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Foundations of Modern Macroeconomics: Third Edition Ben J. Heijdra December 13, 2016

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Page 1: Foundations of Modern Macroeconomics: Third Edition

Foundations of Modern Macroeconomics: Third

Edition

Ben J. Heijdra

December 13, 2016

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Contents

List of Figures xv

List of Tables xxi

List of Intermezzos xxiii

Preface xxv

I Intermediate macroeconomics 1

1 Review of the AD-AS model 31.1 The aggregate labour market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 The demand for labour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 The supply of labour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.3 Aggregate supply in the goods market: Adaptive expectations . . . . . . . 131.1.4 Nominal wage rigidities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2 Aggregate demand: Review of the IS-LM model . . . . . . . . . . . . . . . . . . . 161.2.1 The demand for money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.2 The IS-LM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.2.3 The AD curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.2.4 Effectiveness of fiscal policy . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.3 Schools in macroeconomics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.3.1 Classical economists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.3.2 Keynesians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.3.3 The neo-Keynesian synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . 251.3.4 The monetarists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.3.5 New classical economists . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.3.6 Supply siders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.3.7 New Keynesians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.4 Punchlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2 The open economy 332.1 Some international bookkeeping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2 The IS-LM model for a small open economy . . . . . . . . . . . . . . . . . . . . . . 35

2.2.1 Fixed exchange rates and immobile capital . . . . . . . . . . . . . . . . . . 372.2.2 Fixed exchange rates and perfect capital mobility . . . . . . . . . . . . . . 392.2.3 Flexible exchange rates with perfect capital mobility . . . . . . . . . . . . . 39

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2.2.4 Imperfect capital mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.3 Aggregate supply considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.3.1 The Armington approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.3.2 The extended Mundell-Fleming model . . . . . . . . . . . . . . . . . . . . 51

2.4 Transmission of shocks in a two-country world . . . . . . . . . . . . . . . . . . . . 572.4.1 Nominal wage rigidity in both countries . . . . . . . . . . . . . . . . . . . 612.4.2 Real wage rigidity in both countries . . . . . . . . . . . . . . . . . . . . . . 642.4.3 Real wage rigidity in Europe and nominal wage rigidity in the United States 65

2.5 Punchlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Appendix: Analyzing two-country models . . . . . . . . . . . . . . . . . . . . . . . . . 72

3 Dynamics in aggregate demand and supply 773.1 What is stability? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.2 Adaptive expectations and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.3 Investment, the capital stock, and stability . . . . . . . . . . . . . . . . . . . . . . . 84

3.3.1 Adjustment costs and investment . . . . . . . . . . . . . . . . . . . . . . . 843.3.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.4 Wealth effects and the government budget constraint . . . . . . . . . . . . . . . . 943.4.1 Short-run macroeconomic equilibrium . . . . . . . . . . . . . . . . . . . . . 953.4.2 Money finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983.4.3 Bond finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.5 A first look at hysteresis⋆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.5.1 Alienation of the unemployed . . . . . . . . . . . . . . . . . . . . . . . . . 1023.5.2 History matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.6 Punchlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4 Perfect foresight and economic policy 1094.1 Dynamic investment theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.1.1 The basic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.1.2 Fiscal policy: Investment subsidy . . . . . . . . . . . . . . . . . . . . . . . . 114

4.2 A dynamic IS-LM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.3 Exchange rate expectations and fiscal and monetary policy . . . . . . . . . . . . . 136

4.3.1 The Dornbusch model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374.3.2 Price stickiness and overshooting . . . . . . . . . . . . . . . . . . . . . . . . 141

4.4 Punchlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5 Rational expectations and economic policy 1475.1 What are rational expectations? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.1.1 The basic idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1475.1.2 Do we really believe the idea? . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.2 Applications of REH in macroeconomics . . . . . . . . . . . . . . . . . . . . . . . . 1565.3 Should we take the PIP seriously? . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5.3.1 One-period nominal wage contracts . . . . . . . . . . . . . . . . . . . . . . 1615.3.2 Overlapping wage contracts . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.4 Rational expectations in a small open economy⋆ . . . . . . . . . . . . . . . . . . . 1685.4.1 Unmanaged economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1705.4.2 Introduction to Dynare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1765.4.3 Managed economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

5.5 Punchlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

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6 The government budget deficit 1896.1 Ricardian equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

6.1.1 A simple model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1906.1.2 Distorting taxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1986.1.3 Borrowing restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2036.1.4 Finite lives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2046.1.5 Some further reasons for Ricardian non-equivalence . . . . . . . . . . . . . 2096.1.6 Empirical evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

6.2 The theory of government debt creation . . . . . . . . . . . . . . . . . . . . . . . . 2106.2.1 A simple model of tax smoothing . . . . . . . . . . . . . . . . . . . . . . . 2106.2.2 Implications from the tax smoothing model . . . . . . . . . . . . . . . . . . 217

6.3 Punchlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

7 A closer look at the labour market 2237.1 Some stylized facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2237.2 Standard macroeconomic labour market theory . . . . . . . . . . . . . . . . . . . . 228

7.2.1 Skilled and unskilled labour . . . . . . . . . . . . . . . . . . . . . . . . . . 2287.2.2 The effects of taxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2357.2.3 The Holy Grail of macroeconomics . . . . . . . . . . . . . . . . . . . . . . . 241

7.3 Trade unions and the labour market . . . . . . . . . . . . . . . . . . . . . . . . . . 2427.3.1 Unions in partial equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 2427.3.2 Unions in general equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 2517.3.3 Unions and real wage rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . 253

7.4 Efficiency wages and the labour market . . . . . . . . . . . . . . . . . . . . . . . . 2547.4.1 Efficiency wages in partial equilibrium . . . . . . . . . . . . . . . . . . . . 2557.4.2 Efficiency wages in general equilibrium . . . . . . . . . . . . . . . . . . . . 258

7.5 Punchlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

8 Search in the labour market 2658.1 Search in the labour market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

8.1.1 A canonical two-sided search model . . . . . . . . . . . . . . . . . . . . . . 2668.1.2 Market equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2748.1.3 Comparative static effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2758.1.4 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

8.2 Applications of the canonical search model . . . . . . . . . . . . . . . . . . . . . . 2798.2.1 The effects of taxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2808.2.2 Deposits on workers? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

8.3 Endogenous job destruction⋆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2848.3.1 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2848.3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2858.3.3 Market equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

8.4 Punchlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

9 Dynamic inconsistency in public and private decision making 2959.1 Dynamic inconsistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

9.1.1 A classic tale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2959.1.2 A neoclassical tale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2969.1.3 Reputation as an enforcement mechanism . . . . . . . . . . . . . . . . . . 300

9.2 The voting approach to optimal inflation . . . . . . . . . . . . . . . . . . . . . . . 304

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9.3 Dynamic consistency and capital taxation . . . . . . . . . . . . . . . . . . . . . . . 3079.3.1 The first-best optimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3089.3.2 The second-best problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3109.3.3 Dynamic inconsistency of the optimal tax plan . . . . . . . . . . . . . . . . 313

9.4 Are consumers dynamically inconsistent? . . . . . . . . . . . . . . . . . . . . . . . 3149.4.1 A dynamically consistent consumer . . . . . . . . . . . . . . . . . . . . . . 3159.4.2 A dynamically inconsistent consumer . . . . . . . . . . . . . . . . . . . . . 317

9.5 Punchlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

II Towards advanced macroeconomics 325

10 Money 32710.1 Functions of money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32710.2 Modelling money as a medium of exchange . . . . . . . . . . . . . . . . . . . . . . 330

10.2.1 Setting the stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33010.2.2 Shopping costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33210.2.3 Money in the utility function . . . . . . . . . . . . . . . . . . . . . . . . . . 333

10.3 Money as a store of value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33510.3.1 Overlapping-generations model of money . . . . . . . . . . . . . . . . . . 33810.3.2 Uncertainty and the demand for money . . . . . . . . . . . . . . . . . . . . 341

10.4 The optimal quantity of money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35110.4.1 A basic general equilibrium model . . . . . . . . . . . . . . . . . . . . . . . 35410.4.2 The satiation result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35710.4.3 Critiques of the full liquidity rule . . . . . . . . . . . . . . . . . . . . . . . . 358

10.5 Punchlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

11 New Keynesian economics 36511.1 Reconstructing the “Keynesian” multiplier . . . . . . . . . . . . . . . . . . . . . . 365

11.1.1 A static model with monopolistic competition . . . . . . . . . . . . . . . . 36611.1.2 The short-run balanced-budget multiplier . . . . . . . . . . . . . . . . . . . 37211.1.3 The short-run multiplier in isolation . . . . . . . . . . . . . . . . . . . . . . 37411.1.4 The “long-run” multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37411.1.5 Welfare effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

11.2 Monopolistic competition and money . . . . . . . . . . . . . . . . . . . . . . . . . 38111.3 Sticky prices and the non-neutrality of money . . . . . . . . . . . . . . . . . . . . 383

11.3.1 Menu costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38411.3.2 Quadratic price adjustment costs . . . . . . . . . . . . . . . . . . . . . . . . 39811.3.3 Staggered price contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

11.4 Punchlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

12 Exogenous economic growth – Solow-Swan 40512.1 Stylized facts of economic growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40512.2 The Solow-Swan model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

12.2.1 No technological progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40712.2.2 Technological progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

12.3 Properties of the Solow-Swan model . . . . . . . . . . . . . . . . . . . . . . . . . . 41112.3.1 The golden rule of capital accumulation . . . . . . . . . . . . . . . . . . . . 41112.3.2 Transitional dynamics and convergence . . . . . . . . . . . . . . . . . . . . 41312.3.3 The speed of adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

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12.3.4 Human capital to the rescue . . . . . . . . . . . . . . . . . . . . . . . . . . . 41812.4 Macroeconomic applications of the Solow-Swan model . . . . . . . . . . . . . . . 420

12.4.1 Fiscal policy in the Solow model . . . . . . . . . . . . . . . . . . . . . . . . 42012.5 Two-sector model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42512.6 Punchlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

13 Exogenous economic growth – Ramsey-Cass-Koopmans 43113.1 The Ramsey-Cass-Koopmans model . . . . . . . . . . . . . . . . . . . . . . . . . . 431

13.1.1 The representative household . . . . . . . . . . . . . . . . . . . . . . . . . . 43213.1.2 The representative firm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43713.1.3 The phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

13.2 Properties of the Ramsey-Cass-Koopmans model . . . . . . . . . . . . . . . . . . . 44013.2.1 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44013.2.2 Transitional dynamics and convergence . . . . . . . . . . . . . . . . . . . . 441

13.3 Macroeconomic applications of the Ramsey-Cass-Koopmans model . . . . . . . . 44213.3.1 Fiscal policy in the Ramsey-Cass-Koopmans model . . . . . . . . . . . . . 44213.3.2 Ricardian equivalence once again . . . . . . . . . . . . . . . . . . . . . . . 444

13.4 An open-economy RCK model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44513.4.1 Some model complications . . . . . . . . . . . . . . . . . . . . . . . . . . . 44513.4.2 Model solution and convergence speed . . . . . . . . . . . . . . . . . . . . 44713.4.3 Effects of an investment subsidy . . . . . . . . . . . . . . . . . . . . . . . . 452

13.5 The RCK model with endogenous labour supply . . . . . . . . . . . . . . . . . . . 45413.5.1 Model elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45413.5.2 The phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45613.5.3 Permanent fiscal policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45813.5.4 Temporary fiscal policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464

13.6 The RCK model with search unemployment . . . . . . . . . . . . . . . . . . . . . 47013.6.1 Model elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47013.6.2 Is unemployment efficient? . . . . . . . . . . . . . . . . . . . . . . . . . . . 47613.6.3 Transitional dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

13.7 A monetary RCK model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48113.7.1 Model elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48113.7.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48813.7.3 Monetary neutrality? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

13.8 Punchlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49113.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493Appendix: Phase diagram for the unit-elastic RCK model . . . . . . . . . . . . . . . . . 494

14 Endogenous economic growth 49914.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49914.2 “Capital-fundamentalist” models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

14.2.1 Factor substitutability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50014.2.2 AK models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

14.3 Human capital formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51514.3.1 Steady-state growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51814.3.2 Transitional dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52014.3.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

14.4 Endogenous technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52214.4.1 R&D and expanding input variety . . . . . . . . . . . . . . . . . . . . . . . 522

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14.4.2 R&D and rising input quality . . . . . . . . . . . . . . . . . . . . . . . . . . 53114.5 Punchlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540

15 Overlapping generations in continuous time 54315.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54315.2 Individual behaviour under lifetime uncertainty . . . . . . . . . . . . . . . . . . . 543

15.2.1 Yaari’s lessons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54315.2.2 Realistic mortality profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552

15.3 Macroeconomic consequences of lifetime uncertainty . . . . . . . . . . . . . . . . 55815.3.1 Individual households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55815.3.2 Aggregate households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56015.3.3 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56215.3.4 Government and market equilibrium . . . . . . . . . . . . . . . . . . . . . 56215.3.5 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

15.4 Basic model properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56515.4.1 Fiscal policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56515.4.2 Non-neutrality of government debt . . . . . . . . . . . . . . . . . . . . . . 56515.4.3 Economic growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56715.4.4 Dynamic efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56815.4.5 Small open economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571

15.5 Endogenous labour supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57915.5.1 Model elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58015.5.2 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58115.5.3 Raising the consumption tax . . . . . . . . . . . . . . . . . . . . . . . . . . 58515.5.4 Quantitative evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588

15.6 Punchlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590Appendix: Derivation of the phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . 594

16 Overlapping generations in discrete time 59716.1 The Diamond-Samuelson model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597

16.1.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59816.1.2 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59916.1.3 Market equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60016.1.4 Dynamics and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60116.1.5 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

16.2 Social security and the macroeconomy . . . . . . . . . . . . . . . . . . . . . . . . . 60316.2.1 Pensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60316.2.2 PAYG pensions and endogenous retirement . . . . . . . . . . . . . . . . . . 61616.2.3 The macroeconomic effects of ageing . . . . . . . . . . . . . . . . . . . . . 623

16.3 The tragedy of annuitization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62616.4 Further applications of the DS model . . . . . . . . . . . . . . . . . . . . . . . . . . 630

16.4.1 Human capital accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . 63016.4.2 Public investment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64016.4.3 Endogenous fertility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649

16.5 Punchlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653

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III Stochastic general equilibrium macroeconomics 659

17 Decision making in a stochastic environment 66117.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66117.2 Dynamic programming in a deterministic world . . . . . . . . . . . . . . . . . . . 661

17.2.1 Finite planning horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66217.2.2 Infinite planning horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670

17.3 Dynamic programming in a stochastic world . . . . . . . . . . . . . . . . . . . . . 67217.3.1 Finite planning horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67317.3.2 Infinite planning horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683

17.4 Complete markets and Arrow-Debreu securities . . . . . . . . . . . . . . . . . . . 68517.4.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68517.4.2 Pareto optimal allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68617.4.3 Decentralization with Arrow-Debreu securities . . . . . . . . . . . . . . . 687

17.5 Constructing the representative agent . . . . . . . . . . . . . . . . . . . . . . . . . 69017.6 Punchlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692

18 Dynamic Stochastic General Equilibrium – New Classical models 69518.1 The Lucas research programme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69518.2 Building the unit-elastic RBC model . . . . . . . . . . . . . . . . . . . . . . . . . . 696

18.2.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69718.2.2 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701

18.3 Model analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70318.3.1 Loglinearized model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70418.3.2 The shock process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70818.3.3 Impulse-response functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 70918.3.4 Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71618.3.5 A detour: Dynare comes to the rescue . . . . . . . . . . . . . . . . . . . . . 718

18.4 Extending the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71918.4.1 Employment variability puzzle . . . . . . . . . . . . . . . . . . . . . . . . . 72218.4.2 Productivity puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72618.4.3 Unemployment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726

18.5 Punchlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727Appendix: On the unit-elastic RBC model . . . . . . . . . . . . . . . . . . . . . . . . . . 730

19 Dynamic Stochastic General Equilibrium – New Keynesian models 73519.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73519.2 Building an MBC model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736

19.2.1 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73719.2.2 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74419.2.3 Macroeconomic equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 746

19.3 The canonical New Keynesian DSGE model . . . . . . . . . . . . . . . . . . . . . . 74719.3.1 Preliminary steps (without apologies) . . . . . . . . . . . . . . . . . . . . . 74919.3.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75419.3.3 Interest rate setting, instability, and Taylor rules . . . . . . . . . . . . . . . 757

19.4 Back to the general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75919.4.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76019.4.2 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76219.4.3 Visualizing some shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765

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19.5 Estimation rather than calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77119.5.1 Model features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77119.5.2 Model calibration and estimation . . . . . . . . . . . . . . . . . . . . . . . . 77519.5.3 Main findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776

19.6 Critics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77619.6.1 New Classical critics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77719.6.2 New Keynesian critics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777

19.7 Punchlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779

Mathematical appendix 781A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781A.2 Matrix algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781

A.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781A.2.2 Addition, subtraction, multiplication . . . . . . . . . . . . . . . . . . . . . 781A.2.3 Transposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782A.2.4 Square matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783A.2.5 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785A.2.6 Characteristic roots and vectors . . . . . . . . . . . . . . . . . . . . . . . . . 786A.2.7 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788

A.3 Implicit function theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788A.3.1 Single equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788A.3.2 System of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790A.3.3 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791

A.4 Static optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791A.4.1 Unconstrained optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 791A.4.2 Equality constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792A.4.3 Inequality constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794A.4.4 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796

A.5 Single differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796A.5.1 First-order (constant coefficients) . . . . . . . . . . . . . . . . . . . . . . . . 797A.5.2 First-order (variable coefficients) . . . . . . . . . . . . . . . . . . . . . . . . 798A.5.3 Leibnitz’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799A.5.4 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799

A.6 Systems of differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799A.6.1 The Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800A.6.2 Simple applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803A.6.3 Systems of differential equations . . . . . . . . . . . . . . . . . . . . . . . . 805A.6.4 Hysteretic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810A.6.5 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813

A.7 Difference equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813A.7.1 Basic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814A.7.2 The z-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815A.7.3 Simple application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816A.7.4 The saddle-path model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817A.7.5 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818

A.8 Dynamic optimization in continuous time . . . . . . . . . . . . . . . . . . . . . . . 818A.8.1 From Lagrange to the Optimum Principle . . . . . . . . . . . . . . . . . . . 818A.8.2 Unconstrained . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821A.8.3 (In)equality constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823

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A.8.4 Second-order conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823A.8.5 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823

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List of Figures

1.1 Short-run profit maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 The demand for labour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 The consumption-leisure choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 The supply of labour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Aggregate supply and expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.6 Aggregate supply with downward nominal wage rigidity . . . . . . . . . . . . . . 161.7 The liquidity preference function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.8 Derivation of the LM curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.9 Monetary and fiscal policy in the classical model . . . . . . . . . . . . . . . . . . . 231.10 Monetary and fiscal policy in the Keynesian model . . . . . . . . . . . . . . . . . . 241.11 Monetary and fiscal policy in the neo-Keynesian synthesis model . . . . . . . . . . 261.12 The Laffer curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.1 The degree of capital mobility and the balance of payment . . . . . . . . . . . . . . 372.2 Monetary and fiscal policy with immobile capital and fixed exchange rates . . . . 382.3 Monetary and fiscal policy with perfect capital mobility and fixed exchange rates 402.4 Monetary policy with perfect capital mobility and flexible exchange rates . . . . . 412.5 Fiscal policy with perfect capital mobility and flexible exchange rates . . . . . . . 432.6 Foreign interest rate shocks with perfect capital mobility and flexible exchange rates 442.7 Monetary policy with imperfect capital mobility and flexible exchange rates . . . 462.8 Aggregate supply in the open economy . . . . . . . . . . . . . . . . . . . . . . . . . 542.9 Aggregate demand shocks under wage rigidity . . . . . . . . . . . . . . . . . . . . 562.10 Monetary policy under wage rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . 582.11 Fiscal policy with nominal wage rigidity in both countries . . . . . . . . . . . . . . 622.12 Monetary policy with nominal wage rigidity in both countries . . . . . . . . . . . 632.13 Fiscal policy with real wage rigidity in both countries . . . . . . . . . . . . . . . . 662.14 European fiscal policy with real wage rigidity in Europe and nominal wage rigid-

ity in the United States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.15 US fiscal policy with real wage rigidity in Europe and nominal wage rigidity in

the United States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.16 US monetary policy with real wage rigidity in Europe and nominal wage rigidity

in the United States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.1 Fiscal policy under adaptive expectations . . . . . . . . . . . . . . . . . . . . . . . 803.2 Stability and adaptive expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.3 Adjustment costs of investment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.4 Comparative static effects in the IS-LM model . . . . . . . . . . . . . . . . . . . . . 91

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3.5 The effect on capital of a rise in public spending . . . . . . . . . . . . . . . . . . . . 923.6 Capital accumulation and the Keynesian effects of fiscal policy . . . . . . . . . . . 933.7 The effects of fiscal policy under money finance . . . . . . . . . . . . . . . . . . . . 973.8 Fiscal policy under (stable) bond financing . . . . . . . . . . . . . . . . . . . . . . . 1013.9 Long-run effect of fiscal policy under different financing modes . . . . . . . . . . . 103

4.1 Investment with constant real wages . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.2 Derivation of the saddle path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.3 An unanticipated and permanent increase in the investment subsidy . . . . . . . 1194.4 An unanticipated and permanent increase in the rate of interest . . . . . . . . . . . 1214.5 An anticipated and permanent increase in the rate of interest . . . . . . . . . . . . 1224.6 Investment with full employment in the labour market . . . . . . . . . . . . . . . . 1244.7 An anticipated and permanent abolition of the investment subsidy . . . . . . . . . 1254.8 An unanticipated and temporary increase in the investment subsidy . . . . . . . . 1274.9 A fall in the tax on labour income: investment and employment effects . . . . . . 1304.10 The short-run and long-run labour market effects . . . . . . . . . . . . . . . . . . . 1314.12 Anticipated fiscal policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354.13 Understanding the quasi-reduced-form expressions for y and RS . . . . . . . . . . 1394.14 Phase diagram for the Dornbusch model . . . . . . . . . . . . . . . . . . . . . . . . 1404.15 Fiscal policy in the Dornbusch model . . . . . . . . . . . . . . . . . . . . . . . . . . 1424.16 Monetary policy in the Dornbusch model . . . . . . . . . . . . . . . . . . . . . . . 1434.17 Exchange rate dynamics with perfectly flexible prices . . . . . . . . . . . . . . . . 144

5.1 Monetary policy under adaptive expectations . . . . . . . . . . . . . . . . . . . . . 1485.2 Expectational errors under adaptive expectations . . . . . . . . . . . . . . . . . . . 1495.3 The normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1515.4 Actual and expected price under REH . . . . . . . . . . . . . . . . . . . . . . . . . 1545.5 Actual and expected price under AEH . . . . . . . . . . . . . . . . . . . . . . . . . 1555.6 Wage setting with single-period contracts . . . . . . . . . . . . . . . . . . . . . . . 1625.7 The optimal contract length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1675.8 Price flexibility and the characteristic roots of ∆ . . . . . . . . . . . . . . . . . . . . 172

6.1 Ricardian equivalence experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1936.2 Liquidity restrictions and the Ricardian experiment . . . . . . . . . . . . . . . . . . 2046.3 Overlapping generations in a three-period economy . . . . . . . . . . . . . . . . . 2056.4 Optimal taxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2176.5 Optimal taxation, tax smoothing, and deficit financing . . . . . . . . . . . . . . . . 219

7.1 Postwar unemployment in selected OECD countries . . . . . . . . . . . . . . . . . 2247.2 Unemployment over the centuries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2267.3 The markets for skilled and unskilled labour . . . . . . . . . . . . . . . . . . . . . . 2307.4 The effects of taxation when wages are flexible . . . . . . . . . . . . . . . . . . . . 2397.5 The effects of taxation with a fixed consumer wage . . . . . . . . . . . . . . . . . . 2417.6 Labour demand and supply and the macroeconomic wage equation . . . . . . . . 2427.7 The iso-profit locus and labour demand . . . . . . . . . . . . . . . . . . . . . . . . 2447.8 Indifference curves of the union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2457.9 Wage setting by the monopoly union . . . . . . . . . . . . . . . . . . . . . . . . . . 2477.10 Wage setting in the right-to-manage model . . . . . . . . . . . . . . . . . . . . . . . 2497.11 Wages and employment under efficient bargaining . . . . . . . . . . . . . . . . . . 2507.12 Unions and wage dispersion in a two-sector model . . . . . . . . . . . . . . . . . . 252

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7.13 Unions and unemployment in a two-sector model . . . . . . . . . . . . . . . . . . 2547.14 Efficiency wages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2557.15 A tractable effort function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2577.16 Efficiency wages and unemployment in a two-sector model . . . . . . . . . . . . . 260

8.1 Search equilibrium in the labour market . . . . . . . . . . . . . . . . . . . . . . . . 2758.2 The effects of a higher job destruction rate . . . . . . . . . . . . . . . . . . . . . . . 2768.3 The effects of a payroll tax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2818.4 The effects of a labour income tax . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2828.5 The effects of a deposit on labour . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2838.6 The uniform distribution and job scrapping . . . . . . . . . . . . . . . . . . . . . . 2868.7 Job creation and destruction in the labour market . . . . . . . . . . . . . . . . . . . 2918.8 The effects of a higher unemployment benefit . . . . . . . . . . . . . . . . . . . . . 292

9.1 Consistent and optimal monetary policy . . . . . . . . . . . . . . . . . . . . . . . . 2989.2 Temptation and enforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3039.3 The frequency distribution of the inflation aversion parameter . . . . . . . . . . . 305

10.1 The barter economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32810.2 Money as a store of value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33710.3 Choice set with storage and money . . . . . . . . . . . . . . . . . . . . . . . . . . . 34010.4 Portfolio choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34810.5 Portfolio choice and a change in the expected yield on the risky asset . . . . . . . 35210.6 Portfolio choice and an increase in the volatility of the risky asset . . . . . . . . . . 35310.7 Monetary equilibrium in a perfect foresight model . . . . . . . . . . . . . . . . . . 356

11.1 Government spending multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37311.2 Multipliers and firm entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37611.3 Menu costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

12.1 The Solow-Swan model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41012.2 Per capita consumption and the savings rate . . . . . . . . . . . . . . . . . . . . . . 41312.3 Per capita consumption during transition to its golden rule level . . . . . . . . . . 41412.4 Growth convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41512.5 Conditional growth convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41612.6 Augmented Solow-Swan model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41912.7 Fiscal policy in the Solow-Swan model . . . . . . . . . . . . . . . . . . . . . . . . . 42312.8 Ricardian non-equivalence in the Solow-Swan model . . . . . . . . . . . . . . . . . 424

13.1 Phase diagram of the Ramsey-Cass-Koopmans model . . . . . . . . . . . . . . . . 43913.2 Fiscal policy in the Ramsey-Cass-Koopmans model . . . . . . . . . . . . . . . . . . 44313.3 Investment in the open economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45113.4 An investment subsidy with high mobility of physical capital . . . . . . . . . . . . 45313.5 Phase diagram of the unit-elastic model . . . . . . . . . . . . . . . . . . . . . . . . 45813.6 Effects of fiscal policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46113.7 Phase diagram of the linearized model . . . . . . . . . . . . . . . . . . . . . . . . . 46213.8 Temporary fiscal policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46613.9 The transition function for temporary and permanent shocks . . . . . . . . . . . . 46713.10 Phase diagram for a temporary shock . . . . . . . . . . . . . . . . . . . . . . . . . . 46813.11 Transitional dynamics in the RCK unemployment model . . . . . . . . . . . . . . 487

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A13.1 The χ(L) function characterizing labour market equilibrium . . . . . . . . . . . . 495A13.2 Constructing the capital isocline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495A13.3 Phase diagram of the calibrated unit-elastic model . . . . . . . . . . . . . . . . . . 498

14.1 Difficult substitution between labour and capital . . . . . . . . . . . . . . . . . . . 50214.2 Easy substitution between labour and capital . . . . . . . . . . . . . . . . . . . . . 50314.3 Consumption-capital ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50714.4 Steady-state growth due to public infrastructure . . . . . . . . . . . . . . . . . . . 51114.5 Productive government spending and growth . . . . . . . . . . . . . . . . . . . . . 51214.6 Quality ladders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53214.7 Effective demand facing the quality leader . . . . . . . . . . . . . . . . . . . . . . . 53414.8 Equilibrium in the Rising Input Quality model . . . . . . . . . . . . . . . . . . . . 539

15.1 Cumulative distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54515.2 Density function and survival probability . . . . . . . . . . . . . . . . . . . . . . . 54515.3 Actual and fitted survival fraction for the Dutch cohort born in 1960 . . . . . . . . 55315.4 Features of a realistic mortality profile . . . . . . . . . . . . . . . . . . . . . . . . . 55415.5 Life-cycle profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55615.6 Phase diagram of the Blanchard-Yaari model . . . . . . . . . . . . . . . . . . . . . . 56415.7 Fiscal policy in the Blanchard-Yaari model . . . . . . . . . . . . . . . . . . . . . . . 56615.8 Endogenous growth in the Blanchard-Yaari model . . . . . . . . . . . . . . . . . . 56815.9 Dynamic inefficiency and declining productivity . . . . . . . . . . . . . . . . . . . 57215.10 A patient small open economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57415.11 A productivity shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57515.12 An impatient small open economy . . . . . . . . . . . . . . . . . . . . . . . . . . . 57615.13 A temporary productivity shock in a non-saving nation . . . . . . . . . . . . . . . 57615.14 A temporary productivity shock in the RA model . . . . . . . . . . . . . . . . . . . 57915.15 Phase diagram for the extended Blanchard-Yaari model . . . . . . . . . . . . . . . 58315.16 Factor markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58415.17 Consumption taxation with a dominant GT effect . . . . . . . . . . . . . . . . . . . 58715.18 Consumption taxation with a dominant FS effect . . . . . . . . . . . . . . . . . . . 58815.19 Transitional dynamics in the extended BY model . . . . . . . . . . . . . . . . . . . 591A15.1 Phase diagram of the calibrated extended BY model . . . . . . . . . . . . . . . . . 596

16.1 The unit-elastic Diamond-Samuelson model . . . . . . . . . . . . . . . . . . . . . . 60216.2 Individual effect of a fully-funded pension system . . . . . . . . . . . . . . . . . . 60516.3 Individual effect of a PAYG pension system . . . . . . . . . . . . . . . . . . . . . . 60716.4 PAYG pensions in the unit-elastic model . . . . . . . . . . . . . . . . . . . . . . . . 60916.5 Deadweight loss of taxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62216.6 The effects of ageing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62616.7 Impact and long-run effect of opening an annuity market . . . . . . . . . . . . . . 62816.8 Introducing a perfect annuity market . . . . . . . . . . . . . . . . . . . . . . . . . . 62916.9 Endogenous growth due to human capital formation . . . . . . . . . . . . . . . . . 63416.10 Public and private capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64316.11 Steady-state fertility rate and capital intensity . . . . . . . . . . . . . . . . . . . . . 654

17.1 Value functions and policy functions: Deterministic case . . . . . . . . . . . . . . . 66917.2 Markov process for labour productivity . . . . . . . . . . . . . . . . . . . . . . . . 67517.3 Labour productivity over the life cycle . . . . . . . . . . . . . . . . . . . . . . . . . 67617.4 Value functions: Stochastic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680

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17.5 Policy functions: Stochastic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681

18.1 A shock to technology and the labour market . . . . . . . . . . . . . . . . . . . . . 71018.2 Purely transitory productivity shock . . . . . . . . . . . . . . . . . . . . . . . . . . 71318.3 Permanent productivity shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71518.4 Temporary productivity shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71718.5 Visualizing contemporaneous correlations . . . . . . . . . . . . . . . . . . . . . . . 72318.6 Lottery model and contemporaneous correlations . . . . . . . . . . . . . . . . . . . 725

19.1 Transitory productivity shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76619.2 Transitory money supply shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76919.3 Transitory government consumption shock . . . . . . . . . . . . . . . . . . . . . . 772

A.1 Non-negativity constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795A.2 Piecewise continuous function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802

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List of Tables

2.1 Imperfect capital mobility under fixed and flexible exchange rates . . . . . . . . . 452.2 The extended Mundell-Fleming model . . . . . . . . . . . . . . . . . . . . . . . . . 552.3 Wage rigidity and demand and supply shocks . . . . . . . . . . . . . . . . . . . . . 562.4 A two-country extended Mundell-Fleming model . . . . . . . . . . . . . . . . . . . 60

4.1 The Dornbusch model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.1 A Dynare model file for the sticky-price model . . . . . . . . . . . . . . . . . . . . 1785.2 Output from the Dynare model output file Program05 01.mod file . . . . . . . . 1815.3 A Dynare model file with a monetary policy rule . . . . . . . . . . . . . . . . . . . 1845.4 Selected output from the Dynare model output Program05 02.mod file . . . . . 187

7.1 Taxes and the competitive labour market . . . . . . . . . . . . . . . . . . . . . . . . 238

11.1 A simple macro model with monopolistic competition . . . . . . . . . . . . . . . . 37111.2 A simple monetary monopolistic competition model . . . . . . . . . . . . . . . . . 38211.3 A simplified Blanchard-Kiyotaki model (no menu costs) . . . . . . . . . . . . . . . 38711.4 Menu costs and the markup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39611.5 Menu costs and the elasticity of marginal cost . . . . . . . . . . . . . . . . . . . . . 396

12.1 The Meade-Uzawa growth model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

13.1 The Ramsey-Cass-Koopmans growth model . . . . . . . . . . . . . . . . . . . . . . 43813.2 Convergence speed in the Ramsey-Cass-Koopmans model . . . . . . . . . . . . . 44213.3 The Ramsey-Cass-Koopmans model for the open economy . . . . . . . . . . . . . 44813.4 The unit-elastic RCK model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45613.5 Government consumption multipliers and elasticities . . . . . . . . . . . . . . . . 46513.6 The RCK model with search unemployment . . . . . . . . . . . . . . . . . . . . . . 47513.7 A Matlab file for the Ramsey-Cass-Koopmans model . . . . . . . . . . . . . . . . . 48213.8 Matlab function for the steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . 48413.9 Matlab function for the transition path . . . . . . . . . . . . . . . . . . . . . . . . . 48513.10 The monetary RCK model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

14.1 An AK growth model with inter-firm external effects . . . . . . . . . . . . . . . . . 50614.2 An AK growth model with public capital . . . . . . . . . . . . . . . . . . . . . . . . 50914.3 Growth and human capital accumulation . . . . . . . . . . . . . . . . . . . . . . . . 519

15.1 The Blanchard-Yaari model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562

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15.2 The extended Blanchard-Yaari model . . . . . . . . . . . . . . . . . . . . . . . . . . 58215.3 The linearized extended BY model . . . . . . . . . . . . . . . . . . . . . . . . . . . 58515.4 The birth rate and the GT effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

16.1 Age composition of the population . . . . . . . . . . . . . . . . . . . . . . . . . . . 62416.2 Growth, human capital, and overlapping generations . . . . . . . . . . . . . . . . 632

17.1 Some numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665

18.1 The basic RBC model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70318.2 The log-linearized RBC model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70418.5 The unit-elastic RBC model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71818.4 A Dynare model file for the basic RBC model . . . . . . . . . . . . . . . . . . . . . 720

19.1 The basic MBC model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74819.2 A minimal DSGE model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74919.1 A Dynare model file for the general MBC model . . . . . . . . . . . . . . . . . . . 763

A.1 Commonly used Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 801A.2 Commonly used z-transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816

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List of Intermezzos

1.1 The Cobb-Douglas production function and labour demand . . . . . . . . . . . . . 61.2 The Stone-Geary utility function and labour supply . . . . . . . . . . . . . . . . . . 111.3 Baumol’s transactions theory of the demand for money . . . . . . . . . . . . . . . 17

2.1 Imperfect substitution between domestic and foreign goods: Armington . . . . . 48

3.1 The expected price level under the adaptive expectations hypothesis . . . . . . . . 813.2 The cost of capital to the firm: Modigliani-Miller . . . . . . . . . . . . . . . . . . . 86

4.1 Tobin’s q-theory of investment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.2 Tobin’s q as the present value of discounted marginal products of capital . . . . . 1184.3 Some production theory: the two-factor production function . . . . . . . . . . . . 126

5.1 The method of undetermined coefficients in a rational expectations model . . . . 1585.2 Solving the two-period overlapping wage contract model . . . . . . . . . . . . . . 1655.3 Asymptotic variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.1 The two-period consumption model . . . . . . . . . . . . . . . . . . . . . . . . . . 1946.2 Ricardian equivalence in a small open economy that taxes interest income . . . . 2006.3 Welfare loss of taxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

7.1 Some production theory: the three-factor production model . . . . . . . . . . . . . 231

8.1 Some statistical theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2708.2 Comparative static effects in the matching model of unemployment . . . . . . . . 2768.3 Some tedious derivations for the matching model of unemployment . . . . . . . . 288

9.1 Deriving the optimal spending and taxation rules . . . . . . . . . . . . . . . . . . . 312

10.1 Attitude to risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

11.1 Two-stage budgeting and Dixit-Stiglitz preferences . . . . . . . . . . . . . . . . . . 36711.2 Multipliers and the marginal cost of public funds . . . . . . . . . . . . . . . . . . . 38011.3 The envelope theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

12.1 Deriving the fundamental differential equation for the Solow-Swan model . . . . 40812.2 Some tedious derivations for the Meade-Uzawa model . . . . . . . . . . . . . . . . 428

13.1 Deriving the household’s lifetime budget constraint under perfect foresight . . . . 433

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13.2 Deriving the Euler equation with the method of optimal control . . . . . . . . . . 43513.3 The method of undetermined coefficients in a perfect foresight model . . . . . . . 449

14.1 Loglinearization of an endogenous growth model . . . . . . . . . . . . . . . . . . . 513

15.1 Technical results for the Yaari model . . . . . . . . . . . . . . . . . . . . . . . . . . 549

16.1 Some tedious but important derivations for the Diamond-Samuelson model . . . 60516.2 Government solvency condition under perfect foresight . . . . . . . . . . . . . . . 61416.3 Dynamic consistency of the social planner . . . . . . . . . . . . . . . . . . . . . . . 63816.4 Calvo-Obstfeld two-step procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 646

18.1 Optimal household and firm decisions under uncertainty . . . . . . . . . . . . . . 69918.2 Deriving the firm’s objective function . . . . . . . . . . . . . . . . . . . . . . . . . . 70218.3 Linearization of nonlinear stochastic systems . . . . . . . . . . . . . . . . . . . . . 704

19.1 The continuum approach to product differentiation . . . . . . . . . . . . . . . . . . 73819.2 Some derivations for the New Keynesian DSGE model . . . . . . . . . . . . . . . . 74219.3 Blanchard-Kahn stability and existence conditions . . . . . . . . . . . . . . . . . . 756

A.1 Matrix inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784A.2 Eigenvalues, eigenvectors, and matrix diagonalization . . . . . . . . . . . . . . . . 787A.3 Generalized implicit function theorem: a macroeconomic application . . . . . . . 790A.4 Pegging the nominal interest rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811A.5 Current account dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812