MacroLectures-2013-pp1-22

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Lectures

Macroeconomic TheoryVersion: April 11, 2013

Volker Bhm

Department of EconomicsBielefeld University

P.O. Box 100 131D33501 Bielefeld, Germany

e-mail: [email protected]: www.wiwi.uni-bielefeld.de/boehm/

These Lectures contain the material of the major chapters of the course Macroeconomic Theory whichhas been given at Bielefeld University over many years. As time progressed the course material has changedgradually shifting emphasis from being primarily a course on the microeconomic foundations of macroeconomicsderiving prices, wages, output, and employment for a monetary economy within the framework of temporarygeneral equilibrium theory the short run , to an extended sequential and dynamic analysis investigating theconsequences of stationary fiscal policies on allocations, on the quantity of money as well as on the dynamicevolution of the economy with and without noise.

The analysis consists of a unified general equilibrium approach starting from the competitive temporary equilib-rium model in the spirit of Patinkin, Hicks, Keynes, and others. It proceeds by applying the sequential structureinduced by the principles of intertemporal optimization of finitely lived generations of agents and intertemporalbudget consistency in a closed monetary economy, deriving general structural results for the short run allocationsas well as for the long run dynamics. It contains a detailed presentation and analysis of the dynamics of pricesand expectations and inflation, of employment and output as an outcome of fiscal deficits and surpluses andmoney creation induced by a stationary fiscal policy.

The material contrasts within the same intertemporal structure of a closed monetary economy the allocativeresults and the induced dynamics of the two main competing approaches in macroeconomics: the one postulat-ing full price flexibility to guarantee equilibrium in all markets at all times under perfect foresight or rationalexpectations, versus the so called disequilibrium approach, where trading occurs typically at non market clearingprices and wages while these adjust sluggishly from period to period in response to market disequilibrium signals.

The current version of the Lecture Notes is not yet complete and still undergoing revisions. The material of theoriginal Chapter 3 has been excluded from the current text since it was not presented in class during the pastyears. Chapter 4 presents the center piece of the analysis describing first in great detail the allocative implicationsof the basic macroeconomic model under perfectly competitive markets. Extensions on non competitive price andwage determination (monopolistic equilibria and union bargaining) have been added while aspects of asymmetricinformation or matching in labor markets are not yet included. Chapter 5 and 6 present the central results onthe dynamics under perfect foresight and rational expectations. Chapters 7 through 9 present the results underdisequilibrium including a detailed numerical bifurcation analysis when endogenous business cycles appear.

Later chapters assume familiarity with the material of preceding ones. The appendix collects basic results ondynamical systems in discrete time in an informal presentation, including concepts and results for systems withrandom perturbations. At this stage the material has been made available primarily as supportive material tothose who attend the course and to interested colleagues and researchers. Other interested readers are welcome.

Before citing or referring to these notes, however, please, contact the author. As usual, any suggestions,criticisms, or remarks are very welcome.


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Contents

Preface 6

1 Introduction 111.1 Microeconomics versus Macroeconomics . . . . . . . . . . . . . . . . . . . . . . . 11

2 Microeconomic Foundations 132.1 The Basic Intertemporal Model with Money . . . . . . . . . . . . . . . . . . . . 132.2 Intertemporal Decisions of Consumers . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 The Geometry of the Budget Set and of Demand . . . . . . . . . . . . . 152.2.2 The Intertemporal Utility Index: An Alternative Approach . . . . . . . . 192.2.3 Price Expectations of Consumers . . . . . . . . . . . . . . . . . . . . . . 192.2.4 Intertemporal Decisions of Producers . . . . . . . . . . . . . . . . . . . . 23

2.3 Equilibrium of an Exchange Economy with Money . . . . . . . . . . . . . . . . . 282.3.1 Failure of Existence of Monetary Equilibria: . . . . . . . . . . . . . . . . 33

2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Theory of Non Walrasian Temporary Equilibria 433.1 Rationing Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 Individual Optimization under Rationing The Theory of Effective Demand . . 433.3 Equilibria at non Walrasian Prices . . . . . . . . . . . . . . . . . . . . . . . . . . 433.4 Existence, Uniqueness, and Optimality . . . . . . . . . . . . . . . . . . . . . . . 433.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Models of Monetary Equilibrium 454.1 Markets and Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Competitive Temporary Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2.1 Prices and Wages, Output and Employment . . . . . . . . . . . . . . . . 494.2.2 Aggregate Demand and Aggregate Supply . . . . . . . . . . . . . . . . . 524.2.3 Prices and Wages: A two step geometric analysis . . . . . . . . . . . . . 564.2.4 Isoelastic Functions: An Example . . . . . . . . . . . . . . . . . . . . . . 59

4.3 Noncompetitive Commodity Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 624.3.1 Monopolistic Price Setting: the New Keynesian Approach . . . . . . . . 624.3.2 Oligopolistic price setting . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3.3 Heterogeneity and Product Differentiation . . . . . . . . . . . . . . . . . 674.3.4 Dynamics of Cournot Competition among firms . . . . . . . . . . . . . . 71

4.4 Noncompetitive Wage Determination . . . . . . . . . . . . . . . . . . . . . . . . 72

4.4.1 Monopsonistic Wage Setting by Producers . . . . . . . . . . . . . . . . . 724.4.2 Monopolistic Wage Setting by a Labor Union . . . . . . . . . . . . . . . 744.4.3 Non Competitive Wage Setting versus Wage Bargaining . . . . . . . . . . 80

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4.5 Bargaining in the Labor Market . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.5.1 Efficient Bargaining of Wages and Employment . . . . . . . . . . . . . . 854.5.2 The Role of Union Power and Efficiency . . . . . . . . . . . . . . . . . . 974.5.3 Comparing Bargaining and Competition . . . . . . . . . . . . . . . . . . 984.5.4 Inefficient Redistribution under Efficient Wage Bargaining . . . . . . . . 1014.5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.6 Prices, Wages, and Payoffs: the Isoelastic Case Revisited . . . . . . . . . . . . . 1054.6.1 The Role of Union Power . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.6.2 Union Power and Wages . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.6.3 Comparing Labor Market Scenarios . . . . . . . . . . . . . . . . . . . . . 113

4.7 Wage Rigidities and Unemployment . . . . . . . . . . . . . . . . . . . . . . . . . 1224.7.1 Unemployment Compensation and Unemployment . . . . . . . . . . . . . 1234.7.2 Fixed Costs, the Number of Producers, and the Role of Aggregate Demand1284.7.3 The Efficiency Wage Model . . . . . . . . . . . . . . . . . . . . . . . . . 1284.7.4 Summary and Reappraisal of noncompetitive Wage Models . . . . . . . . 130

4.7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5 Dynamics of Monetary Equilibrium 1335.1 Dynamics of Money, Prices, and Expectations . . . . . . . . . . . . . . . . . . . 133

5.1.1 Perfect Foresight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.1.2 Money Balances and Prices under Perfect Foresight . . . . . . . . . . . . 1405.1.3 Balanced Paths of Inflation and Deflation . . . . . . . . . . . . . . . . . 1455.1.4 Dynamics with Adaptive Expectations . . . . . . . . . . . . . . . . . . . 153

5.2 Dynamics with Random Perturbations . . . . . . . . . . . . . . . . . . . . . . . 1565.2.1 Demand and Supply with Random Productivity . . . . . . . . . . . . . . 156

5.2.2 Dynamics with Random Aggregate Demand . . . . . . . . . . . . . . . . 1665.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6 Fiscal Policy and the Dynamics of Monetary Equilibrium 1736.1 Deficit rules and permanent inflation/deflation . . . . . . . . . . . . . . . . . . . 1736.2 Deficit rules with money and government debt . . . . . . . . . . . . . . . . . . . 1786.3 Random deficits and monetary expansion . . . . . . . . . . . . . . . . . . . . . . 1786.4 Pay as you go pension systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7 The Keynesian Model with Money 1797.1 Disequilibrium: Trading when markets do not clear . . . . . . . . . . . . . . . . 179

7.2 The IS-LM model revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1827.3 The General Model with Constant Labor Supply . . . . . . . . . . . . . . . . . . 1847.4 Temporary Feasible States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1867.5 Money Balances and the Government Budget . . . . . . . . . . . . . . . . . . . 1897.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

8 Dynamics in Disequilibrium Endogenous Business Cycles 1938.1 Price and Wage Adjustment The Law of Supply and Demand . . . . . . . . . . 1938.2 Existence and Uniqueness of Stationary States . . . . . . . . . . . . . . . . . . . 1978.3 The isoelastic case with linear price and wage adjustment . . . . . . . . . . . . 198

8.3.1 Keynesian Stationary States . . . . . . . . . . . . . . . . . . . . . . . . . 2008.3.2 Inflationary Stationary States . . . . . . . . . . . . . . . . . . . . . . . . 2028.3.3 Local stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

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8.3.4 Endogenous cycles and complexity . . . . . . . . . . . . . . . . . . . . . 2058.3.5 Coexisting cycles and Sensitivity on initial conditions . . . . . . . . . . . 2118.3.6 Fiscal policy and the business cycle . . . . . . . . . . . . . . . . . . . . . 213

8.4 Business Cycles with endogenous labor supply . . . . . . . . . . . . . . . . . . . 2158.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

9 Disequilibrium Dynamics with Random Perturbations 2179.1 Random Productivity with Constant Labor Supply . . . . . . . . . . . . . . . . 217

9.1.1 Smooth uniform productivity shocks . . . . . . . . . . . . . . . . . . . . 2219.1.2 Random labor elasticity: perturbation of curvature . . . . . . . . . . . . 2239.1.3 Random Demand Shocks Random Fiscal Policy . . . . . . . . . . . . . 226

9.2 Random Productivity with endogenous labor supply: The Phillips Curve . . . . 2299.2.1 A smooth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2309.2.2 B smooth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

9.3 Aggregate Employment with encoded densities . . . . . . . . . . . . . . . . . . . 241

9.3.1 B smooth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2419.3.2 A smooth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

9.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

10 Government Bonds and Assets 25510.1 Structure of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25510.2 Temporary Feasible States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

10.2.1 Rationing and the bond price . . . . . . . . . . . . . . . . . . . . . . . . 25810.3 Adjustment of Prices and Wages . . . . . . . . . . . . . . . . . . . . . . . . . . . 26210.4 The Complete Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . . . 263

10.4.1 Steady States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26410.4.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26410.4.3 The Government Budget with Money and other assets . . . . . . . . . . 265

11 The Inventory Model 26711.1 B uffer stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

11.1.1 Sequential Trading and Feasible Allocations . . . . . . . . . . . . . . . . 26811.1.2 Money Balances and Inventory Dynamics . . . . . . . . . . . . . . . . . . 27011.1.3 The Complete Dynamical System . . . . . . . . . . . . . . . . . . . . . . 27111.1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

A Optimization and Implicit Functions 277A.1 Solving Constraint Optimization Problems . . . . . . . . . . . . . . . . . . . . . 277A.2 The Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

B Dynamical Systems in Discrete Time 281B.1 Stability in Deterministic Dynamical Systems . . . . . . . . . . . . . . . . . . . 281

B.1.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281B.1.2 One Dimensional Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 282B.1.3 Loss of Stability and Local Bifurcations . . . . . . . . . . . . . . . . . . . 284B.1.4 Two Dimensional Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 289

B.2 Homogeneous Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 291B.2.1 Orbits with balanced Expansion or Contraction . . . . . . . . . . . . . . 291

B.3 Random Dynamical Systems in Discrete Time . . . . . . . . . . . . . . . . . . . 293



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B.3.1 Random dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . 294B.3.2 Random fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295B.3.3 Affine Random Dynamical Systems . . . . . . . . . . . . . . . . . . . . . 297B.3.4 Iterated Function Systems with Probabilities . . . . . . . . . . . . . . . . 298B.3.5 Affine difference equations with discrete i. i. d. noise . . . . . . . . . . 298B.3.6 Univariate random variables: Definitions and computation of moments . 299

C Proofs and Further Results 301C.1 Proofs for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

C.1.1 Proof of Lemma C.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301C.1.2 Proof of Lemma 4.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

C.2 Dynamics of Real Money Balances . . . . . . . . . . . . . . . . . . . . . . . . . 303C.2.1 Proof of Theorem 5.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303C.2.2 Proof of Theorem ?? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

C.3 Stability in the Disequilibrium model . . . . . . . . . . . . . . . . . . . . . . . . 306

C.3.1 Proof of Theorem 8.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

Bibliography 307



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Preface

Research in macroeconomic theory over the past decades has emphasized dynamic aspects inmany different ways, in an attempt to understand, describe, and test the intertemporal aspectsof an economy at the macroeconomic level. Apart from the emphasis on dynamic issues,a common thread of theoretical modeling has been to use proper microeconomic behavioralfoundations embedded in a full general equilibrium setting. This has lead to branches of research

with specific modeling approaches like the representative agent model, competitive businesscycle theory, real business cycle theory with divers modifications and additions. Extensionsor innovations originating from Keynesian theories, which were dominating some time prior tothis development, were largely neglected or discarded, in spite of the fact that there was a surgeof approaches in the 1970s designing monetary intertemporal models with a Keynesian flavor1.

After this period a majority of economists did not continue the line of research to model dise-quilibrium issues further in a dynamic context. In contrast, research concentrated on treatingmacroeconomic questions within a diversity of real models under full intertemporal equilibrium(mostly with) with rational expectations, often extended to include other equilibrium issuesas well. Thus, macroeconomic research diverted from asking questions of possibleendogenous

causes of disequilibrium situations (unemployment and inflation), of instability of steady states,and of permanent fluctuations in monetary economies to a description of the best of all worldswhere the intertemporal development was defined as the solution of a single optimizing agentwith rational expectations.

This essentially meant that the driving forces or causes of business cycles were found in the oc-currence ofexogenous random disturbancesto an economy in full intertemporal equilibrium andunder permanent behavioral optimality of agents. Thus, endogenous explanations of businesscycles or disequilibria were excluded as a possibility from the standard paradigm and exoge-nous random effects constituted the primary origin of fluctuations while monetary issues werediscarded from the agenda.

Much of existing dynamic Keynesian macroeconomics uses postulated aggregate functionalforms which are not derived from microeconomic principles. Some contributions investigatingendogenous causes of business cycles in nonmarket clearing theories applied techniques fromdynamical systems theory quite early (see the survey by Lorenz 1993b). While these modelsshow sufficient evidence of endogenous reasons for disequilibria and cycles, they can be criticizedfor their lack of microeconomic consistency. One such model of the business cycle was originallyproposed by Kaldor (1940). Dana & Malgrange (1984) perform extensive simulations for aKaldor type model. They show that varying just one of the parameters of the model changesa stable fixed point to a closed orbit which, after a further increase of the parameter, becomesunstable itself. Thus, strange attractors or possibly chaos occur, showing the endogenousoccurrence of cycles. In contrast to the so called competitive business cycle theory2, however,

1see for example Barro & Grossman (1976), Benassy (1975b), Malinvaud (1977) to name just a few.2see for example Grandmont (1988)

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these Keynesian models seldom are built on the same level of microeconomic basic principles.They are often considered too much ad hoc and unacceptable to the researchers within theNew Classical Macroeconomics.

Regarding expectations and random perturbations, research of the past years has shown that

intertemporal equilibria in models under rational expectations often cannot be shown to exist,they may turn out to be unstable dynamically or non stationary. In the case of non existence,which may be due to high levels of government demand or other parametric restrictions in themodel, the theory becomes vacuous and inapplicable. Instability implies essentially that theeconomic development of any such model could not be observed in a long run experimentalenvironment since the time series would be diverging. Thus, such models could not serve ina meaningful way as a measuring rod to compare with empirical data since these are oftenassumed to be drawn from a stationary environment. Since equilibria are defined as simulta-neous solutions ofa complete set of infinitely many conditionsto hold at all times, any out ofequilibrium allocations cannot be accounted for, so that the usual critique applies3:

out of equilibrium adjustments of prices and trading are not part of the descriptive frameworkof the model and cannot be analyzed,

the role of non rational expectations cannot be assessed excluding an analysis of the effectsof the correction of beliefs and of learning,

the stability of adjustment processes cannot be examined within the model since it lacks aforwardrecursive structure.

This leaves equilibrium macroeconomics as an extension of Walrasian equilibrium theory underrational expectations and ttonnement without descriptive elements for a theory of price andstock adjustments or for expectations adjustment to achieve learning, and without an oper-ative concept of stability. These arguments seem to imply three good reasons to develop a

disequilibrium macroeconomic theory.1. In order to escape the ttonnement trap, it is necessary to formulateexplicitlythe laws of

motion of those variables as part of the conceptual primitives within the economy. Thismeans, in particular, that the adjustmentof prices has to be explained (among otherthings) as a consequence of trading at non market clearing prices and the adjustmentof expectations has to be given as a result of correcting predictions. Market clearingsituations with constant prices at any one time are the exception rather than the ruledefining particular steady states or stationary situations. Rational expectations or perfectforesight holds only whenever the rational or perfect predictor is available or when anexplicit learning procedure has been successful in the long run.

2. The theory of trading at non Walrasian prices in any one period provides a formal ba-sis independent of whether equilibrium prices and allocations exist, of whether they areunique or not, or whether they are computable analytically or not. The disequilibriumtrading rules can be described in a forward recursive format since they correspond essen-tially to allocation mechanisms generating outcomes from signals. Also, the expectationformation principles correspond to the standard statistical methods based on data fromtime series, which again guarantee a forward recursive mapping.

The complexity of interacting markets with disequilibrium trading calls for high dimen-sional non linear models typically without analytical or even computable solutions fortemporary equilibria. Thus, with an explicit recursive structure for trading, there is no

3This debate is as old as equilibrium theory itself after Walras, who conceptualized equilibrium withoutexplicit dynamics, see for example Chapter 1 in Lindahl (1939).



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conceptual restriction to the degree of heterogeneity, as long as the size of the model doesnot outgrow the computing facilities of available machines. A priori, there is no need foranalytical forms of outcomes in any one period.

3. Finally, once the model is formulated as a dynamical system the appropriate mathemat-

ical tools of the associated theory become available. These have been improved greatlyduring the last decades to deal with non linearities and random perturbations simulta-neously. These so called random dynamical systems are the appropriate mathematicalframework as time series generators for economic models. Their properties can be ana-lyzed in particular using numerical procedures to obtain data to examine stability, prop-erties of stationary orbits, and to carry out a bifurcation analysis. This also provides theonly reliable tool to investigate endogenouscauses of instability and of business cycles.

The basic motivation to pursue a fully explicit dynamic theory for models of the Keynesian typederives its justification from the fact that a consistent extension of the equilibrium paradigm inthe Keynesian spirit can be found to overcome the structural difficulties associated with the fullintertemporal equilibrium approach. The features of Keynesian theory are used and extendedto a consistent microfounded disequilibrium theory for all markets as presented by Barro &Grossman (1976), Malinvaud (1977), Benassy (1975b), and others for a short run analysis.The extension to a general workable dynamic model has to be performed. While embeddingthe disequilibrium features in a proper microeconomic setting of an overlapping generationsmodel, the same natural microeconomic basis as in competitive business cycle theory is usedwith a disequilibrium trading rule. In addition, standard price and wage adjustment principleswhich respond to disequilibrium situations in each market are invoked4 to describe explicitdynamics of prices and wages. In this way, an explicit forward recursive dynamical systemis obtained avoiding the difficulties of the market clearing approach. Moreover, the marketclearing situation under perfect foresight or rational expectations becomes a special case of the

disequilibrium model.These Lecture Notes present first the microeconomic foundations of an intertemporal theory oftrading among overlapping generations of consumers (Chapters 2 and 3), along the lines of theliterature mentioned above. The situation under market clearing contains the standard macroeconomic equilibrium models with atemporal production (Chapter 4) along with an analysisof the dynamics under perfect foresight and rational expectations (Chapter 5 ). Chapter 6discusses the long run consequences of permanent government deficits and of balanced fiscaland government expansion. Chapters 7 and 8 provide the extension of the short run Keynesianstructure to a fully consistent dynamic model. It provides a detailed dynamic analysis showingsurprising results of the occurrence of endogenous business cycles. Random perturbations in

the basic model with money also show new qualitative relationships for stationary time series,for example properties of a long run Phillips curve.

Chapters 10 and 11 describe extensions to include government bonds and inventories. Theseindicate a further richness of results to be studied in more detail. An application of such modelsto questions of exchange rate volatility has been presented in Bhm & Kikuchi (2005). It seemsclear that further extensions and applications within the complete dynamic setting will lead toadditional results which may resolve other macroeconomic issues, which are not tractable usingthe intertemporal equilibrium approach.

4These areexplicit formulations of the heuristic adjustment principles used in ttonnement theory.



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Chapter 1

Introduction

1.1 Microeconomics versus Macroeconomics

What are the essential differences between the two fields of economics?

1. forms of modeling

full generality and heterogeneity of consumers, producers, commodities versusaggregate reduced form models

partial equilibrium general equilibrium

= system of markets and agents which is closed

= interdependence of trade on markets: Output, employment, money

= J. M. Keynes (1936):The General Theory of Employment, Interest, and Money,

2. types of models

real models versus monetary models

certainty versus uncertainty

deterministic versus stochastic (random shocks)

finite horizon versus infinite horizon models

types of markets: spot, forward, and futures markets

a complete system of markets with sufficient information needsno money, no assets, no revision of plans, no reopening of trading

= an interesting macroeconomic model built on market principles with interactingagents requires a modelling approach where

= the system of markets is incomplete

= there must exist some uncertainty

= paper assets/contracts like money, credit, or bonds are essential

= the time horizon is infinite,since positive valuation of assets in equilibrium requires an infinite horizon

3. types of analysis

static versus dynamic analysis

short run medium run long run

= interdependence of markets in short run

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Chapter 2

Microeconomic Foundations

2.1 The Basic Intertemporal Model with Money

The description of the intertemporal trading structure of commodities, agents, and marketsis the one originally proposed by Hicks (1965) and Patinkin (1965). The mathematical pre-sentation here follows Grandmont (1983), which corresponds to an extension of the generalArrow-Debreu economy to an infinite sequence of trading periods. Let time evolve in a discreteway, (. . . , t1, t , t+ 1, . . .); t Z, where a trading period t is interpreted as the shortestconceptual time length during which markets are opened and trading at equilibrium prices isdetermined1. Let 1 denote the number of commodities traded in each period, with onemarket for each commodity. Money serves as a unit of account. Moreover, for simplicity, it isalso the only store of value between periods, no credit is allowed, no other assets are available.

These can be introduced in a straightforward way. Prices of commodities are expressed in termsof money and trading in markets is carried out in exchange for money.

At any date/periodtthe consumption sector of the economy consists of finitely many consumerseach with a finite life time remaining in the economy. LetA= , |A| < , denote the set ofconsumers in the market in period t with the typical consumer denoteda A. For each a A,the integer na 0 denotes the number of periods of economic life ofa remaining after t. Inother words,1 +na describes the number of periods of economic activity of agent a.

Given these general data, consumers are characterized in the usual way by their consumptionsets, their preferences, and their endowments. For the purposes here, assume that consumerscan consume non negative amounts of all commodities in any period, so that the individual

life time consumption sets are equal to the non negative orthant R(1+na)+ . Therefore, the

consumption bundles are given by vectors of the form ca = (c, c) : = (c, c1, . . . , cna) withc R+ and = 1, . . . , n

a.

Let eaR+; = 0, . . . , n

a denote consumer asendowment of commodities in period which

is assumed to be given and known with certainty. Define ea = (e0, e1, . . . , ena) R(1+na)+ to

be his life time endowment of commodities which is a feasible consumption plan under theassumptions made. In addition to his lifetime endowment of commodities consumer a hasan initial non negative quantity of money ma0 0 which may be the result of savings in theprevious period or of a transfer. Intertemporal preferences of each consumera are describedby a continuous utility function ua : R

(1+na)+ R; c

a ua(ca) defined on the consumption

1Hicks refers to a trading week where markets open on Monday and equilibrium prices are determined byFriday. Then trades are carried out and goods and services are exchanged against money.

13


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14 Intertemporal Decisions of Consumers

set R(1+na)

+ .

2.2 Intertemporal Decisions of Consumers

Let (p, pa) := (p, pa1, . . . , pana) R

(1+na)+ denote intertemporalprices, where

p R+ are current market prices,

pa := (pa1, . . . , pana) are expected/predicted prices for future periods by a.

a consumption plan ca := (c, c1, . . . , cna) 0 and

a savings plan ma = (m1, . . . , mna) 0 are called feasible for prices pa 0 if

pc +m1 = pe0+ma0

pac+m+1 = pae+m = 1, . . . , n

a

ca is called an optimal consumption plan for prices pa, if it is a solution to the optimizationproblem

(2.2.1) ca =(ma0, p , pa) := arg max

ca

ua(ca)

pc +m1 = pe0+m

a0

pac+m+1 = pae+m

= 1, . . . , na

.

The mapping (ma0, p , pa) is consumer as demand function (correspondence).

Consumer theory yields the following result for the optimization problem (2.2.1).Proposition 2.2.1Letua be continuous, strictly concave, and strictly monotonically increasing. For every(p, pa)>>0, (ea, ma0) 0, (2.2.1) has a unique solution and(m

a0, p , p

a) is a continuous function.

Let

a(ma0, p , pa)

denote the first components ofa(ma0, p , pa), describing theconsumption demandin the

current period.

a(ma0, p , pa) :=ma0+ pe0 p

a(ma0, p , pa)

denote thefinal money holdingsof the current period,

a(ma0, p , pa) ma0 hissavings.

a and a are the relevant functions to determine behavior in the current period

All other variables are optimal plansto be carried out in the future,

they will be changed in future periods (reoptimizing under new conditions), unless priceexpectations and endowments for future periods are correct

= they are irrelevant for current demand and supply behavior.

Proposition 2.2.2Under the assumptions of Proposition 2.2.1:

1. (ma0

, p , pa) is homogeneous of degree zero in(ma0

, p , pa)

2. a(ma0, p , pa) is homogeneous of degree zero in(ma0, p , p

a)

3. a(ma0, p , pa) is homogeneous of degree 1 in(ma0, p , p

a)



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2.2.1 The Geometry of the Budget Set and of Demand 15

4. which satisfy

p(a(ma0, p , pa) e0) +

a(ma0, p , pa) ma0 = 0

for all(ma0, p , pa) R+ R

(1+na)+ .

These properties can be rewritten in terms of the excess demand functionza(ma0, p , pa) :=a(ma0, p , p

a) e0, implying

pza(ma0, pa) +a(ma0, p , p

a) ma0 = 0,

the value of individual excess demand in the goods market and money market is equal tozero for all current prices.

In other wordsWalras Law holds for every agent a individually in temporary analysis forall(p, pa), i.e. for all current and future expectedprices

but not for current prices palone.

2.2.1 The Geometry of the Budget Set and of Demand

Example 2.2.1 (Exchange Economies)A typical consumer withna = 1, = 1, m0 > 0, (e0, e1) 0Intertemporal consumption demand

max{u(c, c1)|pc +m= pe0+m0, p1c1=m +p1e1}

Income effects/wealth effects:

0

e1

e1+m0/p1

0 e0 e0+m0/p

c

c1

p/p1

(a) Budget set

c

c1

(b) Income expansion path

Figure 2.2.1: The role of money balances; p0,p1 constant

price effects: imply regular offer curve if demand point is regular

with increasing indirect utility for all price changes from initial no trade position

change from demand to supply induced by intertemporal substitution in consumption if and



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only if savings changes from positive to negative, i.e.

M RS

e0, e1+

m0p1

>===0

c

c1

(a) interest on credit

c

c1

(b) binding credit restriction and small interest rate

Figure 2.2.3: The role of credit restrictions and interest rates

changes of expectations imply pure intertemporal substitution effects

and regular offer curve



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2.2.1 The Geometry of the Budget Set and of Demand 17

c

c1

(a) intertemporal substitution with p1

c

c1

(b) Offer curve

Figure 2.2.4: Intertemporal substitution ; p constant

Demand for consumption and money

max

u

c,

m

p1

pc +m= pe0+m0

the indifference curves ofu(c,m/p1) depend on future prices

c

m

(a) Budget set

0

m0

pe0+m0

0 e0 e0+m0/p

c

m

pos. savings

neg. savings

(b) Offer curve of consumption - demand for money

Figure 2.2.5: Budget set and demand for money: p1 constant



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Example 2.2.2 (Consumer worker)withm0 > 0 ande0=e1 = 0; (p, w, p1)

max,c,c1

{u(,c,c1)|pc +m= m0+w, p1c1 = m}

max,c,m

u

,c,

m

p1

|m 0

the indifference curves ofu(,c,m) depend on future pricesp1

m0

lmax

m0/p

m

l

c

w

w/p

(a)

c

(b)

Figure 2.2.6: Budget set of consumerworker

c

m0/p

l lmax

l

c

m=m0

(a)

m0/p

lmax

l

c

m=m0

(b)Figure 2.2.7: Demand and Supply of consumerworker



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2.2.2 The Intertemporal Utility Index: An Alternative Approach 19

2.2.2 The Intertemporal Utility Index: An Alternative Approach

Given expected pricespa = (p1, . . . , pna)of consumer a, one may consider a consumers optimalconsumption and savings decision of the current period in a two step procedure, using the socalledBellman principle2

suppose (c, m1, pa) R+ R+ Rna are given

define the intertemporal utility index

v(c, m1, pa) := max

ca{ua(c, ca) |pac+m+1=p

ae+ m, = 1, . . . , n

a}

where ca = (c1, . . . , cna) is the vector of future consumption plans

v(c, m1, pa) is homogeneous of degree zero in (m1, p

a)

Then, the application of the Bellman principle stipulates that

maxc,m1

{v(c, m1, pa)|pc + m1 pe0+ m0, m1 0}

= maxc,ca

u(c, ca)

pc0+m1 = pe0+ma0 m1 0pac+m+1 = p

ae+m m0, = 1, . . . , n

a

and (m0, p , p

a)(m0, p , p

a)

= arg max

c,m1{v(c, m1, p

a)|pc +m1 pe0+m0, m1 0}

in other words, the two step maximization leads to

the same maximum and

the same set of maximizersin the current period.

NOTE: v is a derived utility function with money and prices as arguments

money in the utility function

which is maximized on a single budget setmaxc,m

{v(c,m,pa)|pc +m pe0+ m0, m 0}

2.2.3 Price Expectations of ConsumersPrice information and expectations formation consists of two parts:

current market prices, common to all agents in period t arecertainandobjective

expected prices for the future areuncertainand subjective

forecasting requires information statistical/econometric techniques

convincing subjective beliefs of relevant aspects of economic evolution

requiresan economic model !

= forecasts may depend on

2see for example Grandmont (1974)



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current prices

past prices

other observable variables

unobservable shocks

A standard assumption for temporary equilibrium analysisAssumption 2.2.1 A forecasting rule

(2.2.3) : R+ Rna p(p) = (1(p), . . . , na(p)) = (p1, . . . , pna)

Special properties, which are imposed sometimes:

1. is only a function of current prices

2. is unit elastic (homogeneous of degree one):

(p) =(p), >0, p R

=prices between any two periods have constant growth factors

3. Static expectations: i(p) =p, i= 1, . . . , n(for example Patinkin 1965)

4. a constant expected rate of inflation >1 implying

i(p) = (1 +)ip, i= 1, . . . , n .

often agents form expectations on the basis of current and past data, for example

let : R(T+1) Rna , whereT is the memory of the forecasting system and is the forecasting rule or forecasting method chosen by agent a

= future expected prices pa = (p, p1, p2, . . . pT) become a function of current and pastdata

since past prices (p1, p2, . . . pT) are given, an analysis ofcurrent demand behavior withrespect tocurrent pricescan ignore them;

= forecasts can be considered as functions of current prices alone, implying

= pa = (p) with : R Rna

= substituting in the demand function one obtains

(m0, p) :=(mo, p, (p)).

similarly, one obtains the intertemporal utility index with expectations function as

V(c,m,p) :=v(c,m, (p)).

Observe that with the expectations formation , the demand function and the utility indexare functions of current data alone,

however, expectations formation has a decisive influence on intertemporal expected maximalutility and on consumer demand

being responsible for strong or weak intertemporal substitution effects.

current prices enter through the expectations,

which destroys any general homogeneity property of demand or intertemporal utility, exceptas stated in the following lemma.



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2.2.3 Price Expectations of Consumers 21

Lemma 2.2.1 If is homogeneous of degree one, then

V(c,m,p) is homogenous of degree zero in(m, p),

(m0, p) is homogeneous of degree zero in(m0, p).

Examples

p

p1

(a) constant, deflationary, inflationary

0

0

p

p1

(b) unit elastic (linear) versus nonlinear

Figure 2.2.8: Forms of price expectations

c

c1

(a)

c

c1

(b)Figure 2.2.9: Role of unit elastic expectations

Expectation functions which are homogeneous of degree one are sometimes calledunit elastic

unit elastic expectation functions induce

pure real balance effects and

no intertemporal substitution effects, i.e.= n = 1;p1 = (p) = (1 +)p

= p/p1= 1/(1 + ) is independent ofp.

= changes ofp induce changes ofm0/p only



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Demand under price variations and expectations in general

Consider the effect of changes of current prices in the general case with commodities andassume that there is a proportional -fold increase of all current prices, i. e. p p. Thisinduces an associated 1/ change of purchasing power of real balances from m to m/. If the

induced demand effect is more or less than the one implied by the pure change of real balances,then one says that there is a positive or a negative intertemporal substitution effect inducedthrough expectations.

More specifically, consider the excess demand function

Z(p, m) :=(p, m) e0.

Then, one may decompose the total effect of a proportional change of all price by into asubstitution effectand a real balance effectwritten as

Z : =Z(p,m) Z(p, m)

=Z(p,m) Zp,

m

intertemporal substitution effect

+ Z

p,

m

Z(p, m)

real balance effect

.(2.2.4)

Thus, the existence of specific expectations function with a direct link from current prices to

00 c

c1

e0 e0+m0/p e0+m0/p

e1

(a) A negative substitution effect

00 c

c1

e0 e0+m0/p e0+ m0/p

e1

(b) A small positive substitution effect

Figure 2.2.10: Decomposition: Intertemporal substitution vs. real balance effects

expected future prices may cause a positive substitution into (an increase of) current demandwhen its price rises. In static demand theory such an effect would correspond to a situation ofa Giffen good, which requires commodities to be inferior. In intertemporal optimization sucheffects may occur even without inferiority.

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