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雷达卡
Economic Growth: Theory and Numerical Solution Methods
This is a book on deterministic and stochastic Growth Theory and the computational methods needed to produce numerical solutions. Exogenous and endogenous growth models are thoroughly reviewed. Special attention is paid to the use of these models for fiscal and monetary policy analysis. Modern Business Cycle Theory, the New Keynesian Macroeconomics, the class of Dynamic Stochastic General Equilibrium models, can be all considered as special cases of models of economic growth, and they can be analyzed by the theoretical and numerical procedures provided in the textbook.
Analytical discussions are presented in full detail. The book is self contained and it is designed so that the student advances in the theoretical and the computational issues in parallel. EXCEL and Matlab files are provided on an accompanying website to illustrate theoretical results as well as to simulate the effects of economic policy interventions.
About the Author
ALFONSO NOVALES is Professor of Economics at the Department of Quantitative Economics at Universidad Complutense (Madrid). He holds a Ph.D. in Economics from University of Minnesota and a Ph.D. in Mathematics from Universidad del Pais Vasco. After graduation, he was Assistant Professor at State University of New York (Stony Brook). He has been President of the Spanish Economic Association and chairman of FEDEA (Fundación de Estudios de Economía Aplicada, Madrid). He has published in Econometrica, Journal of Economic Dynamics and Control, Journal of Macroeconomics, International Journal of Forecasting, Journal of Forecasting, Journal of Time Series Analysis, Journal of Banking and Finance, Economic Modelling, Computational Economics, Journal of International Financial Institutions, Markets and Money, International Journal of Money and Finance, Applied Financial Economics, Applied Economics Letters, and International Journal of Finance, and he has contributed with chapters to books on Macroeconomics and Econometrics. He is the author of Econometria and Estadistica y Econometría, both textbooks edited by McGrawHill in Spanish. His research interests include Economic Policy Evaluation and Financial Econometrics.
JESUS RUIZ is Associate Professor of Economics at the Department of Quantitative Economics at Universidad Complutense (Madrid). He obtained his Ph.D. in Economics from Universidad Complutense in 1997. He has published in Journal of Economic Dynamics and Control, Economic Modelling, Applied Financial Economics, Spanish Economic Review and other Spanish journals, and he has contributed with chapters to books on Computational Economics. His research interests include Economic Policy Evaluation in Dynamic Macroeconomic Models and Environmental Economics.
ESTHER FERNÁNDEZ is Associate Professor of Economics at the Department of Quantitative Economics at Universidad Complutense (Madrid). Her Ph.D. dissertation was selected as the best doctoral dissertation in Economics at Universidad Complutense in 1999. Her research interests include Monetary Theory, Economic Growth and Environmental Economics. She has published at Economic Modelling and Spanish academic journals.
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1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 A Few Time Series Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Structural Macroeconomic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Why are Economic Growth Models Interesting? . . . . . . . . . . . . . . . . . 27
1.4 Numerical Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.5 Synopsis of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2 The Neoclassical Growth Model Under a Constant Savings Rate . . . . . 53
2.3 The Neoclassical Growth Model of Solow and Swan . . . . . . . . . . . . . 5
Accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.4 Solving the Continuous-Time Solow–Swan Model . . . . . . . . . . . . . . . 76
2.5 The Deterministic, Discrete-Time Solow Swan Model . . . . . . . . . . . . 85
2.6 The Stochastic, Discrete Time Version of the Solow–Swan Model . . 95
2.6.1 Numerical Exercise – Solving the Stochastic
Solow–Swan Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3 Optimal Growth. Continuous Time Analysis . . . . . . . . . . . . . . . . . . . . . . 101
3.1 The Continuous-Time Version of the Cass–Koopmans Model . . . . . . 101
3.2 Stability and Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.3 Interpreting the Central Planners’s Model as a Competitive
3.4 A Competitive Equilibrium with Government . . . . . . . . . . . . . . . . . . . 131
3.5 On the Efficiency of Equilibrium with Government . . . . . . . . . . . . . . 138
3.6 The Ricardian Doctrine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
3.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4 Optimal Growth. Discrete Time Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.1 Discrete-Time, Deterministic Cass–Koopmans Model . . . . . . . . . . . . 155
4.2 Fiscal Policy in the Cass–Koopmans Model . . . . . . . . . . . . . . . . . . . . 167
4.2.3 Numerical Exercise – Fiscal Policy Evaluation. . . . . . . . . . . . 179
4.3 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Taxes in Discrete Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5 Numerical Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
5.1 Numerical Solutions and Simulation Analysis . . . . . . . . . . . . . . . . . . . 195
5.2 Analytical Solutions to Simple Growth Models . . . . . . . . . . . . . . . . . . 197
5.3 Solving a Simple, Stochastic Version of the Planner’s Problem . . . . . 203
5.3.1 Solving the Linear-Quadratic Approximation
5.4 Solving the Stochastic Representative Agent’s Problem
with Taxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
5.5 Nonlinear Numerical Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . 238
5.6 Appendix – Solving the Planner’s Model Under Full Depreciation . . 251
5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
6 Endogenous Growth Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
6.1 The AK Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
6.2 The Discrete Time Version of the Model . . . . . . . . . . . . . . . . . . . . . . . 262
6.3 Stability in the AK Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
6.4 Effects from Transitory Changes in Policy Parameters . . . . . . . . . . . . 271
6.5 Dynamic Laffer Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
6.6 Solving the Stochastic, Discrete Time Version of the AK Model . . . . 280
6.7 An Endogenous Growth Model with Productive Public
Expenditures: Barro’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
6.8 Transitional Dynamics in Endogenous Growth: The Jones
and Manuelli Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
6.9 The Stochastic Version of Jones and Manuelli Model . . . . . . . . . . . . . 294
6.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
7 Additional Endogenous Growth Models . . . . . . . . . . . . . . . . . . . . . . . . . . 305
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
7.2 A Variety of Producer Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
7.3 Technological Diffusion and Growth . . . . . . . . . . . . . . . . . . . . . . . . . . 323
7.4 Schumpeterian Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
7.5 Endogenous Growth with Accumulation of Human Capital . . . . . . . . 342
7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
8 Growth in Monetary Economies: Steady-State Analysis
of Monetary Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
8.2 Optimal Growth in a Monetary Economy: The Sidrauski Model . . . . 378
8.4 Two Modelling Issues: Nominal Bonds and the Timing of Real
Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
8.5 Monetary Policy Analysis Under Consumption and Income Taxes . . 401
8.6 Monetary Policy Under Endogenous Labor Supply . . . . . . . . . . . . . . 406
8.7 Optimal Monetary Policy Under Distortionary Taxation
and Endogenous Labor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
8.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
9 Transitional Dynamics in Monetary Economies:
Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
9.2 Stability of Public Debt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
9.3 Alternative Strategies for Monetary Policy: Control of Nominal
Rates vs.Money Growth Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
9.4 Deterministic Monetary Model with the Monetary Authority
9.5 Deterministic Monetary Model with the Monetary Authority
Choosing Nominal Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
9.6 Transitional Effects of Policy Interventions . . . . . . . . . . . . . . . . . . . . . 441
9.7 The Stochastic Version of the Monetary Model . . . . . . . . . . . . . . . . . . 450
9.8 A New Keynesian Monetary Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
9.9 Appendix: In a Log-Linear Approximation, Etπˆ t+1 = ıˆt −rˆt . . . . . . . 491
9.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
10 Mathematical Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
10.1 The Deterministic Control Problem in Continuous Time . . . . . . . . . . 495
10.2 The Deterministic Control Problem in Discrete Time . . . . . . . . . . . . . 499
10.3 First Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
10.4 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
10.5 Some Notes on Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
10.6 Solving a Dynamic Two-Equation System with Complex Roots . . . . 514
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
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