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Models for Dynamic Macroeconomics

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Models for Dynamic Macroeconomics

Fabio-Cesare Bagliano

Giuseppe Bertola

1

Great Clarendon Street, Oxford ox2 6 dp

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First published 2004

First published in paperback 2007

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to specialists than the general public.

Yet, macroeconomics and the problems it attempts to deal with areextremely important, even if they are sometimes difficult to grasp It cannot

be denied that macroeconomic analysis has become more technical over thelast few decades The formal treatment of expectations and of inter-temporalinteractions is nowadays an essential ingredient of any model meant to addresspractical and policy problems But, at the same time, it has also become morepragmatic because modern macroeconomics is firmly rooted in individualagents’ day-to-day decisions To understand and appreciate scientific researchpapers, the modern macroeconomist has to master the dynamic optimizationtools needed to represent the solution of real, live individuals’ problems interms of optimization, equilibrium and dynamic accumulation relationships,expectations and uncertainty The macroeconomist, unlike most microecono-mists, also needs to know how to model and interpret the interactions ofindividual decisions that, in different ways and at different levels, make aneconomy’s dynamic behavior very different from the simple juxtaposition ofits inhabitant’s actions and objectives

This book offers its readers a step-by-step introduction to aspects ofmacroeconomic engineering, individual optimization techniques and modernapproaches to macroeconomic equilibrium modeling It applies the relevantformal analysis to some of the standard topics covered less formally by allintermediate macroeconomics course: consumption and investment, employ-ment and unemployment, and economic growth Aspects of each topic aretreated in more detail by making use of advanced mathematics and settingthem in a broader context than is the case in standard undergraduate text-books The book is not, however, as technically demanding as some othergraduate textbooks Readers require no more mathematical expertise than isprovided by the majority of undergraduate courses The exposition seeks to

vi PREFACE TO PAPERBACK EDITION

develop economic intuition as well as technical know-how, and to preparestudents for hands-on solutions to practical problems rather than providingfully rigorous theoretical analysis Hence, relatively advanced concepts (such

as integrals and random variables) are introduced in the context of economicarguments and immediately applied to the solution of economic problems,which are accurately characterized without an in-depth discussion of thetheoretical aspects of the mathematics involved The style and coverage ofthe material bridges the gap between basic textbooks and modern appliedmacroeconomic research, allowing readers to approach research in leadingjournals and understand research practiced in central banks and internationalresearch institutions as well as in academic departments

How to Use This Book

Models for Dynamic Macroeconomics is suitable for advanced undergraduate

and first-year graduate courses and can be taught in about 60 lecture hours.When complemented by recent journal articles, the individual chapters—which differ slightly in the relative emphasis given to analytical techniquesand empirical perspective—can also be used in specialized topics courses Thelast section of each chapter often sketches more advanced material and may

be omitted without breaking the book’s train of thought, while the chapters’appendices introduce technical tools and are essential reading Some exercisesare found within the chapters and propose extensions of the model discussed

in the text Other exercises are found at the end of chapters and should be used

to review the material Many technical terms are contained in the index, whichcan be used to track down definitions and sample applications of possiblyunfamiliar concepts

The book’s five chapters can to some extent be read independently, butare also linked by various formal and substantive threads to each other and

to the macroeconomic literature they are meant to introduce Discrete-timeoptimization under uncertainty, introduced in Chapter 1, is motivated anddiscussed by applications to consumption theory, with particular attention toempirical implementation Chapter 2 focuses on continuous-time optimiza-tion techniques, and discusses the relevant insights in the context of partial-equilibrium investment models Chapter 3 revisits many of the previouschapters’ formal derivations with applications to dynamic labor demand, inanalogy to optimal investment models, and characterizes labor market equi-librium when not only individual firms’ labor demand is subject to adjustmentcosts, but also individual labor supply by workers faces dynamic adjustment

PREFACE TO PAPERBACK EDITION vii

problems Chapter 4 proposes broader applications of methods introduced bythe previous chapters, and studies continuous-time equilibrium dynamics ofrepresentative-agent economies featuring both consumption and investmentchoices, with applications to long-run growth frameworks of analysis Chapter

5 illustrates the role of decentralized trading in determining aggregate ria, and characterizes aggregate labor market dynamics in the presence of fric-tional unemployment Chapters 4 and 5 pay particular attention to strategicinteractions and externalities: even when each agent correctly solves his or herindividual dynamic problem, modern micro-founded macroeconomic mod-els recognize that macroeconomic equilibrium need not have unambiguouslydesirable properties

equilib-Brief literature reviews at the end of each chapter outline some recentdirections of progress, but no book can effectively survey a literature as wide-ranging, complex, and evolving as the macroeconomic one In the interests

of time and space this book does not cover all of the important analytical andempirical issues within the topics it discusses Overlapping generation dynam-ics and real and monetary business cycle fluctuations, as well as more technicalaspects, such as those relevant to the treatment of asymmetric informationand to more sophisticated game-theoretic and decision-theoretic approachesare not covered It would be impossible to cover all aspects of all relevant topics

in one compact and accessible volume and the intention is to complementrather than compete with some of the other texts currently available.∗ Thepositive reception of the hardback edition, however, would seem to confirmthat the book does succeed in its intended purpose of covering the essentialelements of a modern macroeconomist’s toolkit It also enables readers toknowledgeably approach further relevant research It is hoped that this paper-back edition will continue to fulfil that purpose even more efficiently for anumber of years to come

The first hardback edition was largely based on Metodi Dinamici eFenomeni Macroeconomici (il Mulino, Bologna, 1999), translated by FabioBagliano (ch.1), Giuseppe Bertola (ch 2), Marcel Jansen (chs 3, 4, 5, edited

by Jessica Moss Spataro and Giuseppe Bertola) For helpful comments theauthors are indebted to many colleagues (especially Guido Ascari, Onorato

∗Foundations of Modern Macroeconomics, by Ben J Heijdra and Frederick van der Ploeg (Oxford

University Press, 2002) is more comprehensive and less technical; the two books can to some extent complement each other on specific topics This book o ffers more technical detail and requires less

mathematical knowledge than Lectures on Macroeconomics, by Olivier J Blanchard and Stanley Fischer

(MIT Press, 1989), and o ffers a more up to date treatment of a more limited range of topics It is less

wide ranging than Advanced Macroeconomics, by David Romer (McGraw-Hill 3rd rev edn 2005) but

provides more technical and rigorous hands-on treatment of more advanced techniques By contrast,

Recursive Macroeconomic Theory, by Lars Ljungqvist and Thomas J Sargent (MIT Press, 2nd edn 2004)

o ffers a more rigorous but not as accessible formal treatment of a broad range of topics, and a narrower range of technical and economic insights.

viii PREFACE TO PAPERBACK EDITION

Castellino, Elsa Fornero, Pietro Garibaldi, Giulio Fella, Vinicio Guidi, ClaudioMorana) and to the anonymous reviewers The various editions of the bookhave also benefited enormously from the input of the students and teachingassistants (especially Alberto Bucci, Winfried Koeniger, Juana Santamaria,Mirko Wiederholt) over many years at the CORIPE Master program in Turin,

at the European University Institute, and elsewhere Any remaining errors andall shortcomings are of course the authors’ own

C O N T E N T S

D E T A I L E D C O N T E N T S

DETAILED CONTENTS xi

4 Growth in Dynamic General Equilibrium 130

4.3 Decentralized Production and Investment Decisions 144

5 Coordination and Externalities in Macroeconomics 170

xii DETAILED CONTENTS

Appendix A5: Strategic Interactions and Multipliers 211

L I S T O F F I G U R E S

2.2 Dynamics of q (supposing that ∂ F (·)/∂ K is decreasing in K ) 57

2.3 Dynamics of K (supposing that ∂È(·)/ ∂ K − ‰ < 0) 58

2.6 A hypothetical jump along the dynamic path, and the resulting time

2.7 Dynamic effects of an announced future change of w 64

2.8 Unit profits as a function of the real wage 68

2.10 Dynamic effects of an anticipated fiscal restriction 75

2.12 Installed capital and optimal irreversible investment 79

3.2 Adjustment costs and dynamic labor demand 111

3.3 Nonlinearity of labor demand and the effect of turnover costs on

3.4 The employer’s surplus when marginal productivity is equal to the wage 118

3.5 Dynamic supply of labor from downsizing firms to expanding firms,

3.6 Dynamic supply of labor from downsizing firms to expanding firms,

without employers’ adjustment costs, if mobility costs Í per unit of labor 124

4.3 Effects of an increase in the savings rate 136

4.4 Convergence and steady state with optimal savings 141

5.4 Optimal quantity of money M∗and ex ante probability of

xiv LIST OF FIGURES

5.6 Equilibrium of the labor market with frictional unemployment 198

1 Dynamic

Consumption Theory

Optimizing models of intertemporal choices are widely used by theoreticaland empirical studies of consumption This chapter outlines their basic ana-lytical structure, along with some extensions The technical tools introducedhere aim at familiarizing the reader with recent applied work on consumptionand saving, but they will also prove useful in the rest of the book, when weshall study investment and other topics in economic dynamics

The chapter is organized as follows Section 1.1 illustrates and solves thebasic version of the intertemporal consumption choice model, deriving the-oretical relationships between the dynamics of permanent income, currentincome, consumption, and saving Section 1.2 discusses problems raised byempirical tests of the theory, focusing on the excess sensitivity of consumption

to expected income changes and on the excess smoothness of consumptionfollowing unexpected income variations Explanations of the empirical evi-dence are offered by Section 1.3, which extends the basic model by introducing

a precautionary saving motive Section 1.4 derives the implications of optimalportfolio allocation for joint determination of optimal consumption whenrisky financial assets are available The Appendix briefly introduces dynamicprogramming techniques applied to the optimal consumption choice Biblio-graphic references and suggestions for further reading bring the chapter to aclose

1.1 Permanent Income and Optimal Consumption

The basic model used in the modern literature on consumption and savingchoices is based on two main assumptions:

1 Identical economic agents maximize an intertemporal utility function,

defined on the consumption levels in each period of the optimizationhorizon, subject to the constraint given by overall available resources

2 Under uncertainty, the maximization is based on expectations of future

relevant variables (for example, income and the rate of interest) formed

rationally by agents, who use optimally all information at their disposal.

We will therefore study the optimal behavior of a representative agent who

lives in an uncertain environment and has rational expectations Implications

subject to the constraint (for i = 0 , , ∞)

A t+i +1 = (1 + r t+i ) A t+i + y t+i − c t+i ,

where A t+i is the stock of financial wealth at the beginning of period t + i ; r t+i

is the real rate of return on financial assets in period t + i ; y t+i is labor income earned at the end of period t + i , and c t+i is consumption, also assumed totake place at the end of the period The constraint therefore accounts for theevolution of the consumer’s financial wealth from one period to the next.Several assumptions are often made in order easily to derive empiricallytestable implications from the basic model The main assumptions (some ofwhich will be relaxed later) are as follows

r Intertemporal separability (or additivity over time) The generic utility

function U t(·) is specified as

U t (c t , c t+1 , ) = v t (c t) +v t+1 (c t+1) + .

(withv

t+i > 0 and v

t+i < 0 for any i ≥ 0), where v t+i (c t+i) is the

val-uation at t of the utility accruing to the agent from consumption c t+i

at t + i Since v t+i depends only on consumption at t + i , the ratio of

marginal utilities of consumption in any two periods is independent ofconsumption in any other period This rules out goods whose effects onutility last for more than one period, either because the goods themselvesare durable, or because their consumption creates long-lasting habits

(Habit formation phenomena will be discussed at the end of this chapter.)

r A way of discounting utility in future periods that guarantees

intertempo-rally consistent choices Dynamic inconsistencies arise when the valuation

at time t of the relative utility of consumption in any two future periods,

t + k1and t + k2(with t < t + k1< t + k2), differs from the valuation ofthe same relative utility at a different time t + i In this case the optimal

levels of consumption for t + k1 and t + k2 originally chosen at t may

not be considered optimal at some later date: the consumer would thenwish to reconsider his original choices simply because time has passed,even if no new information has become available To rule out this phe-nomenon, it is necessary that the ratios of discounted marginal utilities

of consumption in t + k1 and t + k2 depend, in addition to c t+k1 and

c t+k2, only on the distance k2− k1, and not also on the moment in timewhen the optimization problem is solved With a discount factor for the