Aspects of Household Heterogeneity in New Keynesian Economics · 2014-04-23 · Aspects of Household Heterogeneity in New Keynesian Economics By Martin Seneca A dissertation submitted

Aspects of HouseholdHeterogeneity in New Keynesian

Economics

By Martin Seneca

A dissertation submitted to

The Faculty of Social Sciences

in partial ful�lment of the requirements for the degree of

Doctor of Philosophy

in

Economics

University of Aarhus

Denmark

This book is chie�y addressed to my fellow economists�J.M. Keynes

Table of contents

PREFACE vii

INTRODUCTIONAND SUMMARY ix

DANSK RESUME xvii

P A R T I� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

CHAPTER 1 3Labour market asymmetries in a monetary union

CHAPTER 2 45Monetary policy and welfare in a monetary union with labourmarket heterogeneity

P A R T II� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

CHAPTER 3 89Rule-of-thumb consumers, productivity and hours

CHAPTER 4 125Fiscal shocks, rule-of-thumb consumers and real rigidities

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Preface

This dissertation is the result of my PhD studies at the School of Eco-nomics and Management at the University of Aarhus. The �rst equationswere derived sometime during the summer of 2004, and it has taken formduring the time I have spent at the University, at Danmarks Nationalbankin Copenhagen, and visiting the Universitat Pompeu Fabra in Barcelona. Iam very grateful to Danmarks Nationalbank for �nancing these studies.I would like to thank a number of people who have helped me along the

way. First and foremost, I would like to thank my main supervisor Torben M.Andersen for his guidance and encouragement. Also, I would like to thank mysecondary supervisor Niels Haldrup for standing by in case of econometricemergencies. And I dare not forget the PhD students at the School: Youare great! A million thanks to Malene Kallestrup for always being there,and to Søren Tang Sørensen for always pointing me to Hayek (and for livingin SW7). I wish I could also thank Henrik Filskov Simonsen; I think oflate nights spent working on proofs for PhD Maths assignments and I amreminded of what really matters in life. At the bank, I would particularly liketo thank Peter Storgaard and Kim Abildgren for many helpful discussions.Many thanks also to Jordi Galí for kindly sponsoring my stay at the UPF.I have greatly bene�ted from the eight months I was part of the stimulatingintellectual environment there. Special thanks go to my co-author FrancescoFurlanetto for our fruitful collaboration, his friendship, and not least forinvaluable encouragement when it was badly needed. Finally, I am gratefulto the assessment committee, John Dri¢ ll, Svend E. Hougaard Jensen andBo Sandemann Rasmussen, for their thorough and constructive work, andfor their many helpful comments and suggestions at the pre-defence. Thereare much better places to thank family and friends than the preface to a PhDdissertation. I will �nd such a place soon.

Martin SenecaAarhus, July 2008

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Introduction and summary

I began my PhD studies shortly after Michael Woodford published hismonumental book �Interest and Prices, Foundations of a Theory of Mone-tary Policy�(Woodford, 2003). This book has been a companion throughoutmy PhD studies, and I have spent many hours reading it, reproducing itsmain equations, and pondering its arguments. It is a book that brings frus-tration as well as enlightenment �much like everything else a PhD studenthas to go through � but it is a book that has had an immense in�uenceon the framing of my thinking about macroeconomics and economic policy.Indeed, the models in this thesis are closely related to the class of modelsconsidered by Woodford (2003), namely the dynamic stochastic general equi-librium (DSGE) models with imperfect competition, nominal rigidities andendogenous monetary policy that have come to dominate much of macroeco-nomics.1

Woodford (2003) refers to this class of models as neo-Wickellian stress-ing that monetary policy is conducted by setting a short-term interest ratein response to macroeconomic developments as suggested by Knut Wick-sell in his �Interest and Prices�more than a century ago (Wicksell, 1898).Goodfriend and King (1997) suggest instead that the models represent anew neo-classical synthesis because they accommodate di¤erent assumptionsabout the frictions that may potentially obstruct the e¢ ciency of markets(as they do in Keynesian economics), hence replacing past methodologicaldisagreements between the ever feuding schools of macroeconomic thoughtby the near consensus view that economic dynamics should be derived fromwell-de�ned decision problems of economic agents as in the neo-classical realbusiness cycle (RBC) literature. The synthesis is �new�to distinguish it fromthe �old� neo-classical synthesis in which the long-run is classical and theshort run Keynesian. In the new synthesis, the classical long run is to be

1The seminal contribution is Yun (1996).

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thought of as an important dynamic benchmark or, in the words of Woodford(2003, p. 9), a �virtual equilibrium of the economy at each point in time �theequilibrium that one would have if wages and prices were not in fact sticky.�Other economists �less afraid, perhaps, of the possible stigma related to

the ideas exposed by John Maynard Keynes in his �General Theory�(Keynes,1936), and not least to the later interpretations thereof �refer to the modelsas New Keynesian. One example is Galí (2008), whose admirably clear expo-sition of the material I have much to thank for appreciating the underlyingideas. Hereby, they emphasise that labour and goods markets may be imper-fectly competitive in these models, and that wages or prices may fail to adjustfully and instantaneously to shocks. But importantly, the DSGE frameworkapplied means that it is a new generation of New Keynesian economics incomparison with the New Keynesian literature of the 1980s as de�ned andexempli�ed by the contributions in the volumes edited by Mankiw and Romer(1991). This early generation of New Keynesian economics developed Key-nesian mechanics without the ad hoc assumption of the old Keynesian ap-proach, but abstracted from issues related to risk, dynamics and/or generalequilibrium e¤ects. As �Neo-New Keynesian Economics�is as much a tonguetwister as �The New Neo-Classical Synthesis�, or perhaps more because ofhabit than anything else, I follow Galí (2008) and others in referring to thecurrent generation of DSGE models with imperfect competition and nominalrigidities simply as �New Keynesian�, and it is in this sense that the inclusionof that term in the title of this thesis should be understood.Despite the rigorous DSGE approach, the equilibrium conditions of the

basic New Keynesian models can be summarised in a very simply form thatis reassuringly similar to the equations of the old ad hoc models. In particu-lar, the dynamics of the simplest models can be expressed in log-linear termsas an expectation-augmented Phillips curve, an IS equation and a monetarypolicy rule. But there are important di¤erences, in particular because ex-plicit attention to dynamic e¤ects makes the models forward-looking. Theexpectations that matter for in�ation dynamics are �rst and foremost cur-rent expectations of future in�ation, and the IS equation gives the relationbetween interest rates and output in a forward-looking manner. Equally im-portant, the basic model can be extended in numerous ways without compro-mising the ambitious methodological standards set by the DSGE approach.The models in this thesis are examples of such extensions.Because of this appeal, DSGE modelling has become increasingly pop-

ular, and notably so even in the time I have spent as a PhD student, not

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only as the comme-il-faut approach in academic macroeconomics, but alsoin policy making institutions. Central banks, in particular, have built and inmany instances estimated medium-sized New Keynesian models often usingrecently developed Bayesian methods in macroeconometrics. These modelsare increasingly the driving forces behind economic policy deliberations atthese central banks, and in many instances they have replaced existing struc-tural macroeconometric models for both policy experiment and forecastingpurposes. This is the case in particular in central banks that have adoptedin�ation targeting.The models in this thesis are probably too stylised to be taken to the

data in a way that can be expected to be meaningful, at least in their currentforms, and no attempt has been made to estimate them formally. But thatis not to say that they are not inspired by the data or in�uenced by theempirical evidence. Observations about structural economic characteristicsand evidence from empirical studies in the literature have served as importantinputs to the motivations for the chapters that follow.The models in this thesis di¤er from most of the New Keynesian litera-

ture by emphasising household heterogeneity, that is, by the assumption thathouseholds in the economy have di¤erent characteristics. As obvious as thismay appear to be to the uninitiated, macroeconomists usually assume thathouseholds are all alike, or more subtly that the economic decisions by house-holds in the economy can be summarised by the decisions of a representativehousehold without much consideration for the conditions under which thisassumption provides a meaningful approximation to reality. In this thesis,no such representative household exits, though the deviations from the rep-resentative household assumption are admittedly small as the representativehousehold approach to aggregation is often maintained.Hence, I by no means claim to provide anything close to an exhaustive

analysis of household heterogeneity, but instead I provide two examples ofa basic kind of household heterogeneity with implications for the propaga-tion of shocks to the economy, and I analyse some of these implications inthe following four chapters. Indeed, in most (though not all) of the analy-sis, households are inherently identical in the sense that they have identicalpreferences, but importantly they face di¤erent constraints in their immedi-ate economic environments, i.e. in the environments in which they have tomake economic decisions, with non-trivial e¤ects on their behaviour. Suchan analysis is a �rst step �or rather a supplement �to the analysis of theimplications of deeper household heterogeneity.

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� � �

The thesis consists of two parts each divided into two self-containingchapters.In part I consisting of chapters 1 and 2, households supply their labour

services in a heterogeneous labour market. In particular, it is assumed thathouseholds in di¤erent countries in a monetary union (or alternatively indi¤erent sectors of a closed economy) are allowed to adjust their wages withdi¤erent frequencies and that the market power with which they can setwages di¤er.Chapter 1 is called �Labour market asymmetries in a monetary union�

and is joint work with Torben M. Andersen. This paper takes a �rst step inanalysing how a monetary union performs in the presence of labour marketasymmetries. Di¤erences in wage �exibility, market power and country sizesare allowed for in a setting with both country-speci�c and aggregate shocks.The implications of asymmetries for both the overall performance of the mon-etary union and the country-speci�c situation are analysed. It is shown thatasymmetries can have important e¤ects, and that there are substantial spill-over e¤ects. Aggregate output volatility is not strictly increasing in nominalrigidity but hump-shaped. A disproportionate share of the consequences ofwage in�exibility may fall on small countries. In the case of country-speci�cshocks a country unambiguously bene�ts in terms of macroeconomic stabil-ity by becoming more �exible, but in general an in�exible country does notnecessarily achieve more output stability by becoming more �exible.The title of Chapter 2 is �Monetary policy and welfare in a monetary

union with labour market heterogeneity�. The paper addresses the ques-tion of how monetary policy should be conducted in a monetary union whenlabour market structures are di¤erent in member countries. As this is thecase in the European Monetary Union, this question should be of particularinterest to European policy makers. The paper develops a dynamic stochas-tic general equilibrium model of a two-country monetary union with labourmarket heterogeneity. Asymmetries in labour market structures are proxiedfor by di¤erent degrees of nominal �exibility. A welfare loss function derivedas a second-order approximation to household utility is evaluated. The re-sults suggest that price in�ation targeting may lead to non-negligible welfarelosses compared to monetary policy alternatives when the important nom-inal rigidity is in the labour market. Welfare may be noticeably improved

xii

in such a case by targeting wage in�ation especially when shocks are highlycorrelated across member countries and when labour markets are very het-erogeneous. Further welfare improvements can be obtained by putting moreweight to �ghting wage in�ation in the more rigid labour market if shocksare less than perfectly correlated.In part II consisting of chapters 3 and 4, households di¤er with respect

to access to capital and �nancial markets. Some households, call them rule-of-thumb consumers, are barred from access to these markets and have nochoice but to consume their current income. Other households, however,have full access to capital and �nancial markets, which means that theysubstitute consumption intertemporally by investing in capital goods or bybuying and selling �nancial assets. As an alternative interpretation, rule-of-thumb consumers may have access to �nancial and capital markets, butnonetheless choose not to take part in them for some reason or other, maybebecause they are myopic, extremely impatient, ignorant or afraid. Underthis interpretation, rule-of-thumb consumers are inherently di¤erent fromtheir optimising neighbours.Chapter 3, �Rule of thumb consumers, productivity and hours�, is joint

work with Francesco Furlanetto. In this chapter we study the transmissionmechanisms of productivity shocks in a model with rule-of-thumb consumers.In the literature, this �nancial friction has been studied only with referenceto �scal shocks. We show that the presence of rule-of-thumb consumers isalso very helpful in accounting for recent empirical evidence on productivityshocks. Rule-of-thumb agents, together with nominal and real rigidities,play an important role in reproducing the negative response of hours and thedelayed responses of output and consumption after a productivity shock.In Chapter 4, �Fiscal shocks, rule-of-thumb consumers and real rigidities�,

which is joint work with Francesco Furlanetto, we show that empiricallyplausible results on the e¤ects of �scal shocks in Galí et al. (2007) rely onan excessive degree of price stickiness and an implausibly large percentageof �nancially constrained agents. We show that it is possible to obtain anempirically appealing consumption multiplier for plausible values of theseparameters if real rigidities are added to the model. Real rigidities in the formof habit persistence, �xed �rm-speci�c capital and Kimball demand curvesinteract in interesting ways with nominal and �nancial rigidities, and we showthat they are useful in the study of �scal shocks in addition to monetary andproductivity shocks as has been shown in the previous literature.The thesis has two main messages �or perhaps, given the risk of succumb-

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ing to the temptation of drawing too strong conclusions from the evidencepresented, the results provide two main indications. The �rst is that con-sumer heterogeneity �either in the form of inherent di¤erences in householdcharacteristics or in the form of asymmetries in the immediate economicenvironment in which individual households have to act �matters for theaggregate economy and for economic policy, and that more research shouldbe devoted to understanding the implications of such heterogeneity both forsocial welfare and for the design of policy. The second message is that theinteraction of nominal, real and �nancial rigidities is important in macroeco-nomic analysis even when a frictionless, perfectly competitive virtual generalequilibrium is taken to be an important benchmark. The extent to which thislatter indication must be modi�ed by advances in the understanding of thepotential limits to �or deviations from standard assumptions on �economicagents�decision making capabilities is, I believe, a particularly interestingavenue for further research in macroeconomics.

xiv

References

[1] Keynes, J.M.. 1936. The general theory of employment, interest, andmoney. Houndmills: Palgrave Macmillan.

[2] Galí, J., 2008. Monetary policy, in�ation and the business cycle. Prince-ton: Princeton University Press.

[3] Galí, J., López-Salido, J.D., Vallés, J., 2007. Understanding the e¤ects ofgovernment spending on consumption. Journal of the European EconomicAssociation, 5(1), 227-270.

[4] Goodfriend, M., King, R.G., 1997. The New Neoclassical Synthesis andthe role of monetary policy. NBER Macroeconomics Annua, 12, 231-283.

[5] Mankiw, N.G., Romer, D., 1991. New Keynesian Economics, Vol. 1 & 2.Cambridge, MA: The MIT Press.

[6] Wicksell, K., 1898. Geldzins und Güterpreise. Jena: Gustav Fischer Ver-lag.

[7] Woodford, M., 2003. Interest and prices. Foundations of a theory of mon-etary policy�, Princeton: Princeton University Press.

[8] Yun, T.. 1996. Nominal price rigidity, money supply endogeneity, andbusiness cycles. Journal of Monetary Economics 37, 345-370.

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Dansk resumé

Denne afhandling består af �re selvstændige kapitler. Afhandlingen erinddelt i to dele bestående af henholdsvis kapital 1-2 og kapitel 3-4.I afhandlingens første del udbyder husholdningerne deres arbejdskraft i

et heterogent arbejdsmarked. Mere speci�kt antages det, at husholdningeri forskellige lande i en møntunion (eller alternativt i forskellige sektorer ien lukket økonomi) justerer deres lønninger med forskellig frekvens og harforskellige grader af markedsmagt i lønfastsættelsen.Kapitel 1 hedder "Labour market asymmetries in a monetary union" og

er skrevet sammen med Torben M. Andersen. Kapitel tager et første skridti analysen af, hvordan en møntunion fungerer, når arbejdsmarkedsstruk-turerne er forskellige på tværs af landegrænser. Forskelle i løn�eksibilitet,markedsmagt og landestørrelse tillades i en model med både landespeci�kkeog fælles stød. Implikationerne af asymmetrier for både den overordnedemakroøkonomiske stabilitet i møntunionen og den landespeci�kke situationanalyseres. Det vises, at asymmetrier kan have vigtige e¤ekter, og at derer betydelige spill-over-e¤ekter. Blandt andet viser analysen, at den ag-gregerede outputvariabilitet ikke er strengt stigende i graden af nominelrigiditet, men snarere pukkelformet. En disproportional andel af konsekvens-erne af manglende løn�eksibilitet kan tilfalde små lande, og et u�eksibel landopnår ikke nødvendigvis større stabilitet ved at blive mere �eksibelt.Titlen på kapital 2 er "Monetary policy and welfare in a monetary union

with labour market heterogeneity". Kapitlet adresserer spørgsmålet om,hvordan pengepolitikken skal indrettes i en møntunion, når arbejdsmarkeds-strukturerne er forskellige i unionens medlemslande. Da dette er tilfældeti Den Europæiske Møntunion, bør dette spørgsmål have særlig interessefor europæiske policymakere. I kapitlet udvikles en dynamisk stokastiskgenerel ligevægtsmodel for at besvare dette spørgsmål. Asymmetrier i ar-bejdsmarkedets strukturer er repræsenteret af forskellige grader af nominellerigiditeter. En velfærdstabsfunktion udledes som en andenordensapproksi-

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mation til husholdningernes nytte. Resultaterne indicerer, at en prisin�a-tionsmålsætning kan føre til ikke-ubetydelige velfærdstab, når den vigtigstenominelle rigiditet er at �nde på arbejdsmarkedet og ikke på varemarkedet.Velfærden i møntunion kan mærkbart forbedres i et sådant tilfælde ved atanvende lønin�ation som mål, især når stød er stærkt korrelerede på tværsaf landegrænser, og når arbejdsmarkederne er meget heterogene. Velfærdenkan yderligere forbedres ved, at centralbanken lægger mere vægt på lønin�a-tionen i landet med det mest rigide arbejdsmarked, når stød er mindre endperfekt korrelerede.I afhandlingens anden del bestående af kapitel 3-4, er husholdningerne

forskellige med hensyn til deres adgang til �nans- og kapitalmarkederne.Nogle husholdninger, kald dem tommel�ngerregelforbrugere, har ikke adgangtil disse markeder og har dermed ingen anden mulighed end at forbruge deresdisponible indkomst i hver periode. Men andre husholdninger har fuld adgangtil �nans- og kapitalmarkederne, hvilket betyder, at de kan substituere for-brug intertemporalt ved at investere i kapital eller ved at købe og sælge�nansielle aktiver. Som en alternativ fortolkningsmulighed kan tommel�n-gerregelforbrugerne tænkes af have adgang til �nans- og kapitalmarkederne,men ikke desto mindre vælge ikke at tage del i dem af en eller anden grund,måske forbi de er myopiske, særdeles utålmodige, ignorante eller bange for attage del i disse markeder. Under denne alternative fortolkning er tommel�n-gerregelforbrugerne forskellige fra optimerende forbrugere i deres personligekarakteristika.Kapitel 3, "Rule-of-thumb consumers, productivity and hours", er skrevet

med Francesco Furlanetto. I dette kapitel undersøger vi transmissionsmekanis-men af produktivitetsstød i en model med tommel�ngerregelforbrugere. Ilitteraturen er denne �nansielle friktion kun blevet undersøgt i relation til�nanspolitiske stød. Vi viser, at tilstedeværelsen af tommel�ngerregelfor-brugere også hjælper med at forklare nylige empiriske resultater om produk-tivitetsstød. Tommel�ngerregelforbrugere, sammen med nominelle og realerigiditeter, er vigtige for at reproducere den negative reaktion af arbejd-stimer og de forsinkede reaktioner af produktion og forbrug efter et produk-tivitetsstød.I kapital 4, "Fiscal shocks, rule-of-thumb consumers and real rigidities",

som er skrevet med Francesco Furlanetto, viser vi, at empirisk plausible re-sultater i J. Galí, J.D. López-Salido og J. Vallés, "Understanding the e¤ectsof governmetn spending on consumption", Journal of the European Eco-nomics Association, 5, pp. 227-270, er baseret på en overdrevet grad af

xviii

prisrigiditet og en uplausibel stor procentdel af �nansielt begrænsede for-brugere. Vi viser, at det er muligt at opnå en empirisk plausibel forbrugs-multiplikator for plausible værdier for disse parametre, hvis reale rigiditeterintroduceres i modellen. Reale rigiditeter i form af vanedannelse i forbruget,fast virksomhedsspeci�k kapital og Kimball-efterspørgselskurver interagererpå interessant vis med nominelle og �nansielle rigiditeter, og vi viser, at deer nyttige i analysen af �skale stød i tillæg til monetære og teknologiske stød,som det tidligere er demonstreret i litteraturen.Afhandlingen har to hovedbudskaber - eller måske snarere, givet risikoen

for at falde for fristelsen til at drage for stærke konklusioner på baggrund afde fremlagte beviser, giver resultaterne to hovedantydninger. Den første er,at forbrugerheterogenitet �enten i form af forskelle i forbrugernes person-lige karakteristika eller i form af asymmetrier i den umiddelbare økonomiskesfære, i hvilken husholdningerne skal agere �er af betydning for samfund-søkonomien og for den økonomiske politik, og at mere forskning skulle dedik-eres til at forstå implikationerne af en sådan heterogenitet både for den sam-lede velfærd og for udformningen af den økonomiske politik. Det andet bud-skab er, at samspillet mellem nominelle, reale og �nansielle rigiditeter ervigtigt i makroøkonomisk analyse, selv når en friktionsløs, fuldkommen kom-petitiv, virtuel generel ligevægt med rimelighed anses for at være et vigtigtbenchmark. Graden med hvilken denne sidste implikation skal modi�ceressom professionen gør fremskridt i forståelsen af de potentielle begrænsningeri �eller afviger fra standardantagelser om �økonomiske agenters evner tilat træ¤e økonomiske beslutninger er efter min mening et særligt interessantemne for den fremtidige makroøkonomiske forskning.

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Part I

Chapter 1

Labour market asymmetries in a monetaryunion�

Torben M. AndersenUniversity of Aarhus, CEPR, EPRU and IZA

Martin SenecaUniversity of Aarhus

July 2008

AbstractThis paper takes a �rst step in analysing how a monetary union

performs in the presence of labour market asymmetries. Di¤erencesin wage �exibility, market power and country sizes are allowed for in asetting with both country-speci�c and aggregate shocks. The implica-tions of asymmetries for both the overall performance of the monetaryunion and the country-speci�c situation are analysed. It is shown thatasymmetries can have important e¤ects, and that there are substantialspill-over e¤ects. Aggregate output volatility is not strictly increasingin nominal rigidity but hump-shaped. A disproportionate share of theconsequences of wage in�exibility may fall on small countries. In thecase of country-speci�c shocks a country unambiguously bene�ts interms of macroeconomic stability by becoming more �exible, but ingeneral an in�exible country does not necessarily achieve more outputstability by becoming more �exible.JEL classi�cation: E30, E52, F41.Key words: wage formation, nominal wage rigidity, staggered con-

tracts, monetary policy, monetary union, business cycles, shocks

�For useful comments, we thank seminar participants at Danmarks Nationalbank, Uni-versitat Pompeu Fabra, Kiel Institute for the World Economy, SAE 2007 in Granada, andthe Royal Economic Society�s 2008 Conference at Warwick. The paper has appeared asCEPR Discussion Paper No. 6800 and Danmarks Nationalbank Working Paper No. 53.

5

1 Introduction

In the run-up to the establishment of the European Monetary Union, muchfocus was on whether the potential member countries ful�lled the conditionsfor an optimal currency area. According to the traditional theory on opti-mal currency areas, participation in a currency union and the implied loss ofautonomy in monetary policy require that labour markets are �exible in ad-justing to country-speci�c shocks (see e.g. de Grauwe, 2005). The �exibilitycan either be in terms of wage adjustment or labour mobility. In a secondwave of the literature, it was stressed that the conditions for an optimalcurrency area are endogenous since participation in a currency union a¤ectsmarket fundamentals (via further integration), incentives in wage formationand possibly the incentive to undertake structural reforms (see e.g. Rose andFrankel, 1998, and Calmfors, 2001). Whether the countries constitute an op-timal currency union is an ex-post rather than an ex-ante question. In eithercase, it is presumed that a monetary union eventually will be characterizedby symmetries.However, perceiving a currency union as a homogeneous area is in most

cases misleading, and for the European Monetary Union (EMU) in particu-lar this assumption does not seem appropriate. European countries are fairlyheterogeneous, re�ecting di¤erent institutional, political and historical devel-opments. The homogeneity assumption did not hold neither in an ex anteand nor in an ex post sense so far since the incidence and nature of reformsundertaken in recent years do not seem to indicate that these di¤erencesare about to be eliminated. Di¤erences in business cycle situations and thetensions these have created in the assessment of the common monetary pol-icy of the ECB reveal that countries are a¤ected by di¤erent shocks and/orstructures.This paper takes a �rst step in analysing the role of asymmetries for

business cycle �uctuations within a currency union, focussing on the role ofasymmetric sizes, structures and shocks. One reason why such asymmetriesmay be important and why the homogeneity assumption underlying stan-dard approaches is potentially misleading arises from the observation thatthe common monetary policy tends to react to common or aggregate shockswithin the area. Hence, asymmetries are potentially more important sincethey cannot easily be countered via the common monetary policy. Asym-metries across countries may arise directly from country-speci�c shocks, buteven aggregate or common shocks may create asymmetric e¤ects when they

6

interact with di¤erences in size and structures across the member countries.This, in turn, implies that aggregate measures of the performance of theunion, like in�ation and output (gaps) to which the common monetary pol-icy reacts, may critically depend on the asymmetries. To further complicatematters, the "asymmetries" are transmitted across member countries viatrade links. It is an implication that the performance of the monetary unionis not in general well described by a "representative" country approach.The framework used in this paper is based on recent intertemporal ap-

proaches in both open economy macroeconomics and in the closed-economyNew Keynesian literature on monetary policy.1 To focus on the interactionswithin the monetary union, it is considered to be a closed area where themember countries share a common monetary policy and engage in trade witheach other. The focus is positive in the sense of exploring the consequences ofasymmetries (size, structure and shocks) for the business cycle performanceat the country and aggregate level for a given monetary policy setting. Thespeci�c structural asymmetries analyzed apply to the labour market, and dif-ferences in nominal adjustment and the degree of competition in the labourmarket are allowed for. These two types of structural dimensions can beseen as examples of structural di¤erences in the real and the nominal dimen-sion. They are further motivated by the fact that it is well-established thatthere are such di¤erences across European countries (member or potentialmember countries of EMU), see e.g. the OECD (various issues), EuropeanCommission (2006) and Arpaia and Pichelmann (2007).2 The speci�c shocksconsidered are supply (productivity) shocks. Modern business cycle the-ory has studied such shocks extensively and therefore constitutes a naturalbenchmark for the analysis of the implications of asymmetries. However,the main mechanisms are not dependent on whether shocks are originatingon the supply or the demand side. Finally, we model monetary policy bya simple Taylor-rule in accordance with much recent research on monetarypolicy.3

Whether participation in a monetary union strengthens the incentive toundertake reforms can not really be addressed without having analysed how

1Seminal contributions are Obstfeld and Rogo¤ (1995) and Yun (1996), respectively.For monographic expositions, cf. Obstfeld and Rogo¤ (1996) and Woodford (2003).

2Recent work has also pointed to the importance of downward nominal wage rigiditiesand cross-country di¤erences in this form of rigidity, see Holden and Wulfsberg (2008).

3See, for instance, Smets and Wouters (2003) for an estimated Taylor rule for the euroarea, and Galí and Gertler (2007) for a recent discussion.

7

the monetary union works in the presence of given asymmetries. The de-bate seems to take it for granted that more �exibility is good both froma country and an overall monetary union perspective. Is it necessarily thecase that less �exible countries su¤er from a disproportionate share of eco-nomic problems and therefore have the largest incentives to undertake re-forms? Is it only asymmetries across large member countries that matter,or are the structural characteristics of smaller member countries also impor-tant? Surprisingly, these questions have not been much researched. We shedsome light on these questions by considering the implications of asymmet-ric structures both for the aggregate performance of a monetary union andcountry-speci�c performance. Interestingly, we �nd that there may be a con-�ict between what may seem best from a country perspective and from anoverall monetary union perspective. Moreover, changes in structural parame-ters may release spill-over e¤ects between member countries, suggesting thatnon-cooperative structural policy making does not necessarily imply struc-tures which are optimal seen from both an overall monetary union and acountry-speci�c perspective.Other forms of asymmetries or heterogeneities in a monetary union have

been addressed in the literature. One important issue is the interdependencybetween monetary and �scal policy when the former is centralised and thelatter decentralised. This naturally leads to potential asymmetries in �scalpolicy, which raises questions concerning interdependencies between national�scal policies as well as between the aggregate �scal stance and the monetarypolicy (see, e.g., Lombardo and Sutherland, 2004, Beetsma and Jensen, 2005,and Andersen, 2005). Also, Benigno (2002) analyses, from a normative pointof view, how monetary policy should be designed when member countrieshave di¤erent degrees of nominal price rigidities, and it is shown that thecentral bank should attach more weight to in�ation in countries characterisedby more nominal inertia. Beetsma and Jensen (2004, 2005) allow for labourmarket asymmetries in their analysis of the interactions between monetaryand �scal policy in a monetary union. Dellas and Tavlas (2004) present athree-country model allowing for asymmetries in nominal wage �exibility, and�nd that countries with a high degree of nominal wage rigidity are better o¤ina monetary union. Similarly, Hallett and Jensen (2001) �nd that countrieswish to join a monetary union only when its markets are relatively more�exible. Andersen (2008) also analyses the implications of labour marketasymmetries in a monetary union in an intertemporal model, but in a settingwith one-period contracts.

8

The paper is organised as follows. The model structure is laid out in sec-tion 2, and the equilibrium processes for output and in�ation are determinedin section 3. Section 4 considers the shock transmission in a symmetric base-line example. The implications of various forms of asymmetries are exploredin section 5 both from a unionwide and country-speci�c perspective. Section6 o¤ers a few concluding comments.

2 Amonetary union with heterogeneous labourmarkets

Consider a monetary union where the central bank has the monetary author-ity over I separate and otherwise independent countries (or more generallyregions) indexed by i. In particular, the central bank sets the nominal in-terest rate Rt earned on risk-free nominal bonds throughout this monetaryunion between periods t and t+1. The union is closed to the outside world.4

Each country i is populated by a continuum of households h 2 [0; 1] and hasa continuum of �rms f 2 [0; 1]. Countries may be of di¤erent sizes, wherethe relative size of country i is given by vi 2 [0; 1] such that

PIi=1 vi = 1. All

�rms in a given country produce the same internationally traded consump-tion good, di¤erent from those produced in other countries (a specialisedproduction structure). For simplicity, product markets are assumed to beperfectly competitive and prices to be �exible. Labour markets, in contrast,are imperfectly competitive and have nominal rigidities in the form of nom-inal wage contracts. Labour market structures are allowed to di¤er acrosscountries in terms of degrees of market power in wage setting, and degrees ofnominal rigidity are di¤erent across countries. There is no mobility of labouracross borders.

2.1 Firms

2.1.1 Labour demand

In each period t, each household h in country i supplies a di¤erentiatedlabour service Nit (h). The labour used in production in country i, Lit, is

4This means that the model could also be interpreted as a closed-economy model of asingle country with I sectors, which are potentially asymmetric in terms of structures andshocks.

9

assumed to be an aggregate of the continuum of labour services supplied bythe households:

Lit =

�Z 1

0

Nit (h)�i�1�i dh

� �i�i�1

(1)

where �i > 1 is the elasticity of substitution between labour services.Each household determines its wage rate, Wit(h), taking into account

how �rms� labour demand depends on the wage (a right-to-manage struc-ture). That is, given wages, actual employment is determined by labourdemand. The demand for household h�s labour service is determined by thecost minimization problems of the country�s �rms, which minimise costs,taking households�wage rates, Wit(h), as given. The representative �rmminimizes Z 1

0

Wit(h)Nit (h) dh (2)

with respect to Nit (h) subject to (1). This leads to a demand for householdh�s labour service given by

Nit (h) =

�Wit (h)

Wit

���iLit (3)

where Wit is the wage index de�ned by

Wit =

�Z 1

0

Wit (h)1��i dh

� 11��i

(4)

This wage index has the property that the minimum cost of acquiring Litunits of aggregate labour is given by WitLit. It follows that the labour de-mand elasticity is �i, and hence the market power of wage setters is inverselyrelated to �i. The demand for household h�s labour service is a decreasingfunction of the household�s relative wage.

2.1.2 Pro�t maximisation

The representative �rm in country i produces output Yit according to theproduction function

Yit =1

L itU

1� it (5)

where Uit is the stochastic period-t productivity of �rms in country i, and0 < < 1 is the degree of returns to scale. Real capital is disregarded to

10

simplify, but decreasing returns can be interpreted as arising from a secondfactor of production in �xed supply.Product markets are perfectly competitive, and the representative �rm

in country i maximizes pro�ts, which it distributes to households. There areno nominal price rigidities, and the �rm takes the price of its product, Pit,as given. The pro�t maximization problem yields a demand for aggregatedlabour services given by

Lit =

�Wit

Pit

�� 11�

Uit (6)

Inserting in (5) gives the supply relation

Yit =1

�Wit

Pit

� �1

Uit (7)

In logs and measuring variables as deviations from steady state5, we have

yit = � (pit � wit) + uit (8)

where � � = (1� ). The supply shock is assumed to be generated by theprocess

uit = �uuuit�1 + "it (9)

where �1 < �uu < 1 and "it~iid N (0; �2i ). The correlation of the innovationsCorr("i; "j) is denoted �". Hence, di¤erent situations can easily be charac-terized by varying this correlation coe¢ cient. If �" = 1, it follows that allinnovations are identical across regions, i.e., "it = "t 8i, which corresponds toan aggregate shock. If �" = 0, the shocks are idiosyncratic or country-speci�cshocks.

5In order to solve the model, it is written in log-deviations from the non-stochasticsteady state. Steady-state values are indicated by omission of time subscripts, and lower-case letters denote (log-)deviations from steady-state values of corresponding upper-casevariables (xt � dXt=X � ln (Xt=X)). Throughout, aggregate log-variables are de�nedas weighted averages of country-speci�c log-variables, i.e., for any variable x, we havext =

PIi=1 viXit. In general, a log-linearization around a steady-state of Xt =

PIi=1 ViXit

gives this average where vi =VjXj

X . Symmetry of the steady state implies vi = Vi.

11

2.2 Households

2.2.1 Consumption and bond holdings

Household h in country i has the utility function

Et

1X�=0

����

� � 1Cit+� (h)��1� � 1

1 + �Nit+� (h)

1+�

�(10)

where Et is an operator representing expectations over all states of the econ-omy conditional on period-t information, � 2 (0; 1) is the subjective discountfactor, and Cit+� (h) is a real consumption index for period t + � , Nit+� islabour supply in period t+ � . � > 0 is the elasticity of intertemporal substi-tution of consumption and � > 0.6

The consumption index is de�ned over the di¤erentiated commoditiesproduced in the union�s member countries. Speci�cally,

Cit (h) =

"IXj=1

vjCijt (h)��1�

# ���1

(11)

where � > 0, vj is the relative size of country j (as noted above), and Cijt (h)represents consumption of country j�s commodity by household h in countryi. In every period t, this household chooses Cijt (h) for a given level of realconsumption by minimizing

IXj=1

PjtCijt (h)

subject to (11). This yields a demand for country j�s product by householdh in country i given by

Cijt (h) =

�PjtvjPt

���Cit (h) (12)

6Real money balances could be included in the utility function in order to analysemoney demand. However, the central bank�s policy instrument is the interest rate, whileit passively supplies the money demanded by households. Thus, as long as money entersadditively separably in the utility function, nothing will change in what follows sincethe inclusion of money will only add a money demand relation recursively determiningmoney demand as a function of the variables of interest. See, e.g., Woodford (2003) for adiscussion.

12

when Pt is the price index de�ned by

Pt =

"IXj=1

vj

�Pjtvj

�1��# 11��

(13)

This price index has the property that the minimum cost of acquiring Citunits of real consumption is given by PtCit. From (12) it follows that � is theprice elasticity of demand for Cijt.Asset markets are assumed to be complete (see discussion below), i.e.,

available �nancial assets completely span the possible states of the economy.This assumption leads to the following period-t �ow budget constraint for ahousehold in country i:

Et [Qt;t+1Bit (h)] + PtCit (h) = Bit�1 (h) +Wit (h)Nit (h) + �it (h) (14)

The right-hand side gives available resources as the sum of initial �nancialwealth, Bit�1 (h), labour income, Wit (h)Nit (h), and nominal pro�t income,�it (h). The left-hand side represents the allocation of resources to consump-tion, PtCit (h), and bond-holdings, Et [Qt;t+1Bit (h)], where Qt;t+1 is the assetpricing kernel.7

Given existing wage contracts, the household maximises expected utility(10) subject to the sequence of budget constraints (14) and (implicitly) asolvency condition. De�ning the net risk-free nominal interest rate Rt by therelation (1 +Rt)

�1 = E [Qt;t+1], the �rst-order conditions determining theoptimal choice of consumption and bond-holdings can be combined to yieldthe Euler equation

Cit (h)� 1� = � (1 +Rt)Et

�Cit+1 (h)

� 1�PtPt+1

�(15)

summarising the household�s intertemporal consumption decisions.8

7The asset-pricing kernel is the period-t price of a claim to one unit of currency in statest+1 in period t+1 divided by the probability of that state occurring conditional on period-t information, Prt

�st+1

�. The bond Bit is a random variable paying Bit

�st+1

�units of

currency in state st+1 in period t + 1. At time t, the household chooses the completespeci�cation of this random variable in all states st+1. It follows that Et [Qt;t+1Bit] is theallocation of resources to a portfolio of bonds.

8Note that it is an implication of (15) that monetary policy a¤ects aggregate demandin all countries symmetrically.

13

2.2.2 Wage setting

To model nominal rigidities, it is assumed that wages are set by householdsin a staggered fashion with random duration of wage contracts analogous tothe mechanism in Calvo (1983). In particular, in every period each householdin country i is allowed to reset the wage rate it demands for its labour servicewith a �xed probability (1� �i). Hence, the wage rate set by household hat time t, W �

it (h), is the prevailing wage rate for the household at time t+ � ,i.e., Wit+� (h) = W

�it (h), with probability �

�i , and the expected duration of

a contract given by (1� �i)�1. For given wages, employment is determinedfrom the demand side.When a household resets its wage, it does so to maximise expected utility

(10) subject to the demand for its labour (3), its budget constraint (14) andthe price setting mechanism just described. For a household changing itswage rate at time t, this is equivalent to maximizing the following functionwith respect to W �

it (h) subject to (3) and (14):9

Et

1X�=0

(�i�)�

��

� � 1Cit+� (h)��1� � 1

1 + �Nit+� (h)

1+�

�(16)

The �rst-order condition becomes

Et

1X�=0

(�i�)�

���i

1� �iNit+� (h)

� + Cit+� (h)� 1�W �it (h)

Pt+�

�Nit+� (h)

�= 0

(17)It follows that the monopolistically competitive household sets its wage rateso that the marginal utility of income from an extra unit of labour e¤ort is aconstant mark-up over the marginal disutility in discounted expected valueterms. This captures the standard result in wage bargaining models that themarket power of wage setters depends on the elasticity of labour demand(see, e.g., Blanchard and Fisher (1989)).In the special case with �exible wages where households are allowed to

reset the wage each period, the �rst-order condition collapses to

W (h)�itPt

=�i

�i � 1N (h)�it

C (h)� 1�

it

=�i

�i � 1MRS (h)it (18)

9This di¤ers from (10) in that implicit terms representing states where the wage to beset is not the prevailing wage are excluded.

14

where MRSit is the marginal rate of substitution between consumption andleisure.As shown in appendix A, the �rst-order condition for the representative

household�s wage-setting problem (17), the household labour demand relation(3) and the law of motion of the wage index (25) can be used to derive thefollowing wage-setting equation for country i (in logs):

!it = �i [mrsit � (wit � pt)] + �Et!it+1 (19)

where !it = wit�wit�1 is wage in�ation, mrsit = �lit+��1ct is the marginalrate of substitution, and �i is a decreasing function of the Calvo parameter�i and of the elasticity of substitution between labour services �i:

�i =(1� �i) (1� �i�)�i (1 + ��i)

(20)

For later reference, note that �i depends on both the parameter charac-terising wage adjustment, �i, and the parameter determining the mark-upor market power in wage formation, �i.

2.2.3 Risk-sharing

Staggered wage setting implies that households in a given country are notidentical, and there is no representative household. In the general case, in-dividual decisions will depend on initial wealth, which implies that decisionswill be path dependent, cf. Obstfeld and Rogo¤ (1995) and Corsetti andPesenti (2001). This causes both substantial technical problems and prob-lems with multiplicity of equilibria. This has been overcome in the literatureeither by imposing assumptions precluding wealth transfers or by assumingthat risk-sharing arrangements are in place. We choose the latter approachhere. By assuming that all households in the monetary union have enteredthe world with the same level of wealth, the complete-markets assumptionimplies that they will choose the same consumption levels, i.e., risk-sharingis complete both within and between the member countries of the monetaryunion.10 While overcoming technical problems, this assumption also servesthe purpose of focusing on the implications of supply side asymmetries acrossmember countries in a monetary union without mixing them with demand

10This follows from the �rst-order conditions of the utility-maximisation problems as allhouseholds face the same asset-pricing kernel.

15

and wealth e¤ects. Risk-sharing implies that Cit (h) = Cit for all h, andtherefore the h index can be dropped in what follows.Aggregate demand for good j can now be de�ned as the weighted sum

over countries i of (12):

Djt =IXi=1

viCijt =

�PjtvjPt

���Ct (21)

where

Ct =

IXi=1

viCit (22)

is de�ned as aggregate unionwide consumption. International risk-sharingimplies that Cit = Cjt for all i; j. Hence, Cit = Ct for all i.In logs, the Euler equation (15) becomes

Etct+1 = ct + � (rt � Et�t+1) (23)

where rt � log(1+Rt) and �t+1 = pt+1�pt is in�ation, and aggregate demandfor commodity j (21)

djt = �� (pjt � pt) + ct (24)

In addition, the complete-markets assumption implies that the fraction(1��i) of households in country i changing their wage rates at time t choosethe same rate W �

it. The remaining fraction �i of households continue withthe wage rate prevailing at time t� 1 where the distribution of wage rates isunchanged. Hence, the law of motion of the aggregate wage index in countryi is given by

Wit =

�Z 1

0

Wit (h)1��i dh

� 11��i

=h�iW

1��iit�1 + (1� �i) (W �

it)1��ii 11��i (25)

2.3 Monetary policy

The aim of this paper is to consider the implications of labour market asym-metries for a given monetary policy. Therefore, a standard monetary policyreaction function is speci�ed, namely a so-called Taylor rule, cf. Taylor

16

(1993). Speci�cally, it is assumed that the interest rate is determined by (inlogs)11

rt = k��t + kyyt (26)

where yt = yt� �yt is the output gap. The level of �potential�output, �yt, usedin this de�nition is the level of output under �exible wages. As shown inappendix B, it is given by

�yt = �ut (27)

where� =

1 + �

1 + � (1 + �) + ���1(28)

Note that the level of output under �exible wages is independent of monetarypolicy.

3 Equilibrium in�ation and output

Market clearing requires that demand for each good equals its supply. Thatis, for all commodities i = 1; 2; : : : ; I the equilibrium conditions read (in logs)

dit = yit (29)

This implies that there is no aggregate net wealth accumulation or decumu-lation:

ct = yt (30)

Wages are determined according to (25), and for these wages employment isdemand determined from (6). Finally, the interest rate is given by (26).The steady state is symmetric so that Bi = 0, Ci = C = Yi = Y ,

R = ��1 � 1, v�1i Pi = P and Wi = W . The equilibrium process for theendogenous variables can be explicitly characterized by solving the model bythe undetermined coe¢ cients method (see Appendix C).12

11Empirically interest rate smoothing is important, we disregard it here to simplify theexposition and focus on the role of asymmetries.12The minimal state representation of the equilibrium is followed, cf. McCallum

(1983,1999).

17

Output and in�ation in country i are given by

yit =IXj=1

vjbij0 ujt +

IXj=1

vjbij1 ujt�1 +

IXj=1

vjbij2 yjt�1 (31)

and

�it =JXj=1

vjcij0 ujt +

JXj=1

vjcij1 ujt�1 +

JXj=1

vjcij2 yjt�1 (32)

Note that the system is in ARMA (1; 1) form. Aggregate output and in�ationfollows straightforwardly by aggregation of (31) and (32).

3.1 Numerical illustrations

To illustrate the model�s properties, we shall later present numerical simula-tions. They are made for a two-country version of the model, i.e., the casewhere I = 2 and i 2 f1; 2g. Numerical results are particularly useful sincethe complexity of the model makes it di¢ cult to extract analytical results.Restricting attention to the two-country case avoids unnecessary complica-tions while illustrating the main properties of the model.The log-linear version of the model is solved numerically using DYNARE,13

and the solution is used to calculate moments of the variables of interest.These moments are taken as measures of the performance of the macro-economic variables. Simulations are performed for country-speci�c shocks,imperfectly correlated shocks and unionwide shocks, i.e., for the cases �" = 0,�" = 0:5 and �" = 1.

14

4 Symmetry and shock transmission

To set the scene for the subsequent discussion of heterogeneities, it is useful toconsider the symmetric case, i.e., the case where all countries are of equal sizes(vi = v for all i) and have the same structural parameters (�i = � and �i = �for all i). However, shocks are allowed to di¤er. This structurally symmetric

13For documentation, see www.cepremap.cnrs.fr/dynare/.14The choice of the values for the remaining parameters is inspired by estimated para-

meters for the euro area in Smets and Wouters (2003). The baseline values are �i = 0:75,�i = 4, �

2i = 1, � = 1, � =

23 , � = 1, � = 4, = 0:7, � = 0:95, k� = 1:5, ky = 0:5.

18

case serves the purpose of clarifying the basic mechanisms operating in themodel.

4.1 Shock transmission

Consider the transmission of a productivity shock speci�c to country i. Onimpact, since prices are fully �exible, this shock will tend to both increasethe output of commodity i, yi, and to decrease it�s price, pi, cf. (8). Hence,the terms of trade change to the disfavour of country i in the sense thatthe relative price of its export good decreases. However, this works to shiftdemand from goods produced in other parts of the monetary union towardscountry i�s commodity, cf. (24). In turn, this shift in demand causes the priceof foreign products (pj; j 6= i) to fall, but not by as much as the price ofcommodity i.15 Consequently, the output level in the other country decreases,i.e., a country-speci�c shock induces a negative correlation in country-speci�coutputs via the terms of trade e¤ect. The terms of trade e¤ect plays a crucialrole in the di¤erence between the volatility of country-speci�c and aggregateoutput since from (24) we have

V ar(yi) = V ar(y) + �2V ar(pi � p)� 2�Cov(pi � p; y)

Note that in the limiting case of perfectly correlated shocks (aggregateshocks), there are no terms of trade changes, and hence V ar(yi) = V ar(y)and country-speci�c outputs are perfectly correlated. Similar reasoning ap-plies to in�ation; that is, country-speci�c prices (in�ation) are positivelycorrelated, and more so the larger the correlation in the shocks.

4.2 Output and in�ation volatility

As mentioned above, the implications of various structural factors are as-sessed in terms of the standard deviations and correlations of the two keyvariables - output and in�ation. Figure 1 shows the standard deviation forcountry-speci�c as well as aggregate output and in�ation as a function of� = �1 = �2 for various levels of the correlation of shocks.

15Note that the decrease in prices triggers a monetary expansion which, in turn, increasesactivity in both countries. If the response is su¢ ciently strong, it is possible that outputincreases in both countries. In the simulations reported, the parameter values ensure thatthis does not happen; i.e. the direct e¤ects of the shocks described in the text dominate.

19

First, note a basic smoothing e¤ect in the sense that aggregate output isless volatile than country-speci�c outputs (except in the limiting case of per-fectly correlated shocks where they are equal). This is a direct consequenceof smoothing by aggregation since

�2(y) = (v1)2�2(y1) + (v2)

2�2(y2) + 2v1v2� (y1; y2) (33)

and hence in the symmetric case (�2(y1) = �2(y2) = �2(yi) and v1 = v2 =1=2), we have

�2(y) =1

2�2(yi) [1 + � (y1; y2)] � �2(y)

This expression also shows that the negative correlation in output for themember countries (for �" < 1) described above contributes to lower aggregatevolatility. Since country-speci�c in�ation rates are positively correlated, thesmoothing e¤ect for aggregate in�ation is less strong.Second, considering the e¤ects of changing the degree of nominal rigidity,

�, it is found that country-speci�c output volatility is strictly increasingin the degree of nominal rigidity if shocks are not too highly correlated.However, aggregate output variability is hump-shaped for all three types ofshocks. Consequently, there is a critical level of nominal �exibility, ��, whereaggregate output variability is increasing in � for � < ��, and decreasing in� for � > ��. In the numerical illustration, the critical value �� is close toone half, corresponding to expected contract lengths of two periods.To understand the mechanism generating this hump, note that since out-

put is generated by an ARMA(1; 1) process, the unconditional variance (inthe case of common shocks) is given as

V AR (yt) =b0 + b

21 + 2b0b1b21� b22

1

1� �2u�2" (34)

where the coe¢ cients are all functions of the nominal rigidity parameter, �,i.e., b0 = b0 (�), b1 = b1 (�) and b2 = b2 (�). It is seen that the variancedepends on the properties of the shock, but also on the endogenous responsescaptured by the impact e¤ect (b0) and the persistence-generating mechanisms(b1,b2). Note that b0 is increasing in �, while b1 and b2 are decreasing in �.It follows that stronger nominal rigidities (higher �) tend to increase outputvariability by increasing the impact e¤ects of shocks, and to lower variabilityby reducing the persistence in the response to shocks. These two counter-acting e¤ects create the hump-shaped relation, where output variability is at�rst increasing and then decreasing in the nominal rigidity (�).

20

Figure 1 also shows the volatility of country-speci�c and unionwide in�a-tion. In both cases, the variability of the in�ation rate increases when � islow, and at some level it remains almost invariant to changes in �. Hence,while in�ation volatility is quite sensitive to changes in the degree of nominalwage rigidity when wages are �exible (low �), it is relatively insensitive tochanges in the degree of nominal wage rigidity when the starting point is onewith rigid wages (high �).Finally, considering the importance of imperfect competition (the � para-

meter), we also �nd (�gures not shown) that aggregate output is less volatilethan country-speci�c output (and identical for perfect correlation of shocks).In this case, more competition (higher �) implies less volatility of both aggre-gate output and in�ation, but the e¤ect levels o¤when converging to perfectcompetition (high �). However, country-speci�c output is more volatile, thehigher �, unless the shocks are highly correlated. This is due to the impliedreduction in the weight, �i, put on the current relation between the marginalrate of substitution and the real wage in wage setting, cf. (20). This reductionin �i leads to a weaker instantaneous reaction of wages and hence, trough itse¤ect on marginal costs, of output. This implies a larger adjustment burdenon prices and thus the terms of trade.

5 Asymmetries in size and structure

We now turn to an analysis of the implications of labour market asymme-tries between the member countries of the monetary union. To clarify theissues, this proceeds in three steps. First, we consider asymmetries arisingfrom di¤erent country sizes (v1 6= v2), maintaining symmetric labour marketstructures across countries. Second, we consider structural asymmetries,while maintaining identical country sizes, with respect to the degree of nomi-nal wage rigidity (�1 6= �2) and the degree of competition (�1 6= �2). Finally,we intersect the two dimensions of asymmetry, size and structure.

21

5.1 Di¤erent country sizes

To consider the e¤ects of asymmetric country sizes, note �rst that it followsfrom (33) that (using v1 + v2 = 1)

@�2(y)

@v1= 2v1�

2(y1)� 2(1� v1)�2(y2) + 2 [1� 2v1]� (y1; y2)

= �2(yi)(1� 2v1)2 [� (y1; y2)� 1]

If all structural parameters for the two countries are the same (implying�2(y1) = �

2(y2) = �2(yi)), we have

@�2(y)

@v1= �2(yi)(1� 2v1)2 [� (y1; y2)� 1]

and hence

@�2(y)

@v1= 0 for � = 1

@�2(y)

@v1> 0 for v1 >

1

2and � < 1

@�2(y)

@v1< 0 for v1 <

1

2and � < 1

i.e. asymmetric size exaberates aggregate output volatility unless the corre-lation of shocks is one. Similar reasoning applies to in�ation volatility.Figure 2 shows the standard deviations of output and in�ation as a func-

tion of � = �1 = �2 for di¤erent sizes of country 1 (the pattern is symmetricfor country 2). This con�rms that the level of aggregate output volatility isgenerally higher when countries are of asymmetric sizes. Moreover, aggregateoutput volatility is hump-shaped in nominal rigidity (�), as in the case withsymmetric country sizes. However, the volatilities of the country-speci�c out-put levels di¤er. Output volatility is generally higher the smaller the country,and an increase in nominal wage rigidity (an increase in �) leads to largerincreases in output volatility in the small country than in the large country(unless shocks are highly correlated). This suggests that nominal rigiditiesmay be more problematic for small countries than for large countries.For in�ation, we �nd also that the level of volatility is generally higher due

to asymmetric country sizes. Both aggregate and country-speci�c in�ationtend to be increasing in nominal wage rigidity, but the e¤ect levels o¤ when

22

nominal rigidities reach a certain level. Country size matters less for country-speci�c in�ation than for country-speci�c output.Qualitatively, the e¤ects of variations in the degree of imperfect competi-

tion are the same when countries have asymmetric sizes as when they are ofequal sizes, but the level of volatility is generally larger with asymmetricallysized countries.

5.2 Asymmetric labour market structures

Next, we turn to the role of asymmetry with respect to structural factors.This issue is complicated by the fact that changing the structural parame-ters for one country has implications not only for the speci�c country butalso at the aggregate level, and hence there is a risk of mixing up e¤ectsarising from asymmetries with e¤ects arising from changing the aggregateproperties of the currency union. To overcome this problem and to focus onthe role of asymmetries for given aggregate structural characteristics of themonetary union, we keep aggregate variables unchanged in this subsectionin the sense that the weighted average of the coe¢ cients across countries iskept constant.16

Consider �rst the degree of wage �exibility. Figure 3 shows the behaviourof the standard deviations of output and in�ation for di¤erent values of �1when the average degree of nominal wage rigidity is restricted to be 0:5, i.e.,

�� = v1�1 + v2�2 = 0:5

For equally sized countries, i.e., for v1 = v2 = 0:5, this implies

�2 = 1� �1

Hence, �2 goes from 1 to 0 as �1 goes from 0 to 1. In other words, the labourmarket in country 1 becomes relatively less �exible and the labour market incountry 2 more �exible when moving from left to right on the x-axis.We �nd that the volatility in aggregate output is somewhat lower with

asymmetries in nominal rigidities than with symmetric labour market struc-tures, i.e., asymmetries in nominal wage �exibility contribute to lowering

16In the next subsection, we �x the structural parameters of country 2 while allowingthose of country 1 to vary in order to analyze the incentives for unilateral reform in country1.

23

aggregate output volatility.17 Similarly, for in�ation we �nd that its volatil-ity is reduced due to asymmetries. Interestingly, the form of the volatilityof country-speci�c output as a function of the degree of nominal rigiditynow changes and becomes hump-shaped. This suggests interesting spill-overe¤ects in the structural parameters between the two countries. For country-speci�c output, we �nd that small asymmetries may imply a larger volatilityin domestic output, while larger asymmetries may cause output volatility tobe lower. This suggests that if countries have fairly asymmetric structures,the direction in which there is an incentive to change structural characteris-tics via reforms may be ambiguous.Similarly, we consider the role of asymmetries in the parameters char-

acterising the degree of monopolistic competition in labour markets, �i fori = 1; 2. The average elasticity of substitution of labour services is restrictedto be 4, i.e., � = 4, and hence

�2 = 8� �1

It turns out (�gures not shown) that this form of asymmetry has essentiallyno e¤ect on the volatility of domestic and aggregate output or on in�ation.

5.3 Unilateral changes in structural characteristics

While the above sheds light on the implications of structural asymmetries, itdoes not directly clarify the incentives for structural reforms. Such reformsare unilateral and thus lead to changes in structural parameters in one coun-try, leaving structural parameters in other countries unchanged. Assumingthat the model�s labour market parameters (�i; �i) can be a¤ected throughvarious structural policies, it is therefore of interest to determine the direc-tion in which a country has an incentive to lead its reforms when the countryaims at stabilizing its national output and in�ation.In �gure 4, we present standard deviations as a function of �1 when �2 is

�xed at 0.75. If shocks are uncorrelated, a lower value of �1 decreases both theoutput and in�ation volatility of country 1. Therefore, in this case a countryhas an incentive to implement reforms that make wages more �exible. Such

17At the unionwide level, the standard deviations are symmetric in the degree of nominalrigidity attaining a maximum at �1 = �2 = 0:5. This is the point where the two countriesare identical. The symmetry, of course, arises because the countries have the same size sothat the restriction �� = 0:5 implies that �2 = 1� �1.

24

reforms also tend to stabilize output in country 2, but not necessarily at theaggregate level. In addition, if �1 is initially very high, the stabilization ofcountry-speci�c output may be at the expense of a higher volatility of thecountry-speci�c in�ation.The picture is less clear if shocks are correlated. In this case, the volatili-

ties of output and in�ation in country 1 are hump-shaped in nominal rigidityin country 1. Hence, it is not necessarily in the interest of country 1 toimplement labour market reforms inducing more �exibility. Moreover, anintermediary level of nominal �exibility in country 1 may bring about thehighest output volatility for country 1, but the lowest level in country 2.As suggested by the results in the previous subsection, we �nd that uni-

lateral reforms aimed at changing the degree of monopolistic competition inlabour markets have only very limited e¤ect on the volatility of output andin�ation. It follows that no incentives to change the degree of competitionin labour markets follow from a concern over stabilization of output andin�ation.

5.4 Di¤erent country-sizes and labour market struc-tures

Finally, we turn to the interaction between the various forms of asymmetries.The preceding analysis has suggested that asymmetries in size and nominal�exibility are the more important both with respect to the aggregate andcountry-speci�c performance. In the following, a case is considered in whichdi¤erent country sizes (one country is small, the other large) interact withdi¤erent degrees of nominal rigidities. The interaction between country sizesand degrees of imperfect competition, and the interaction between degreesof nominal rigidities and degrees of competition have also been considered.However, results are not reported as no new insights are gained by investi-gating these combinations of asymmetries. The former combination is closeto the case with di¤erent country sizes only, and the latter to the case withasymmetries in nominal wage rigidity being the only deviation from the sym-metric baseline example.Consider the case where the weight of the small country is v1 = 0:3 and

aggregate nominal rigidity is � = 0:75. Figure 5 shows volatility of outputand in�ation. We �nd that more nominal rigidity in the small country �incombination with less nominal rigidity in the large country �has a hump-

25

shaped e¤ect on aggregate output. This is the same type of result as foundabove. However, while the volatility of country 2 output is strictly increasingin the nominal rigidity in country 1, the volatility of country 1�s output ishump-shaped if shocks are su¢ ciently correlated. This suggests that theremay be an important negative externality from the small to the large countryin the sense that more nominal rigidity in the small country may increaseoutput volatility in the large country and decrease it in the small country.For in�ation �both country-speci�c and aggregate �the volatility is hump-shaped. Hence, asymmetries lower in�ation variability in this case.

6 Concluding remarks

In this paper, we have analyzed the consequences of labour market asym-metries in a monetary union with focus on degrees of nominal wage rigidityand of monopolistic competition in wage setting. We have considered bothaggregate and country-speci�c shocks and how they are propagated acrossmember countries that may not be equal in size. Moments of country-speci�cas well as unionwide output and in�ation �uctuations have been calculated.These moments are taken as measures of the macroeconomic performance inthe monetary union.First, our results indicate that asymmetry in the sizes of member countries

may in itself be an impediment to macroeconomic stability in a monetaryunion. In particular, the level of output and in�ation volatility is generallyhigher when countries are of di¤erent sizes than when they are equal insize. In a monetary union consisting of a large core and a small periphery,the output in the periphery is generally more volatile than in the core. Inaddition, the periphery is more sensitive to changes in the degree of nominalrigidities.Second, asymmetry in the degrees of nominal rigidity may smooth ag-

gregate output and in�ation volatility, while asymmetry in the degree ofmonopolistic competition has essentially no e¤ect on the volatility of themacroeconomic variables of interest. Thus, the present analysis suggeststhat structural asymmetries alone are no hindrance to macroeconomic sta-bility at the unionwide level. At the country level, however, the picture isless clear; our results indicate that there are non-trivial spill-over e¤ects fromasymmetries in nominal rigidities.Third, when shocks are country-speci�c, i.e., when there is no correla-

26

tion between shocks hitting the countries in the monetary union, a countryunambiguously bene�ts in terms of macroeconomic stability by pursuing uni-laterally structural labour market reforms that reduce wage rigidities. Foraggregate shocks hitting the whole monetary union with the same force, how-ever, results are ambiguous.Forth, we �nd that structural labour market reforms have di¤erent e¤ects

on macroeconomic stability at the country level than at the aggregate level.Hence, there is risk of a �reform de�cit�from the unionwide perspective. Anindividual member country may not have an incentive to reform its labourmarket unilaterally, while such reforms may be bene�cial for the monetaryunion as a whole.Given that only a few unambiguous results can be established, the in-

centives to undertake reform from the point of view of individual membercountries, and the desirability of such reform from the point of view of themonetary union as a whole, depend crucially on the structural characteristicsof national labour markets. An interesting topic for future research, then,is to estimate this model on data for a monetary union such as the EMUto identify and quantify the important asymmetries. As a �nal remark, weemphasize that, though we implicitly assume that macroeconomic stabilityis desirable, our statements about the volatility of macroeconomic variablescannot literally be interpreted as statements about welfare. Hence, in futureresearch, we hope to address this issue �along with normative issues concern-ing monetary policy responses to structural asymmetries �in more explicitterms.

27

A Log-linearization

Write the �rst-order condition (17) as

W �itEt

1X�=0

(�i�)� C

� 1�

it+�

Pt+�Nit+� = Et

1X�=0

(�i�)� �

(� � 1)N1+�it+� (35)

Taking the di¤erential with respect to W �it+� , Cit+� , Pt+� and Nit+� , evalu-

ating at the steady-state values �W , C, P and N respectively �dividingthrough by W and rearranging gives the following log-linear approximationaround the steady state:

��it = (1� �i�)Et1X�=0

(�i�)� ��nit+� + ��1cit+� + pt+�� (36)

= (1� �i�)Et1X�=0

(�i�)� �� (lit+� � � (��it � wit+� )) + ��1cit+� + pt+��

where the second equality follows by using a log-linear version of (3) to replacenit+� . Rearranging gives

��it =(1� �i�)1 + ��

��lit � ��wit + ��1cit + pt

�+ (�i�)Et�

�t+1 (37)

Similarly, a log-linear approximation to (25) is given by

wit = �iwit�1 + (1� �i) ��it (38)

Subtracting wit from both sides of (37) and using (38) to eliminate ��it gives

!it =(1� �i) (1� �i�)�i (1 + ��)

��lit + �

�1cit � (wit � pt)�+ �Et!it+1 (39)

where !it = wit � wit�1.

B Flexible wage equilibrium

Suppose wages are �exible as well as prices. In this case, the wage equationbecomes

wit = pt + �lit + ��1ct (40)

28

Substituting out lit by a linear version of the production function (5) gives

wit = pt + �

�1

yit �

1�

uit

�+ ��1ct (41)

implying

wt = pt + �

�1

yt �

1�

ut

�+ ��1yt (42)

Inserting this in aggregated supply

yt = � (pt � wt) + ut (43)

gives

�yt = �

����1

�yt �

1�

ut

�� ��1t �y

�+ ut (44)

or�yt =

1 + �

1 + � (1 + �) + ���1ut (45)

C Sticky wage equilibrium

Imposing the equilibrium condition means cit = ct = yt. Using this, the wageequation (19) and the aggregate supply relation (8) can be combined to givethe �AS�relation

�it + ��1 [(uit � uit�1)� (yit � yit�1)]

= �i���+ (1 + �) ��1 + ��1

�yit +

�� � ��1

�yt � (1 + �) ��1uit

�+��Et�it+1 + �

�1 [(�u � 1)uit � (Etyit+1 � yit)]�

(46)

where a log-linear version of (5) has been used to substitute out lit. Similarly,combining the Euler equation (23), the Taylor rule (26) and the intratemporaldemand function (24) gives the �IS�relation

yit � yit�1 (47)

= �� (�it � �t) + Etyt+1�� (k��t + ky (yt � �ut)� Et�t+1)� yt�1

29

Hence, two equations summarise the dynamics of output and in�ation foreach country. Disturbances to the system follow from the stochastic process

uit = �uuit�1 + "it (48)

We guess that output and in�ation in country i take the forms:

yit =Xj

vjbij0 ujt +

Xj

vjbij1 ujt�1 +

Xj

vjbij2 yjt�1 (49)

and�it =

Xj

vjcij0 ujt +

Xj

vjcij1 ujt�1 +

Xj

vjcij2 yjt�1 (50)

These conjectures imply the following expressions for aggregate outputand in�ation:

yt =Xi

Xj

vivjbij0 ujt +

Xi

Xj

vivjbij1 ujt�1 +

Xi

Xj

vivjbij2 yjt�1 (51)

�t =Xi

Xj

vivjcij0 ujt +

Xi

Xj

vivjcij1 ujt�1 +

Xi

Xj

vivjcij2 yjt�1 (52)

In addition, expectations become

Etyit+1 =Xj

vj�bij0 �u + b

ij1

�ujt +

Xj

vjbij2 yjt (53)

Et�it+1 =Xj

vj�cij0 �u + c

ij1

�ujt +

Xj

vjcij2 yjt (54)

Etyt+1 =Xi

Xj

vivj�bij0 �u + b

ij1

�ujt +

Xi

Xj

vivjbij2 yjt (55)

Et�t+1 =Xi

Xj

vivj�cij0 �u + c

ij1

�ujt +

Xi

Xj

vivjcij2 yjt (56)

To verify our conjectures, we �nd values of the coe¢ cients�bij0 ; b

ij1 ; b

ij2 ; c

ij0 ; c

ij1 ; c

ij2

�that satisfy the restrictions imposed by the log-linear model. Inserting theconjectures in (46) gives

30

Xj

vjcij0 ujt +

Xj

vjcij1 ujt�1 +

Xj

vjcij2 yjt�1

+��1 (uit � uit�1)� ��1 X

j

vjbij0 ujt +

Xj

vjbij1 ujt�1 +

Xj

vjbij2 yjt�1 � yit�1

!

= �i��+ (1 + �) ��1 + ��1

� Xj

vjbij0 ujt +

Xj

vjbij1 ujt�1 +

Xj

vjbij2 yjt�1

!

+�i�� � ��1

� Xn

Xj

vnvjbnj0 ujt +

Xn

Xj

vnvjbnj1 ujt�1 +

Xn

Xj

vnvjbnj2 yjt�1

!� (1 + �) ��1�iuit + �

Xj

vj�cij0 �u + c

ij1

�ujt

+�Xn

vncin2

Xj

vjbnj0 ujt +

Xj

vjbnj1 ujt�1 +

Xj

vjbnj2 yjt�1

!+���1 (�u � 1)uit � ���1

Xj

vj�bij0 �u + b

ij1

�ujt

����1Xj

vjbij2

Xn

vnbjn0 unt +

Xn

vnbjn1 unt�1 +

Xn

vnbjn2 ynt�1

!

+���1

Xj

vjbij0 ujt +

Xj

vjbij1 ujt�1 +

Xj

vjbij2 yjt�1

!(57)

Equating coe¢ cients on uit gives the restriction

vicii0 + �

�1 � ��1vibii0= �i

��+ (1 + �) ��1 + ��1

�vib

ii0 + �i

�� � ��1

�Xj

vjvibji0

� (1 + �) ��1�i + �vi�cii0 �u + c

ii1

�+ �

Xj

vjvicij2 b

ji0

+���1 (�u � 1)� ���1vi�bii0 �u + b

ii1

�����1

Xj

vjvibij2 b

ji0 + ��

�1vibii0 (58)

31

Equating coe¢ cients on ujt where j 6= i gives

cij0 � ��1bij0

= �i��+ (1 + �) ��1 + ��1

�bij0 + �i

�� � ��1

�Xn

vnbnj0

+��cij0 �u + c

ij1

�+ �

Xn

vncin2 b

nj0

����1�bij0 �u + b

ij1

�� ���1

Xn

vnbin2 b

nj0 + ��

�1bij0 (59)

Equating coe¢ cients on uit�1:

vicii1 � ��1 � ��1vibii1

= �i��+ (1 + �) ��1 + ��1

�vib

ii1

+�i�� � ��1

�Xn

vnvibni1 + �

Xn

vnvicin2 b

ni1

����1Xn

vnvibin2 b

ni1 + ��

�1vibii1 (60)

Equating coe¢ cients on ujt�1where j 6= i:

cij1 � ��1bij1

= �i��+ (1 + �) ��1 + ��1

�bij1

+�i�� � ��1

�Xn

vnbnj1 + �

Xn

vncin2 b

nj1

����1Xn

vnbin2 b

nj1 + ��

�1bij1 (61)

Equating coe¢ cients on yit�1 gives

vicii2 � ��1

�vib

ii2 � 1

�= �i

��+ (1 + �) ��1 + ��1

�vib

ii2 + �i

�� � ��1

�Xn

vnvibni2

+�Xn

vnvicin2 b

ni2 � ���1

Xn

vnvibin2 b

ni2 + ��

�1vibii2 (62)

32

and on yjt�1where j 6= i:

vjcij2 � ��1vjb

ij2

= �i��+ (1 + �) ��1 + ��1

�vjb

ij2 + �i

�� � ��1

�Xn

vnvjbnj2

+�Xn

vnvjcin2 b

nj2 � ���1

Xn

vnvjbin2 b

nj2 + ��

�1vjbij2 (63)

Inserting conjectures in (47) givesXj

vjbij0 ujt +

Xj

vjbij1 ujt�1 +

Xj

vjbij2 yjt�1 � yit�1

= �� X

j

vjcij0 ujt +

Xj

vjcij1 ujt�1 +

Xj

vjcij2 yjt�1

!

+(� � �k�) X

n

Xj

vnvjcnj0 ujt +

Xn

Xj

vnvjcnj1 ujt�1 +

Xn

Xj

vnvjcnj2 yjt�1

!+Xn

Xj

vnvj�bnj0 �u + b

nj1

�ujt

+Xm

Xn

vmvnbmn2

Xj

vjbnj0 ujt +

Xj

vjbnj1 ujt�1 +

Xj

vjbnj2 yjt�1

!

��ky

Xn

Xj

vnvjbnj0 ujt +

Xn

Xj

vnvjbnj1 ujt�1 +

Xn

Xj

vnvjbnj2 yjt�1

!+ �ky�

Xj

vjujt

+�Xn

Xj

vnvj�cnj0 �u + c

nj1

�ujt

+�Xm

Xn

vmvncmn2

Xj

vjbnj0 ujt +

Xj

vjbnj1 ujt�1 +

Xj

vjbnj2 yjt�1

!�Xj

vjyjt�1 (64)

Equating coe¢ cients on ujt gives

33

bij0

= ��cij0 + (� � �k�)Xn

vncnj0 +

Xn

vn�bnj0 �u + b

nj1

�+Xm

Xn

vmvnbmn2 bnj0 � �ky

Xn

vnbnj0 + �ky�vj + �

Xn

vn�cnj0 �u + c

nj1

�+�Xm

Xn

vmvncmn2 bnj0 (65)

On ujt�1:

bij1

= ��cij1 + (� � �k�)Xn

vncnj1 +

Xm

Xn

vmvnbmn2 bnj1

��kyXn

vnbnj1 + �

Xm

Xn

vmvncmn2 bnj1 (66)

yit�1:

vibii2 � 1

= ��vicii2 + (� � �k�)Xn

vnvicni2 +

Xm

Xn

vmvnbmn2 vib

ni2

��kyXn

vnvibni2 + �

Xm

Xn

vmvncmn2 vib

ni2 � vi (67)

and �nally, yjt�1 where j 6= i:

bij2

= ��cij2 + (� � �k�)Xn

vncnj2 +

Xm

Xn

vmvnbmn2 bnj2

��kyXn

vnbnj2 + �

Xm

Xn

vmvncmn2 bnj2 � 1 (68)

The restrictions (58)-(63) and (65)-(68) constitute a system of 6I2 equa-tions determining the 6I2 coe¢ cients in the conjectures. Indeed, this systemis recursive. The 2I2 restrictions from equating coe¢ cients on yit�1 may becombined to solve for fbij2 ; c

ij2 gij , which may then be used in the 2I2 restric-

tions from uit�1 to solve for {bij1 ; c

ij1 gij. Finally, these coe¢ cients may be used

in the restrictions from equation coe¢ cients on uit to �nd the remaining 2I2

coe¢ cients fbij0 ; cij0 gij.

34

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37

Figure 1: Standard deviations of country-speci�c and aggregate output andin�ation as functions of nominal rigidity (�); symmetric structures (�1 =�2 = �; �1 = �2 = �; v1 = v2 = 0:5) and di¤erent levels of shock correlations(�").

38

Figure 2: Standard deviations of country 1 and aggregate output and in-�ation as functions of country size (v1); symmetric structures (�1 = �2 =�; �1 = �2 = �;) and country-speci�c shocks (�" = 0).

39

Figure 3: Standard deviations of country 1 and aggregate output and in�a-tion as functions of country 1 nominal rigidity (�1); average nominal rigidity� = 0:5; symmetry wrt. to competition and size (�1 = �2 = �; v1 = v2 = 0:5)for di¤erent levels of shock correlations (�").

40

Figure 4: Standard deviations of country-speci�c and aggregate output andin�ation as functions of a unilateral change in country 1 nominal rigidity (�1);symmetry wrt. to competition and size (�1 = �2 = �; v1 = v2 = 0:5;�2 =0:75) for di¤erent levels of shock correlations (�")

41

Figure 5: Standard deviations of country 1 and 2, and aggregate output andin�ation as functions of country 1 nominal rigidity (�1); average nominalrigidity � = 0:75, asymmetric sizes (v1 = 0:3; v2 = 0:7), symmetry wrt. tocompetition (�1 = �2 = �) for di¤erent levels of shock correlations (�").

42

Chapter 2

Monetary policy and welfare in a monetaryunion with labour market heterogeneity�

Martin SenecaUniversity of Aarhus

July 2008

Abstract

How should monetary policy be conducted in a monetary unionwhen labour market structures di¤er across member countries as isarguably the case in the euro area? This paper develops a dynamicstochastic general equilibrium model of a two-country monetary unionwith labour market heterogeneity to answer this question. Asymme-tries in labour market structures are proxied for by di¤erent degreesof nominal �exibility. A welfare loss function derived as a second-order approximation to household utility is evaluated. Results sug-gest that price in�ation targeting may lead to non-negligible welfarelosses compared to monetary policy alternatives when the importantnominal rigidity is in the labour market. Welfare may be noticeablyimproved in such a case by targeting wage in�ation, especially whenshocks are highly correlated across member countries and labour mar-kets are very heterogeneous. Noticeable welfare improvements can beobtained by putting more weight on �ghting wage in�ation in the morerigid labour market if shocks are less than perfectly correlated.JEL classi�cation: E32Key words: In�ation targeting, monetary policy, nominal wage

rigidity, monetary union, business cycles, shocks

�I thank, without implicating, Torben M. Andersen for very useful comments on a veryearly draft of this paper.

47

1 Introduction

This paper addresses the question of how monetary policy should be con-ducted in a monetary union in which labour market structures are di¤erentacross member countries. It it well known that such labour market asymme-tries are characteristic for the European Monetary Union (EMU), not leastin terms of labour market �exibility, cf. for instance Arpaia and Pichelmann(2007) or Holden and Wulfsberg (2008) for recent documentation. Whilethese labour market asymmetries are frequently mentioned in discussions oflabour market policies, surprisingly little research e¤ort has been devoted tounderstanding the implications of labour market asymmetries for monetarypolicy. In particular, little is known about how monetary policy should bedesigned in a monetary union with labour market asymmetries.1

This paper takes a �rst step in providing answers to this question. Itdevelops a dynamic stochastic general equilibrium (DSGE) model of a two-country monetary union with monopolistic competition in labour marketsand nominal wage rigidities. Labour market asymmetries are modelled byallowing the degrees of monopolistic competition and of nominal rigidityto di¤er across the monetary union�s member countries. Disturbances tothe union may be common across the member countries or idiosyncratic.The paper proceeds by evaluating the welfare of alternative monetary policyrules that the central bank may contemplate, emphasising the potential needto deal with labour market asymmetries across the union. Following theapproach of Rotemberg and Woodford (1999), welfare is evaluated using aquadratic loss function derived from second-order Taylor approximations tothe levels of expected household utility. This exercise is a �rst important stepin understanding how monetary policy is best designed when monetary policyis common across structurally diverse, interdependent economic regions.The current practice in most central banks pursuing an independent mon-

etary policy is to emphasise the stabilisation of goods price in�ation based onthe development of a consumer price index. This includes the European Cen-tral Bank (ECB) responsible for monetary policy in the EMU. This practice�nds theoretical justi�cation in the New Keynesian literature in which nomi-

1Beetma and Jensen (2004) allow for labour market asymmetries in a model of a mon-etary union but focus on monetary and �scal policy interaction. Dellas and Tavlas (2005)present a three-country model allowing for asymmetries in nominal wage �exibility, and�nd that countries with a high degree of nominal wage rigidity are better o¤ in a monetaryunion.

48

nal goods price rigidity is a distinguishing feature, see for instance Woodford(2003). In the basic New Keynesian model, nominal price rigidities lead toan ine¢ cient allocation of resources in the economy unless the constraintsrepresented by nominal rigidities do not bind. This happens in the modelwhen in�ation is stabilised.As argued by Woodford (2003), allowing for nominal wage rigidities is

probably not crucial if the objective of the analysis is to construct a positivemodel of the co-movements of output and in�ation. But it is by no meansobvious that nominal price rigidities are more important than nominal wagerigidities from an empirical perspective. Christiano et al. (2005), for instance,provide evidence of the contrary using US data. Nor does it appear to beempirically realistic to abstract from nominal wage rigidities. Though Smetsand Wouters (2003) estimate a higher degree of nominal price rigidity thannominal wage rigidity for the euro area, their estimates suggest that nominalwage rigidities are substantial; the expected duration of wage contracts isestimated to be about one year.Furthermore, nominal wage rigidities matter in important ways for a

welfare-theoretic assessment of the proper goals of monetary policy as shownby Erceg et al. (2000). In a closed-economy with both nominal price andwage rigidities, they show that the optimal monetary policy places greaterweight on stabilising in�ation in the more rigid variable, prices or wages.That is, from a welfare-theoretic perspective, central banks should stabilisethe nominal variables that fail to adjust so as to make the adjustment con-straint non-binding, preventing misallocation in the markets characterised byrigidities. Indeed, they �nd that in�ation targeting of the sort that is oftenconsidered to be a good approximation to actual central bank behaviour mayinduce substantial welfare losses when the important nominal rigidity is inthe labour market as opposed to the goods market. A central bank operat-ing with a seemingly empirically successful sticky price model may thereforeseriously misjudge the welfare implications of its actions if sizeable nominalwage rigidities are present.In a monetary union, the assessment of the welfare implications of alter-

native monetary policy prescriptions is further complicated by the fact thatmember countries often have very di¤erent characteristics along a number ofdimensions of importance for the economic decision making of agents in theeconomy. Benigno (2004) addresses this question for the case of structuraldi¤erences in goods markets. Speci�cally, he shows that optimal policy ina two-country monetary union in which nominal price rigidities di¤er across

49

the two member countries is such that a higher weight is given to �ghtinggoods price in�ation in the country with the highest degree of nominal pricerigidity. Moreover, Lombardo (2006) assesses the welfare losses of simplemonetary policy rules, including goods price in�ation targeting allowing forasymmetries also in the degree of monopolistic competition in goods mar-kets. He shows that the union�s central bank should put a higher weight onthe more competitive country. In fact, if the two countries di¤er su¢ cientlyin terms of market power in goods markets, Benigno�s (2004) result may beoverturned as the central bank might optimally assign a higher weight to thecountry with more �exible prices if competition is su¢ ciently �erce in thiscountry. This indicates that asymmetries in structural characteristics haveimportant implications for the design of monetary policy.As shown by Andersen and Seneca (2008), di¤erences in structural fea-

tures in labour markets across a monetary union have non-trivial implicationsfor the propagation of shocks across the union and for the incentives for struc-tural labour market reforms. This indicates that labour market asymmetriesmay be important for monetary policy, and providing a better understand-ing of these implications therefore seems an urgent task that should be ofparticular interest to policy makers in Europe, where such labour marketasymmetries are present. To do so, this paper abstracts from other asym-metries that may characterise the monetary union so as to isolate the e¤ectsstemming from labour market asymmetries. The interesting question of howthese asymmetries interact with other potential asymmetries is left for fu-ture research. Hence, structural characteristics in goods markets, preferencesetc. will be symmetric across the union. In addition, goods prices will beassumed to be perfectly �exible. While this is a strong assumption, it servesthe purpose of keeping a strict labour market perspective. Thus, what is al-lowed to di¤er are characteristics of the labour markets in which householdso¤er their labour services. That is, in an important market for householdwelfare, households face di¤erent immediate economic environments as boththe degree of market power in their wage setting and the expected durationof their wage contracts - or more generally employment contracts - may di¤eracross the monetary union, though the emphasis will be on asymmetries inwage rigidities.In this paper, the relative welfare under alternative simple monetary

policy rules is investigated under the assumption that taxes are designedto o¤-set the distortions from monopolistic competition in the economy�ssteady state. This serves to focus attention on distortions induced by dif-

50

fering degrees of nominal wage rigidity. The monetary policies consideredare, �rst, �exible monetary policy rules according to which the central bankmay respond to price in�ation, union-wide or national wage in�ation, and thewelfare-relevant output gap. Second, strict targeting rules are considered thatlead to successful stabilisation of either aggregate price in�ation, aggregateor country-speci�c wage in�ation, or the output gap.An important question for this paper to answer is whether in�ation tar-

geting by a monetary union�s central bank similar to the one used by theECB leads to substantial welfare losses when the important nominal rigidi-ties are in the labour market, and, most importantly, when labour marketsare characterised by di¤ering degrees of nominal rigidity.The results suggest that this may well be the case. In particular, when

prices are �exible and wages rigid, �exible as well as strict price in�ationtargeting regimes lead to non-negligible welfare losses. Welfare may be no-ticeably improved by targeting wage in�ation. The mistake made by tar-geting price in�ation rather than wage in�ation is found to be larger asshocks become more highly correlated across the monetary union�s membercountries, and as the degree of heterogeneity in the labour market structureincreases. Finally, further welfare improvements can be obtained by puttingmore weight on �ghting wage in�ation in the country with a more rigid labourmarket, especially if shocks are idiosyncratic and labour market structureshighly asymmetric.The paper is organised as follows. Section 2 presents the model and

derives its log-linear representation. Section 3 presents the welfare functionused to evaluate alternative monetary policy rules. Section 4 presents thewelfare analysis for a calibrated version of the model. Section 5 concludes.

2 The model

The model economy consists of two countries in a monetary union. Eachcountry has a large number of households and a large number of �rms. Thereis one central bank responsible for monetary policy throughout the union.In particular, the central bank sets the risk-free interest rate Rt earned onone-period risk-free bonds in the union�s single �nancial market. There is noactive �scal policy for stabilisation purposes, and the union is closed to theoutside world.Labour is immobile across borders, and national labour markets are char-

51

acterised by monopolistic competition and nominal rigidities in the form ofCalvo (1983) wage contracts of random duration. With monopolistic com-petition, each household supplies a di¤erentiated labour service and has acertain degree of market power in setting the wage it demands for this ser-vice. Given the wage chosen, the household stands ready to supply the workhours demanded by �rms. These assumptions lead to a downward-slopingdemand curve for each household�s labour service. Although both the degreeof market power in wage setting and the degree of the nominal rigidities areallowed to di¤er between the two countries, emphasis is put on implicationsof di¤erences in the expected duration of wage contracts. Such di¤erencesare taken to represent heterogeneity in labour market structure across themonetary union.All �rms in a given country are assumed to produce the same interna-

tionally traded good in a competitive market, but this good is di¤erentiatedfrom the goods produced by �rms abroad. This leads to a downward-slopingdemand curve for each country�s product. To focus on the labour market,prices of goods are assumed to be perfectly �exible. It is assumed that theweight of each product in the consumption bundle is the same in both coun-tries. Hence, there is no home bias in consumption. The weight assigned toeach product is interpreted as the relative size of the country producing it.

2.1 Firms

The representative �rm in country i 2 f0; 1g produces output Yit accordingto the production function

Yit = AitN it (1)

where Ait is the stochastic period-t productivity of �rms in country i, and0 < < 1 is the degree of returns to scale. Wit represents aggregate wagesin country i paid for the aggregate labour input into production, Nit, as de-scribed below. Real capital is disregarded to simplify, but decreasing returnscan be interpreted as arising from a second factor of production in �xedsupply.The representative �rm in country i maximises pro�ts, which it distrib-

utes to households. There are no nominal price rigidities, and the �rm takesthe price of its product, Pit, as given. The pro�t maximisation problem yields

52

a demand for aggregated labour services de�ned by the relation

(1� tei )Wit

Pit= AitN

�1it (2)

when tei is a �xed-rate employment subsidy paid to �rms (and �nanced bylump-sum taxes, cf. below).2 The labour demand relation equates the realwage (as perceived by �rms) to the marginal product of labour.

2.2 Households

In each period t, each household h in country i supplies a di¤erentiatedlabour service, Nit (h). The labour used in production in country i, Nit, isassumed to be an aggregate of the continuum of labour services supplied bythe households:

Nit =

�Z 1

0

Nit (h)�i�1�i dh

� �i�i�1

(3)

where �i > 1 is the elasticity of substitution between labour services.Each household sets the wage rate it demands for its labour service as

described below and satis�es �rms�labour demand at the chosen wage. Thatis, given existing wage contracts, household h�s labour e¤ort is determinedby demand. This demand for household h�s labour service is determined bythe cost minimisation problems of the country�s �rms, which minimise coststaking households�wage rates, Wit(h), as given. This leads to a demand forhousehold h�s labour service given by

Nit (h) =

�Wit (h)

Wit

���iNit (4)

when Wit is the wage index with the property that the minimum cost of Nitunits of aggregate labour is given by WitNit. It follows that the demandfor household h�s labour service is a decreasing function of the household�srelative wage with elasticity �i. Hence, �i is inversely related to the degreeof market power in wage setting.

2The employment subsidy is used to neutralise the distorting e¤ect from monopolisticcompetition in the steady state around which the model is log-linearised. This facilitatesthe welfare analysis of monetary policy alternatives emphasising implications of nominalwage rigidities.

53

Household h in country i 2 f1; 2g has the utility function

Et

1X�=0

����

� � 1Cit+� (h)��1� � 1

1 + �Nit+� (h)

1+�

�(5)

where Et is an operator representing expectations over all states of the econ-omy conditional on period-t information, � 2 (0; 1) is the subjective discountfactor, and Cit (h) is a real consumption index. � > 0 is the elasticity ofintertemporal substitution of consumption, and � > 0 is the inverse of theFrisch labour elasticity.The consumption index is de�ned over the di¤erentiated commodities

produced in the union�s member countries. Speci�cally,

Cit =hv1�1 C

��1�

i1t + v1�2 C

��1�

i2t

i ���1

(6)

where � > 0, vj is the relative size of country j 2 f1; 2g ; and Cijt representsconsumption of country j�s commodity by households in country i. In everyperiod t, households choose Cijt for a given level of real consumption tominimise consumption expenditures. This yields a demand for country j�sproduct in country i given by

Cijt = vj

�PjtPt

���Cit (7)

when Pt is the price index de�ned by

Pt =�v1P

1��1t + v2P

1��2t

� 11�� (8)

This price index has the property that the minimum cost of Cit units of realconsumption is given by PitCit.Asset markets are assumed to be complete, i.e., available �nancial assets

completely span the possible states of the economy.3 This assumption leadsto the following period-t �ow budget constraint for a household in country i:

Et [Qt;t+1Bit (h)] + PtCit (h) + Tit = Bit�1 (h) +Wit (h)Nit (h) + �it (9)

3Note that this is likely to decrease the potential welfare improvements through sta-bilisation policy as it provides an insurance mechanism for households. In particular,households share consumption risk through these complete markets so that consumptionlevels are equalised across the monetary union.

54

The right-hand side gives available resources as the sum of initial �nancialwealth, Bit�1 (h), labour income, Wit (h)Nit (h), and nominal pro�t income,�it. The left-hand side represents the allocation of resources to consumption,PtCit (h), bond-holdings, Et [Qt;t+1Bit (h)], where Qt;t+1 is the asset pricingkernel, and to a lump-sum tax, Tit, used by the government to �nance anemployment subsidy paid to �rms.Households choose real consumption and wages to optimise expected util-

ity (5) subject to the sequence of budget constraints (9) and labour demand(4).4 De�ning the net risk-free nominal interest rate Rt by the relation

(1 +Rt)�1 = E [Qt;t+1] (10)

the �rst-order conditions determining the optimal choice of consumption andbond holdings can be combined to yield the Euler equation

C� 1�

t = � (1 +Rt)Et

�C� 1�

t+1

PtPt+1

�(11)

where the international risk-sharing property that consumption is equalisedacross households and countries has been used.5

Wages are set by households in a staggered fashion. Following Calvo(1983), each household in country i is allowed to reset the wage rate it de-mands for its labour service with a �xed probability (1� �i). Hence, thewage rate set by household h at time t, W �

it (h), is the prevailing wage ratefor the household at time t + � , i.e., Wit+� (h) = W �

it (h), with probability��i , and the expected duration of a contract is given by (1� �i)

�1. Thecomplete-markets assumption implies that the fraction (1 � �i) of house-holds in country i changing their wage rates at time t choose the same rateW �it. The remaining fraction �i of households continue with the wage rate

prevailing at time t � 1 where the distribution of wage rates is unchanged.Hence, the law of motion of the aggregate wage index in country i is given

4Implicitly, optimisation is also subject to a solvency condition that may be used totransform the sequence of �ow budget constraints into a single life-time budget constraint.This has no e¤ect on the �rst-order conditions, and since a log-linearised version of themodel around its steady state is analysed here, the bounded stochastic shock processesspeci�ed below will ensure solvency at all times and in all states.

5Throughout, aggregate variables are indicated by the omission of country subindicesand are given as averages of national variables with relative country sizes as weights.

55

by

Wit =

�Z 1

0

Wit (h)1��i dh

� 11��i

=h�iW

1��iit�1 + (1� �i) (W �

it)1��ii 11��i (12)

When a household resets its wage, it does so to maximise expected utility(5) subject to the demand for its labour (4), its budget constraint (9) and theprice setting mechanism just described. For a household changing its wagerate at time t, this is equivalent to maximizing the following function withrespect to W �

it (h) subject to (4) and (9):6

Et

1X�=0

(�i�)�

��

� � 1Cit+� (h)��1� � 1

1 + �Nit+� (h)

1+�

�(13)

The �rst-order condition can be written as

Et

1X�=0

(�i�)�

���i

1� �iNit+� (h)

� + C� 1�

t+�

W �it (h)

Pt+�

�Nit+� (h)

�= 0 (14)

It follows that the monopolistically competitive household sets its wage rateso that the marginal utility of income from an extra unit of labour e¤ort is aconstant mark-up over the marginal disutility in discounted expected valueterms.Note that in the special case with �exible wages in which all households

are allowed to reset the wage each period, the �rst-order condition collapsesto

Wit

Pt=

�i�i � 1

N�itC

1�t (15)

where Nit = Nit (h) and Wit = W�it (h) for all h. That is, wages are set so as

to equalise the real wage (as perceived by the household) to a mark-up overthe marginal rate of substitution.

2.3 Log-linear representation

Welfare will be evaluated using solutions to the model in log-linear form inwhich variables are expressed in log-deviations from the steady state with

6This di¤ers from (5) in that implicit terms representing states where the wage to beset is not the prevailing wage are excluded.

56

stable wages.7

A log-linear version of the production function (1) is given by

yit = ait + nit (16)

Technology, ait, is assumed follow a �rst-order autoregressive process

ait = �aait�1 + "it (17)

where the innovations, "it, are N (0; �2i ) and may be correlated across coun-tries as governed by the correlation parameter �" 2 [0; 1] :These innovationsare assumed to be the only shocks to the monetary union.A log-linearisation of the labour demand relation gives

wit � pit = ait � (1� )nit (18)

Note that the employment subsidy drops out of the log-linear labour demandrelation. Hence, it has no e¤ect on the dynamic responses to shocks in thelog-linearised economy.Log-linearisations of the �rst-order condition from the households�wage

setting problem (14) and the law of aggregate wages (12) can be combinedto yield a New Keynesian Phillips curve for wage in�ation:8

!it = �i���1ct + �nit � (wit � pt)

�+ �Et!it+1 (19)

where !it = wit � wit�1 is wage in�ation in country i, and �i is a decreas-ing function of the Calvo parameter �i and of the elasticity of substitutionbetween labour services �i:

�i =(1� �i) (1� �i�)�i (1 + ��i)

(20)

The supply side of the model is thus summarised by equations (16)-(19)for i 2 f0; 1g. Note that (19) collapses to the labour supply relation

wit � pt = ��1ct + �nit (21)

7Steady state variables are indicated by the omission of time subscripts, and lower-case letters denote log-deviations from steady-state values of corresponding upper-casevariables.

8See Andersen and Seneca (2008, app. A) for details on the derivation.

57

in the virtual equilibrium with full wage �exibility, i.e. for �i = 0.The demand side is represented by a log-linear version of the Euler equa-

tion (11) given asct = Etct+1 � � (rt � Et�t+1) (22)

where �t = pt�pt�1 is price in�ation, and a demand relation for the productproduced in each country i

yit = �� (pit � pt) + yt (23)

The latter is found by summing (7) over countries i, log-linearising, andimposing the equilibrium condition

ydit = yit (24)

where ydit is the aggregate demand in the monetary union for products pro-duced in country i. Note that monetary policy a¤ects the economy throughthe aggregate demand relation (22) only. In this sense, its e¤ect on the twocountries is symmetric.

2.4 Monetary policy

The model is closed by specifying the monetary policy reaction function. Twotypes of monetary policy rules are considered: Explicit �exible targeting rulesand implicit strict targeting rules. Both types are within the class of simplemonetary policy rules in the sense that they are operational rules that maybe considered realistic in an actual monetary policy regime rather than theoutcome of a Ramsey problem that the central bank may attempt to solve.9

With a �exible targeting rule, the central bank sets the risk-free interestrate in response to a vector of endogenous variables. Attention is restrictedto rules that may be stated in log-linear form as

rt = k��t + k1!1t + k2!2t + kyyt (25)

where yt = yt � ynt is the output gap given as output in excess of the levelof output in the virtual �exible wage equilibrium. As shown in appendixA, this "natural" level of output can be found by combining equations (16),(18), (21) and (23). It reads

ynt =1 + �

(1� ) + �+ ��1at

9For a thorough discussion of optimal vs. simple policy rules, see Woodford (2003).

58

For k1 = k2 = 0, (25) reduces to the monetary policy rule speci�ed byTaylor (1999), at least up to the de�nition of the output gap. Similarly, fork1 = k2 = ky = 0, the rule reduces to a very simple rule according to whichthe central bank is only concerned about price in�ation. More generally,however, (25) allows the central bank to respond not only to average wagein�ation (k1 = k2 > 0), but also to place di¤erent weights on wage in�ationin the two countries (0 < k1 6= k2 > 0). In the welfare analysis below, itis of particular interest to investigate if welfare is improved by allowing thecentral bank to do so.With a strict targeting rule, the central bank is assumed to successfully

stabilise union-wide price in�ation, national wage in�ation in country 1, na-tional wage in�ation in country 2, aggregate wage in�ation, or the outputgap. Given these restrictions imposed on the equilibrium dynamics, the pathsof the central bank�s policy instrument will be complicated functions of theshocks to the economy. The evaluations of these rules are important bench-marks in an analysis of the relative importance of alternative target variablesfor monetary policy.

3 The welfare function

Following the approach of Rotemberg and Woodford (1999), the social wel-fare measure used to assess alternative monetary policy rules is derived fromthe utility functions of households. The welfare function gives the welfarelosses experienced by households when the economy deviates from its e¢ -cient path expressed as a percentage of steady-state consumption. As shownin appendix B, under the assumption that the distortions associated withmonopolistic competition have been eliminated in the steady-state by appro-priate choices for the employment subsidies in the two countries (see furtherdiscussion below), the average period welfare loss can be expressed as

Lt = �yV AR (yt) + �sE�s2t�+

2Xi=1

�!iV AR�!2jt�+ t:i:p: (26)

where yt = yt � ynt is the output gab, st = st � ~snt is a terms-of-trade gapfor the terms of trade de�ned as st = p2t � p1t, and t:i:p: represents termsindependent of policy that may safely be ignored when comparing monetarypolicy alternatives. The function parameters are given as

59

�y =1

2

���1 +

(1� ) + �

��s =

1

2

(1 + �) �2

v1v2

�!i =1

2

vi�i �i (1 + ��i)

(1� �i) (1� �i�)=1

2

vi�i

�i

Welfare will be evaluated by the solutions to the model in log-linear formas described above. Given the loss function derived here, this gives a validwelfare ranking of policy alternatives only when the steady state is e¢ cient;i.e. when the distortion from monopolistic competition that leads to anine¢ ciently low output level (regardless of whether nominal rigidities mightbe present) has been eliminated, cf. Kim and Kim (2003) and Woodford(2003, ch. 6). Otherwise, �uctuations around the steady state would havean asymmetric impact on welfare as positive shocks would push the economycloser towards the e¢ cient level and so actually improve welfare, and negativeshocks would push the economy a long way away from the e¢ cient level andso reduce welfare considerably. To take account of such e¤ects, a second-orderapproximation to the model�s structural relations is required.The e¢ cient allocation is characterised by the e¢ ciency conditions that

the marginal rate of substitution equals the marginal product of labour ineach country.10 Combining (2) and (15) gives

AitN �1it

PitPt=(1� tei ) �i�i � 1

C1�t N

�it (27)

which reduces to

AiN �1i =

(1� tei ) �i�i � 1

C1�N�

i (28)

in the steady state. Hence, any remaining gap between the marginal rate ofsubstitution and the marginal product of labour as a consequence of marketpower in wage setting can be eliminated in the steady state by setting tei =��1i . This assumption will be maintained in the welfare analysis.

10Formally, this can be veri�ed as the outcome of a sequence of static social planner�sproblems maximising a weighted average of the instantaneous utility functions of a generichousehold in each country subject to the economy�s resource constraints.

60

From (26), welfare losses can be seen to be driven by variation in fourvariables, namely the union-wide output gap, the terms-of-trade gap, and thetwo national wage in�ation levels. Variation in these variables are related tothe remaining distortions in the economy. The sources of these remainingine¢ ciencies are the nominal rigidities represented by staggered wage con-tracts of random duration, and an international externality similar to theone identi�ed by Corsetti and Pesenti (2001).The wage in�ation and aggregate output gap �uctuations appear in (26)

for the same reasons that they appear in loss functions for closed economieswith nominal wage rigidity, cf. for instance Erceg et al. (2000) and Galí(2008, ch. 6). With nominal wage rigidities, wages are not fully adjustedin response to shocks to the economy, and the average mark-up of the realwage (as perceived by the household) over the marginal rate of substitu-tion will generally di¤er from the desired one prevailing in the �exible wageequilibrium. In addition, the staggered wage setting implied by the randomduration of contracts under the Calvo wage-setting scheme leads to a relativewage distortion and so to a suboptimal allocation of hours across households.Only when the economy is stabilised at a point where households have no in-centives to change their wages, given the prevailing wages and current shocksto the economy, will the constraint represented by the nominal rigidity be-come unbinding. In this case, wages are stabilised and the �exible wageallocation is attained. The only di¤erence to the closed economy is that theloss function weighs the contribution from wage in�ation in each of the twocountries according to its size and structural characteristics.In the limiting case where v1 = 1 and v2 = 0, (26) reduces to the loss

function for a closed economy with sticky wages and �exible prices (see forinstance Galí, 2008, ch. 6):

Lt = �yV AR (yt) + �!1V AR�!21t�

(29)

Similarly, when the two countries are identical and shocks are common toboth countries, there are no terms-of-trade movements and the loss functionbecomes

Lt = �yV AR (yt) + �!V AR�!2t�

(30)

where �! = �!1 = �!2 and !t = !1t = !2t. In this particular case, themonetary union is isomorphic to a standard New Keynesian economy with�exible prices and sticky wages, and the optimal policy takes a very simpleform. The central bank should simply keep wages stable, and it follows

61

from (19) that the �exible wage equilibrium will automatically be attained.Moreover, given that an employment subsidy ensures e¢ ciency in the steadystate, (27) implies that this �exible wage allocation will also be e¢ cient.This result can be generalised further to the case with structural het-

erogeneity. This is because wage changes are the only mechanism throughwhich common shocks can be propagated di¤erently in the two countries. Thestructural features that are allowed to di¤er across countries are all containedin the composite parameter �i in (19). Hence, if wage in�ation is somehowsuccessfully stabilised in each of the two countries, common shocks will in-duce identical behaviour of other national variables. This means that therecan be no terms-of-trade changes, and so the only remaining potential sourceof welfare loss would be the output gap. By a similar argument as above,however, zero wage in�ation is associated with a zero output gap. Hence,in the case of common shocks and structural heterogeneity, optimal mone-tary policy would be one that successfully stabilises wage in�ation across themonetary union.11

In the general case with less than perfectly correlated shocks, the centralbank faces a trade-o¤ between stabilisation of all the variables entering thesocial welfare function. This is because the central bank needs to consideran international externality in addition to the distortions created by thenominal rigidities themselves. This extra distortion is re�ected in the terms-of-trade gap term in (26). Notice also from (27) that, as a consequence ofmovements in the terms of trade, the "natural" �exible wage equilibrium isgenerally ine¢ cient even in the case of an employment subsidy ensuring thee¢ ciency of the steady state around which the model�s structural relationsare approximated. Consequently, the terms-of-trade gap entering (26) isde�ned not simply as the deviation of the terms of trade from its "natural"level under �exible wages, but rather as the deviation from a multiple of this"natural" level. Only in the special case where the intertemporal elasticityof substitution is one will the �exible wage equilibrium be e¢ cient in thegeneral case (see the appendix for details).The international externality arises because movements in the terms of

trade works to shift demand between goods produced in each of the two coun-tries. For instance, an increase in the terms of trade de�ned as st = p2t� p1t11This leaves open the question of whether the central bank is able to do so given that

it has one instrument to its disposal, the e¤ect of which goes through aggregate demand,cf. (22). See the welfare analysis below for further discussion.

62

works to shift demand from goods produced in country 2 to goods producedin country 1. This decreases the disutility from work in country 2 withoutreducing the utility from consumption, which stay constant as a consequenceof international risk sharing. Hence, a positive technology shock in coun-try 1, which increases output in country 1 and reduces the relative price ofits product, e¤ectively allows country 2 to act as a monopolist increasingits relative price and reducing its output. This means that more hours canbe devoted to leisure without reducing the opportunity for consumption ofother goods.12 The �ip-side of this, of course, is that households in country1 work more without increasing their consumption. The presence of nominalrigidities works to amplify this e¤ect on impact. This follows since wageadjustment costs will prevent workers in country 1 from increasing wages asmuch as they would have done without the restrictions of the Calvo wage-setting mechanism. As a consequence, �rms demand more labour and theincrease in output is larger than in the case of �exible wages. This, in turn,ampli�es the terms-of-trade e¤ect through the equilibrium e¤ect on prices.As noted above, the weights with which the output gap, the terms-of-

trade gap and the wage in�ation �uctuations enter the loss function dependon the model�s structural parameters. Considering the weight on the outputgap �rst, note that

@�y�< 0;

@�y < 0;

@�y�> 0 (31)

That is, the weight on the output gap is decreasing in � and , the elasticityof intertemporal substitution and the degree of returns to scale, respectively,and increasing in �, the inverse of the Frisch labour elasticity. This is becausea reduction in � or , or an increase in �, increases the ine¢ ciency gapbetween the marginal product of labour and the marginal rate of substitutionfor any given deviation from the �exible wage equilibrium. To see this,consider (27) augmented with an ine¢ ciency gap, gapit, de�ned residually so

12In this economy with perfect risk sharing, consumption stays constant. In a moregeneral setting, consumption of other goods may fall depending on the degree of substi-tutability, but as long as goods produced in the two countries are substitutes (� > 1),the reduction in utility from a fall in consumption will be smaller than the reduction indisutility from the change in workload. For � = 1, the two e¤ects would cancel out. SeeDe Paoli (2007) for a discussion in relation to a small open economy with sticky pricesand �exible wages.

63

as to make the relation hold in the sticky wage equilibrium

AitN �1it

PitPt= gapitC

1�t N

�it (32)

and notice that a given deviation of consumption and hours from their "nat-ural" values would increase the value of gapit needed to make this relationhold if � or are decreased, or � increased.Similarly,

@�s�> 0;

@�s�> 0;

@�s < 0 (33)

which means that the weight on the terms-of-trade gap is increasing in � and�, and decreasing in . Intuitively, an increase in � increases the e¤ect ofterms-of-trade movements on demand, a reduction in increases the changein hours needed to adjust to the change in demand, and an increase in �ampli�es the utility e¤ect of such changes. Moreover, this weight is decreas-ing in the degree of asymmetry in size as re�ected in v1v2 (which takes itsmaximum for v1 = v2 = 0:5).Finally, note that

@�!ivi

> 0;@�!i�i

> 0;@�!i�i

> 0;@�!i�

> 0;@�!i

> 0 (34)

The �rst of these derivatives re�ects the e¤ect of relative size on the ag-gregate welfare function as described above, e¤ectively because the centralbank weighs welfare according to the mass of economic activity in the twocountries. The second of the derivatives illustrates that an increase in thedegree of nominal wage rigidity in a country increases the country�s contri-bution to the monetary union�s welfare loss. This indicates that the centralbank should be more concerned about wage in�ation in the country withthe most rigid labour market as re�ected in the expected duration of wagecontracts. The third of the derivates suggests that the central bank shouldbe more concerned about wage in�ation in the country with the most com-petitive labour markets. In other words, the welfare implications of givennominal wage rigidities may be larger if the labour market is very compet-itive.13 Finally, an increase in � increases the e¤ect of any given nominal13This e¤ect is similar to the �nding of Lombardo (2006) that a central bank in a two-

country monetary union with nominal price rigidities should be more concerned aboutprice in�ation arising in the member country with the most competitive goods market.

64

rigidity by reducing the slope of the Phillips curve for wage in�ation, whilean increase in ampli�es the e¤ect of a given suboptimal allocation of hoursacross households through the aggregate labour input.

4 Welfare analysis

This section evaluates a number of monetary policy alternatives using theloss function derived in the previous section. The loss function is evaluatedusing a solution to the log-linearised model in section 2 when the model�sparameters are set at values commonly used in the literature. The use of thelog-linear structural relations means that the welfare analysis is validly rank-ing monetary policy alternatives according to their ability to counter distor-tionary e¤ects from nominal rigidities. The analysis abstracts from welfareimprovements that could be made by an appropriately designed monetarypolicy to undo the steady state distortions from monopolistic competition.Instead, it is assumed that �scal policy neutralises this distortion in thesteady state around which the model is log-linearised.

4.1 Calibration

The model�s parameters are set to values that are within the ranges consid-ered in the literature and not too far from those estimated for the euro area,for instance by Smets and Wouters (2003). It is important to note, however,that the objective here is to illustrate particular features of the economy,rather than to build a full empirical model.Considering the parameters governing preferences �rst, the value for the

subjective discount factor is set to � = 0:99, corresponding to a steady-state interest rate of four per cent with the interpretation that a periodcorresponds to a quarter. In the baseline calibration, utility is assumed tobe logarithmic in consumption (� = 1), and the Frisch elasticity of labour isset to unity (� = 1). The elasticity of substitution between goods producedin the two countries is set to � = 4 in the baseline calibration. This isprobably in the high end of empirical estimates, but it serves to highlight thee¤ects in play in the model. A lower value is considered in the sensitivityanalysis, and though this variable has a substantial in�uence on the level ofwelfare losses, the qualitative implications for the choice between monetarypolicy alternatives are robust to the changes in � considered. In the baseline

65

calibration, the weights to the two countries�goods in the real consumptionindex are equalised to one half, i.e. v1 = v2 = 0:5.The baseline calibration of the supply-side parameters is given by = 0:7,

�1 = �2 = 4; �u = 0:9 and �1 = �2 = 0:15. This corresponds to an economywith a labour share of 70 per cent and a wage mark-up of approximately 33per cent. The technology shocks considered are temporary but very persis-tent. In the welfare analysis below, the size of these shocks as determinedby the standard deviation are crucial in determining the levels of the welfarelosses incurred under alternative monetary policy rules. For a quantitativeassessment of the level of welfare losses it is therefore important to considershocks of a size that is empirically plausible. However, this is not absolutelycrucial for the purposes of this paper because the emphasis is on the contri-bution of labour market heterogeneity to the relative levels of welfare lossesunder alternative monetary policy rules. A more important quali�cation re-gards the absence of other shocks that may a¤ect resource allocations undernominal rigidities. There is considerable controversy in the macroeconomicliterature concerning the relative importance of di¤erent shocks in driving thebusiness cycle (see for example Galí and Rabanal, 2005), and a quantitativeassessment of the level of welfare losses would need to take the contributionof other shocks (e.g. demand shocks) to the �uctuations in endogenous vari-ables into account. Concerning the correlation of shocks across countries,three cases are considered. In the �rst, �" = 0 and shocks are uncorrelated.This corresponds to purely country-speci�c shocks. In the second, �" = 0:5and shocks are correlated put imperfectly so. And �nally in the third, �" = 1and shocks are perfectly correlated across countries. This corresponds tocommon shocks to the monetary union.Concerning the degree of nominal rigidity, nine cases are considered that

di¤er with respect to the average degree of nominal rigidity in the monetaryunion and the degree of heterogeneity across countries. The average degreeof nominal rigidity is measured by the average expected duration of wagecontracts under the Calvo wage setting scheme:

AED =v1

1� �1+

v21� �2

(35)

Similarly, the degree of heterogeneity in nominal rigidities is measured bythe relative expected duration de�ned as

RED =1� �11� �2

(36)

66

Welfare is evaluated for combinations of AED 2 f3; 4; 5g and RED 2f1; 2; 3g.Finally, six combinations of values for the parameters in the �exible tar-

geting rules are considered. First, a price in�ation rule with (k�; k1; k2; ky) =(1:5; 0; 0; 0), a symmetric wage in�ation rule with (k�; k1; k2; ky) = (0; 1:5; 1:5; 0),and an asymmetric wage in�ation targeting rule with (k�; k1; k2; ky) = (0; 1:1; 2; 0)so that the central bank puts more weight on the member country withthe more rigid labour market (which is always country 2). Second, each ofthese rules are combined with a positive parameter on the output gap; i.e.three additional rules are considered with (k�; k1; k2; ky) = (1; 5; 0; 0; 0:5=4),(k�; k1; k2; ky) = (0; 1:5; 1:5; 0:5=4), and (k�; k1; k2; ky) = (0; 1:1; 2; 0:5=4).The values of the parameters are of the size suggested by Taylor (1993).

4.2 Main results

Results from the baseline calibration are presented in tables 1-3 for REDgiven by 1, 2 and 3 quarters, respectively. That is, table 1 presents resultsfor a symmetric monetary union, table 2 presents results for an asymmetricunion where wage contracts are expected to be twice as long in country 2 asin country 1, and table 3 presents results for a highly heterogeneous monetaryunion with wage contracts thrice as long in country 2 as in country 1. Ineach table, the left panel gives results for AED = 3 corresponding to anaverage expected duration of wage contracts of three quarters, the middlepanel gives results for AED = 4 equivalent to an average expected durationof one year, and the right panel for AED = 5. In each panel, the �rst columnshows results for country-speci�c shocks with �" = 0, the second columnfor imperfectly correlated shocks with �" = 0:5, and the third column forcommon shocks with �" = 1.Consider the symmetric case in table 1 �rst. When shocks are common,

welfare losses can be eliminated if the central bank avoids targeting pricein�ation �at least up to the fourth decimal place of a percentage of steadystate consumption. In this �exible-price economy, when the central bankdoes not interfere with the price adjustment process, movements in wagesthat would lead to an ine¢ cient allocation can be avoided. All the centralbank needs to do is to stand ready to respond to deviations from zero wagein�ation so as to ensure determinacy; i.e. to rule out the possibility of endingup in a situation with inherent instability. The welfare losses resulting fromboth a �exible and a strict in�ation rule are high compared to the other cases

67

considered, and they are increasing in the average degree of nominal rigidity.This indicates that if shocks are common to the monetary union and theimportant nominal rigidity is in the labour market, a central bank responsiblefor monetary policy in this union may in�ict non-negligible welfare losses onthe economy if its key target variable is price in�ation.When shocks are idiosyncratic, the welfare losses from in�ation targeting

policies are smaller than in the case of common shocks. Again, welfare lossesare increasing the average expected duration of contracts. The e¢ cient al-location can no longer be achieved, but welfare may be improved by lettingthe central bank respond to wage in�ation instead of price in�ation. In thiscase, the lowest welfare loss is obtained by targeting aggregate wage in�ation,and a �exible wage in�ation rule results in the same welfare loss as a strictaggregate wage in�ation rule. In addition, the same welfare loss is achievedby allowing the central bank to respond to the output gap in addition toaggregate wage in�ation in the �exible rule and to strictly target the out-put gap. This suggests that these alternatives may be close to achieving thebest monetary policy can do within the class of monetary policy rules con-sidered. Not surprisingly, given the symmetry of labour market structures inthis case, welfare losses increase if the central bank is more concerned aboutwage in�ation in one of the two member countries. Note also that strictlytargeting wage in�ation in one country increases welfare losses in comparisonwith targeting aggregate wage in�ation. This is because stabilisation in onecountry comes at the expense of greater �uctuations in the other country.Results for imperfectly correlated shocks are in between the two cases with

idiosyncratic and common shocks, respectively. Welfare losses are higher un-der price in�ation targeting than when shocks are uncorrelated, but they arenoticeably smaller under wage in�ation targeting regimes. Again, the resultssuggest that the central bank should target wages either by following a �exi-ble wage in�ation rule responding to aggregate wage in�ation, or by strictlytargeting aggregate wage in�ation or the output gap. Consequently, as thegap between welfare losses under price and wage in�ation targeting increasesin the correlation of shocks, it appears that the mistake made by targetingprice in�ation rather than wage in�ation when the important nominal rigid-ity relates to wages and not goods prices is more serious when shocks arehighly correlated. The reason for this is the international externality work-ing through the terms of trade, which is absent when shocks are commonto the two countries. By dampening price in�ation, the central bank alsodampens the terms-of-trade movements. This in itself has a bene�cial e¤ect

68

on welfare o¤-setting some of the welfare losses resulting from the interfer-ence with the price adjustment process that would otherwise facilitate wagestability.Table 2 presents results for the case with labour market heterogeneity in

the sense that wage contracts can be expected to last twice as long in country2 as in country 1. As discussed above, the optimal monetary policy in caseof common shocks and labour market heterogeneity is one that successfullystabilises in�ation in each of the two member countries. From table 2, it isseen that the optimal policy can be implemented as long as the central bankdoes not target price in�ation but rather stands ready to react to deviationsfrom the zero wage in�ation target. Note also that the mistake made bytargeting price in�ation is larger in the case with labour market heterogeneitythan in a symmetric monetary union. As before, welfare losses under pricein�ation targeting are increasing in the average degree of nominal rigidity.When shocks are uncorrelated, welfare losses are all positive. Price in-

�ation targeting either through a �exible rule or through strict targetingleads to considerably higher welfare losses than the other policy alternativesconsidered. Thus, the symmetric �exible wage in�ation rule performs betterthan the �exible price in�ation rule. Moreover, welfare may be noticeablyimproved by putting more weight on wage in�ation in the more rigid countryand less on the more �exible one. In addition, responding to the output gapas well further improves welfare. Notice also that strictly targeting wagein�ation in the more rigid country comes close to achieving the same wel-fare loss as the asymmetric �exible rules, while the best policy response isone that stabilises the output gap completely. This indicates that furtherimprovements in welfare may be achieved by carefully selecting the values ofparameters in the �exible targeting rule. It this connection, it is importantto note that it may not be possible for an actual central bank to respondto the output gap as it is de�ned here. The output gap entering the lossfunction is the welfare relevant output gap de�ned as output in deviationsfrom the virtual �exible wage output. This generally makes the output gapunobservable, and targeting an output gap derived, say, by �ltering an out-put series will be a very poor approximation to targeting yt, cf. for instanceSbordone (2002).Results for shocks with �" = 0:5 fall in between results for idiosyncratic

and common shocks with heterogeneous labour markets as well. Again, thebest performing monetary policy under the baseline calibration is strict out-put gap targeting, and wage in�ation targeting performs noticeably better

69

than price in�ation targeting. But although welfare is improved by takingthe di¤erences in wage rigidity into account, the bene�t from doing so issmaller in this case than in the case with idiosyncratic shocks.Table 3 presents results for a highly asymmetric monetary union with

a relative expected duration of wage contracts set to three. The main re-sults discussed above go through to this case. For common shocks, welfarelosses can be eliminated by abstaining from price in�ation targeting. Foridiosyncratic shocks, wage in�ation targeting is better than price in�ationtargeting, and welfare can be further improved by assigning a higher weightto the country in which labour markets are more rigid. In addition, alsotargeting the output gap improves welfare, and the lowest welfare loss isobtained by strictly targeting the output gap. Again, welfare losses are in-creasing in the degree of nominal rigidity. The main di¤erence is that thewelfare improvement obtained by following an asymmetric wage in�ation ruleis higher in this case, where labour market structures are very asymmetric.Similarly, strictly targeting wage in�ation in the more rigid country becomesmore appealing as labour markets are more heterogeneous, and this is secondonly to the strict output targeting rule in this case.

4.3 Sensitivity analysis

Table 4 presents a sensitivity analysis for the case withAED = 4 andRED =2 (cf. the middle panel in table 2). Alternative values are considered for �,� and �, in particular � = 2=3, � = 0:2 and � = 1:5.14

The left panel of table 4 presents results for � = 2=3, keeping the re-maining parameters at the baseline values. This change in the value of theintertemporal elasticity of substitution has no e¤ect on the main results pre-sented above. The main di¤erence is that welfare losses are larger under theprice in�ation targeting regimes. A contributing factor to this result is thecurvature e¤ect from changing this parameter discussed in relation to the lossfunction. A lower � increases the ine¢ ciency gap and hence welfare lossesfor a given deviation from the �exible wage equilibrium. Hence, a monetarypolicy that fails to close this ine¢ ciency gap will result in higher welfarelosses when � is low. Notice, however, that for � = 2=3, the �exible wagetargeting rules are unable to stabilise wages enough to completely eliminate

14Other values of these parameters have been considered, but the signs of derivatives ofchanges are the same as those discussed here.

70

welfare losses up to the fourth decimal place when shocks are common.The middle panel of table 4 gives results for � = 0:2, keeping the remain-

ing parameters at the baseline values (so that now again � = 1). In thiscase, welfare losses are generally lower. A higher labour elasticity decreasesthe curvature e¤ect contributing to this result, now along with the labourresponse e¤ects working through the slope of the Phillips curve and withthe direct utility e¤ects of ine¢ ciency as discussed in section 3.1. The mainstory that wage in�ation targeting is better than price in�ation targeting,and that welfare may be improved by following an asymmetric policy rule, iscon�rmed in this case. Moreover, the best monetary policy alternative con-sidered is now the one that leads the central bank to stabilise wage in�ationin the more rigid country.The right panel of table 4 gives results for � = 1:5 keeping the remaining

parameters at the baseline values. The main e¤ect of this change in the inter-national elasticity of substitution is to lower the welfare losses by reducingthe international externality. As the bene�t from targeting price in�ationgoes through the terms of trade, it follows that the largest relative reductionin welfare losses occurs under the wage targeting policy regimes. Again, anasymmetric wage targeting rule is preferable when shocks are idiosyncratic,and welfare losses can be eliminated by targeting wage in�ation when shocksare common. It is clear, however, that once wages are targeted rather thanprices, di¤erences across the monetary policy alternatives are small.

5 Conclusion

This paper has addressed the question of how monetary policy should beconducted in a monetary union in which labour market structures are di¤er-ent across member countries. The analysis has maintained the assumptionthat the important nominal rigidities are in the labour market rather thanin the goods market, and alternative simple monetary policy rules have beenevaluated under this assumption that prices are �exible and wages sticky.The results indicate that price in�ation targeting may result in non-

negligible welfare losses if nominal wage rigidities are important and nominalprice rigidities are not. It is found that the mistake made by targeting pricein�ation under these assumptions is larger when shocks are common to themonetary union than when they are speci�c to the member countries dueto an international externality. Also, the mistake is larger if the monetary

71

union�s member countries are characterised by di¤erent degrees of nominalwage rigidity; i.e. when labour market structures are asymmetric.Finally, the results suggest that the central bank can improve the welfare

of the monetary union�s citizens by putting more weight on wage in�ation inthe country with the more rigid labour market. The welfare improvementsfrom doing so are larger when labour markets are very heterogeneous, andwhen shocks are idiosyncratic rather than correlated.An interesting topic for further research is to further characterise the

optimal monetary policy responses in a monetary union of the kind describedin this paper.

72

A Natural equilibrium

In the virtual or "natural" equilibrium with �exible wages, the supply sideis summarised by the production function

yit = ait + nit (37)

, nit = �1 (yit � ait)

and the labour demand relation

wit � pit = ait � (1� )nit (38)

while the labour supply relation is

wit � pt = ��1yt + �nit (39)

and the relative demand

yit = �� (pit � pt) + yt (40)

, pit � pt = ���1 (yit � yt)

all for i 2 f1; 2g.Combining (37) and (38) gives

wit � pit = ait � (1� )nit= ait � (1� ) �1 (yit � ait)

=1

ait �

1�

yit

and after rearranging

yit =1

1� ait �

1� (wit � pit)

Substituting (39) and (40) into this relation gives

yit =1

1� ait �

1� ����1yt + �

�1 (yit � ait)�+ ��1 (yit � yt)

�=

1

1� �(1 + �) ait +

���1 � ��1

� yt �

��+ ��1

�yit�

73

Rearranging this expression gives the equilibrium value of country-speci�coutput:

yit

�1 +

�+ ��1

1�

�=

1

1� �(1 + �) ait +

���1 � ��1

� yt�

, yit(1� ) + �+ ��1

1� =1

1� �(1 + �) ait +

���1 � ��1

� yt�

, ynit =1 + �

(1� ) + �+ ��1ait +

���1 � ��1

�

(1� ) + �+ ��1ynt (41)

where a superscript n has been added to indicate equilibrium values in this"natural" equilibrium with �exible wages.Aggregate output in the �exible wage equilibrium is found by aggregating

(41) over countries and rearranging:

ynt =1 + �

(1� ) + �+ ��1at +

���1 � ��1

�

(1� ) + �+ ��1ynt

,(1� ) + �+ ��1 �

���1 � ��1

�

(1� ) + �+ ��1ynt =

1 + �

(1� ) + �+ ��1at

, ynt =1 + �

(1� ) + �+ ��1at (42)

The equilibrium terms of trade under �exible wages can be found bycombining (40) with (41):

snt = p2t � p1t = ���1 (y2t � y1t)

, snt = �1 + �

(1� ) � + �� + (a2t � a1t) (43)

B Welfare function

This section derives a second order approximation to the average welfarelosses incurred by a generic household. The derivation follows the approachof Galí (2008).In each country i 2 f1; 2g, there is a continuum of households indexed by

h 2 [0; 1] with an instantaneous utility function

Uit (h) = U (Cit (h) ; Nit (h)) (44)

74

where Cit (h) = Ct owing to perfect risk-sharing.A second-order Taylor expansion of Ut (h) around the steady state U �

U(C;N) is given as

Uit (h) = U + Uc (Ct � C) + Un (Nit (h)�N)

+1

2Ucc (Ct � C)2 +

1

2Unn (Nit (h)�N)2

+Ucn (Ct � C) (Nit (h)�N) +R [(Ct; Nit (h)) ; (C;N)]

where R (:; :) is a remainder term satisfying

R [(Ct; Nit (h)) ; (C;N)]k(Ct � C;Nit (h)�N)k3

! 0

as (Ct � C;Nit (h)�N) ! (0; 0). Noting that Ucn = 0 by assumption, itfollows that a second-order Taylor approximation expressed in terms of log-deviations from the steady state is given as

Uit (h)� U � UcC

�Ct � CC

�+ UnN

�Nit (h)�N

N

�+1

2UccC

2

�Ct � CC

�2+1

2UnnN

2

�Nit (h)�N

N

�2� UcC

�ct +

1

2c2t

�+ UnN

�nit (h) +

1

2nit (h)

2

�+1

2UccC

2c2t +1

2UnnN

2nit (h)2

= UcC

�yt +

1� ��12

y2t

�+ UnN

�nit (h) +

1 + �

2nit (h)

2

�where the second (approximate) equality makes use of the second-order ap-proximationsCt � C

�1 + ct +

12c2t�andNjt (h) � N

�1 + njt (h) +

12njt (h)

2�,and the third equality of the de�nitions ��1 = �UccC=Uc and � = UnnN=Unas well as of the equilibrium condition ct = yt. Integrating over householdsgives Z 1

0

(Uit (h)� U) dh � UcC

�yt +

1� ��12

y2t

�+UnN

�Ehnit (h) +

1 + �

2Ehnit (h)

2

�75

Noting that

nit +1

2n2it � Ehnit (h) +

1

2Ehnit (h)

2

andEhnit (h)

2 � n2it + �2iV ARh (wit (h))up to a second-order approximation, we can write this asZ 1

0

(Uit (h)� U) dh � UcC

�yt +

1� ��12

y2t

�+UnN

�nit +

1 + �

2n2it +

�2i�

2V ARh (wit (h))

�From the de�nition of aggregate employment, it follows that nit = yit �ait + dit where

dit ��i2V ARh (wit (h))

Using this givesZ 1

0

(Uit (h)� U) dh � UcC

�yt +

1� ��12

y2t

�+UnN

�yit +

1 + �

2 (yit � ait)2 +

(1 + �i�) �i2

V ARh (wit (h))

�+t:i:p:

where t:i:p: represents terms that are independent of policy (here terms inait) and so irrelevant for the evaluation of monetary policy alternatives.Under the assumption of an e¢ cient steady state (so that �Un=Uc =

Y=N), we can express the deviations of utility in percentage terms of steady-state consumption as follows:

Uit � UUcC

� yt +1� ��12

y2t � yit �1 + �

2 (yit � ait)2

�(1 + �i�) �i2

V ARh (wit (h))

where terms independent of policy have been dropped for convenience. Ag-

76

gregating over countries gives

Ut � UUcC

=1� ��12

y2t �1 + �

2

2Xi=1

vi (yit � ait)2 (45)

�2Xi=1

vi(1 + �i�) �i

2V ARh (wit (h))

By inserting the expression (41) for country-speci�c output, the �rst twoterms of this relation become15

1� ��12

y2t �1 + �

2

2Xi=1

vi (�� (pit � pt) + yt � ait)2

=1� ��12

y2t �1 + �

2

2Xi=1

vi�(� (pit � pt) + ait)2 + y2t � 2 (� (pit � pt) + ait) yt

�= �1

2

���1 +

(1� ) + �

�y2t +

1 + �

ytat

�1 + �2

2Xi=1

vi�(� (pit � pt) + ait)2

�+ t:i:p:

After using (42) to replace at (and dropping t:i:p:), this expression can bewritten as

�12

���1 +

(1� ) + �

�y2t +

(1� ) + �+ ��1

ytynt

�1 + �2

2Xj=1

vj (� (pit � pt) + ait)2

= �12

"���1 +

(1� ) + �

�(yt � ynt )

2 +1 + �

2Xj=1

vj (� (pit � pt) + ait)2#

15Second-order terms from a second-order equivalent to (41) would only add third-orderterm to (45) due to the quadratic terms in this relation.

77

Since

2Xi=1

vi (� (pit � pt) + ait)2

=

2Xi=1

vi (� (pit � pt) + ait)!2

+v1v2 [(� (p2t � pt) + a2t)� (� (p1t � pt) + a1t)]2

= v1v2 [� (p2t � p1t) + (a2t � a1t)]2 + t:i:p:

= v1v2�2

�st �

(1� ) + �+ ��1

1 + �snt

�2+ t:i:p:

where st � p2t � p1t represents the terms-of-trade, we can write (45) as

Ut � UUcC

= �12

����1 +

(1� ) + �

�(yt � ynt )

2 +(1 + �) �2

v1v2 (st � ~snt )

2

��

2Xi=1

vi(1 + �i�) �i

2V ARh (wit (h)) (46)

where

~snt =(1� ) + �+ ��1

1 + �snt

Taking the expected value then gives the following second-order approxima-tion to the consumer�s lifetime utility:

Wt = Et

1X�=t

��Ut � UUcC

= �12Et

1X�=t

������1 +

(1� ) + �

�(yt � ynt )

2 +(1 + �) �2

v1v2 (st � ~snt )

2

+2Xi=1

vi (1 + �i�) �iV ARh (wit (h))

)(47)

By applying proposition 6.3 in Woodford (2003), we may write

V ARh (wit (h)) � �iV ARh (wit�1 (h)) +�i

1� �i!2t

78

which is accurate up to a remainder term of third order. Integrating thisexpression forward and taking the discounted value yields the second-orderaccurate relation

1X�=t

��V ARh (wit (h)) ��i

(1� �i) (1� �i�)

1X�=t

��!2t + t:i:p:

Consequently, Wt may be written as

Wt = �Et1X�=t

��

"�y (yt � ynt )

2 + �s (st � ~snt )2 +

2Xi=1

�!i!2jt

#(48)

where

�y =1

2

���1 +

(1� ) + �

��s =

1

2

(1 + �) �2

v1v2

�!i =1

2

vi�i �i (1 + ��i)

(1� �i) (1� �i�)=1

2

vi�i

�i

Finally it follows that the average period welfare loss is

Lt = �yV AR (yt) + �sE�s2t�+

2Xi=1

�!iV AR�!2jt�

(49)

where yt = yt � ynt is the welfare relevant output gap and st = st � ~snt is aterms-of-trade gap.

79

References

[1] Arpaia, A., Pichelmann, K., 2007. Nominal and real wage �exibility inEMU. European Commision European Economy Economic Paper 281.

[2] Andersen, T.M., Seneca, M., 2008. Labour market asymmetries in amonetary union. CEPR Discussion Paper 6800.

[3] Beetsma, R., Jensen, H., 2004. Mark-up �uctuations and �scal stabi-lization policy in a monetary union. Journal of International Economics,67(2), 320-352.

[4] Benigno, P., 2004. Optimal monetary policy in a currency area. Journalof International Economics 63, 293-320.

[5] Calvo, G.A., 1983. Staggered prices in a utility-maximizing framework.Journal of Monetary Economics 12, 383-398.

[6] Christiano, L.J., Eichenbaum, M., Evans, L.E., 2005. Nominal rigidi-ties and the dynamic e¤ects of a shock to monetary policy. Journal ofPolitical Economy 113(1).

[7] Corsetti, G., Pesenti, P., 2001. Welfare and Macroeconomic Interdepen-dence. The Quarterly Journal of Economics 116(2), 421-445.

[8] De Paoli, B., 2007. Monetary policy and welfare in a small open economy.London School of Economics, mimeo.

[9] Dellas, H., Tavlas, G., 2005. Wage rigidity and monetary union. Eco-nomic Journal, 115, 907-927.

[10] Erceg, C.J., Henderson, D.W., Levin, A.T., 2000. Optimal monetarypolicy with staggered wage and price contracts. Journal of MonetaryEconomics 46, 281-313.

[11] Galí, J., 2008. Monetary policy, in�ation and the business cycle. Prince-ton: Princeton University Press.

[12] Galí, J., Rabanal, P., 2005. Technology shocks and aggregate �uctu-ations: How well does the RBC model �t postwar US data? NBERMacroeconomic Annual, 225-288.

80

[13] Holden, S., Wulfsberg, F., 2008. Downward nominal wage rigidity in theOECD. The B.E. Journal of Macroeconomics 8.

[14] Kim, J., Kim, S., 2003. Spurious welfare reversals in international busi-ness cycle models. Journal of International Economics 60, 471-500.

[15] Lombardo, G., 2006. In�ation targeting rules and welfare in an asym-metric currency area. Journal of International Economics 68, 424-442.

[16] Rotemberg, J., Woodford, M., 1999. Interest-rate rules in an esti-mated sticky-price model. In J.B. Taylor (ed.), Monetary Policy Rules,Chocago: University of Chicago Press.

[17] Sbordone, A., 2002. Prices and unit labor costs: A new test of pricestickiness. Journal of Monetary Economics 49, 265-292.

[18] Smets, F., Wouters, R., 2003. An estimated dynamic stochastic generalequilibrium models of the euro area. Journal of the European EconomicAssociation 1(5).

[19] Taylor, J.B, 1993. Discretion versus policy rules in practice. Carnegie-Rochester Conference Series on Public Policy 39, 195-214.

[20] Woodford, M., 2003. Interest and prices. Foundations of theory of mon-etary policy. Princeton: Princeton University Press.

81

Table1:WelfarelossesforRED=1

LAED=3

AED=4

AED=5

Flexiblerules

�"=0

�"=0:5

�"=0

�"=0

�"=0:5

�"=1

�"=0

�"=0:5

�"=1

(1:5;0;0;0)

0:1659

0:2008

0:2358

0:2565

0:3266

0:3967

0:3346

0:4362

0:5378

(0;1:5;1:5;0)

0:0480

0:0240

0:0000

0:0581

0:0291

0:0000

0:0656

0:0328

0:0000

(0;1:1;2;0)

0:0488

0:0244

0:0000

0:0590

0:0295

0:0000

0:0665

0:0332

0:0000

(1:5;0;0;0:125)

0:1318

0:1497

0:1676

0:1678

0:1936

0:2194

0:1835

0:2096

0:2357

(0;1:5;1:5;0:125)

0:0480

0:0240

0:0000

0:0581

0:0291

0:0000

0:0656

0:0328

0:0000

(0;1:1;2;0:125)

0:0486

0:0243

0:0000

0:0588

0:0294

0:0000

0:0662

0:0331

0:0000

LStrictrules

�t=0

0:1816

0:2243

0:2671

0:2375

0:2981

0:3587

0:2815

0:3567

0:4318

!t=0

0:0480

0:0240

0:0000

0:0581

0:0291

0:0000

0:0656

0:0328

0:0000

!1t=0

0:0637

0:0319

0:0000

0:0770

0:0385

0:0000

0:0866

0:0433

0:0000

!2t=0

0:0637

0:0319

0:0000

0:0770

0:0385

0:0000

0:0866

0:0433

0:0000

y t=0

0:0480

0:0240

0:0000

0:0581

0:0291

0:0000

0:0656

0:0328

0:0000

Note:BenchmarkcalibrationandRED=1.Flexiblerulesrefertomonetarypolicyrulesoftheform

(25)

andareindicatedby(k�;k1;k2;ky).

82

Table2:WelfarelossesforRED=2

LAED=3

AED=4

AED=5

Flexiblerules

�"=0

�"=0:5

�"=0

�"=0

�"=0:5

�"=1

�"=0

�"=0:5

�"=1

(1:5;0;0;0)

0:1745

0:2184

0:2623

0:2764

0:3612

0:4460

0:3667

0:4892

0:6116

(0;1:5;1:5;0)

0:0459

0:0229

0:0000

0:0556

0:0278

0:0000

0:0629

0:0315

0:0000

(0;1:1;2;0)

0:0423

0:0211

0:0000

0:0524

0:0262

0:0000

0:0601

0:0301

0:0000

(1:5;0;0;0:125)

0:1329

0:1570

0:1810

0:1697

0:2021

0:2345

0:1855

0:2181

0:2508

(0;1:5;1:5;0:125)

0:0449

0:0224

0:0000

0:0545

0:0273

0:0000

0:0618

0:0309

0:0000

(0;1:1;2;0:125)

0:0419

0:0210

0:0000

0:0521

0:0260

0:0000

0:0598

0:0299

0:0000

LStrictrules

�t=0

0:1673

0:2035

0:2397

0:2209

0:2741

0:3273

0:2634

0:3306

0:3978

!t=0

0:0533

0:0266

0:0000

0:0640

0:0320

0:0000

0:0720

0:0360

0:0000

!1t=0

0:0770

0:0385

0:0000

0:0892

0:0446

0:0000

0:0976

0:0488

0:0000

!2t=0

0:0443

0:0221

0:0000

0:0581

0:0290

0:0000

0:0687

0:0343

0:0000

y t=0

0:0403

0:0202

0:0000

0:0511

0:0256

0:0000

0:0592

0:0296

0:0000

Note:BenchmarkcalibrationandRED=2.Flexiblerulesrefertomonetarypolicyrulesoftheform

(25)

andareindicatedby(k�;k1;k2;ky).

83

Table3:WelfarelossesforRED=3

LAED=3

AED=4

AED=5

Flexiblerules

�"=0

�"=0:5

�"=0

�"=0

�"=0:5

�"=1

�"=0

�"=0:5

�"=1

(1:5;0;0;0)

0:1829

0:2370

0:2912

0:3000

0:4029

0:5057

0:4092

0:5590

0:7088

(0;1:5;1:5;0)

0:0437

0:0218

0:0000

0:0528

0:0264

0:0000

0:0598

0:0299

0:0000

(0;1:1;2;0)

0:0369

0:0185

0:0000

0:0464

0:0232

0:0000

0:0539

0:0269

0:0000

(1:5;0;0;0:125)

0:1323

0:1633

0:1943

0:1698

0:2100

0:2502

0:1859

0:2266

0:2673

(0;1:5;1:5;0:125)

0:0414

0:0207

0:0000

0:0501

0:0251

0:0000

0:0569

0:0285

0:0000

(0;1:1;2;0:125)

0:0357

0:0179

0:0000

0:0452

0:0226

0:0000

0:0527

0:0264

0:0000

LStrictrules

�t=0

0:1487

0:1763

0:2039

0:1989

0:2423

0:2857

0:2390

0:2956

0:3521

!t=0

0:0580

0:0290

0:0000

0:0693

0:0347

0:0000

0:0776

0:0388

0:0000

!1t=0

0:0822

0:0411

0:0000

0:0938

0:0469

0:0000

0:1015

0:0507

0:0000

!2t=0

0:0306

0:0153

0:0000

0:0443

0:0221

0:0000

0:0550

0:0275

0:0000

y t=0

0:0301

0:0150

0:0000

0:0415

0:0208

0:0000

0:0502

0:0251

0:0000

Note:BenchmarkcalibrationandRED=3.Flexiblerulesrefertomonetarypolicyrulesoftheform

(25)

andareindicatedby(k�;k1;k2;ky).

84

Table4:Sensitivityanalysis

L�=2=3

�=0:2

�=1:5

Flexiblerules

�"=0

�"=0:5

�"=0

�"=0

�"=0:5

�"=1

�"=0

�"=0:5

�"=1

(1:5;0;0;0)

0:3382

0:4545

;05708

0:1335

0:1568

0:1802

0:2254

0:3329

0:4404

(0;1:5;1:5;0)

0:0554

0:0281

0:0009

0:0440

0:0220

0:0000

0:0054

0:0027

0:0000

(0;1:1;2;0)

0:0526

0:0267

0:0008

0:0430

0:0215

0:0000

0:0051

0:0025

0:0000

(1:5;0;0;0:125)

0:2173

0:2738

0:3302

0:1020

0:1101

0:1181

0:1216

0:1773

0:2330

(0;1:5;1:5;0:125)

0:0545

0:0276

0:0007

0:0437

0:0218

0:0000

0:0053

0:0026

0:0000

(0;1:1;2;0:125)

0:0523

0:0265

0:0007

0:0429

0:0214

0:0000

0:0051

0:0025

0:0000

LStrictrules

�t=0

0:2597

0:3320

0:4044

0:1235

0:1408

0:1582

0:1637

0:2400

0:3163

!t=0

0:0634

0:0317

0:0000

0:0454

0:0227

0:0000

0:0060

0:0030

0:0000

!1t=0

0:0876

0:0438

0:0000

0:0494

0:0247

0:0000

0:0081

0:0041

0:0000

!2t=0

0:0574

0:0287

0:0000

0:0410

0:0205

0:0000

0:0059

0:0030

0:0000

y t=0

0:0511

0:0256

0:0000

0:0423

0:0211

0:0000

0:0050

0:0025

0:0000

Note:Parametersotherthanthosementionedareattheirbaselinevalues.Flexiblerulesrefertomonetary

policyrulesoftheform

(25)andareindicatedby(k�;k1;k2;ky).

85

Part II

Chapter 3

Rule-of-thumb consumers, productivity andhours�

Francesco FurlanettoNorges Bank

Martin SenecaUniversity of Aarhus

July 2008

Abstract

In this paper we study the transmission mechanisms of productiv-ity shocks in a model with rule-of-thumb consumers. In the literature,this �nancial friction has been studied only with reference to �scalshocks. We show that the presence of rule-of-thumb consumers is alsovery helpful in accounting for recent empirical evidence on produc-tivity shocks. Rule-of-thumb agents, together with nominal and realrigidities, play an important role in reproducing the negative responseof hours and the delayed responses of output and consumption aftera productivity shock.JEL Classi�cation: E32.Keywords: rule-of-thumb consumers, productivity shocks, nominal

rigidities, real rigidities

�For useful comments, we thank T.M. Andersen, J. Galí, J. Imbs, A. Ripatti, T. Sveenand seminar participants at the 3rd DYNARE Conference in Paris, ASSET 2007 in Padova,SAE 2007 in Granada, CEA 2008 in Vancouver, HEC Lausanne and University of Aarhus.F. Furlanetto thanks the Swiss National Research Fund for �nancial support. The opinionsexpressed are solely those of the authors and do not necessarily re�ect the views of NorgesBank. The paper has appeared as Danmarks Nationalbank Working Paper No. 48 andNorges Bank Working Paper 2007/5.

91

1 Introduction

Recent research on �scal policy in dynamic stochastic general equilibrium(DSGE) models has shown that deviations from Ricardian equivalence areinstrumental in generating empirically plausible responses to governmentspending shocks. In particular, Galí et al. (2007) show that private con-sumption may rise after a positive shock to government spending if so-calledrule-of-thumb consumers, who simply consume their current disposable in-come each period, are allowed to co-exist with intertemporally optimisingconsumers.1 In the model, optimising consumers decrease their consumptionfollowing a government spending shock because they correctly anticipate adecline in life-time income as a consequence of taxation. But rule-of-thumbconsumers increase their consumption if current disposable income increases.This happens in the model when the government �nances the increase in itsspending at least partially through the issuance of bonds, under assumptionsof sticky prices and an imperfectly competitive labour market. In this case,if a su¢ ciently large fraction of households follow a rule of thumb, aggregateconsumption rises.A number of papers have further studied the implications of rule-of-thumb

behaviour for �scal policy in DSGE models, and rule-of-thumb consumershave become a standard ingredient in DSGE models at policy-making insti-tutions, in particular at central banks.2 But as far as we know, the implica-tions of rule-of-thumb behaviour have not been investigated beyond the �scalpolicy dimension so far. This is potentially important since rule-of-thumbconsumers represent a substantial deviation from the standard optimisingframework of DSGE models. In the baseline calibration in Galí et al. (2007),50 per cent of households have no access to �nancial and capital markets andso cannot smooth consumption intertemporally. The market incompletenessintroduced by this assumption may be suspected to have potentially size-able e¤ects on the model�s propagation of shocks to variables other than

1To our knowledge, the idea that a fraction of consumers consume their current incomeseach period, while the remaining fraction optimise intertemporally, was �rst put forwardby Hall (1978) as an alternative to the permanent income hypothesis. Campbell andMankiw (1989, 1991) reject the permant income hypothesis against this alternative, andMankiw (2000) suggests that rule-of-thumb consumers should be included in models builtfor the analysis of �scal policy issues.

2Papers include Andersen (2005), Bilbiie (2005), Coenen and Straub (2005), Colciago(2007), Erceg et al. (2006), Forni et al. (2007), Furlanetto (2007), Furlanetto and Seneca(2007), Galí et al. (2004) and Natvik (2006).

92

government spending. This is an important objection as counterfactual re-sponses to other kinds of shocks may question the plausibility of introducingrule-of-thumb consumers even for analysing �scal policy issues.The purpose of this paper is to test this conjecture for the case of shocks

to productivity. Hence, we analyse the impact of rule-of-thumb consump-tion behaviour on the propagation of technology shocks in the frameworkdeveloped by Galí et al. (2007).3 Considering the recent debate in macro-economics on the importance of technology shocks for business cycle �uctu-ations, it seems particularly important to study the performance of the classof DSGE models with rule-of-thumb consumers in response to these shocks.On one hand, beginning with the seminal papers by Kydland and Prescott(1982) and Prescott (1986), real business cycle (RBC) theory suggests thattechnology shocks are the most important driving force behind business cy-cle �uctuations. On the other hand, a number of later papers, particularlyGalí (1999), have challenged this claim based on empirical evidence on theimpulse responses of macroeconomic variables. In this paper, we contributeto this debate by shedding light on how �nancial frictions in the form ofrule-of-thumb behaviour may a¤ect the transmission of technology shocks inthe economy.Galí (1999), and more recently Francis and Ramey (2005), provide evi-

dence on responses to technology shocks in the US by identifying such shocksin an estimated vector autoregression (VAR) through long-run restrictions.In both studies, a positive technology shock has a signi�cant negative e¤ecton hours worked - in stark contrast with the predictions of the RBC liter-ature. Furthermore, in both studies output does not respond to the shockon impact, but it increases with a lag. Since output and hours are stronglypositively correlated in the data, it follows that technology shocks cannot bethe main driving force behind business cycle �uctuations.Of course, these claims have not stood unchallenged. Christiano et al.

(2004) and McGrattan (2004) argue that Galí�s (1999) results are sensitiveto small changes in the speci�cation of the empirical model. When hours areintroduced in levels, and not in �rst di¤erences as in Galí (1999), these au-thors obtain a positive response of hours. However, in recent papers Fernald

3We have also considered the e¤ects of monetary, preference and cost-push shocks.The model with rule-of-thumb consumers delivers results very similar to the model with-out them as long as wages are sticky. Wage rigidity e¤ectively shuts down the mechanismsthrough which rule-of-thumb behaviour may change the propagation of these shocks. Re-sults are available upon request.

93

(2007) and Canova et al. (2007) show that once low-frequency movementsin hours are taken into account, the negative response of hours is robust; seealso Galí and Rabanal (2005) for a discussion. In addition, Gambetti (2005)con�rms that hours fall using a bayesian VAR with time-varying coe¢ cients.Consequently, we consider the evidence from the VAR literature to favourthe view that hours decrease on impact of a technology shock.An alternative empirical approach is taken by Basu et al. (2006). They

use a sophisticated growth accounting framework to correct Solow residualsfor the in�uences of increasing returns, imperfect competition, variable factorutilisation and sector compositional e¤ects. Somewhat surprisingly, perhaps,this approach leads to results that are very similar to those of the VARliterature. In particular, Basu et al. (2006) estimate a signi�cant decline inhours on impact of a technology shock, while they �nd a zero impact responseof output.4

How can a theoretical model deliver a decline in hours after a technol-ogy shock? Galí (1999) shows how nominal rigidities, a key feature of NewKeynesian models, can lead to such a response. However, Dotsey (2002)shows that this is true only if monetary policy is modelled as an exoge-nous money growth rule; when monetary policy follows a Taylor (1993) rule,hours increase as in the baseline RBC model. Francis and Ramey (2005)show that an RBC model augmented with real rigidities (habit persistencein consumption and capital adjustment costs) can generate a negative re-sponse of hours without relying on nominal rigidities.5 In this paper, weshow that a �nancial friction represented by rule-of-thumb behaviour a¤ectsthe model�s transmission mechanism in a way that makes it easier to obtaina decline in hours on impact of a productivity shock. In addition, we showhow rule-of-thumb behaviour interacts with the nominal and real rigiditiesthat have previously been considered in literature as potential explanationsof the negative response of hours.The Galí et al. (2007) model is characterised by three rigidities, namely

4The evidence in Basu et al. (2006) is based on macrodata for the US. Interestingly,using �rm-level data for Italy and Sweden, respectively, Marchetti and Nucci (2005) andCarlsson and Smedsaas (2006) �nd that �rms reduce the input of labour on impact of apositive technology shock. See also Carlsson (2003).

5Galí and Rabanal (2005) estimate a New Keynesian model using Bayesian techniques,and they �nd that both nominal and real rigidities are important, while Galí et al. (2003)detect signi�cant di¤erences across periods in the Federal Reserve�s responses to technologyshocks, reconciling the results of Galí (1999) and Dotsey (2002).

94

price stickiness, capital stickiness (due to capital adjustment costs) and the�nancial rigidity barring a fraction of households, the rule-of-thumb con-sumers, from access to �nancial and capital markets. We extend this frame-work in two steps.In the �rst step, we extend the model with nominal wage rigidity as in

Furlanetto (2007). Sticky wages have been shown to be important in orderto generate plausible dynamics in macroeconomic variables in response toa wide variety of shocks, cf. Christiano et al. (2005). Moreover, Liu andPhaneuf (2005) show that sticky wages, in combination with sticky prices,are important in order to explain the dynamics of hours and wages followinga productivity shock. The Galí et al. (2007) model extended with nominalwage rigidity is our baseline model.In the second step, we introduce a �fth rigidity, namely consumption

stickiness in the form of habit persistence in consumption. Habit formationhas recently received a lot of attention in the literature, e.g. by Francis andRamey (2005), Galí and Rabanal (2005) and Fève (2004). As we shall see,this extension allows us to explain empirical evidence on key macroeconomicvariables besides hours and wages. In addition, it allows us to analyse the roleplayed by many of the frictions studied in the literature and their interactionwith rule-of-thumb consumers. Building on the terminology of McGrattan(2004), we refer to the Galí et al. (2007) model extended with both nominalwage rigidity and habit persistence as the quintuple-sticky model.This paper�s �rst key result is that the model with rule-of-thumb con-

sumers is able to reproduce a sizeable decline in hours in keeping with theempirical evidence. Like Galí and Rabanal (2005), we �nd that a model withthree types of rigidities (sticky prices, sticky wages and capital adjustmentcosts) can reproduce a negative response of hours - even under endogenousmonetary policy in the form of a Taylor rule. But we show that this responseis very small. As shown in �gure 1, a one per cent increase in technology leadsto a 0.2 per cent decline in hours worked on impact under our preferred cal-ibration. In our baseline model with rule-of-thumb consumers, hours declinemore: -0.6 per cent. Thus, the model�s response coincides with the estimatesin Basu et al. (2006) and Francis and Ramey (2005).The intuition is the following. A positive shock to technology means that

�rms can produce a given level of output with fewer hours. Because pricesare sticky, the level of output is determined by demand. This means thathours will go down if demand does not increase su¢ ciently after the shock.As to government spending shocks, rule-of-thumb and optimising households

95

react to technology shocks in di¤erent ways. Optimising consumers correctlyanticipate an increase in life-time income and so they increase their consump-tion. This works to o¤set the decline in hours through aggregate demand.Rule-of-thumb consumers, in contrast, see current income go down becauseof combined e¤ects of sticky prices and wages, and this makes them con-sume less. This curbs the aggregate demand e¤ect, and hours decline moreas a result when some households consume according to a rule of thumb.In a nutshell, our �nancial rigidity ampli�es the impact of nominal and realrigidities, making the transmission more contractionary.While the literature has studied the response of hours to technology shock

in great detail, less attention has been devoted to the responses of othermacroeconomic variables. Both Basu et al. (2006) and Francis and Ramey(2005) �nd not only a decline in hours on impact of a technology shock, butalso a zero response of both output and consumption. Surprisingly, this ad-ditional evidence suggesting a delayed expansion in output and consumptionfollowing a technology improvement is yet to be explained in the theoreticalliterature.The second key result of this paper is that the quintuple-sticky model can

reproduce the zero impact-responses of output and consumption found inBasu et al. (2006) and Francis and Ramey (2005) in addition to a decline inhours worked. In the model, habit persistence works to smooth consumption,in e¤ect delaying the full response of consumption to shocks. We stress,however, that the presence of all the �ve rigidities considered is crucial toobtain this result.Consequently, rule-of-thumb agents are instrumental not only in obtain-

ing a large negative response of hours, but also in reproducing delayed re-sponses of output and consumption as in the empirical evidence. Thus, rule-of-thumb consumers, representing a substantial deviation from the standardoptimising DSGE framework, do not worsen the performance of the model.In contrast, they can be very helpful in replicating important empirical reg-ularities. This implies that researchers may safely rely on rule-of-thumb con-sumers in �scal policy analyses in the sense that rule-of-thumb consumptionbehaviour generates reasonable responses to other shocks.The paper has the following structure. In section 2 we brie�y present the

baseline model, and in section 3 we present impulse responses to technologyshocks from this version of the model. In section 4 we discuss the quintuple-sticky model, we compare our results to other papers in the literature, andwe present a sensitivity analysis. Section 5 concludes.

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2 A DSGE model with rule-of-thumb con-sumers

The model is a standard New Keynesian model augmented with capital andrule-of-thumb consumers as in Galí et al. (2007), and with sticky wages as inFurlanetto (2007).6 The economy consists of a continuum of �rms, a contin-uum of households, a continuum of labour unions, a central bank responsiblefor monetary policy, and a government collecting lump-sum taxes.7

There is monopolistic competition in both goods and labour markets. Inparticular, there is a continuum of di¤erentiated intermediate goods and acontinuum of di¤erentiated labour services. In the goods market, this leadsto a downward-sloping demand curve for each intermediate good, and in thelabour market it leads to a downward-sloping demand curve for each labourtype.A fraction � of households are rule-of-thumb consumers - or �spenders�in

the terminology of Mankiw (2000). These consumers simply consume theirrespective disposable incomes each period. The remaining fraction (1� �)of households are optimisers - or �savers�- who have access to both �nancialand capital markets. Hence, they choose plans for consumption, investmentand bond holdings to maximise life-time utility. Wages are set by unions thateach represent a di¤erentiated type of labour service supplied by households.Wage rigidity is introduced by assuming adjustment costs as in Rotemberg(1982).Each �rm produces one of the di¤erentiated intermediate goods. It does

so by combining rented capital with a homogenous labour input constructedas a Dixit and Stiglitz (1977) aggregate of the di¤erentiated labour servicessupplied by households. The �rm sets its price according to a Calvo (1983)price-setting mechanism and stands ready to satisfy demand at the chosenprice.Each period begins by the realisation of shocks to the economy. We

concentrate on technology shocks and abstract from other types of shocksthat may a¤ect the economy.

6In the appendix we further extend the model with habit formation in consumption.7We abstract from �scal policy as the model�s propagation of government spending

shocks has been thoroughly analysed in the literature, cf. references above.

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2.1 Households

Households have identical instantaneous utility functions

U it =

(Cit)1�� � 11� �

� (Nit )1+'

1 + '(1)

where i 2 fo; rg denotes the household�s type, i.e. optimising or rule-of-thumb. Ci

t is the household�s real consumption at time t (implicitly a Dixitand Stiglitz, 1977, index of intermediate goods), N i

t is the hours worked bythe household in period t, ' > 0 is the inverse of the Frisch labour elasticity,and � > 0 is the coe¢ cient of relative risk aversion and, at the same time,the inverse of the elasticity of intertemporal substitution.An optimising household maximises expected life-time utility given by

Et

1Xk=0

�kU ot+k

where Et is an operator representing expectations over all states of the econ-omy conditional on period-t information, and � 2 (0; 1) is the subjectivediscount factor. Maximisation is subject to a sequence of �ow budget con-straints (and implicitly a no-Ponzi game condition):

Pt (Cot + It) + Et (�t;t+1Bt+1) =WtN

ot +Rk

tKt +Bt � PtTot � Ft

where It is real investment, Wt is the nominal wage, Rkt is the nominal rental

rate on the stock of capital owned by the household at the beginning of periodt, Kt, and T ot is the real lump-sum tax paid by optimising consumers. Theright-hand side gives available resources as the sum of labour income, WtN

ot ,

income from renting capital to �rms, RktKt, initial �nancial wealth, Bt, less

nominal lump-sum taxes paid to the government, PtT ot , and less a nominalunion membership fee, Ft. On the left-hand side, resources are allocated toconsumption, investment and a portfolio of bonds, Et (�t;t+1Bt+1). �t;t+1 isthe stochastic discount factor. Hence, the gross risk-free interest rate is givenby the relation 1 +Rt = (Et�t;t+1)

�1.The household�s capital evolves according to

Kt+1 = (1� �)Kt + �

�ItKt

�Kt

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where � is the rate of depreciation, and � (:) is an adjustment cost functionsatisfying � (�) = �, �0 > 0, �0 (�) = 1 and �00 � 0.The optimisation problem, according to which the household chooses

plans for consumption, bond holdings and investment, gives rise to the fol-lowing �rst-order conditions that we state in log-linear form:8

cot = Etcot+1 �

1

�(rt � Et�t+1) (2)

kt+1 = (1� �) kt + �it (3)

qt = � (rt � Et [�t+1]) + [1� � (1� �)]Et�rkt+1 � pt

�+ �Et [qt+1] (4)

it � kt = �qt (5)

where � � �1= (�00 (�) �). Here, (2) is the Euler equation, (3) is the capitalaccumulation equation, while (4) and (5) represent the dynamics of Tobin�sq, denoted qt, and its relation to investment, respectively.A rule-of-thumb household faces the simple budget constraint

PtCrt = WtN

rt � PtT

r � Ft

where Crt is the household�s real consumption at time t, N

rt is the hours

worked by the household in period t, and Ft is a nominal union membershipfee. As a rule-of-thumb household simply consumes its current income, con-sumption follows directly from the budget constraint. A �rst-order log-linearapproximation around the steady state with constant consumption equalisedacross households gives

crt =WN

PC(wt + nt) (6)

where omission of time subscripts indicate steady-state variables. Note thattaxes drop out of the �rst-order approximation because we abstract fromgovernment spending shocks. Also, the union membership fee drops outbecause the fee is assumed to be a quadratic function of wage in�ation,which is zero in the steady state, cf. below.Aggregate variables are given as simple weighted averages:

ct = �crt + (1� �) cot (7)

nt = �nrt + (1� �)not (8)

8For details on the derivation we refer the reader to GLV (2007). Lowercase variablesdenote log-deviations from the steady state of the corresponding uppercase variables.

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2.2 Firms

Each �rm produces according to the technology

Yt = AtK t N

1� t

where Yt is output, At is a technology shock, and 0 � � 1. Each period, a�rm is allowed to set a new price, P �t , with a �xed probability (1� �p) as inCalvo (1983). It does so to maximise the value of the �rm to its owners, theoptimising households,

1Xk=0

�kEt��t;t+k

�P �t Yt+kjt �Wt+kNt+kjt �Rk

t+kKt+kjt��

where subscript t+ kjt indicates the value of the variable at time t+ k for a�rm that has last reset its price in period t. Maximisation is subject to thedownward-sloping demand curve it faces as a consequence of monopolisticcompetition.As is wellknown, the optimality conditions from this problem imply the

New Keynesian Phillips curve

�pt = �Et��pt+1

�+ �pmct (9)

where � = (1� ��p) (1� �p) ��1p , �

pt = pt � pt�1 is price in�ation, and where

mct is real marginal costs given by

mct = (wt � pt)� (yt � nt) (10)

In addition, cost minimisation implies that relative factor inputs satisfy thecondition

kt � nt = (wt � pt)��rkt � pt

�(11)

Up to a �rst-order approximation, production is given by

yt = at + kt + (1� )nt (12)

2.3 Labour Unions

The economy has a continuum of unions z 2 [0; 1] each representing a con-tinuum of workers. A fraction (1� �) are optimising, and fraction � arerule-of-thumb consumers. Each union sets the wage rate for its members,

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who stand ready to satisfy �rms�demand for their labour services at thechosen wage. The workers in a union provide the same type of labour (ir-respective of their consumption behaviour) di¤erentiated from the type oflabour services provided by members of other unions. The labour servicesupplied by each union, N (z), is a simple aggregate of its members�labourservices. In turn, the labour entering the production function of any �rm isa Dixit and Stiglitz (1977) aggregate of the labour services provided by theunions in the economy.Each period, a representative union choosesWt (z) to maximise the present

value of an average of its members�current and future period utility func-tions, that is,

maxWt(z)

Et

1Xk=0

�t+k��U r

t+k + (1� �)U ot+k

�subject to the labour demand functions and the budget constraints of itsmembers, thus taking the e¤ect of the wage decision on the income of itsmembers into account. Wage adjustments are assumed to be costly. Inparticular, it is assumed that the wage adjustment cost is a quadratic functionof the increase in the wage demanded by the union as modelled in Rotemberg(1982) for prices demanded by �rms. For simplicity, the adjustment cost isproportional to the aggregate wage bill in the economy (this parallels thespeci�cation of price adjustment costs in Ireland, 2003). Though the wagebargaining process is not explicitly modelled, one way of thinking of thiscost is that unions have to negotiate wages each period and that this activitydemands economic resources; the larger the increase in wages obtained, themore e¤ort unions would have needed to put into the negotiation process.Each member of the union covers an equal share of the wage adjustment costby paying a union membership fee. Hence, the nominal fee paid by a memberof union z at time t is given by

Ft (z) =�w2

�Wt (z)

Wt�1 (z)� 1�2

WtNt

where the size of the adjustment costs is governed by the parameter �w.The optimality conditions imply a New Keynesian Phillips curve for wage

in�ation given by

�wt = �Et��wt+1

�+ �w (mrst � (wt � pt)) (13)

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where mrst is the average marginal rate of substitution given by

mrst = �ct + 'nt (14)

and the slope coe¢ cient �w is

�w ="w � 1�w

The derivation is given in the appendix.9

In the special case where �w = 0, the model e¤ectively collapses to themodel in Galí et al. (2007). Firms do not discriminate between consumertypes in their labour demand, and so it follows from the unions�problemsthat nrt = not = nt.

2.4 Monetary policy

The central bank controls the risk-free interest rate, which it sets accordingto a simple Taylor rule

rt = r + ���t (15)

This speci�cation implies that monetary policy is endogenous. The centralbank responds to in�ation, which is endogenously determined in the economy.

2.5 Equilibrium

Market clearing requires that

Yt = Ct + It +G+ Ft

where G = T is government spending. In log-linear form, this becomes

yt =C

Yct +

I

Yit (16)

9Instead of wage adjustment costs, we may assume that a union is allowed to reset itswage rate each period with a �xed probability (1� �w) as in Calvo (1983). But to undo theimplications of the implied heterogeneity across unions, each household must be assumedto provide all types of labour simultaneously in this case, or alternatively a risk-sharingarrangement between unions must be in place. This follows since rule-of-thumb consumersare barred from sharing risk through �nancial markets. Results, however, are very similar.In particular we would get a Phillips curve with �w = (1� ��w) (1� �w) ��1w (1 + '"w)

�1

where "w is the wage elasticity of labour demand.

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The only shocks to the economy that we consider are technology shocks.They evolve according to an autoregressive process of order one:

at = �aat�1 + ea;t (17)

It follows that the equilibrium dynamics are summarised by (2)-(17).

3 Impulse response analysis

In this section, we present impulse responses of key variables in a calibratedversion of the model. To facilitate comparison, our calibration follows Galíet al. (2007). Hence, we consider a time period to be a quarter, and we set� = 0:5, � = � = 1, �p = 0:75, = 0:3, � = 0:025 and � = 0:99. In addition,we set G=Y = 0:20 with the implication that I=Y = �� (1=� + �)�1 ��1p =0:18, C=Y = 0:62 and WN=PC = (1� �)Y=C�p, under the assumptionthat steady-state price mark-ups

��p � 1

�are 20 per cent, cf. Galí et al.

(2007). However, we deviate from the calibration in Galí et al. (2007) insetting ' = 1 instead of ' = 0:2, a value we consider to be unrealisticallylow. Galí et al. (2007) need to set a high value for the labour elasticity toensure determinacy of the equilibrium. But the introduction of wage rigiditiesincreases the range of values of ' for which the equilibrium is determinate,cf. Colciago (2007). This allows us to set a more realistic value. Finally weset "w = 4 and �w = 174:7. This corresponds to a steady-state wage mark-upof approximately 33 per cent, and a degree of price rigidity corresponding to�w = 0:75 under the alternative Calvo (1983) wage-setting scheme, i.e. anaverage duration of wage contracts of four quarters.We are interested in the implications of introducing rule-of-thumb con-

sumers into the New Keynesian model, and so we compare the responsesunder the baseline calibration above with a calibration in which � = 0, cor-responding to a version of the model without rule-of-thumb consumers.Figure 2 presents responses to a one standard deviation technology shock

�a = 0:9. Dashed lines are responses from the baseline model presented in theprevious section, whereas solid lines are responses from the model withoutrule-of-thumb consumers.Comparing the dashed and the solid lines, it is clear that the introduction

of rule-of-thumb consumers is not without consequence for the responses toa technology shock. In particular, hours decline more following a positiveproductivity shock in the economy with rule-of-thumb behaviour in keeping

103

with the empirical evidence in Basu et al. (2006) and Francis and Ramey(2005). Indeed, with our baseline calibration, hours go down by -0.6 per centin the period when the technology shock hits the economy. This coincideswith the estimate in Basu et al. (2006) and Francis and Ramey (2005).The transmission is as follows. The increase in productivity lowers �rms�

marginal costs. If prices were �exible, �rms would lower their prices andincrease supply. But since prices are sticky, some �rms cannot do so andthe reduction in the overall price level is limited. This means that outputincreases less than it would had prices been �exible. In addition, hoursdecline because the improvement in technology allows �rms to produce thesame output as before with less labour. The monetary policy authority reactsto the reduction in prices by a measured reduction of the nominal interestrate.The fall in the interest rate makes it optimal for consumers to consume

more in the current period. Optimising consumers realise this, and theyalso correctly anticipate that the productivity shock leads to an increase inpermanent income. These two forces make optimising consumers increasetheir consumption.Rule-of-thumb consumers behave di¤erently, however. As their horizon

is static, neither the increase in permanent income nor the reduction in realinterest rates a¤ects their consumption decisions. Instead, they choose con-sumption on the basis of current income, which is determined by currenthours in production and the real wage. As noted above, hours decline be-cause prices are sticky, but real wages respond little as a consequence of stickywages. Hence, the decline in hours is larger than the increase in real wages,and current income declines. This makes rule-of-thumb agents consume less.The e¤ect on aggregate consumption depends on the relative importance

of optimising and rule-of-thumb consumers in the economy, and on the size oftheir responses to the shocks. Aggregate consumption may still rise despiterule-of-thumb behaviour, but the presence of households that do not opti-mise intertemporally has an important contractionary e¤ect. The aggregatedemand e¤ect that could potentially o¤set the initial reduction in hours issmaller because rule-of-thumb consumers decrease their consumption. From�gure 2 we see this e¤ect clearly: The model with rule-of-thumb consumersexhibits a smaller increase in aggregate consumption and output than themodel without rule-of-thumb behaviour, and it exhibits a larger decline inhours.This leads us to the �rst key result of this paper. A model with rule-

104

of-thumb consumers, interacting with nominal and real rigidities, can betterexplain the empirical evidence provided by Basu et al. (2006) and Francisand Ramey (2005). This is so even though the shock is too expansionarycompared to the data.We note that sticky wages is an essential assumption needed to obtain

this result. In a model with �exible wages the increase in the real wage wouldbe larger than the decrease in hours and rule-of-thumb agents would increasetheir level of consumption. There would be no contractionary e¤ect fromrule-of-thumb behaviour in this case.Indeed, it is important to stress that all four frictions - sticky prices,

sticky wages, rule-of-thumb behaviour and capital adjustment costs - areessential to subdue the expansionary e¤ect of the shock. Sticky prices areneeded for a decline in hours, and sticky wages are needed for this to lead toa reduction in the current income of rule-of-thumb consumers. A su¢ cientlyhigh fraction of rule-of-thumb consumers, then, is needed for this reductionin current income to have an e¤ect on the aggregate economy. And �nally,capital adjustment costs are needed to dampen investment, an increase inwhich would otherwise o¤set the contractionary e¤ect from the response ofrule-of-thumb consumers.Finally, we note that the real wage increase in the model with sticky wages

perfectly �ts the empirical evidence on the real wage response provided inLiu and Phaneuf (2005), whereas the response of the real wage in the modelwith �exible wages would be excessively procyclical. In our opinion, this factis further con�rmation that sticky wages is a sensible assumption.From the analysis in this section, we conclude that the introduction of

rule-of-thumb consumers, a considerable change to the standard DSGE set-up, does not lead to counterfactual responses to productivity shocks. On thecontrary, (to the extent that a productivity shock leads to a decline in hourson impact) we �nd that the model�s transmission mechanism is improved.It is important to stress, however, that we perform a conditional analysis

in the spirit of Galí (1999) and not an unconditional exercise as is typicalin the RBC literature. That is, it is not our goal to reproduce the uncondi-tional moments found in the data. Indeed, given the response of hours, ourone-shock model would perform very badly in such an exercise. As in Galí(1999), Galí and Rabanal (2005) and Francis and Ramey (2005), produc-tivity shocks are not the main driving force of aggregate �uctuations in our

105

model.10 Nevertheless, we believe that our conditional analysis is relevant inorder to evaluate the e¤ects of rule-of-thumb behaviour. Even if productivityshocks are not the main driving force behind the business cycle, they stillrepresent a source of �uctuations in the economy that needs to be consideredin detail, especially because of the prominent role played by these shocks inthe RBC literature.

4 Quintuple stickiness

In this section we present results from the model in section 2 extended withhabit formation in consumption. That is, we let utility today depend not onconsumption today by itself, but on consumption today relative to consump-tion in the previous period. This makes optimising households look back aswell as forward when making consumption decisions. In addition, unions takethe e¤ect of habit on the utility of its members into account when settingwages. Thus, the introduction of habit formation in consumption changesthe Euler equation of optimising consumers and the wage-setting equation.Details are given in the appendix.Our model�s quintuple stickiness makes our analysis more comprehensive

than previous studies of technology shocks. In particular, we model the cap-ital accumulation process explicitly, and we introduce endogenous monetarypolicy by letting the model�s central bank respond to in�ation developments.In comparison, Galí and Rabanal (2005) ignore investment dynamics in theirmodel, while Francis and Ramey (2005) let monetary policy be exogenous.Finally, we consider the implication of credit constraints by allowing for rule-of-thumb behaviour.The analysis of the quintuple-sticky model serves two purposes. First,

the model helps us explain the empirical evidence on key macroeconomicvariables besides hours worked. Second, it allows us to analyse the rolesplayed by many of the frictions studied in the literature on technology shocksand the interaction of these frictions with rule-of-thumb consumers. Weconsider each of these issues in turn.10To improve the unconditional performance of the model, we should include shocks to

other variables such as demand shocks or possibly investment-speci�c technology shocksas in Fisher (2006).

106

4.1 Output, consumption and investment

The model presented in the previous section can easily reproduce the declinein hours after a technology shock found in the empirical literature. In ad-dition, the response of the real wage seems plausible given the empiricalevidence. But the model fares less well when considering other key macroeco-nomic variables for which we have empirical evidence. In particular, output,consumption and investment all increase on impact of a shock to technologyin the model. This is in contrast to the evidence in Basu et al. (2006),who �nd that output and consumption change little on impact of a tech-nology shock before increasing in the periods following the shock, whereasnon-residential investment falls sharply on impact before rising. Francis andRamey (2005) �nd similar responses for output and consumption, whereasthe response of investment is statistically insigni�cant in their analysis.Figure 3 presents the second key result in this paper. The quintuple-

sticky model with rule-of-thumb consumers (dashed lines) can reproduce thezero impact responses of output and consumption. This is because habitpersistence slows down the response of optimising consumers. With habitformation in consumption, optimising consumers need time to appreciatethe increased scope for consumption given to them by the positive shock totechnology. This leads to a hump-shaped response of optimising household�sconsumption, further restraining the expansion in the economy.Now, perhaps the contractionary e¤ects are even too strong in our quin-

tuple-sticky model with rule-of-thumb consumers; hours go down more thanone per cent, and aggregate consumption actually declines. However, as weshow below, we may undo this excess contraction by modifying the baselinecalibration, e.g. by lowering the percentage of rule-of-thumb consumers orthe degree of price stickiness.11 Here we keep the parameter values chosen byGalí et al. (2007) to facilitate comparison. An estimated model could delivermore precise guidance on the parameter values needed to replicate the em-pirical results. Our objective here is simply to show that the quintuple-stickymodel with rule-of-thumb consumers delivers a very contractionary propaga-tion of technology shocks, and that a �nancial friction in this form providesan additional explanation, along with nominal and real rigidities, of why aproductivity shock may lead to a decline in hours on impact.

11Indeed, given the empirical evidence, both these parameters may appear to be un-comfortably high in our baseline calibration. For a discussion, see Furlanetto and Seneca(2007).

107

Figure 3 also shows responses to a technology shock for the model withoutrule-of-thumb consumers (solid lines). We see that rule-of-thumb behaviouris crucial in order to replicate the zero impact responses found in empiricalstudies. Without this friction, both output and consumption increase onimpact of the shock.Turning to investment, Basu et al. (2006) �nd a signi�cantly negative

response of this variable after a productivity shock. Given our analysis,this is puzzling. Indeed, investment increases after a positive technologyshock in all versions of the model. In particular, the positive response ofinvestment is not related to the presence of rule-of-thumb consumers. Basuet al. (2006) argue that their evidence on investment is compatible with asticky price model in the case where monetary policy is exogenous. Once weallow for an endogenous reaction from the monetary authority, the responseof investment is always both positive and fairly large. We believe that ourassumption about monetary policy is the more reasonable one, however, andthe evidence provided by Galí et al. (2003) supports this claim.The identifying assumption often used in the empirical VAR literature

is that a technology shock has a permanent e¤ect on labour productivity.Therefore, we check whether a permanent technology shock delivers the sameresults as the temporary (but highly persistent) shock considered above. Thisis con�rmed in �gure 4. In particular, the impact responses of hours, the realwage, consumption and output are in line with the estimated responses inBasu et al. (2006) and Francis and Ramey (2005).

4.2 Interacting frictions and sensitivity analysis

It is important to note that all the frictions in the quintuple-sticky model areneeded to obtain the results just considered. In �gure 5, we show impulseresponses to a (temporary, highly persistent) technology shock for the modelwith nominal rigidities only (sticky wages and sticky prices) and for the modelwith real rigidities only (capital adjustment costs and habit persistence), inboth cases without rule-of-thumb consumers. We see that the model withnominal rigidities only (solid lines) performs poorly. Consumption, output,hours and investment all increase sharply following the technology shock.This is because, without capital adjustment costs, a technology shock leadsto an investment boom that more than o¤sets the contractionary e¤ect fromother frictions in the model. In keeping with the theoretical results in Francisand Ramey (2005), the model with real rigidities only (dashed lines) is able

108

to reproduce a decline in hours, but output, consumption and investmentresponses are too expansionary compared to those estimated by Basu et al.(2006) and Francis and Ramey (2005).Figure 5 con�rms that real, nominal and �nancial frictions are all impor-

tant to obtain a sizeable negative response of hours and a zero impact re-sponse of output and consumption. The same message comes from the morecareful sensitivity analysis for the quintuple-sticky model that we present in�gure 6. There, we plot the impact responses of output, consumption andhours for a large spectrum of parameter values to our temporary, but highlypersistent technology shock. If lines are �at, it means that impact responsesare not a¤ected by the speci�c parameter considered.12

From �gure 6 we see that the labour supply elasticity and the habit per-sistence parameter for constrained agents do not in�uence impact responses.The other panels con�rm that impact responses are declining in the percent-age of rule-of-thumb consumers, in the degree of wage and price rigidity, andin the degree of habit persistence in optimising consumption. Notice thatthe excess contraction in consumption disappears if � is close to 0.25 insteadof 0.5 as in the baseline calibration, or if we let �p be in the region of 0.6instead of 0.75. The results for the coe¢ cient in the Taylor rule suggest thatthe central bank will have to be very aggressive to overturn the results. Ona similar note, if � is very low, optimising consumers respond strongly tomonetary policy and the contractionary e¤ect from rule-of-thumb behaviourcarries less weight at the aggregate level. Finally, when � is high it meansthat capital adjustment costs are low, in which case all the impact responsesbecome positive. Again, this is because investment overreacts when pricesare sticky and capital adjustment costs are low. Thus, �gure 6 con�rms thatall �ve frictions in the model are needed to curb the expansionary e¤ects ofpositive shocks to technology.

5 Conclusion

The introduction of rule-of-thumb consumers into the New Keynesian DSGEmodel has proven to be a useful way to explain responses to �scal shocks. Thepurpose of this paper is to check whether the introduction of this substantial�nancial friction a¤ects the transmission mechanism of productivity shocks.

12The analysis is partial in the sense that we vary one parameter at a time, while theremaining parameters are �xed at the values chosen for the baseline calibration.

109

We �nd that rule-of-thumb consumers, in combination with real and nom-inal rigidities, can explain a sizeable decline in hours worked after a positiveproductivity shock as suggested by the empirical evidence in Galí (1999),Francis and Ramey (2005) and Basu et al. (2006).Moreover, we show that within our quintuple-sticky business cycle frame-

work, only a combination of nominal rigidities, real rigidities and limitedaccess to �nancial markets can reproduce a sizeable negative e¤ect on hoursand a zero impact response of output and consumption.In addition, our quintuple-sticky model is a useful laboratory in which to

compare results from many other papers in the literature. A model with realrigidities alone can explain a negative e¤ect on hours, but the inclusion ofa �nancial friction interacting with nominal rigidities is essential in order toreproduce a zero impact e¤ect on output and consumption. A model withnominal rigidities alone cannot explain a decline in hours.We conclude that the transmission mechanism for technology shocks is

improved by including rule-of-thumb consumers in the model. Thus, ouranalysis suggests that researchers may safely build rule-of-thumb consumersinto their models to reproduce empirically plausible responses to �scal policyshocks without having to fear that the model becomes less realistic in otherdimensions. Indeed, this �nancial friction may be an additional explanation,along with nominal and real rigidities, of why hours may decline following aproductivity shock as the empirical literature suggests. An important topicfor future empirical research is to investigate the relative importance of thesefrictions. In future work we therefore plan to estimate the quintuple-stickymodel using Bayesian techniques.

110

A Appendix

Derivation of the wage schedule. The �rst-order condition to the union�sproblem becomes

0 = �@U r

t

@Crt

@Crt

@Wt (z)+ (1� �)

@U ot

@Cot

@Cot

@Wt (z)� (Nt (z))

' @Nt (z)

@Wt (z)

+�Et

��@U r

t+1

@Crt+1

@Crt+1

@Wt (z)+ (1� �)

@U ot+1

@Cot+1

@Cot+1

@Wt (z)

�Since the demand for union z�s type of labour service is given by the usualrelation (implied by the Dixit-Stiglitz aggregator)

Nt (z) =

�Wt (z)

Wt

��"wNt

we have@Nt (z)

@Wt (z)= �"w

Nt (z)

Wt (z)

and from the budget constraints we get

@Cit

@Wt (z)=1

Pt

�(1� "w)Nt (z)� �w

�Wt (z)

Wt�1 (z)� 1�

WtNt

Wt�1 (z)

�and

@Cit+1

@Wt (z)=

1

Pt+1�w

�Wt+1 (z)

Wt (z)� 1�

W 2t+1

(Wt (z))2Nt+1

for i 2 fo; rg.Inserting these expressions in the �rst-order condition, imposing symme-

try so that Wt (z) =Wt and Nt (z) = Nt for all z, and rearranging gives

0 =

��@U r

t

@Crt

+ (1� �)@U o

t

@Cot

�Wt

Pt[(1� "w)� �w (�

wt � 1)�wt ] + "wN

't

+�Et

���@U r

t+1

@Crt+1

+ (1� �)@U o

t+1

@Cot+1

��w��wt+1 � 1

��wt+1

Wt+1

Pt+1

Nt+1

Nt

�where �wt = Wt=Wt�1 and

@U it

@Cit

=�Cit

���111

when the instantaneous utility is given by (1). Log-linearising gives (13) inthe text.Habit persistence. With habit persistence in consumption, the instan-

taneous utility function of a household is given by

U it =

�Cit � hi �C

it�1�1�� � 1

1� �� (N

it )1+'

1 + '

where i 2 fo; rg and �Cit�1 denotes aggregate consumption by households of

type i at time t. The degree of habit in consumption is governed by the pa-rameter hi. With this speci�cation, habit formation is external with respectto the household itself in the sense that the household ignores the e¤ect ofits current consumption choice on the lagged consumption term that entersthe utility function next period. But habit formation is internal with respectto the type of household since the lagged consumption term is aggregateconsumption by the class of households to which the household belongs asopposed to aggregate consumption by all households in the economy. In thelimiting case where hi = 0, there is no habit formation for a household oftype i.With habit formation, the marginal utility of consumption becomes

@U it

@Cit

=�Cit � hi �C

it�1���

=�Cit � hiC

it�1���

where the last equality follows from the fact that all households of a giventype are identical so that Ci

t =�Cit for all t. Using this expression in the

union�s �rst-order condition and log-linearising gives (13) in the text onlythat now (14) must be replaced by

mrst = �r�crt � hrc

rt�1�+ �o

�cot � hoc

ot�1�+ 'nt (18)

where

�r = ��

1� hr

(1� ho)�

� (1� ho)� + (1� �) (1� hr)

�

and

�o = �(1� �)

1� ho

(1� hr)�

� (1� ho)� + (1� �) (1� hr)

�

Note that for hr = ho = 0 this is identical to (14) in the text. Forhr = ho = h > 0, we get �r = �= (1� h) and �o = (1� �) = (1� h). Wegenerally assume that hr = 0 and ho > 0.

112

Habit persistence also changes the optimising household�s stochastic dis-count factor, which is derived from its �rst-order conditions with respect toconsumption and bond holdings. That is,

�t;t+k = �

�Cot � hoC

ot�1

Cot+k � hoCo

t+k�1

��PtPt+k

Taking expectations of this equation with k = 1 gives the Euler equation foroptimising consumption with habit persistence. The log-linear representationis given by

cot =ho

1 + hocot�1 +

1

1 + hocot+1 �

1� ho1 + ho

1

�(rt � Et�t+1) (19)

With habit formation, this equation replaces (2) in the text. Note that theyare identical when ho = 0.

113

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114

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[18] Fernald, J., 2007. Trend breaks, long run restrictions and contractionarytechnology improvements. Federal Reserve Bank of San Francisco Work-ing Paper, 2005-21.

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[20] Fisher, J., 2006. The dynamic e¤ects of neutral and investment-speci�ctechnology shocks. Journal of Political Economy 114, 413-451.

[21] Forni, L., Monteforte, L., Sessa, L., 2007. The estimated general e¤ectsof �scal policy: the case of the Euro area. Manuscript, Banca d�Italia.

[22] Francis, N., Ramey, V., 2005. Is the technology-driven business cyclehypothesis dead? Shocks and aggregate �uctuations revised. Journal ofMonetary Economics 52, 1379-1399.

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115

[24] Furlanetto, F., Seneca, M., 2007. Fiscal shocks and real rigidities. Man-uscript, Norges Bank and University of Aarhus.

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116

[35] Mankiw, N.G., 2000. The savers-spenders theory of �scal policy. Amer-ican Economic Review 90, 120-125.

[36] Marchetti, D.J., Nucci, F., 2005. Price stickiness and the contractionarye¤ect of technology shocks. European Economic Review 49, 1137-1163.

[37] McGrattan, E., 2004. Comment on Galí and Rabanal�s. NBER Macro-economics annual 2004.

[38] Natvik, G.J., 2006. Government spending and the Taylor principle.Norges Bank working paper 2006/11.

[39] Prescott, E., 1986. Theory ahead of business cycle measurement. Min-neapolis Fed Quarterly Review 10, 9-22.

[40] Rotemberg, J., 1982. Monopolistic price adjustment and aggregate out-put. Review of Economic Studies 49, 517-531.

[41] Taylor, J.B., 1993. Discretion versus policy rules in practice. Carnegie-Rochester Conference Series on Public Policy 39, 195-214.

117

0 5 10 15−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0hours

time

per

cent

no rule−of−thumbrule−of−thumb

Figure 1: The response of hours to a technology shocks (ρa = 0.9) in thebaseline model (with and without rule-of-thumb consumers).

118

0 5 10 15−0.5

0

0.5rule−of−thumb consumption

0 5 10 150

1

2optimising consumption

no rule−of−thumbrule−of−thumb

0 5 10 150

0.5

1

1.5aggregate consumption

0 5 10 15−1

−0.5

0hours

0 5 10 150

0.5

1

1.5investment

0 5 10 150

0.5

1output

0 5 10 150

0.5

1real wage

0 5 10 15−3

−2

−1

0annual interest rate

0 5 10 15−2

−1

0annual inflation

Figure 2: The responses of key variables to a technology shock (ρa = 0.9) inthe baseline model (with and without rule-of-thumb consumers).

119

0 5 10 15−1

−0.5

0

0.5rule−of−thumb consumption

0 5 10 150

0.5

1optimising consumption

0 5 10 15−0.5

0

0.5

1aggregate consumption

0 5 10 15−2

−1

0hours

no rule−of−thumbquintuple stickiness

0 5 10 150

0.5

1

1.5investment

0 5 10 15−0.5

0

0.5output

0 5 10 150

0.5

1real wage

0 5 10 15−3

−2

−1

0annual interest rate

0 5 10 15−2

−1

0annual inflation

Figure 3: The response of key variables to a technology shock (ρa = 0.9) inthe quintuple-sticky model.

120

0 5 10 15−0.5

0

0.5

1rule−of−thumb consumption

0 5 10 150

1

2optimising consumption

0 5 10 15−1

0

1

2aggregate consumption

0 5 10 15−1.5

−1

−0.5

0hours

no rule−of−thumbquintuple stickiness

0 5 10 150

1

2investment

0 5 10 150

0.5

1

1.5output

0 5 10 150

0.5

1

1.5real wage

0 5 10 15−2

−1

0

1annual interest rate

0 5 10 15−2

−1

0

1annual inflation

Figure 4: The responses of key variables to a technology shock (ρa = 1) inthe quintuple-sticky model.

121

0 5 10 150

0.5

1

1.5aggregate consumption

0 5 10 15−1

0

1

2hours

nominal rigidities onlyreal rigidites only

0 5 10 150

5

10investment

0 5 10 150

1

2

3output

0 5 10 150

0.5

1

1.5real wage

0 5 10 15−4

−2

0annual interest rate

0 5 10 15−3

−2

−1

0annual inflation

Figure 5: The responses of key variables to a technology shock (ρa = 0.9) ina version with nominal rigidities only (sticky prices and sticky wages) andin a version with real rigidities only (habit formation and capital adjustmentcosts).

122

0 0.5 1

−2

0

2

lampda

0 0.5 1

−2

0

2

thetaw

0 0.5 1

−2

0

2

thetap

5 10 15

−2

0

2

eta

0 0.5 1

−2

0

2

ho

0 0.5 1

−2

0

2

hr

0 0.5 1 1.5 2

−2

0

2

phi

1 2 3 4 5

−2

0

2

phipi

0 1 2 3 4 5

−2

0

2

sigma

hoursconsumptionoutput

Figure 6: Sensitivity analysis: Impact-responses of hours, consumption andoutput as a function of parameter values.

123

Chapter 4

Fiscal shocks, rule-of-thumb consumers andreal rigidities�

Francesco FurlanettoNorges Bank

Martin SenecaUniversity of Aarhus

July 2008

Abstract

In this paper we show that empirically plausible results on the ef-fects of �scal shocks in Galí, López-Salido and Vallés (2007) rely ona high degree of price stickiness and a large percentage of �nanciallyconstrained agents. Real rigidities in the form of habit persistence,�xed �rm-speci�c capital and Kimball demand curves interact in in-teresting ways with nominal and �nancial rigidities and allow us toreproduce the same consumption multiplier as Galí et al. (2007) un-der only two and a half quarters of price stickiness, instead of four,and only 30 per cent of constrained agents instead of 50 per cent.Therefore, real rigidities are useful in the study of �scal shocks in ad-dition to monetary and productivity shocks as has been shown in theprevious literature.JEL Classi�cation: E32, E62.

�We thank (without implicating) Maarten Dossche, Jordi Galí, Gisle Natvik andTommy Sveen for useful comments and discussion. The paper has appeared as DanmarksNationalbank Working Paper No. 55 and Norges Bank Working Paper 2008/10.

127

1 Introduction

One of the most important developments in modern macroeconomics hasbeen the replacement of traditional ad hoc models with dynamic stochasticgeneral equilibrium (DSGE) models in economic policy analysis. In the NewKeynesian DSGE literature, the bulk of the contributions have focused onmonetary policy issues. But recently, a number of authors have begun toinvestigate the responses of key macroeconomic variables to �scal shocks inthe class of DSGE models with imperfect competition and nominal rigidities.In a noteworthy example, Galí et al. (2007) show that nominal rigidities

in combination with deviations from Ricardian equivalence can explain em-pirically observed responses to government spending shocks, while responsesin the baseline real business cycle (RBC) model are in contrast with theempirical evidence. In particular, a number of recent empirical papers sug-gest that private consumption increases following a positive shock to gov-ernment consumption.1 While the RBC model predicts a decline in privateconsumption following such a shock, cf. Baxter and King (1993), privateconsumption may rise after a positive shock to government spending in thesticky-price model of Galí et al. (2007) if so-called rule-of-thumb consumers,who simply consume their current disposable income each period, are al-lowed to co-exist with intertemporally optimising consumers.2 In the model,intertemporally optimising consumers decrease their consumption followinga government spending shock because they correctly anticipate a decline infuture income as a consequence of taxation. But rule-of-thumb consumersincrease their consumption because their current income increases. Underthe necessary auxiliary assumptions of sticky prices, monopolistic competi-tion in the labour market and de�cit �nancing, if a su¢ ciently large fractionof households behave according to a rule of thumb, aggregate consumptionrises.A potential weakness of the rule-of-thumb theory of consumption is that

both the degree of nominal rigidity and the fraction of rule-of-thumb con-sumers needed to generate a positive response of consumption is uncomfort-

1See, for example, Blanchard and Perotti (2002), Caldara and Kamps (2007), Fatasand Mihov (2002), Galí et al. (2007) and Perotti (2005, 2008).

2Alternative approaches with similar objectives can be found in Bouakez and Rebei(2007), Linnemann and Shabert (2005), Linnemann (2006), López-Salido and Rabanal(2006) and Ravn, Schmitt-Grohé and Uribe (2007).

128

ably high given the recent empirical literature. In the baseline calibration inGalí et al. (2007), the expected duration of prices is set at one year, and halfthe consumers in the economy choose how much to consume by following asimple rule of thumb. Recent microeconomic evidence, however, points totwo or three quarters of expected price duration, e.g. Bils and Klenow (2004)and Nakamura and Steinsson (2007), and several studies arrive at estimatesof the percentage of rule-of-thumb consumers that are much lower than the50 per cent originally suggested by Mankiw (2000). For instance, Campbelland Mankiw (1991) obtain 35 per cent for the US and 20 per cent for theUK, while Banerjee and Batini (2003) �nd 26 per cent for the US and 15 percent for the UK.The values of these parameters are crucial in the Galí et al. (2007) model.

Once they are lowered to more realistic values (say, 2.5 quarters of price stick-iness and 30 per cent of constrained agents as in our benchmark), the main re-sult in Galí et al. (2007), i.e. that a model with rule-of-thumb consumers cangenerate a positive response of consumption following a government spendingshock, is lost.The main objective of this paper is to reconcile the evidence on these

structural characteristics of the economy with the empirical responses of pri-vate consumption to a government spending shock. We show that this canbe done by adding a number of what we consider to be realistic featuresto the model developed by Galí et al. (2007) to lower its dependence onprice stickiness and households that do not take part in �nancial marketsso as to smooth consumption.3 The features we consider are real rigiditiesin the form of habit persistence in consumption, non-constant elasticities ofdemand, and �xed �rm-speci�c capital.4 Each of these rigidities has proven

3The idea that a fraction of households follow the simple rule of thumb that theyconsume their current disposable income each period, while the remaining fraction solvean intertemporal optimisation problem, was �rst put forward in the empirical consumptionliterature as an alternative to the permanent income hypothesis, see in particular Hall(1978) and Campbell and Mankiw (1989). We emphasise the interpretation that somehouseholds follow a rule of thumb because a �nancial friction bars them from participatingin �nancial and capital markets. Alternatively, rule-of-thumb consumers may choose notto do so because of myopia or extreme impatience.

4We refer to all these three features as real rigidies to separate them conceptionallyfrom the nominal rigidities that act as direct impediments to the adjustment of nomi-nal variables, and from the �nancial constraint represented by rule-of-thumb consumers.Hence, our de�nition includes both the rigidities that work as direct impediments to theadjustment of real variables, and the �real rigidities�of Ball and Romer (1990), the pres-

129

to be very useful in DSGE analyses in explaining empirical regularities of thetransmission of other shocks, especially monetary shocks, see e.g. Christianoet al. (2005), Smets and Wouters (2003) and Woodford (2003), and pro-ductivity shocks, e.g. Francis and Ramey (2005) and Furlanetto and Seneca(2007). But their implications for the propagation of �scal shocks have notbeen thoroughly analysed so far. This, in itself, provides a second motivationfor this paper. Before giving a preview of the results, we brie�y discuss eachof these rigidities in turn.The idea that habits may in�uence households�consumption behaviour

grew out of the attempts in the mid-20th century empirical demand theoryto explain the importance of lagged dependent variables in estimated de-mand functions, see e.g. Brown (1952), or the discussions of this literaturein Deaton and Muellbauer (1980) and Deaton (1992). More recently, habitformation has been introduced into policy-oriented general equilibrium mod-els following the speci�cation in the asset pricing model by Abel (1990), inwhich utility today depends on consumption today relative to consumptionin previous periods. For an example, see Christiano et al. (2005). In ourmodel, habit persistence is important because it smooths the negative re-sponse of optimising households to a government spending shock. Hence, asmaller fraction of rule-of-thumb consumers is needed to generate a plausibleresponse of aggregate consumption.The second source of real rigidity that we introduce into the model is de-

mand functions with non-constant elasticity of demand of the sort suggestedby Kimball (1995). This represents a modi�cation of the formalisation of mo-nopolistic competition by Dixit and Stiglitz (1977) that has become standardin macroeconomics following the seminal paper by Blanchard and Kiyotaki(1987). The relative demand for an individual good is still decreasing inthe relative price, but the elasticity �and hence the desired mark-up overmarginal costs of the price-setting �rm that produces it �now depend on itsrelative output. This induces a potential source of strategic complementarityin price setting in the model as discussed by Kimball (1995) and Woodford(2003, ch. 3). If the elasticity of demand falls with relative output, for in-stance, a �rm that reduces its price will moderate its price reduction becausethe increase in demand it induces increases the desired mark-up. In thiscase, the �rm is more reluctant to change prices away from the level charged

ence of which characterises an economy with strategic complementarity in price setting,cf. the discussion in Woodford (2003, ch. 3).

130

by other �rms in the economy that may not be changing their prices in anygiven period. In this way, the Kimball demand speci�cation ampli�es thee¤ect of any nominal price rigidity that prevents some �rms from adjustingprices. This makes it possible to obtain realistic dynamics of key macroeco-nomic variables with lower degrees of nominal price stickiness as emphasisedby Eichenbaum and Fisher (2007) and Levin et al. (2007).Firm-speci�c capital is a relatively recent addition to the DSGE literature

pioneered by Christiano (2005), Sveen and Weinke (2005) and Woodford(2005). The standard assumption in the literature is that �rms rent perfectlymobile capital from households in a rental market. With �rm-speci�c capital,in contrast, the economy�s capital stock is owned by �rms, and capital cannotbe instantaneously reallocated across �rms to equalise marginal costs. Asargued, for instance, by Danthine and Donaldson (2002), the �rm-speci�ccapital assumption is probably the more appealing one in terms of realism.For our purposes, the important implication of �rm-speci�c capital is

that it increases the strategic complementarity in price setting as describedby Sveen and Weinke (2005) and Woodford (2005). For simplicity, we followCoenen et al. (2007) by abstracting from the endogenous accumulation of�rm-speci�c capital.5 Instead, we assume that each �rm is endowed with a�xed level of the capital good as in Sbordone (2002), resulting in a productionprocess with decreasing returns to labour. With this speci�cation, we retainthe important implication of �rm-speci�c capital that �rms cannot reallocatecapital instantaneously across �rms to equalise marginal costs.As already mentioned, non-constant elasticities of demand and �xed �rm-

speci�c capital help us to reduce the degree of price stickiness in the model,according to the mechanism explained in Sveen and Weinke (2005), Eichen-baum and Fisher (2007) and Woodford (2005). However, these frictions arealso useful to lower the percentage of rule-of-thumb consumers in the modeland this e¤ect is new in the literature. In fact, they imply a lower in�ationresponse to a given change in the marginal cost which translates into a lowerresponse by the monetary policy authority. A lower increase in the interestrate pushes-up optimising consumption, making the model less dependent onrule-of-thumb consumers.Introducing the rigidities just described in the model developed by Galí

et al. (2007) gives us this paper�s main result: Real rigidities are useful

5Similarly, some authors abstract from the endogenous capital accumulation processunder the rental market assumption, e.g. Erceg, Henderson and Levin (2000).

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not only in accounting for the economy�s responses to monetary policy andproductivity developments as has been emphasised in the existing literature,but also in accounting for responses to �scal policy shocks. In particular, wearrive at an empirically plausible increase in private consumption followinga government spending shock for a much lower degree of price rigidity and amuch lower fraction of rule-of-thumb consumers than in Galí et al. (2007).With habit formation in consumption, �xed �rm-speci�c capital and Kimballdemand, we obtain the same consumption multiplier as in Galí et al. (2007)with two and a half quarters of expected price duration (as opposed to four)and 30 per cent of rule-of-thumb consumers (as opposed to 50). Thus, asin Furlanetto and Seneca (2007), we �nd an important role for the interac-tion of nominal, real and �nancial rigidities in realistically accounting for theempirical evidence on the response of a key macroeconomic variable to em-pirically important disturbances to the economy. Importantly, rule-of-thumbconsumers remain essential to generate this result. An alternative perspec-tive, then, is that the rule-of-thumb theory becomes more appealing in asetting that is probably more realistic than the one in which it was originallyintroduced.The paper is organised as follows. In section 2, we present the model,

and in section 3 the results. Section 5 gives a few concluding remarks.

2 The model

The model is a standard New Keynesian dynamic stochastic general equi-librium model augmented with habit persistence in consumption, Kimball(1995) demand curves and rule-of-thumb consumers. Except for the presenceof real rigidities, the model is identical to Galí et al. (2007). The economyconsists of a continuum of �rms, a continuum of households, a continuumof labour unions, a central bank responsible for monetary policy, and a gov-ernment collecting lump-sum taxes and issuing bonds to �nance its expendi-tures. There is monopolistic competition in both goods and labour markets.In particular, there is a continuum of di¤erentiated intermediate goods and acontinuum of di¤erentiated labour services. In the goods market, this leadsto a downward-sloping demand curve for each intermediate good, and in thelabour market it leads to a downward-sloping demand curve for each labourtype.A fraction � of households are rule-of-thumb consumers - or �spenders�in

132

the terminology of Mankiw (2000). These consumers simply consume theirrespective disposable income each period. The remaining fraction 1 � � ofhouseholds are optimisers - or �savers�- that have access to �nancial markets.Hence, they choose plans for consumption and bond holdings to maximiselifetime utility. Consumers are assumed to form habits in consumption. Thatis, the utility a household obtains from a given level of consumption in agiven period depends on the level of consumption in that period relative tothe level of consumption in the previous period. Wages are set by unions eachrepresenting a di¤erentiated type of labour service supplied by households.Wages are assumed to be �exible. That is, each union sets a new wage forits members each period to maximise an average of their utilities taking thee¤ect of this wage on the members�budget constraints into account.Each �rm produces one of the di¤erentiated intermediate goods. It does

so by combining capital with a homogenous labour input constructed asa Dixit-Stiglitz aggregate of the di¤erentiated labour services supplied byhouseholds. The �rm sets its price according to a Calvo (1983) price-settingmechanism and stands ready to satisfy demand at the chosen price. Theelasticity of the demand it faces depends on the level of output produced asin Kimball (1995). In particular, the elasticity of demand falls with the levelof output. This is known to increase the degree of strategic complementarityin price-setting, cf. Woodford (2003, ch 3).We consider two alternative assumptions concerning the structure of the

capital market. Under the �rst assumption, the economy�s capital stock isowned by the optimising households. In this case, �rms rent the capitalthey employ in production in a common rental market, and capital can bereallocated across �rms instantaneously. We allow for endogenous accumula-tion of capital under this assumption by letting households choose how muchto invest in new capital each period. But we also assume that it is costlyto adjust the capital stock. Consequently, the aggregate stock of capital is�xed in the limiting case where the capital adjustment cost goes to in�nity.Rule-of-thumb consumers do not take part in the capital market. Under thesecond assumption, the capital stock is owned by �rms, and capital cannotbe reallocated across them. That is, capital is speci�c to individual �rms.For simplicity, we abstract from endogenous capital accumulation and as-sume that the capital stock is �xed under this assumption. To encompassthese two alternative assumptions on the structure of the capital market inthe model, we de�ne a dummy variable � taking the value 1 under the rental

133

market assumption and 0 when capital is �rm-speci�c6, i.e.

� =

�1 if capital is owned by households0 if capital is owned by �rms

Each period begins by the realisation of shocks to the economy. Weconcentrate on �scal spending shocks and abstract from other shocks thatmay a¤ect the economy.

2.1 Households

The instantaneous utility function of a household is given by

U it =

�Cit � hi �Cit�1

�1�� � 11� � � (N

it )1+'

1 + '(1)

where i 2 fo; rg denotes the type of household �optimising or rule-of-thumb�and �Cit�1 denotes aggregate consumption by households of type i at time t.The degree of habit in consumption is governed by the parameter hi. Withthis speci�cation, habit formation is external with respect to the householditself in the sense that the household ignores the e¤ect of its current consump-tion choice on the lagged consumption term that enters the utility functionin the next period. But habit formation is internal with respect to the typeof household since the lagged consumption term is aggregate consumptionby the class of households to which the household belongs as opposed to ag-gregate consumption by all households in the economy. In the limiting casewhere hi = 0, there is no habit formation for a household of type i.An optimising household maximises expected lifetime utility given by

E0

1Xt=0

�tU ot (2)

where Eo is an operator representing expectations over all states of the econ-omy conditional on period-0 information, and � 2 (0; 1) is the subjective

6Nothing, in principle, prevents this variable from taking intermediate values. Thiswould correspond to an economy in which a share of the capital stock is owned by house-holds and rented to �rms, while the remaining share is �rm-speci�c. We do not pursuethis possibility here, though few things would change in the speci�cation of the model.

134

discount factor. Maximisation is subject to a sequence of �ow budget con-straints (and implicitly a no-Ponzi game condition):

PtCot + Et (�t;t+1Bt+1) =WtN

ot +Bt � PtT ot + �

�RktK

ot � PtIot

�(3)

where Wt is the nominal wage, Pt is the aggregate price index and T ot isthe real lump-sum tax paid by optimising consumers. The left-hand sidegives the allocation of resourses to consumption and a portfolio of bonds,Et (�t;t+1Bt+1), where �t;t+1 is the stochastic discount factor so that the risk-free interest rate is given by the relation 1 + Rt = (Et�t;t+1)

�1. The right-hand side gives available resources as the sum of labour income,WtN

ot , initial

�nancial wealth, Bt, less nominal lump-sum taxes paid to the government,PtT

ot . Finally, under the assumption that the economy�s capital stock is

owned by households, the household receives rent for its capital, RktKot , where

Rkt is the rental rate of the capital it owns, Kot , and allocates resources to

investment, PtIot . Under this assumption, the household�s capital evolvesaccording to

Kot+1 = (1� �)Ko

t + �

�IotKot

�Kot (4)

where � is the rate of depreciation, and � (:) is an adjustment cost functionsatisfying � (�) = �, �0 > 0, �0 (�) = 1 and �00 � 0.The optimisation problem, according to which the household chooses

plans for consumption and bond holdings, gives rise to a modi�ed versionof the well-known Euler equation which we state in log-linear form7:

cot =1

1 + hocot�1 +

1

1 + hoEtc

ot+1 �

1� ho1 + ho

1

�(rt � Et�t+1) (5)

Because of habit formation in consumption, the Euler equation now containsa term in lagged consumption. Note that this equation reduces to the stan-dard Euler equation for ho = 0. For � = 0, i.e. under the assumption that�rms own the capital stock, this is the only �rst-order condition for optimis-ing consumers. For � = 1, i.e. with a rental market for capital, the optimisinghousehold also chooses investment. As shown by Galí et al. (2007), the �rst-order conditions to this problem represent the dynamics of Tobin�s q and itsrelation to investment, and their log-linear forms are given by

qt = � (rt � Et [�t+1]) + [1� � (1� �)]Et�rkt+1 � pt

�+ �Et [qt+1] (6)

7In general, lower case variables denote log-deviations from corresponding uppercasevariables. Omission of time subscripts indicates steady-state variables.

135

it � kt = �qt (7)

where � = �1= (�00 (�) �).8A rule-of-thumb household does not take part in �nancial or capital mar-

kets, and thus faces the following simple budget constraint regardless of theassumption on the ownership of capital:

PtCrt = WtN

rt � PtT rt (8)

Here, Crt is the household�s real consumption at time t, and Nrt is the hours

worked by the household in period t. As a rule-of-thumb household simplyconsumes its current income, consumption follows directly from the budgetconstraint. A �rst-order log-linear approximation around the steady statewith constant consumption equalised across households gives

crt =WN

PC(wt + nt)�

Y

Ctrt (9)

where omission of time subscripts indicates steady-state variables.9

Aggregate variables are given as simple weighted averages:

ct = �crt + (1� �) cot (10)

nt = �nrt + (1� �)not (11)

andtt = �t

rt + (1� �) tot (12)

2.2 Labour unions

The economy has a continuum of unions z 2 [0; 1] each representing a con-tinuum of workers, a fraction (1� �) are optimising, and a fraction � arerule-of-thumb consumers. Each union sets the wage rate for its members,who stand ready to satisfy �rms�demand for their labour services at the

8Note that it and kt are the log-deviations from corresponding steady-state valuesof aggregate investment and capital, respectively, de�ned as Kt = (1� �)Ko

t and It =(1� �) Iot .

9We maintain the assumption that consumption is equalised across agents in the steadystate to facilitate comparability with Galí et al. (2007). For an alternative approach, seeNatvik (2008).

136

chosen wage. The workers in a union provide the same type of labour (ir-respective of their consumption behaviour) di¤erentiated from the type oflabour services provided by members of other unions. The labour servicesupplied by each union, N (z), is a simple aggregate of its members�labourservices. In turn, the labour entering the production function of any �rm is aDixit-Stiglitz aggregate of the labour services provided by the unions in theeconomy. Hence, the labour demand for a union�s labour services is given by

Nt (z) =

�Wt (z)

Wt

��"wNt (13)

where Wt (z) is the wage set by the union, and "w is the elasticity of labourdemand.Each period, a representative union choosesWt (z) to maximise the present

value of an average of its members�current and future period utility func-tions, that is,

maxWt(z)

Et

1Xk=0

�t+k��U rt+k + (1� �)U ot+k

�(14)

subject to the labour demand functions and the budget constraints of itsmembers, thus taking the e¤ect of the wage decision on the income of itsmembers into account.The �rst-order condition can be expressed in the form of Galí et al.

(2007): ��

MRSrt+1� �MRSot

�=

"w"w�1

Wt

Pt(15)

where, now, the marginal rate of substitution is given byMRSit = (Cit � hiCit)

�N't

for i 2 fo; rg because of habit formation in consumption. As shown byFurlanetto and Seneca (2007), log-linearising this expression gives

wt � pt = �r�crt � hrcrt�1

�+ �o

�cot � hocot�1

�+ 'nt (16)

where

�r = ��

1� hr(1� ho)�

� (1� ho)� + (1� �) (1� hr)�

and

�o = �(1� �)1� ho

(1� hr)�

� (1� ho)� + (1� �) (1� hr)�

137

2.3 Goods demand

The economy has a continuum of �rms j 2 [0; 1], each of which producesa di¤erentiated product, Yt (j). The �nal good used in private and publicconsumption is an index of this continuum of intermediate goods. FollowingKimball (1995) it is de�ned implicitly by the relationshipZ 1

0

G (Xt (j)) dj = 1 (17)

where Xt (j) = Yt (j) =Yt is relative demand, and G (:) is a function satisfyingG (1) = 1, G 0 > 0 and G 00 < 0.For a given level of consumption and investment, and for given prices,

Pt (j), expenditure minimisation leads to the following demand for �rm j�sproduct

Xt (j) = ~G�Pt (j)Yt�t

�(18)

where ~G (:) is the inverse function of G 0 (:) and �t is the Lagrange multiplierfrom the minimisation problem. If we de�ne the price de�ator Pt implicitlyby

PtYt =

Z 1

0

Pt (j)Yt (j) dj (19)

we have

�t = PtYt

�Z 1

0

G 0 (Xt (j))Xt (j) dj

��1(20)

Note that the assumption that G 00 < 0 implies that this demand function isdownward-sloping. It follows that the price elasticity of demand is given by

� (Xt (j)) = �G 0 (Xt (j))

G 00 (Xt (j))Xt (j)(21)

In log-linear terms, the demand function becomes

yt (j) = ��� (pt (j)� pt) + yt (22)

where �� = � (1).In the special case where

G (Xt (j)) = (Xt (j))��1� (23)

138

(17) reduces to the more common Dixit-Stiglitz aggregator, which leads toa constant elasticity of substitution since, in this case, � (Xt (j)) = �� for allXt (j). As is well-known, this leads to a constant desired mark-up of price-setting �rms given by �p = ��=

��� � 1

�. In the general Kimball speci�cation,

we allow the demand elasticity and hence the desired mark-up to vary withthe level of output. For future reference de�ne

� (Xt (j)) =@� (Xt (j))

@Pt (j)

Pt (j)

� (Xt (j))(24)

This is the own price elasticity of the elasticity of demand. In the steadystate we have � (1) = ��. In the analysis, we employ the case where �� > 0,i.e., the case where the elasticity of demand is increasing in the price set bythe �rm, or equivalently decreasing in its relative output. This is known toincrease the strategic complementarity in price setting as discussed in section1.

2.4 Firms

Firm j produces according to the technology

Yt (j) = ~Kt (j)�Nt (j)

1�� (25)

where ~K (j) the capital used as input by �rm j, Nt (j) is the labour employedby the �rm, and 0 < � < 1. When the capital is owned by the �rms, weassume that all �rms have identical endowments of capital and we normalisethis level to 1. Denoting the household-owned capital employed in productionby �rm j by Kt (j), we have in general that ~Kt (j) = (Kt (j))

�. Note thatreal marginal costs are given by

MCt (j) =Wt=Pt

(1� �)�~Kt (j) =Nt (j)

�� (26)

When �rms rent capital from households, i.e. when � = 1, cost minimi-sation implies that �rm j will choose factor inputs such that

Wt

Rkt=1� ��

Kt (j)

Nt (j)(27)

Since all �rms have to pay the same wage for the labour they employ, andthe same rental rate for the capital they rent, it follows that marginal costs

139

are equalised across �rms under this assumption. In contrast, when � = 0and capital is �rm-speci�c, marginal costs will generally be di¤erent across�rms.We now turn to the �rms�price-setting decisions. Each �rm is allowed

to set a new price, P �t , with a �xed probability (1� �) as in Calvo (1983).This implies that the expected duration of prices is given by (1� �)�1.The�rm�s decision is made to maximise the value of the �rm to its owners, theoptimising households, given by

1Xk=0

Et f�t;t+k [P �t Yt+k (j)�(Yt+k (j))]g (28)

where (:) is the cost function, subject to its production function (25) andto the demand for its product given by (18).10

The following �rst-order condition represents the price-setting equation:

1Xk=0

�kpEt f�t;t+kYt+k (j) [P �t (1� � (Xt+k (j)))]g

=1Xk=0

�kpEt f�t;t+kYt+k (j) [� (Xt+k (j))Pt+kMCt+k (j)]g (29)

where MCt (j) is �rm j�s real marginal cost given by (26).From the log-linearisation of (29) we may derive the following New Key-

nesian Phillips curve for price in�ation

�t = �Et�t+1 + �mct (30)

where the slope parameter � is given by

� =(1� ��) (1� �)

�

�1 +

���� � 1

+ (1� �) �

1� ���

��1(31)

The derivation is sketched in appendix A. Note that � is declining in both �(the degree of nominal rigidity) and �� (the curvature of the demand parame-ter). Also �j�=0 < �j�=1. That is, the New Keynesian Phillips curve is �atterwith �xed �rm-speci�c capital than with rental capital.10With rental capital, the cost function is the value function from the cost minimisation

problem. With �xed �rm-speci�c capital, the cost function is simply Wt+kNt+k (j) wherethe production function is used to substitute for Nt+k (j).

140

2.5 Economic policy

The speci�cation of economic policy follows Galí et al. (2007). The centralbank controls the risk-free interest rate, which it sets according to a simpleTaylor rule

rt = r + ���t (32)

The government budget constraint is

PtTt +R�1t Bt+1 = Bt + PtGt (33)

the linearisation of which becomes

bt+1 = � (bt + gt � tt) (34)

where bt = (Bt=Pt�1 �B=P )Y , gt = (Gt�G)=Y and tt = (Tt�T )=Y . Fiscalpolicy is given by the rule

tt = �bbt + �ggt (35)

Government spending (normalised by steady-state output and expressedin deviations from steady state) evolves exogenously according to the follow-ing �rst-order autoregressive process

gt = �ggt�1 + "t (36)

where 0 < �g < 1 and "t is white noise with variance �2". With this speci�ca-

tion, the government �nances the exogenous disturbances to its spending inany given period partly through taxes, partly through the issuance of bonds.

2.6 Equilibrium

Market clearing requires that

Yt = Ct + It +Gt (37)

In log-linear form, this becomes

yt =C

Yct +

I

Yit + gt (38)

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3 The consumption multiplier

As in Galí et al. (2007), we analyse the e¤ects of government spending shocksemphasising the response of private consumption. Speci�cally, we focus onthe impact response of aggregate private consumption following a shock togovernment spending normalised to one per cent of the level of output in thesteady state. We refer to this impact response as the consumption multiplier.As shown by Galí et al. (2007), this impact multiplier is signi�cantly abovezero in the data.

3.1 The model without real rigidities

To set the scene, �gure 1 shows the consumption multiplier as a function ofthe fraction of rule-of-thumb consumers, �, and as a function of the degreeof price rigidity, �, in the model analysed by Galí et al. (2007). This isequivalent to the model in section 2 when ho = hr = �" = 0 and � = 1. Thatis, it is a version of the model with a rental market for capital, without habitformation in consumption, and with a constant elasticity of demand. Thecalibration of the remaining parameters follows the baseline calibration inGalí et al. (2007). Hence, we consider a time period to be one quarter, andwe set � = 0:025, � = 0:33, � = � = 1, � = 0:99, � = 0:5, g = 0:2, �� = 1:5,�b = 0:33, �g = 0:1, �� = 6, �g = 0:9 and ' = 0:2. Finally, in the baselinecalibration � = 0:5 and � = 0:75. Note for future reference that this baselinecalibration gives a value of the consumption multiplier of approximately 1.2.Consider the solid lines �rst. These lines show the consumption multiplier

in the Galí et al. (2007) model as a function of � (left panel) and � (rightpanel) with the other parameters remaining as under the baseline calibration.We see that, keeping � �xed at 0:75, the consumption multiplier is positiveonly for values of � larger than 0.3. Similarly, keeping � �xed at 0.5, themultiplier is positive only for values of � above a critical value between 0.5and 0.6 corresponding to between two and three quarters of expected pricestickiness. Hence, if we lower one of these two key parameters from the valuechosen under the baseline calibration to one that is more realistic given theempirical evidence described in section 1, the consumption multiplier is nolonger positive.Considering the dashed lines, we see that by lowering one of the two

parameters to a more plausible value �� = 0:6 and � = 0:3 respectively �we make it harder to obtain a positive consumption multiplier for all values

142

of the other parameter. For � = 0:6, the fraction of rule-of-thumb consumersneeds to be close to 0.5 to drive the consumption multiplier above zero, andfor � = 0:3, the expected duration of prices must be longer than a year.Moreover, under our preferred calibration in which � = 0:6 and � = 0:3 atthe same time, the consumption multiplier is seen to be negative.In sum, these pictures show that the positive response of consumption is

a fragile result in two crucial dimensions. It relies on implausibly high valuesfor the degree of nominal rigidity and the percentage of constrained agents.Our contribution is to provide a solution to this problem by reconciling asizeable increase in consumption as in Gali et al. (2007) with reasonablevalues for the degree of nominal rigidity and the �nancial friction. We dothis by adding real rigidities to the model.

3.2 Adding real rigidities

Motivated by the previous sensitivity analysis of the model in Galí et al.(2007), we now present responses from the model augmented with habit per-sistence, Kimball demand and �xed �rm-speci�c capital. We set the fractionof rule-of-thumb consumers, �, to 0.3 inspired by the empirical evidence dis-cussed in section 1, and we set the degree of habit persistence of optimisinghouseholds, ho, equal to 0.85, a value which is within the range of values con-sidered in the literature.11 However, we let the degree of habit persistenceof rule-of-thumb households be zero, that is, hr = 0. This is to facilitate theinterpretation that rule-of-thumb households are inherently di¤erent fromoptimising households by having an entirely static horizon.The calibration of the curvature of the Kimball demand function, rep-

resented by �", is more di¢ cult. As noted by Dossche et al. (2006), thereis no agreement on what a plausible value might be for this parameter inthe literature; estimates range from 1.3 (Bergin and Feenstra, 2000) to 471(Kimball, 1995). In this section we therefore calibrate �" by �xing values forthe slope of the New Keynesian Phillips curve, �, and the degree of nominalrigidity, �. This allows us to recover a value of �" implied by the expression for� given in (31). We set � at 0.6, cf. section 1, while we �x � at 0.03 based onthe reduced-form evidence on the slope of the New Keynesian Phillips curvein Galí et al. (2005) and Levin et al. (2007). The implied value of �" is 25.

11It falls between the value estimated by Christiano et al. (2005) and the one consideredby Woodford (2003, ch. 5).

143

It is possible that 25 is still too high a value for ��, at least accordingto the evidence provided by Dossche et al. (2006). They suggest that avalue around 4 is more reasonable, though they �nd evidence of considerablevariation across sectors. We note that we would need a higher value of ��(around 40) if we had kept the rental capital assumption. This illustratesthat di¤erent real rigidities may interact in the economy in a way that allowsus to consider reasonable values for other parameters representing real and�nancial rigidities.12 Similarly, if we are slightly less ambitious in bringingdown the expected duration of prices, we may obtain a value of � = 0:03 with�� = 4 in the version of our model with �rm-speci�c capital. This requiresus to accept an expected duration of prices of slightly more than 3 quartersinstead of our benchmark 21

2, but still in the range of the plausible values

according to Nakamura and Steinsson (2007).Note that our calibration of � implies a much �atter New Keynesian

Phillips curve than in Galí et al. (2007), where � = 0:0858. In the modelwithout real rigidities, we would need a Calvo parameter of 0.85 to gener-ate a slope of 0:03, clearly an unrealistic value given the empirical evidenceavailable.Figure 2 presents impulse responses to key macroeconomic variables under

this calibration along with responses from the model by Galí et al. (2007).13

The main result of our paper is that the responses of consumption are nearlyidentical in the two models. In both cases, we obtain a consumption multi-plier of approximately 1.2. Hence, the introduction of real rigidities in theform of habit persistence in consumption, Kimball demand and �xed �rm-speci�c capital allows us to generate the same consumption multiplier as inGalí et al. (2007) with an expected price duration of two and a half quarters(instead of four) and with only 30 per cent of �nancially constrained agents(instead of 50). The crucial di¤erence between the two models is that, in themodel with real rigidities, both the fraction of rule-of-thumb consumers andthe degree of price rigidity are more in line with the empirical evidence.Part of the explanation for our result is that, in the model with real

12The model�s equilibrium dynamics for variables other than investment is not a¤ectedby the choice of assumption concerning the structure of the capital market. We thereforeomit reporting of the impulse responses for the rental capital case with Kimball demandand habit formation.13The responses reported here are in percentage deviations from steady state and so

they di¤er slightly from the ones reported in Galí et al. (2007), which are normalised bysteady-state output.

144

rigidities, habit persistence works to mitigate the contractionary e¤ect fromRicardian households by smoothing their response to the shock. Rule-of-thumb households still respond by increasing their consumption since thepartial bond �nancing of the government spending shock makes current in-come go up. But with habit formation in consumption, optimising house-holds need time to adjust to the lower level of consumption called for by thereduction in lifetime income that results from current and future taxation.This makes them reduce consumption less on impact of the shock. Thoughrule-of-thumb consumers now weigh less in the aggregate, the net e¤ect onaggregate consumption is therefore unchanged.This is not the only e¤ect in play, however. With a relatively �at New

Keynesian Phillips curve, a positive shock to government spending that in-creases �rms�marginal costs by increasing aggregate demand in the econ-omy has a smaller e¤ect on in�ation through the price-setting process. Thismakes the central bank respond by increasing interest rates less than in aneconomy with a steeper Phillips curve. This further moderates the nega-tive consumption response of optimising consumers. It is the combination ofhabit formation in consumption and a less responsive demand e¤ect throughmonetary policy that allows us to generate the same consumption multiplieras in Galí et al. (2007) for a lower percentage of rule-of-thumb consumers.Importantly, the introduction of real rigidities that are known to increase

the strategic complementarities in price setting, cf. Woodford (2003), allowsus to reduce the slope of the Phillips curve without increasing the degree ofnominal rigidity. In contrast, our analysis is consistent with �xing � at 0.6in keeping with microeconomic evidence on the frequency of price changes.Note also from �gure 2 that the responses of the other aggregate variablesare also nearly identical in the models. The only exception, of course, isinvestment, which is constant by assumption in the model with �rm-speci�ccapital.14

The importance of habit formation for the consumption response canbe seen from �gure 3, in which we report the consumption multiplier asa function of ho keeping � = 0:03 (left panel), and � keeping ho = 0:85

14As argued by Furlanetto (2007), the model in Galí et al. (2007) exhibits a counter-factually large response of the real wage. However, once he introduces a nominal wagerigidity that smoothes the wage response, the increase in consumption is con�rmed. Wehave also considered a version of the model with real rigidities augmented with nominalwage rigidities. Results are qualitatively similar to the ones reported here. For sake ofcompleteness, they are reported in appendix B.

145

(right panel) when � = 0:3 (in contrast to the baseline � = 0:5). Remainingparameters are at their baseline values. On the left panel it is seen thatreducing the degree of habit persistence lowers the impact response of privateconsumption following a shock to government spending. In the extreme casewithout habit persistence, even if we allow for curvature in the demand curvesby setting �� = 25 so that � = 0:03 when � = 0:6, the consumption multiplieris small (albeit positive).The right panel in �gure 3 shows the consumption multiplier as a function

of �, the slope of the New Keynesian Phillips curve. As noted in section 2,this slope is inversely related to ��, meaning that � goes from 0 to 0.1 as ��goes from in�nity to 0.1. That is, �� declines as we move from left to right onthe graph. When � = 0:3 in the model with habit formation, we see that �has to be close to 0.03 to generate a consumption multiplier close to 1.2. Inparticular, increasing the slope of the New Keynesian Phillips curve reducesthe multiplier. For � = 0:0858 as in the baseline calibration of Galí et al.(2007), we see that the multiplier falls to approximately 0.8 even when habitpersistence curbs the contractionary e¤ect from the 70 per cent of householdsthat optimise intertemporally.To summarise, we have shown that the empirically realistic consumption

multiplier obtained by Galí et al. (2007) with 50 per cent of rule-of-thumbconsumers and 4 quarters of expected price stickiness, can be obtained forconsiderably lower values of these parameters once real rigidities are addedto the model. Habit formation, which directly smoothes the adjustment ofprivate consumption of intertemporally optimising households, reduces thenegative response of optimising consumers for a given monetary policy re-sponse. When combined with real rigidities that amplify the implicationsof nominal rigidities, the contractionary response of monetary policy to the�scal expansion is reduced even for considerably lower degrees of nominalrigidities. This further reduces the negative consumption response of opti-mising households. The combination of these e¤ects allows us to generatethe same positive consumption multiplier as in Galí et al. (2007) with a per-centage of rule-of-thumb consumers given by 30 and an expected duration ofprice rigidities given by two and a half periods.

146

4 Concluding remarks

This paper shows that the rule-of-thumb theory of consumption does notrely on a high degree of nominal rigidity or a large �nancial friction when ac-counting for the conditional responses to government spending shocks. Whenempirically plausible real rigidities are added to the model, they interact withnominal and �nancial rigidities in ways that allow us to specify more rea-sonable parameter values for all the rigidities at work in the model. Hence,we believe that this paper complements the analysis in Galí et al. (2007) byshowing how the rule-of-thumb consumption theory becomes more appealingonce realistic features are added to the model.Interestingly, the same combination of real rigidities that we apply has

been used in the previous literature to replicate conditional responses to othershocks, especially monetary shocks and technology shocks. Habit persistencehas been used to reproduce the hump-shaped response of output and con-sumption on the impact of a monetary shock, while Kimball demand curvesand �rm-speci�c capital have been used to reconcile the microeconomic ev-idence on the degree of price rigidity with the macroeconomic evidence onthe slope of the New Keynesian Phillips curve, cf. references in section 1.In a companion paper to this one, Furlanetto and Seneca (2007) show thatthe interaction of nominal, real and �nancial rigidities is also very helpful inaccounting for the responses of hours worked following a productivity shock.Thus, at a more general level, this paper contributes to this literature by

showing how nominal and real rigidities may interact with a �nancial frictionin ways that generate plausible dynamics following empirically important dis-turbances to the economy. We believe this is a further indication that, whilethe simple basic real business cycle framework is an important benchmarkboth conceptually and methodologically, a realistic model of the economy islikely to be one in which many frictions and rigidities interact. Providingfurther evidence on how this may occur � and not least further empiricalevidence on the relative importance of these rigidities and frictions along thelines of Coenen and Straub (2005) and Forni, Monteforte and Sessa (2007) �is, we believe, an important topic for further research in macroeconomics.

147

A Appendix

The �rst-order condition to the price-setting problem is:

1Xk=0

�kpEt f�t;t+kYt+k (j) [P �t (1� � (Xt+k (j)))� � (Xt+k (j))Pt+kMCt+k (j)]g = 0

We log-linearise this �rst-order condition to get

0 = Et

1Xk=0

(�p�)k ��1� ��� p�t � �1� ���mct+k (j)� �1� ��� pt+k � �� (p�t � pt+k)�

where we have substituted in log-linearisations of (21) and (18). Since

mct+k (j) = mct+k � (1� �)�

1� ��� (p�t � pt+k)

where mct+k is the average marginal cost in log-linear terms, we get

1

1� �p�

�1 +

���� � 1

+ (1� �) ���

1� �

�(p�t � pt�1)

= Et

1Xk=0

(�p�)k

��1 +

���� � 1

�(pt+k � pt�1) +mct+k � (1� �)

���

1� � (pt�1 � pt+k)�

=

�1 +

���� � 1

+ (1� �) ���

1� �

��t +mct

+1

1� ��

�1 +

���� � 1

+ (1� �) ���

1� �

�Et�p�t+1 � pt

�+

��

1� ��

�1 +

���� � 1

+ (1� �) ���

1� �

��t

As shown by Eichenbaum and Fisher (2007), the price index implies that

p�t � pt�1 =�t1� �

Using this gives

�t1� � = (1� ��)�t + (1� ��)

�1 +

���� � 1

+ (1� �) ���

1� �

��1mct

+��

1� �Et�t+1 + ���t

148

Rearranging gives the New Keynesian Phillips curve in the text:

�t =(1� ��) (1� �)

�p

�1 +

���� � 1

+ (1� �) ���

1� �

��1mct + �Et�t+1

B Appendix

An unpleasant feature of the model presented in the previous section is that,independently of the presence of real rigidities, it implies a large increase inthe real wage which is counterfactual. Many empirical studies �Blanchardand Perotti (2002), Perotti (2005), Fatas and Mihov (2002) among manyothers ��nd a zero response or at most a tiny positive response, in general notstatistically signi�cant. Furlanetto (2007) shows that by introducing stickywages in the model, it is possible to reconcile a plausible conditional responseof real wages and a positive and sizeable response of private consumption onthe impact of a government spending shock. In other words, the Galí et al.(2007) result does not rely on the large counterfactual response of real wages,as one might intuitively think, but is con�rmed in a more general setting withwage rigidities. For sake of completeness, we want to show that real rigiditiescan substitute for nominal and �nancial rigidities, also in a framework withsticky wages. As shown in Furlanetto and Seneca (2007), with sticky wagesand habit formation in consumption, equation (15) is substituted by thefollowing equation for wages

�wt = �Et��wt+1

�+ �w (mrst � (wt � pt)) (39)

where mrst is the average marginal rate of substitution given by

mrst = �r�crt � hrcrt�1

�+ �o

�cot � hocot�1

�+ 'nt (40)

and the slope coe¢ cient �w is

�w ="w � 1�w

Here, �w governs the size of wage adjustment costs à la Rotemberg (1982).15

We calibrate "w equal to 4 and �w equal to 454.5. This choice yields the

15Instead of wage adjustment costs, we may assume that a union is allowed to reset itswage rate each period with a �xed probability 1 � �w as in Calvo (1983). But to undothe implications of the implied heterogeneity across unions, a risk-sharing arrangement

149

same New Keynesian Phillips curve for wages as in a Calvo setting à laErceg, Henderson and Levin (2000) with four quarters of wage stickiness.A second criticism that can be raised to the Galí et al. (2007) model

concerns the calibration of the inverse of the labor supply elasticity '. Galíet al. (2007) are forced to set it at 0.2 to make the model determinate.However, the determinacy region is larger under sticky wages and thereforewe can raise ' to more plausible values. We set ' equal to 3, consistentwith a labor supply elasticity of 1/3, as in Galí and Monacelli (2005) andconsistent with a considerable microeconomic evidence. In �gure 4 we plotthe impulse responses for the model in Galí et al. (2007) augmented withsticky wages along with a model further extended with real rigidities as insection 2 (Kimball demand and habit consumption, while keeping the rentalcapital assumption).We see that the model with real rigidities can reproduce approximately

the same multiplier as the model without real rigidities under only 30 percentof constrained agents. Thus, once again, real rigidities can substitute fornominal rigidities and �nancial frictions. Note also that real wages respondvery little in both cases due to wage adjustment costs.

between unions must be in place. This follows since rule-of-thumb consumers are barredfrom sharing risk through �nancial markets. Results, however, are very similar. In partic-ular we would get a Phillips curve with �w = (1� ��w) (1� �w) ��1w (1 + '"w)

�1 where�w is the Calvo parameter for wage setting.

150

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Figure 1: Impact consumption multiplier in the model by Galí et al. (2007)as function of �, the fraction of rule-of-thumb consumers (left panel), and �,the degree of price rigidity (right panel). Remaining parameters at baselinevalues.

155

Figure 2: Impulse responses to a government spending shock normalised toone per cent of steady-state output for � = 0:5 and � = 0:75 in the Galí etal. (2007) model (dashed lines), and for � = 0:3 and � = 0:6 in an extendedversion of the model with real rigidities (solid lines).

156

Figure 3: Impact consumption multiplier as a function of ho, the degree ofhabit persistence of optimising households (left panel), and �, the slope ofthe New Keynesian Phillips curve, for � = 0:3 in the Galí et al. (2007) modelaugmented with real rigidities.

157

Figure 4: Impulse responses to a government spending shock normalised toone per cent of steady-state output for � = 0:5 and � = 0:75 as in the Galíet al. (2007) model augmented with sticky wages (dashed lines), and for� = 0:3 and � = 0:6 in an extended version with real rigidities in addition tosticky wages (solid lines).

158