve the planners problem for arbitrary2 (0; 1):4 Attach La-grange multipliers t2 to the resource constraints (and ignore the non-negativityconstraints oncit since they never bind, due to the period utility function satisfy-ing the Inada conditions). The reason why we divide by2 will become apparentin a moment.

4Note that for = 0 and = 1 the solution to the problem is trivial. For = 0 we havec1t = 0 and c2t = 2 and for = 1 we have the reverse.


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The rst order necessary conditions are

tc1t

= t2

(1 ) t

c2t= t

2

Combining yields

c1tc2t

=1

(2.12)

c1t = 1c2t (2.13)

i.e. the ratio of consumption between the two agents equals the ratio of thePareto weights in every periodt: A higher Pareto weight for agent1 resultsin this agent receiving more consumption in every period, relative to agent2:Using the resource constraint in conjunction with(2:13) yields

c1t + c2t = 2

1c2t + c2t = 2

c2t = 2(1 ) = c2t ( )c1t = 2 = c1t ( )

i.e. the social planner divides the total resources in every period according to thePareto weights. Note that the division is the same in every period, independent

of the agents endowments in that particular period. The Lagrange multipliersare given by

t =2 t

c1t= t

(if we wouldnt have done the initial division by2 we would have to carry the12 around from now on; the results below wouldnt change at all).

Hence for this economy the set of Pareto ecient allocations is given by

P O = ff (c1t ; c2t )g1t =0 : c1t = 2 and c2t = 2(1 ) for some 2 [0; 1]g

How does this help us in nding the competitive equilibrium for this economy?Compare the rst order condition of the social planners problem for agent1

tc1t

= t2

or t

c1t= t

2


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with the rst order condition from the competitive equilibrium above (see equation(2:6)):

tc1t

= 1 pt

By picking 1 = 12 and pt = t these rst order conditions are identical. Sim-

ilarly, pick 2 = 12(1 ) and one sees that the same is true for agent2: So forappropriate choices of the individual Lagrange multipliersi and prices pt theoptimality conditions for the social planners problem and for the householdmaximization problems coincide. Resource feasibility is required in the com-petitive equilibrium as well as in the planners problem. Given that we founda unique equilibrium above but a lot of Pareto ecient allocations (for eachone), there must be an additional requirement that a competitive equilibriumimposes which the planners problem does not require.

In a competitive equilibrium households choices are constrained by thebud-get constraint; the planner is only concerned with resource balance. The laststep to single out competitive equilibrium allocations from the set of Paretoecient allocations is to ask which Pareto ecient allocations would be aord-able for all households if these holds were to face as market prices the Lagrangemultipliers from the planners problem (that the Lagrange multipliers are the ap-propriate prices is harder to establish, so lets proceed on faith for now). Denethe transfer functionst i ( ); i = 1 ; 2 by

t i ( ) = Xt t cit ( ) eitThe number t i ( ) is the amount of the numeraire good (we pick the period0

consumption good) that agenti would need as transfer in order to be able toaord the Pareto ecient allocation indexed by: One can show that thet i asfunctions of are homogeneous of degree one5 and sum to0 (see HW 1).

Computing t i ( ) for the current economy yields

t1( ) = Xt t c1t ( ) e1t= Xt t 2 e1t=

21

21 2

t2( ) =2(1 )

1 2

1 2

To nd the competitive equilibrium allocation we now need to nd the Paretoweight such that t1( ) = t2( ) = 0 ; i.e. the Pareto optimal allocation that

5In the sense that if one gives weightx to agent 1 and x (1 ) to agent 2, then thecorresponding required transfers arext 1 and xt 2 :


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both agents can aord with zero transfers. This yields

0 = 21

21 2

=1

1 + 2 (0; 0:5)

and the corresponding allocations are

c1t1

1 + =

21 +

c2t1

1 + =

2 1 +

Hence we have solved for the equilibrium allocations; equilibrium prices aregiven by the Lagrange multiplierst = t (note that without the normalizationby 12 at the beginning we would have found the same allocations and equilibriumprices pt =

t

2 which, given that equilibrium prices are homogeneous of degree0; is perfectly ne, too).

To summarize, to compute competitive equilibria using Negishis methodone does the following

1. Solve the social planners problem for Pareto ecient allocations indexedby Pareto weight

2. Compute transfers, indexed by, necessary to make the ecient allocationaordable. As prices use Lagrange multipliers on the resource constraintsin the planners problem.

3. Find the Pareto weight(s)^ that makes the transfer functions0:

4. The Pareto ecient allocations corresponding to are equilibrium allo-cations; the supporting equilibrium prices are (multiples of) the Lagrangemultipliers from the planning problem

Remember from above that to solve for the equilibrium directly in generalinvolves solving an innite number of equations in an innite number of un-knowns. The Negishi method reduces the computation of equilibrium to a nite

number of equations in a nite number of unknowns in step 3 above. For aneconomy with two agents, it is just one equation in one unknown, for an economywith N agents it is a system of N 1 equations inN 1 unknowns. This is whythe Negishi method (and methods relying on solving appropriate social plan-ners problems in general) often signicantly simplies solving for competitiveequilibria.


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2.2.5 Sequential Markets Equilibrium

The market structure of Arrow-Debreu equilibrium in which all agents meet onlyonce, at the beginning of time, to trade claims to future consumption may seemempirically implausible. In this section we show that the same allocations asin an Arrow-Debreu equilibrium would arise if we let agents trade consumptionand one-period bonds in each period. We will call a market structure in whichmarkets for consumption and assets open in each period Sequential Markets andthe corresponding equilibrium Sequential Markets (SM) equilibrium.6

Let r t +1 denote the interest rate on one period bonds from periodt to periodt +1 : A one period bond is a promise (contract) to pay1 unit of the consumptiongood in periodt + 1 in exchange for 11+ r t +1 units of the consumption good inperiod t: We can interpret q t 11+ r t +1 as the relative price of one unit of theconsumption good in periodt + 1 in terms of the periodt consumption good.Let a it +1 denote the amount of such bonds purchased by agenti in periodt andcarried over to periodt + 1 : If a it +1 < 0 we can interpret this as the agent takingout a one-period loan at interest rater t +1 : Householdis budget constraint inperiod t reads as

cit +a it +1

(1 + r t +1 )eit + a

it (2.14)

orcit + q t a

it +1 e

it + a

it

Agents start out their life with initial bond holdingsa i0 (remember that period0 bonds are claims to period0 consumption). Mostly we will focus on thesituation in whicha i0 = 0 for alli; but sometimes we want to start an agent o with initial wealth(a i0 > 0) or initial debt (a i0 < 0): We then have the following

denitionDenition 7 A Sequential Markets equilibrium is allocations f cit ; a it +1 i =1 ;2g

1t =1 ;

interest rates f r t +1 g1t =0 such that

1. For i = 1 ; 2; given interest rates f r t +1 g1t =0 f cit ; a it +1 g1t =0 solves

maxf c it ;a it +1 g

1t =0

1

Xt =0 t ln(cit ) (2.15)s.t.

cit +a it +1

(1 + r t +1 )eit + a

it (2.16)

cit 0 for all t (2.17)a it +1 A

i (2.18)6In the simple model we consider in this section the restriction of assets traded to one-period

riskless bonds is without loss of generality. In more complicated economies (with uncertainty,say) it would not be. We will come back to this issue in later chapters.


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2. For all t 0

2

Xi =1 cit =2

Xi =1 eit2Xi =1 a it +1 = 0

The constraint (2:18) on borrowing is necessary to guarantee existence of equilibrium. Suppose that agents would not face any constraint as to howmuch they can borrow, i.e. suppose the constraint(2:18) were absent. Supposethere would exist a SM-equilibriumf cit ; a it +1 i =1 ;2g

1t =1 ; f r t +1 g1t =0 : Without con-

straint on borrowing agenti could always do better by setting

ci0 = ci0 + "1 + r 1a i1 = a

i1 "

a i2 = ai2 (1 + r 2)"

a it +1 = ait +1

t

Yt =1 (1 + r t +1 )"i.e. by borrowing" > 0 more in period0; consuming it and then rolling over theadditional debt forever, by borrowing more and more. Such a scheme is oftencalled a Ponzi scheme. Hence without a limit on borrowing no SM equilibriumcan exist because agents would run Ponzi schemes.

In this section we are interested in specifying a borrowing limit that preventsPonzi schemes, yet is high enough so that households are never constrainedin the amount they can borrow (by this we mean that a household, knowingthat it can not run a Ponzi scheme, would always nd it optimal to choosea it +1 > Ai ): In later chapters we will analyze economies in which agents faceborrowing constraints that are binding in certain situations. Not only are SMequilibria for these economies quite dierent from the ones to be studied here,but also the equivalence between SM equilibria and AD equilibria will breakdown.

We are now ready to state the equivalence theorem relating AD equilibriaand SM equilibria. Assume thata i0 = 0 for alli = 1 ; 2:

Proposition 8 Let allocations f cit i =1 ;2g1t =0 and prices f ^ pt g1t =0 form an Arrow-

Debreu equilibrium. Then there exist Ai

i =1 ;2 and a corresponding sequen-tial markets equilibrium with allocations f ~cit ; ~a it +1 i =1 ;2g1t =0 and interest rates

f ~r t +1 g1t =0 such that ~cit = c

it for all i; all t

Reversely, let allocations f cit ; a it +1 i =1 ;2g1t =0 and interest rates f r t +1 g1t =0 form


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a sequential markets equilibrium. Suppose that it satises

ait +1 > A

i

for all i; all tr t +1 > 0 for all t

Then there exists a corresponding Arrow-Debreu equilibrium f ~cit i =1 ;2g1t =0 ; f ~ pt g1t =0

such that cit = ~c

it for all i; all t

Proof. Step 1: The key to the proof is to show the equivalence of the budgetsets for the Arrow-Debreu and the sequential markets structure. Normalize^ p0 = 1 and relate equilibrium prices and interest rates by

1 + r t +1 =^ pt

^ pt +1(2.19)

Now look at the sequence of sequential markets budget constraints and assumethat they hold with equality (which they do in equilibrium, due to the nonsa-tiation assumption)

ci0 +a i1

1 + r 1= ei0 (2.20)

ci1 +a i2

1 + r 2= ei1 + a

i1 (2.21)

...

cit +a it +1

1 + r t +1= eit + a

it (2.22)

Substituting fora i1 from (2:21) in (2:20) one gets

ci0 +ci1

1 + r 1+

a i2(1 + r 1) (1 + r 2)

= ei0 +ei1

(1 + r 1)

and, repeating this exercise, one gets7

T

Xt =0 cit

Qtj =1 (1 + r j ) +a iT +1

QT +1j =1 (1 + r j ) =T

Xt =0 eit

Qtj =1 (1 + r j )Now note that (using the normalization p0 = 1 )t

Yj =1 (1 + rj ) =

^ p0^ p1

^ p1^ p2

^ pt 1^ pt =

1^ pt (2.23)

7We dene0

Yj =1 (1 + r j ) = 1


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Taking limits with respect tot on both sides gives, using(2:23)

1

Xt =0 ^ pt cit + limT !1 aiT +1

QT +1j =1 (1 + r j )=

1

Xt =0 ^ pt eitGiven our assumptions on the equilibrium interest rates we have

limT !1

a iT +1

QT +1j =1 (1 + r j )lim

T !1

Ai

QT +1j =1 (1 + r j )= 0

and hence1

Xt =0 ^ pt cit1

Xt =0 ^ pt eitStep 2: Now suppose we have an AD-equilibriumf c

it i =1 ;2g

1t =0 , f ^ pt g

1t =0 :We want to show that there exist a SM equilibrium with same consumption

allocation, i.e.~cit = c

it for alli; all t

Obviouslyf ~cit i =1 ;2g1t =0 satises market clearing. Dening as asset holdings

~a it +1 =1

X =1 ^ pt + cit + eit + ^ pt +1

we see that the allocation satises the SM budget constraints (remember1 +~r t +1 = ^ p t^ p t +1 ) Also note that

~a it +1 >1

X =1 ^ pt + eit +

^ pt +1

1

Xt =0 ^ pt eit > 1so that we can take

Ai =1

Xt =0 ^ pt eitThis borrowing constraint, equalling the value of the endowment of agenti atAD-equilibrium prices is also called the natural debt limit. This borrowing limitis so high that agenti; knowing that she cant run a Ponzi scheme, will neverreach it.

It remains to argue that f ~cit i =1 ;2g1t =0 maximizes utility, subject to the

sequential markets budget constraints and the borrowing constraints. Takeany other allocation satisfying these constraints. In step 1. we showed thatthis allocation satises the AD budget constraint. If it would be better thanf ~cit = cit g1t =0 it would have been chosen as part of an AD-equilibrium, which itwasnt. Hencef ~cit g1t =0 is optimal within the set of allocations satisfying the SMbudget constraints at interest rates1 + ~r t +1 = ^ p t^ p t +1 :


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Step 3: Now supposef cit ; a it +1 i 2 I g1t =1 and f r t +1 g1t =0 form a sequential

markets equilibrium satisfying

a it +1 > Ai for alli; all t

r t +1 > 0 for allt

We want to show that there exists a corresponding Arrow-Debreu equilibriumf ~cit i 2 I g

1t =0 ; f ~ pt g1t =0 with

cit = ~cit for alli; all t

Again obviouslyf ~cit i 2 I g1t =0 satises market clearing and, as shown in step

1, the AD budget constraint. It remains to be shown that it maximizes utilitywithin the set of allocations satisfying the AD budget constraint. For~ p0 = 1and ~ pt +1 = ~ p t1+ r t +1 the set of allocations satisfying the AD budget constraint

coincides with the set of allocations satisfying the SM-budget constraint (forappropriate choices of asset holdings). Since in the SM equilibrium we have theadditional borrowing constraints, the set over which we maximize in the AD caseis larger, since the borrowing constraints are absent in the AD formulation. Butby assumption these additional constraints are never binding(a it +1 > Ai ):Then from a basic theorem of constrained optimization we know that if theadditional constraints are never binding, then the maximizer of the constrainedproblem is also the maximizer of the unconstrained problem, and hencef ~cit g1t =0is optimal for householdi within the set of allocations satisfying her AD budgetconstraint.

This proposition shows that the sequential markets and the Arrow-Debreumarket structures lead to identical equilibria, provided that we choose the noPonzi conditions appropriately (equal to the natural debt limits, for example)and that the equilibrium interest rates are suciently high.8 Usually the analy-sis of our economies is easier to carry out using AD language, but the SMformulation has more empirical appeal. The preceding theorem shows that wecan have the best of both worlds.

For our example economy we nd that the equilibrium interest rates in theSM formulation are given by

1 + r t +1 =pt

pt +1=

1

or

r t +1 = r =1

1 =

i.e. the interest rate is constant and equal to the subjective time discount rate= 1 1:

8This assumption can be suciently weakened if one introduces borrowing constraints of slightly dierent form in the SM equilibrium to prevent Ponzi schemes. We may come backto this later.


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2.3. APPENDIX: SOME FACTS ABOUT UTILITY FUNCTIONS 23

2.3 Appendix: Some Facts about Utility Func-

tionsThe utility function

u(ci ) =1

Xt =0 t ln(cit ) (2.24)described in the main text satises the following assumptions that we will oftenrequire in our models:

1. Time separability: total utility from a consumption allocationci equalsthe discounted sum of period (or instantaneous) utilityU (cit ) = ln( cit ): Inparticular, the period utility at timet only depends on consumption inperiod t and not on consumption in other periods. This formulation rules

out, among other things, habit persistence.2. Time discounting: the fact that < 1 indicates that agents are impatient.

The same amount of consumption yields less utility if it comes at a latertime in an agents life. The parameter is often referred to as (subjective)time discount factor. The subjective time discount rate is dened by = 11+ and is often, as we will see, intimately related to the equilibriuminterest rate in the economy (because the interest rate is nothing else butthe market time discount rate).

3. Homotheticity: Dene the marginal rate of substitution between consump-tion at any two datest and t + s as

MRS (ct + s ; ct ) =@u(c)@ct + s@u(c)

@ct

The functionu is said to be homothetic if MRS (ct + s ; ct ) = MRS ( c t + s ; c t )for all > 0 and c: It is easy to verify that foru dened above we have

MRS (ct + s ; ct ) =

t + s

c t + st

c t

=

t + s

c t + st

c t

= MRS ( c t + s ; c t )

and henceu is homothetic. This, in particular, implies that if an agentslifetime income doubles, optimal consumption choices will double ineach period (income expansion paths are linear).9 It also means that consump-tion allocations are independent of the units of measurement employed.

4. The instantaneous utility function or felicity functionU (c) = ln( c) is con-tinuous, twice continuously dierentiable, strictly increasing (i.e.U 0(c) >

9In the absense of borrowing constraints and other frictions which we will discuss later.


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0) and strictly concave (i.e.U 00(c) < 0) and satises the Inada conditions

limc& 0 U 0(c) = + 1

limc% + 1

U 0(c) = 0

These assumptions imply that more consumption is always better, but anadditional unit of consumption yields less and less additional utility. TheInada conditions indicate that the rst unit of consumption yields a lot of additional utility but that as consumption goes to innity, an additionalunit is (almost) worthless. The Inada conditions will guarantee that anagent always choosesct 2 (0; 1 ) for allt

5. The felicity functionU is a member of the class of Constant Relative Risk

Aversion (CRRA) utility functions. These functions have the followingimportant properties. First, dene as(c) = U 00 (c )c

U 0 (c ) the (Arrow-Pratt)coecient of relative risk aversion. Hence(c) indicates a householdsattitude towards risk, with higher (c) representing higher risk aversion.For CRRA utility functions (c) is constant for all levels of consumption,and for U (c) = ln( c) it is not only constant, but equal to (c) = = 1 :Second, the intertemporal elasticity of substitutionis t (ct +1 ; ct ) measuresby how many percent the relative demand for consumption in periodt +1 ;relative to demand for consumption in periodt; c t +1c t declines as the relativeprice of consumption int + 1 to consumption int; q t = 11+ r t +1 changes byone percent. Formally

is t (ct +1 ; ct ) =

d( c t +1c t )c t +1

c t

d 11+ r t +11

1+ r t +1

=

d( c t +1c t )d 11+ r t +1

c t +1c t1

1+ r t +1

But combining(2:6) and (2:7) we see that

U 0(ct +1 )U 0(ct )

=pt +1 pt

=1

1 + r t +1

which, forU (c) = ln( c) becomes

ct +1ct

=1

1

1 + r t +1

1

and thusd c t +1c td 11+ r t +1

=1

11 + r t +1

2


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2.3. APPENDIX: SOME FACTS ABOUT UTILITY FUNCTIONS 25

and therefore

is t (ct +1 ; ct ) =

d( c t +1c t )d 11+ r t +1

c t +1c t1

1+ r t +1

=1 1

1+ r t +1

2

1 11+ r t +1

2 = 1

Therefore logarithmic period utility is sometimes also called isoelastic util-ity.10 Hence for logarithmic period utility the intertemporal elasticity sub-stitution is equal to (the inverse of) the coecient of relative risk aversion.

10 In general CRRA utility functions are of the form

U (c) =c1 1

1

and one can easily compute that the coecient of relative risk aversion for this utility functionis and the intertemporal elasticity of substitution equals 1 :

In a homework you will show that

ln( c) = lim! 1

c1

11

i.e. that logarithmic utility is a special case of this general class of utility functions.


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Chapter 3

The Neoclassical GrowthModel in Discrete Time

3.1 Setup of the ModelThe neoclassical growth model is arguably the single most important workhorsein modern macroeconomics. It is widely used in growth theory, business cycletheory and quantitative applications in public nance.

Time is discrete and indexed byt = 0 ; 1; 2; : : : In each period there are threegoods that are traded, labor servicesn t ; capital serviceskt and a nal outputgoodyt that can be either consumed,ct or invested,i t : As usual for a completedescription of the economy we have to specify technology, preferences, endow-ments and the information structure. Later, when looking at an equilibrium of this economy we have to specify the equilibrium concept that we intend to use.

1. Technology: The nal output good is produced using as inputs labor andcapital services, according to the aggregate production functionF

yt = F (kt ; n t )

Note that I do not allow free disposal. If I want to allow free disposal, Iwill specify this explicitly by dening an separate free disposal technology.Output can be consumed or invested

yt = i t + ct

Investment augments the capital stock which depreciates at a constantrate over time

kt +1 = (1 )kt + i t

We can rewrite this equation as

i t = kt +1 kt + k t

27


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28CHAPTER 3. THE NEOCLASSICAL GROWTH MODEL IN DISCRETE TIME

i.e. gross investmenti t equals net investmentkt +1 kt plus depreciationk t : We will require thatkt +1 0; but not that i t 0: This assumes that,since the existing capital stock can be disinvested to be eaten, capital isputty-putty. Note that I have been a bit sloppy: strictly speaking thecapital stock and capital services generated from this stock are dierentthings. We will assume (once we dene the ownership structure of thiseconomy in order to dene an equilibrium) that households own the capitalstock and make the investment decision. They will rent out capital to therms. We denote both the capitalstock and the ow of capital services bykt : Implicitly this assumes that there is some technology that transformsone unit of the capital stock at periodt into one unit of capital servicesat period t: We will ignore this subtlety for the moment.

2. Preferences: There is a large number of identical, innitely lived house-holds. Since all households are identical and we will restrict ourselvesto type-identical allocations1 we can, without loss of generality assumethat there is a single representative household. Preferences of each house-hold are assumed to be representable by a time-separable utility function(Debreus theorem discusses under which conditions preferences admit acontinuous utility function representation)

u (f ct g1t =0 ) =1

Xt =0 t U (ct )3. Endowments: Each household has two types of endowments. At period0

each household is born with endowmentsk0 of initial capital. Furthermoreeach household is endowed with one unit of productive time in each period,

to be devoted either to leisure or to work.4. Information: There is no uncertainty in this economy and we assume that

households and rms have perfect foresight.

5. Equilibrium: We postpone the discussion of the equilibrium concept to alater point as we will rst be concerned with an optimal growth problem,where we solve for Pareto optimal allocations.

3.2 Optimal Growth: Pareto Optimal Alloca-tions

Consider the problem of a social planner that wants to maximize the utility of the representative agent, subject to the technological constraints of the economy.Note that, as long as we restrict our attention to type-identical allocations, an

1Identical households receive the same allocation by assumption. In the next quarter I)or somebody else) may come back to the issue under which conditions this is an innocuousassumption,


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3.2. OPTIMAL GROWTH: PARETO OPTIMAL ALLOCATIONS 29

allocation that maximizes the utility of the representative agent, subject to thetechnology constraint is a Pareto ecient allocation and every Pareto ecientallocation solves the social planner problem below. Just as a reference we havethe following denitions

Denition 9 An allocation f ct ; kt ; n t g1t =0 is feasible if for all t 0

F (kt ; n t ) = ct + kt +1 (1 )ktct 0; kt 0; 0 n t 1k0 k0

Denition 10 An allocation f ct ; kt ; n t g1t =0 is Pareto ecient if it is feasible and there is no other feasible allocation f ct ; kt ; n t g1t =0 such that

1

Xt =0 t U (ct ) >

1

Xt =0 t U (ct )

3.2.1 Social Planner Problem in Sequential FormulationThe problem of the planner is

w(k0) = maxf c t ;k t ;n t g 1t =0

1

Xt =0 t U (ct )s:t: F (kt ; n t ) = ct + kt +1 (1 )kt

ct 0; kt 0; 0 n t 1k0 k0

The functionw(k0) has the following interpretation: it gives thetotal lifetime utility of the representative household if the social planner choosesf ct ; kt ; n t g1t =0optimally and the initial capital stock in the economy isk0 : Under the assump-tions made below the functionw is strictly increasing, since a higher initialcapital stock yields higher production in the initial period and hence enablesmore consumption or capital accumulation (or both) in the initial period.

We now make the following assumptions on preferences and technology.Assumption 1: U is continuously dierentiable, strictly increasing, strictly

concave and bounded. It satises the Inada conditionslimc& 0 U 0(c) = 1 andlimc!1 U 0(c) = 0 : The discount factor satises 2 (0; 1)

Assumption 2: F is continuously dierentiable and homogenous of de-gree 1; strictly increasing in both arguments and strictly concave. Furthermore

F (0; n) = F (k; 0) = 0 for all k;n > 0: Also F satises the Inada conditionslimk & 0 F k (k; 1) = 1 and limk !1 F k (k; 1) = 0 : Also 2 [0; 1]From these assumptions two immediate consequences for optimal allocations

are that n t = 1 for all t since households do not value leisure in their utilityfunction. Also, since the production function is strictly increasing in capital,k0 = k0 : To simplify notation we denef (k) = F (k; 1)+(1 )k; for allk: The


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function f gives the total amount of the nal good available for consumptionor investment (again remember that the capital stock can be eaten). Fromassumption 2 the following properties of f follow more or less directly:f iscontinuously dierentiable, strictly increasing and strictly concave,f (0) = 0 ;f 0(k) > 0 for allk; limk & 0 f 0(k) = 1 and limk !1 f 0(k) = 1 :

Using the implications of the assumptions, and substituting forct = f (kt )kt +1 we can rewrite the social planners problem as

w(k0) = maxf k t +1 g 1t =0

1

Xt =0 t U (f (kt ) kt +1 ) (3.1)0 kt +1 f (kt )

k0 = k0 > 0 given

The only choice that the planner faces is the choice between letting the consumereat today versus investing in the capital stock so that the consumer can eat moretomorrow. Let the optimal sequence of capital stocks be denoted byf kt +1 g1t =0 :The two questions that we face when looking at this problem are

1. Why do we want to solve such a hypothetical problem of an even more hy-pothetical social planner. The answer to this questions is that, by solvingthis problem, we will have solved for competitive equilibrium allocationsof our model (of course we rst have to dene what a competitive equilib-rium is). The theoretical justication underlying this result are the twowelfare theorems, which hold in this model and in many others, too. We

will give a loose justication of the theorems a bit later, and postponea rigorous treatment of the two welfare theorems in innite dimensionalspaces until the next quarter.

2. How do we solve this problem?2 The answer is: dynamic programming.The problem above is an innite-dimensional optimization problem, i.e.we have to nd an optimal innite sequence(k1 ; k2 ; : : :) solving the prob-lem above. The idea of dynamic programing is to nd a simpler maximiza-tion problem by exploiting the stationarity of the economic environmentand then to demonstrate that the solution to the simpler maximizationproblem solves the original maximization problem.

To make the second point more concrete, note that we can rewrite the prob-

2Just a caveat: innite-dimensional maximization problems may not have a solution evenif the u and f are well-behaved. So the functionw may not always be well-dened. In ourexamples, with the assumptions that we made, everything is ne, however.


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lem above as

w(k0) = maxf k t +1 g

1t =0 s.t.

0 k t +1 f (k t ) ; k 0 given

1

Xt =0 t U (f (kt ) kt +1 )

= maxf k t +1 g

1t =0 s.t.

0 k t +1 f (k t ) ; k 0 given(U (f (k0) k1) + 1Xt =1 t 1U (f (kt ) kt +1 ))

= maxk 1 s.t.

0 k 1 f (k 0 ) ; k 0 given

8>:U (f (k0) k1) + 264 maxf k t +1 g 1t =10 k t +1 f (k t ) ; k 1 given

1

Xt =1 t 1U (f (kt ) kt +1 )3759>=>;

= maxk 1 s.t.

0 k 1 f (k 0 ) ; k 0 given

8>:

U (f (k0) k1) + 264

maxf k t +2 g

1t =0

0 k t +2 f (k t +1 ) ; k 1 given

1

Xt =0

t U (f (kt +1 ) kt +2 )3759>=>;Looking at the maximization problem inside the[ ]-brackets and comparingto the original problem(3:1) we see that the [ ]-problem is that of a social

planner that, given initial capital stockk1 ; maximizes lifetime utility of therepresentative agent from period1 onwards. But agents dont age in our model,the technology or the utility functions doesnt change over time; this suggeststhat the optimal value of the problem in[ ]-brackets is equal tow(k1) and hencethe problem can be rewritten as

w(k0) = max0 k 1 f (k 0 )

k 0 given

f U (f (k0) k1) + w (k1)g

Again two questions arise:

2.1 Under which conditions is this suggestive discussion formally correct? Wewill come back to this in a little while.

2.2 Is this progress? Of course, the maximization problem is much easiersince, instead of maximizing over innite sequences we maximize over just one number,k1 : But we cant really solve the maximization problem,because the functionw(:) appears on the right side, and we dont knowthis function. The next section shows ways to overcome this problem.

3.2.2 Recursive Formulation of Social Planner ProblemThe above formulation of the social planners problem with a function on the

left and right side of the maximization problem is called recursive formulation.Now we want to study this recursive formulation of the planners problem. Sincethe function w(:) is associated with the sequential formulation, let us changenotation and denote byv(:) the corresponding function for the recursive formu-lation of the problem. Remember the interpretation of v(k): it is the discountedlifetime utility of the representative agent from the current period onwards if the


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social planner is given capital stockk at the beginning of the current period andallocates consumption across time optimally for the household. This functionv(the so-called value function) solves the following recursion

v(k) = max0 k 0 f (k )

f U (f (k) k0) + v (k0)g (3.2)

Note again that v and w are two very dierent functions;v is the valuefunction for the recursive formulation of the planners problem andw is thecorresponding function for the sequential problem. Of course below we want toestablish that v = w, but this is something that we have to prove rather thansomething that we can assume to hold! The capital stockk that the plannerbrings into the current period, result of past decisions, completely determineswhat allocations are feasible from today onwards. Therefore it is called thestate variable: it completely summarizes the state of the economy today (i.e.

all future options that the planner has). The variablek0

is decided (or controlled)today by the social planner; it is therefore called the control variable, becauseit can be controlled today by the planner.3

Equation (3:2) is a functional equation (the so-called Bellman equation): itssolution is a function, rather than a number or a vector. Fortunately the math-ematical theory of functional equations is well-developed, so we can draw onsome fairly general results. The functional equation posits that the discountedlifetime utility of the representative agent is given by the utility that this agentreceives today,U (f (k) k0), plus the discounted lifetime utility from tomorrowonwards, v (k0): So this formulation makes clear the planners trade-o: con-sumption (and hence utility) today, versus a higher capital stock to work with(and hence higher discounted future utility) from tomorrow onwards. Hence, fora givenk this maximization problem is much easier to solve than the problem of picking an innite sequence of capital stocksf kt +1 g1t =0 from before. The onlyproblem is that we have to do this maximization for every possible capital stockk; and this posits theoretical as well as computational problems. However, it willturn out that the functional equation is much easier to solve than the sequentialproblem (3:1) (apart from some very special cases). By solving the functionalequation we mean nding a value functionv solving(3:2) and an optimal policyfunction k0 = g(k) that describes the optimalk0 for the maximization part in(3:2); as a function of k; i.e. for each possible value thatk can take. Again weface several questions associated with equation(3:2):

1. Under what condition does a solution to the functional equation(3:2) existand, if it exist, is unique?

2. Is there a reliable algorithm that computes the solution (by reliable wemean that it always converges to the correct solution, independent of theinitial guess forv

3These terms come from control theory, a eld in applied mathematics. Control theory isused in many technical applications such as astronautics.


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3. Under what conditions can we solve(3:2) and be sure to have solved(3:1);i.e. under what conditions do we havev = w and equivalence between theoptimal sequential allocationf kt +1 g1t =0 and allocations generated by theoptimal recursive policyg(k)

4. Can we say something about the qualitative features of v and g?

The answers to these questions will be given in the next two sections: theanswers to 1. and 2. will come from the Contraction Mapping Theorem, tobe discussed in Section 4.3. The answer to the third question makes up whatRichard Bellman called the Principle of Optimality and is discussed in Section5.1. Finally, under more restrictive assumptions we can characterize the solutionto the functional equation(v; g) more precisely. This will be done in Section 5.2.In the remaining parts of this section we will look at specic examples where wecan solve the functional equation by hand. Then we will talk about competitive

equilibria and the way we can construct prices so that Pareto optimal alloca-tions, together with these prices, form a competitive equilibrium. This will beour versions of the rst and second welfare theorem for the neoclassical growthmodel.

3.2.3 An ExampleConsider the following example. Let the period utility function be given byU (c) = ln( c) and the aggregate production function be given byF (k; n ) =k n1 and assume full depreciation, i.e. = 1 : Then f (k) = k and thefunctional equation becomes

v(k) = max0 k 0 k

f ln (k k0) + v (k0)g

Remember that the solution to this functional equation is an entire functionv(:): Now we will apply several methods to solve this functional equation.

Guess and Verify

We will guess a particular functional form of a solution and then verify that thesolution has in fact this form (note that this does not rule out that the functionalequation has other solutions). This method works well for the example at hand,but not so well for most other examples that we are concerned with. Let usguess

v(k) = A + B ln(k)

whereA and B are coecients that are to be determined. The method consistsof three steps:

1. Solve the maximization problem on the right hand side, given the guessfor v; i.e. solve

max0 k 0 k

f ln (ka k0) + (A + B ln(k0))g


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Obviously the constraints onk0 never bind and the objective function isstrictly concave and the constraint set is compact, for any givenk: Therst order condition is sucient for the unique solution. The FOC yields

1k k0

=Bk0

k0 =Bk

1 + B

2. Evaluate the right hand side at the optimumk0 = Bk1+ B : This yields

RHS = ln ( ka k0) + (A + B ln(k0))

= lnk

1 + B+ A + B ln

Bk1 + B

= ln(1 + B ) + ln(k) + A + B lnB

1 + B+ B ln (k)

3. In order for our guess to solve the functional equation, the left hand side of the functional equation, which we have guessed to equal LHS= A+ B ln(k)must equal the right hand side, which we just found. If we can ndcoecients A; B for which this is true, we have found a solution to thefunctional equation. Equating LHS and RHS yields

A + B ln(k) = ln(1 + B ) + ln(k) + A + B lnB

1 + B+ B ln (k

(B (1 + B ))ln( k) = A ln(1 + B ) + A + B lnB

1 + B(3.3

But this equation has to hold forevery capital stock k. The right hand

side of (3:3) does not depend onk but the left hand side does. Hencethe right hand side is a constant, and the only way to make the left handside a constant is to makeB (1 + B ) = 0 : Solving this forB yieldsB = 1 : Since the left hand side of (3:3) is 0; the right hand side betteris, too, forB = 1 : Therefore the constantA has to satisfy

0 = A ln(1 + B ) + A + B lnB

1 + B

= A ln1

1 + A +

1

ln( )

Solving this mess forA yields

A =1

1

1 ln( ) + ln(1 )

We can also determine the optimal policy functionk0 = g(k) as

g(k) =Bk

1 + B= k


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Hence our guess was correct: the functionv (k) = A + B ln(k); with A; B asdetermined above, solves the functional equation, with associated policy func-tion g(k) = k : Note that for this specic example the optimal policy of thesocial planner is to save a constant fraction of total outputk as capital stockfor tomorrow and and let the household consume a constant fraction(1 )of total output today. The fact that these fractions do not depend on the levelof k is very unique to this example and not a property of the model in general.Also note that there may be other solutions to the functional equation; we have just constructed one (actually, for the specic example there are no others, butthis needs some proving). Finally, it is straightforward to construct a sequencef kt +1 g1t =0 from our policy functiong that will turn out to solve the sequentialproblem (3:1) (of course for the specic functional forms used in the example):start from k0 = k0 ; k1 = g(k0) = k 0 ; k2 = g(k1) = k 1 = ( )1+ k

2

0 andin generalkt = ( )P

t 1j =0

j

kt

0 : Obviously, since0 < < 1 we have that

limt !1

kt = ( )1

1

for all initial conditionsk0 > 0 (which, not surprisingly, is the unique solutionto g(k) = k).

Value Function Iteration: Analytical Approach

In the last section we started with a clever guess, parameterized it and used themethod of undetermined coecients (guess and verify) to solve for the solutionv of the functional equation. For just about any other than the log-utility,Cobb-Douglas production function case this method would not work; even yourmost ingenious guesses would fail when trying to be veried.

Consider the following iterative procedure for our previous example

1. Guess an arbitrary functionv0(k): For concreteness lets takev0(k) = 0for all

2. Proceed recursively by solving

v1(k) = max0 k 0 k

f ln (k k0) + v 0(k0)g

Note that we can solve the maximization problem on the right hand sidesince we knowv0 (since we have guessed it). In particular, sincev0(k0) = 0for allk0 we have as optimal solution to this problem

k0 = g1(k) = 0 for allk

Plugging this back in we get

v1(k) = ln ( k 0) + v 0(0) = ln k = ln k


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3. Now we can solve

v2(k) = max0 k 0 k f ln (k k0) + v 1(k

0)g

since we knowv1 and so forth.

4. By iterating on the recursion

vn +1 (k) = max0 k 0 k

f ln (k k0) + v n (k0)g

we obtain a sequence of value functionsf vn g1n =0 and policy functionsf gn g1n =1 : Hopefully these sequences will converge to the solutionv andassociated policyg of the functional equation. In fact, below we willstate and prove a very important theorem asserting exactly that (undercertain conditions) this iterative procedure converges for any initial guess

and converges to the correct solution, namelyv :In the rst homework I let you carry out the rst few iterations in this

procedure. Note however, that, in order to nd the solutionv exactly youwould have to carry out step2: above a lot of times (in fact, innitely manytimes), which is, of course, infeasible. Therefore one has to implement thisprocedure numerically on a computer.

Value Function Iteration: Numerical Approach

Even a computer can carry out only a nite number of calculation and canonly store nite-dimensional objects. Hence the best we can hope for is anumerical approximation of the true value function. The functional equa-tion above is dened for allk 0 (in fact there is an upper bound, butlets ignore this for now). Because computer storage space is nite, we willapproximate the value function for a nite number of points only.4 For thesake of the argument suppose thatk and k0 can only take values inK =f 0:04; 0:08; 0:12; 0:16; 0:2g: Note that the value functionsvn then consists of 5 numbers, (vn (0:04); vn (0:08); vn (0:12); vn (0:16); vn (0:2))

Now let us implement the above algorithm numerically. First we have to pickconcrete values for the parameters and : Let us pick = 0 :3 and = 0 :6:

1. Make an initial guessv0(k) = 0 for allk 2 K

2. Solvev1(k) = max

0 k 0 k 0 : 3k 0 2K

ln k0:3 k0 + 0 :6 0

4In this course I will only discuss so-called nite state-space methods, i.e. methods inwhich the state variable (and the control variable) can take only a nite number of values.Ken Judd, one of the world leaders in numerical methods in economics teaches an exellentsecond year class in computational methods, in which much more sophisticated methods forsolving similar problems are discussed. I strongly encourage you to take this course at somepoint of your career here in Stanford.


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This obviously yields as optimal policyk0(k) = g1(k) = 0 :04 for allk 2 K(note that since k0 2 K is required,k0 = 0 is not allowed). Plugging thisback in yields

v1(0:04) = ln(0 :040:3 0:04) = 1:077v1(0:08) = ln(0 :080:3 0:04) = 0:847v1(0:12) = ln(0 :120:3 0:04) = 0:715v1(0:16) = ln(0 :160:3 0:04) = 0:622v1(0:2) = ln(0 :20:3 0:04) = 0:55

3. Lets do one more step by hand

v2(k) = 8>:

max0 k 0 k 0 : 3

k0

2K

ln k0:3 k0 + 0 :6v1(k0)9>=>;Start with k = 0 :04 :v2(0:04) = max

0 k 0 0:04 0 : 3k 0 2K

ln 0:040:3 k0 + 0 :6v1(k0)

Since 0:040:3 = 0 :381 all k0 2 K are possible. If the planner choosesk0 = 0 :04; then

v2(0:04) = ln 0:040:3 0:04 + 0 :6 ( 1:077) = 1:723

If he choosesk0 = 0 :08; then

v2(0:04) = ln 0:040:3 0:08 + 0 :6 ( 0:847) = 1:710

If he choosesk0

= 0 :12; thenv2(0:04) = ln 0:040:3 0:12 + 0 :6 ( 0:715) = 1:773

If k0 = 0 :16; then

v2(0:04) = ln 0:040:3 0:16 + 0 :6 ( 0:622) = 1:884

Finally, if k0 = 0 :2; then

v2(0:04) = ln 0:040:3 0:2 + 0 :6 ( 0:55) = 2:041

Hence fork = 0 :04 the optimal choice isk0(0:04) = g2(0:04) = 0 :08 andv2(0:04) = 1:710: This we have to do for allk 2 K : One can already seethat this is quite tedious by hand, but also that a computer can do thisquite rapidly. Table 1 below shows the value of

k0:3 k0 + 0 :6v1(k0)

for dierent values of k and k0: A in the column fork0 that this k0 isthe optimal choice for capital tomorrow, for the particular capital stockktoday


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Table 1

k0k 0:04 0:08 0:12 0:16 0:2

0:04 1:7227 1:7097 1:7731 1:8838 2:04070:08 1:4929 1:4530 1:4822 1:5482 1:64390:12 1:3606 1:3081 1:3219 1:3689 1:44050:16 1:2676 1:2072 1:2117 1:2474 1:30520:2 1:1959 1:1298 1:1279 1:1560 1:2045

Hence the value functionv2 and policy functiong2 are given by

Table 2

k v2(k) g2(k)0:04 1:7097 0:080:08 1:4530 0:080:12 1:3081 0:080:16 1:2072 0:080:2 1:1279 0:12

In Figure 3.2.3 we plot the true value functionv (remember that for this ex-ample we know to ndv analytically) and selected iterations from the numericalvalue function iteration procedure. In Figure 3.2.3 we have the corresponding

policy functions.We see from Figure 3.2.3 that the numerical approximations of the valuefunction converge rapidly to the true value function. After 20 iterations theapproximation and the truth are nearly indistinguishable with the naked eye.Looking at the policy functions we see from Figure 2 that the approximatingpolicy function do not converge to the truth (more iterations dont help). This isdue to the fact that the analytically correct value function was found by allowingk0 = g(k) to take any value in the real line, whereas for the approximationswe restricted k0 = gn (k) to lie in K: The function g10 approximates the truepolicy function as good as possible, subject to this restriction. Therefore theapproximating value function will not converge exactly to the truth, either.The fact that the value function approximations come much closer is due to thefact that the utility and production function induce curvature into the valuefunction, something that we may make more precise later. Also note that we weplot the true value and policy function only onK, with MATLAB interpolatingbetween the points inK, so that the true value and policy functions in the plotslook piecewise linear.


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0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-3

-2.5

-2

-1.5

-1

-0.5

0

Capital Stock k Today

V a

l u e

F u n c

t i o n

Value Function: True and Approximated

V0

V1

V2

V10 True Value Function

3.2.4 The Euler Equation Approach and TransversalityConditions

We now relate our example to the traditional approach of solving optimizationproblems. Note that this approach also, as the guess and verify method, willonly work in very simple examples, but not in general, whereas the numericalapproach works for a wide range of parameterizations of the neoclassical growthmodel. First let us look at a nite horizon social planners problem and then atthe related innite-dimensional problem

The Finite Horizon Case

Let us consider the social planner problem for a situation in which the repre-sentative consumer lives forT < 1 periods, after which she dies for sure and


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0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

Capital Stock k Today

P o

l i c y

F u n c

t i o n

Policy Function: True and Approximated

g1

g2

g10

Dirk Krueger - Macroeconomics Theory

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