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### Summary

### Summary

For many years it was fashionable to treat macroeconomics and microeconomics as separate subjects without looking too deeply at the relationship between the two. But in the 1970s there occurred an episode of high inflation and high unemployment, which was inconsistent with orthodox theory. As a result, macroeconomists began to pay much greater attention to the microfoundations of their subject.

In this book Roger E. A. Farmer takes a somewhat controversial point of view, arguing for the future of macroeconomics as a branch of applied general equilibrium theory. His main theme is that macroeconomics is best viewed as the study of equilibrium environments in which the welfare theorems break down. This approach makes it possible to discuss the role of government policies in a context in which policy may serve some purpose.

Since the publication of the first edition in 1993, self-fulfilling prophecies has become a major competitor to the real business-cycle view of economic fluctuations. The second edition has been updated in three ways: (1) problems are included at the end of every chapter, and a study guide containing sample answers to all of the problems is available; (2) a new chapter discusses research from the past five years on business fluctuations in multisector models; and (3) the chapter on representative agent growth models now includes an appendix that explains the transversality condition.

### Author Notes

Roger E. A. Farmer is Professor of Economics at the University of California, Los Angeles. He also teaches at the European University Institute in Florence, Italy.

### Table of Contents

Preface to Second Edition | p. XV |

1 Introduction | p. 1 |

1.1 Equilibrium Theory as an Approach to Macroeconomics | p. 1 |

1.2 Preview of the Argument | p. 3 |

1.3 Example | p. 4 |

1.3.1 Regular Case | p. 6 |

1.3.2 Irregular Case | p. 7 |

1.4 Concluding Remarks | p. 9 |

1.5 Problems | p. 9 |

2 Linear Difference Equations: Part 1 | p. 11 |

2.1 Introduction | p. 11 |

2.2 Linearizing Nonlinear Models | p. 11 |

2.2.1 Nonlinear Models to Represent Economies | p. 11 |

2.2.2 Linearizing Autonomous Equations | p. 12 |

2.2.3 Linearizing Nonautonomous Equations | p. 13 |

2.2.4 Example of Linearization: The Solow Model | p. 14 |

2.3 Solving First-Order Linear Models | p. 17 |

2.3.1 First-Order Deterministic Equations | p. 18 |

2.3.2 First-Order Stochastic Equations | p. 18 |

2.3.3 Sequences of Probability Distributions | p. 21 |

2.3.4 The Solow Model Revisited | p. 22 |

2.4 Solving Higher-Order Linear Models | p. 25 |

2.4.1 Eigenvalues and Eigenvectors | p. 27 |

2.4.2 Higher-Order Deterministic Equations | p. 29 |

2.4.3 Diagonalizing Systems of Nonstochastic Equations | p. 32 |

2.4.4 Stochastic Vector Difference Equations | p. 33 |

2.4.5 Example of a Vector System--The Behavior of the Solow Residual | p. 34 |

2.5 Concluding Remarks | p. 38 |

2.6 Problems | p. 39 |

3 Linear Difference Equations: Part 2 | p. 41 |

3.1 Introduction | p. 41 |

3.2 Linear Rational Expectations Models | p. 42 |

3.2.1 Optimal Growth as an Illustration of a Rational Expectations Model | p. 44 |

3.3 Solving Linear Rational Expectations Models | p. 47 |

3.3.1 Different Types of Rational Expectations Models | p. 47 |

3.3.2 Case of a Regular Equilibrium | p. 49 |

3.3.3 Solving a Rational Expectations Model by Iterating into the Future | p. 50 |

3.3.4 Completing the Solution by Solving Equations that Depend on the Past | p. 51 |

3.3.5 Optimal Growth Model and the Solow Model Compared | p. 52 |

3.4 Cross-equation Restrictions and the Lucas Critique | p. 53 |

3.4.1 Dynamics of a Monetary Model--A Second Example | p. 53 |

3.4.2 Solving the Example Explicitly | p. 55 |

3.4.3 Lucas Critique and the Cross-equation Restrictions | p. 57 |

3.4.4 Irregular Solutions | p. 57 |

3.5 Concluding Remarks | p. 58 |

3.6 Problems | p. 59 |

4 General Equilibrium Theory under Certainty | p. 63 |

4.1 Introduction | p. 63 |

4.2 Idea of Equilibrium | p. 67 |

4.3 Theory of Consumer Choice | p. 68 |

4.3.1 Assumptions about Preferences | p. 68 |

4.3.2 Consumer's Problem | p. 70 |

4.4 Excess Demand Functions | p. 71 |

4.4.1 Individual Excess Demand Functions | p. 71 |

4.4.2 Aggregate Excess Demand Functions | p. 72 |

4.5 Equilibria and Their Properties | p. 74 |

4.5.1 Some Definitions and Development of Notation | p. 75 |

4.5.2 Geometry of Equilibrium | p. 76 |

4.5.3 Debreu-Sonnenschein-Mantel Theorem | p. 78 |

4.6 General Equilibrium Theory and Efficient Allocations of Resources | p. 79 |

4.6.1 First Welfare Theorem | p. 79 |

4.6.2 Second Welfare Theorem | p. 80 |

4.7 Concluding Remarks | p. 82 |

4.8 Problems | p. 83 |

5 Infinite Horizon Economies and Representative Agents | p. 85 |

5.1 Introduction | p. 85 |

5.2 Representative Agent Economy | p. 88 |

5.2.1 Assumptions about Structure | p. 88 |

5.2.2 Assumptions about Preferences | p. 90 |

5.2.3 Budget Sets and Market Structure | p. 92 |

5.2.4 Consumer's Problem | p. 94 |

5.3 Competitive Equilibrium and the Planner's Problem | p. 95 |

5.3.1 Competitive Equilibrium | p. 95 |

5.3.2 Planner's Problem | p. 96 |

5.4 Using the Representative Agent Model to Explain Time Series Data | p. 99 |

5.4.1 Removing the Trend from Data | p. 99 |

5.4.2 Simple RBC Model and Its Implications | p. 102 |

5.4.3 Regression on a Common Trend: What the Model Tells Us to Do | p. 103 |

5.4.4 Hodrick-Prescott Filter: What RBC Economists Actually Do | p. 106 |

5.4.5 Calibration and Summary Statistics: How RBC Theorists Measure Success | p. 108 |

5.5 Concluding Remarks | p. 109 |

5.6 Appendix Transversality Condition | p. 110 |

5.7 Problems | p. 112 |

6 Infinite Horizon Economies and Overlapping Generations | p. 115 |

6.1 Introduction | p. 115 |

6.2 Structure of the Overlapping Generations Economy | p. 116 |

6.3 Consumer's Problem | p. 117 |

6.3.1 Problem of a Young Agent | p. 117 |

6.3.2 Problem of an Old Agent | p. 118 |

6.4 Example of a Pareto Inferior Equilibrium | p. 119 |

6.4.1 Case of Early Endowments | p. 119 |

6.4.2 Case of Late Endowments | p. 120 |

6.5 Institutions that May Improve Allocations | p. 120 |

6.6 Set of Equilibria in the Overlapping Generations Model | p. 122 |

6.6.1 Equilibria as Solutions to Difference Equations | p. 122 |

6.6.2 Stationary Equilibria | p. 124 |

6.6.3 Classifying Economies by Types of Stationary Equilibria | p. 125 |

6.6.4 Dynamic Equilibria | p. 127 |

6.7 Some Questions about the Model | p. 128 |

6.7.1 Why Does the First Welfare Theorem Break Down? | p. 129 |

6.7.2 When Does Indeterminacy Occur? | p. 129 |

6.7.3 When Are Equilibria Efficient? | p. 130 |

6.8 More General Examples of Overlapping Generations Economies | p. 132 |

6.8.1 Kehoe-Levine Approach | p. 134 |

6.8.2 Indeterminacy in the OG Model | p. 135 |

6.9 Concluding Remarks | p. 135 |

6.10 Problems | p. 136 |

7 Infinite Horizon Economies with Nonconvexities | p. 141 |

7.1 Introduction | p. 141 |

7.2 Growth Model with Increasing Returns | p. 142 |

7.2.1 Equations That Characterize an Equilibrium | p. 142 |

7.2.2 Behavior of the Representative Family | p. 143 |

7.2.3 Interpretation 1: Externalities in Production | p. 144 |

7.2.4 Interpretation 2: Monopolistic Competition | p. 145 |

7.3 Empirical Evidence for Increasing Returns | p. 149 |

7.4 Equilibria in the Increasing Returns Economy | p. 152 |

7.4.1 Finding a Balanced Growth Path | p. 152 |

7.4.2 Approximate Linear Model | p. 154 |

7.5 Comparing the Theoretical Properties of RA and IR Models | p. 155 |

7.5.1 Why Does the IR Model Display Different Dynamics? | p. 155 |

7.5.2 RA Model: An Example of a Regular Equilibrium | p. 157 |

7.5.3 IR Model: An Example of an Irregular Equilibrium | p. 158 |

7.6 Comparing Some Empirical Predictions of RA and IR Models | p. 160 |

7.6.1 Contemporaneous Statistics | p. 160 |

7.6.2 Dynamic Responses--The Impulse Response Function | p. 163 |

7.7 Concluding Remarks | p. 164 |

7.8 Problems | p. 166 |

8 Some Recent Developments | p. 171 |

8.1 Introduction | p. 171 |

8.2 New Evidence against Big Increasing Returns | p. 171 |

8.3 Nonseparable Preferences | p. 173 |

8.3.1 Households | p. 173 |

8.3.2 Technology | p. 174 |

8.3.3 Solving the Consumer's Problem | p. 175 |

8.3.4 Equations That Characterize Equilibrium | p. 176 |

8.3.5 Example | p. 178 |

8.4 Two-Sector Models | p. 180 |

8.4.1 Technology | p. 180 |

8.4.2 Production Possibilities Frontier | p. 181 |

8.4.3 Behavior of the Representative Family | p. 183 |

8.4.4 Indeterminacy and the Two-Sector Model | p. 185 |

8.4.5 Procyclical Consumption | p. 185 |

8.4.6 Calibrated Two-Sector Model | p. 187 |

8.5 Concluding Remarks | p. 187 |

8.6 Problems | p. 188 |

9 General Equilibrium Theory and Uncertainty | p. 191 |

9.1 Introduction | p. 191 |

9.2 Debreu's Formulation of the Problem | p. 191 |

9.2.1 Preferences under Uncertainty | p. 191 |

9.2.2 Budget Constraints | p. 196 |

9.3 Arrow's Formulation of the Problem | p. 197 |

9.3.1 Trade in Financial Securities | p. 197 |

9.3.2 Complete and Incomplete Markets | p. 199 |

9.3.3 Multiple Budget Constraints and Incomplete Markets | p. 201 |

9.4 Infinite Horizon Economies with Uncertainty | p. 202 |

9.4.1 Asset Pricing in Lucas Tree Economies | p. 202 |

9.4.2 Digression on Market Structure | p. 204 |

9.4.3 Asset Pricing in the Representative Agent Case | p. 206 |

9.5 Concluding Remarks | p. 208 |

9.6 Problems | p. 209 |

10 Sunspots | p. 211 |

10.1 Introduction | p. 211 |

10.2 Do Sunspots Matter? | p. 212 |

10.2.1 Complete and Incomplete Participation | p. 212 |

10.2.2 Setting up the Environment | p. 213 |

10.2.3 Sunspot Theorems | p. 215 |

10.3 Example of a Macroeconomic Model Where Sunspots Matter | p. 217 |

10.3.1 Description of the Environment | p. 217 |

10.3.2 Set of Equilibria | p. 220 |

10.3.3 Supporting Sunspot Equilibria with Beliefs | p. 223 |

10.3.4 Sunspots, Bubbles, and Regular Equilibria | p. 226 |

10.4 Concluding Remarks | p. 228 |

10.5 Problems | p. 229 |

11 Macroeconomic Models of Money | p. 231 |

11.1 Introduction | p. 231 |

11.1.1 Rate of Return Dominance and Legal Restrictions Theory | p. 231 |

11.1.2 Some Quick Fixes to Rate of Return Dominance | p. 232 |

11.2 Models of Money | p. 233 |

11.2.1 Budget Sets: the Opportunity Cost of Holding Money | p. 234 |

11.2.2 Objective Functions: Cash in Advance and Its Relationship to Money in the Utility Function | p. 237 |

11.3 Dynamics of a Cash-in-Advance Model | p. 239 |

11.3.1 Different Types of Monetary Policy | p. 239 |

11.3.2 Government's Budget Constraints | p. 240 |

11.3.3 Typology of Policy Regimes | p. 242 |

11.4 Equilibrium under Interest Rate Control | p. 243 |

11.4.1 Policy Mix A: Fixed Interest Rates and Zero Debt | p. 243 |

11.4.2 Equilibrium of the Real Economy under Interest Rate Control | p. 246 |

11.4.3 Equilibrium Rates of Change of Nominal Variables under Interest Rate Control | p. 247 |

11.4.4 Indeterminacy of the Nominal Scale of the Economy under Interest Rate Control | p. 248 |

11.4.5 Economics of Indeterminacy under Interest Rate Control | p. 249 |

11.5 Equilibrium under a Fixed Money Growth Rate Rule | p. 250 |

11.5.1 Policy Mix B: Fixed Money Growth Rate and Zero Debt | p. 251 |

11.5.2 Equilibrium of the Real Economy with a Fixed Money Growth Rate | p. 251 |

11.5.3 Example of Indeterminate Equilibria in a Simple Economy | p. 252 |

11.6 Concluding Remarks | p. 254 |

11.7 Problems | p. 255 |

12 Applied Monetary Theory | p. 259 |

12.1 Introduction | p. 259 |

12.2 Monetary Facts: What There Is to Explain | p. 260 |

12.3 Simple Monetary Model: Using Equilibrium Theory to Explain the Facts | p. 263 |

12.3.1 Modeling the Exchange Process | p. 265 |

12.3.2 Formalizing the Exchange Technology | p. 266 |

12.3.3 How to Describe an Equilibrium | p. 267 |

12.4 How Do Equilibria Behave? | p. 270 |

12.4.1 Choosing Functional Forms | p. 270 |

12.4.2 What Do Equilibria Look Like? | p. 271 |

12.4.3 Alternative Views of the Money-Income Correlation? | p. 275 |

12.4.4 What Does All of This Have to Do with Sticky Prices? | p. 276 |

12.5 Concluding Remarks | p. 277 |

12.6 Problems | p. 278 |

Notes | p. 281 |

Bibliography | p. 289 |

Index | p. 295 |