Economic Dynamics in Discrete Time, 2e

by Miao

| ISBN: 9780262043625 | Copyright 2020

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A unified and comprehensive introduction to the analytical and numerical tools for solving dynamic economic problems; substantially revised for the second edition.

This book offers a unified, comprehensive, and up-to-date treatment of analytical and numerical tools for solving dynamic economic problems. The focus is on introducing recursive methods—an important part of every economist's set of tools—and readers will learn to apply recursive methods to a variety of dynamic economic problems. The book is notable for its combination of theoretical foundations and numerical methods. Each topic is first described in theoretical terms, with explicit definitions and rigorous proofs; numerical methods and computer codes to implement these methods follow. Drawing on the latest research, the book covers such cutting-edge topics as asset price bubbles, recursive utility, robust control, policy analysis in dynamic New Keynesian models with the zero lower bound on interest rates, and Bayesian estimation of dynamic stochastic general equilibrium (DSGE) models.

This second edition has been substantially updated. Responding to renewed interest in modeling with multiple equilibria, it incorporates new material on this topic throughout. It offers an entirely new chapter on deterministic nonlinear systems, and provides new material on such topics as linear planar systems, chaos, bifurcations, indeterminacy and sunspot solutions, pruning nonlinear solutions, the bandit problem, rational inattention models, bequests, self-fulfilling prophecies, the cyclical behavior of unemployment and vacancies, and the long-run risk model. The exposition of each chapter has been revised and improved, and many new figures, Matlab codes, and exercises have been added. A student solutions manual can be purchased separately.

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Contents (pg. vii)
Preface to the Second Edition (pg. xvii)
Acknowledgments (pg. xix)
I. Dynamical Systems (pg. 1)
1. Deterministic Linear Systems (pg. 3)
1.1 Some Basic Concepts (pg. 3)
1.2 Scalar First-Order Linear Difference Equations (pg. 6)
1.3 Lag Operators (pg. 10)
1.4 Scalar Second-Order Linear Difference Equations (pg. 11)
1.5 Planar Linear Systems (pg. 13)
1.6 Phase Diagrams (pg. 16)
1.7 Higher-Dimensional Linear Systems (pg. 18)
1.8 Exercises (pg. 29)
2. Deterministic Nonlinear Systems (pg. 31)
2.1 Linear Approximation (pg. 31)
2.2 Local Stability (pg. 33)
2.3 Lyapunov Function (pg. 38)
2.4 Cycles and Chaos (pg. 41)
2.4.1 Periodic Solutions (pg. 41)
2.4.2 Bifurcations (pg. 43)
2.4.3 Chaos (pg. 49)
2.5 Numerical Solutions Using Dynare (pg. 55)
2.6 Exercises (pg. 62)
3. Stochastic Difference Equations (pg. 65)
3.1 First-Order Linear Systems (pg. 65)
3.2 Scalar Linear Rational Expectations Models (pg. 67)
3.2.1 Lag Operators (pg. 67)
3.2.2 Method of Undetermined Coefficients (pg. 70)
3.3 Multivariate Linear Rational Expectations Models (pg. 71)
3.3.1 Blanchard–Kahn Method (pg. 71)
3.3.2 Klein Method (pg. 73)
3.3.3 Sims Method (pg. 75)
3.4 Nonlinear Rational Expectations Models (pg. 81)
3.5 Numerical Solutions Using Dynare (pg. 85)
3.6 Indeterminacy and Sunspot Equilibria (pg. 94)
3.7 Pruning Nonlinear Solutions (pg. 96)
3.8 Exercises (pg. 99)
4. Markov Processes (pg. 103)
4.1 Markov Chains (pg. 104)
4.1.1 Classification of States (pg. 108)
4.1.2 Stationary Distribution: Finite State Space (pg. 111)
4.1.3 Countable-State Markov Chains (pg. 118)
4.2 General Markov Processes (pg. 128)
4.3 Convergence (pg. 132)
4.3.1 Strong Convergence (pg. 132)
4.3.2 Weak Convergence (pg. 137)
4.4 Markov Chain Monte Carlo Algorithms (pg. 140)
4.5 Exercises (pg. 143)
5. Ergodic Theory and Stationary Processes (pg. 147)
5.1 Ergodic Theorem (pg. 147)
5.2 Application to Stationary Processes (pg. 152)
5.3 Application to Stationary Markov Processes (pg. 159)
5.4 Exercises (pg. 164)
II. Dynamic Optimization (pg. 165)
6. Markov Decision Process Model (pg. 167)
6.1 Model Setup (pg. 167)
6.2 Examples (pg. 173)
6.3 Exercises (pg. 181)
7. Finite-Horizon Dynamic Programming (pg. 183)
7.1 A Motivating Example (pg. 183)
7.2 Measurability Problem (pg. 187)
7.3 Principle of Optimality (pg. 189)
7.4 Optimal Control (pg. 196)
7.5 Maximum Principle (pg. 202)
7.6 Applications (pg. 206)
7.6.1 Secretary Problem (pg. 206)
7.6.2 A Consumption–Saving Problem (pg. 207)
7.7 Exercises (pg. 209)
8. Infinite-Horizon Dynamic Programming (pg. 211)
8.1 Principle of Optimality (pg. 211)
8.2 Bounded Rewards (pg. 220)
8.3 Unbounded Rewards (pg. 222)
8.3.1 Negative Dynamic Programming (pg. 222)
8.3.2 Weighted Contraction Approach (pg. 225)
8.4 Optimal Control (pg. 229)
8.5 The Maximum Principle and Transversality Conditions (pg. 233)
8.6 Euler Equations and Transversality Conditions (pg. 236)
8.7 Exercises (pg. 243)
9. Applications (pg. 247)
9.1 Option Exercise (pg. 247)
9.2 Discrete Choice (pg. 250)
9.3 Multi-Armed Bandit (pg. 252)
9.4 Consumption and Saving (pg. 258)
9.4.1 Deterministic Income (pg. 261)
9.4.2 Stochastic Income (pg. 268)
9.5 Consumption/Portfolio Choice (pg. 276)
9.6 Inventory (pg. 278)
9.6.1 Finite-Horizon Problem (pg. 280)
9.6.2 Infinite-Horizon Problem (pg. 284)
9.7 Investment (pg. 289)
9.7.1 Neoclassical Theory (pg. 289)
9.7.2 Q Theory (pg. 291)
9.7.3 Augmented Adjustment Costs (pg. 293)
9.8 Exercises (pg. 299)
10. Linear-Quadratic Models (pg. 301)
10.1 Controlled Linear State-Space System (pg. 301)
10.2 Finite-Horizon Problems (pg. 305)
10.3 Infinite-Horizon Limits (pg. 308)
10.3.1 Value Function Iteration (pg. 312)
10.3.2 Policy Improvement Algorithm (pg. 312)
10.3.3 Lagrange Method (pg. 313)
10.4 Optimal Policy under Commitment (pg. 314)
10.5 Optimal Discretional Policy (pg. 320)
10.6 Robust Control (pg. 324)
10.6.1 Belief Distortions and Entropy (pg. 324)
10.6.2 Two Robust Control Problems (pg. 326)
10.6.3 Recursive Formulation (pg. 327)
10.6.4 Linear-Quadratic Model with Gaussian Disturbances (pg. 328)
10.6.5 Relative Entropy and Normal Distributions (pg. 330)
10.6.6 Modified Certainty Equivalence Principle (pg. 330)
10.7 Exercises (pg. 331)
11. Control under Partial Information (pg. 335)
11.1 Filters (pg. 335)
11.1.1 Kalman Filter (pg. 335)
11.1.2 Smoothing (pg. 344)
11.1.3 Hidden Markov Chain (pg. 344)
11.1.4 Hidden Markov-Switching Model (pg. 346)
11.2 Control Problems (pg. 347)
11.3 Linear-Quadratic Control (pg. 351)
11.4 Rational Inattention (pg. 353)
11.4.1 Information Theory (pg. 353)
11.4.2 Linear-Quadratic-Gaussian Models (pg. 355)
11.5 Exercises (pg. 358)
12. Numerical Methods (pg. 361)
12.1 Numerical Integration (pg. 361)
12.1.1 Gaussian Quadrature (pg. 361)
12.1.2 Multidimensional Quadrature (pg. 363)
12.2 Discretizing AR(1) Processes (pg. 364)
12.2.1 Tauchen (1986) Method (pg. 364)
12.2.2 Tauchen–Hussey (1991) Method (pg. 365)
12.2.3 Simulating a Markov Chain (pg. 366)
12.3 Interpolation (pg. 367)
12.3.1 Orthogonal Polynomials (pg. 369)
12.3.2 Splines (pg. 372)
12.3.3 Multidimensional Approximation (pg. 375)
12.4 Perturbation Methods (pg. 377)
12.5 Projection Methods (pg. 380)
12.6 Numerical Dynamic Programming (pg. 385)
12.6.1 Discrete Approximation Methods (pg. 386)
12.6.2 Smooth Approximation Methods (pg. 388)
12.7 Exercises (pg. 391)
13. Structural Estimation (pg. 393)
13.1 Generalized Method of Moments (pg. 393)
13.1.1 Estimation (pg. 394)
13.1.2 Asymptotic Properties (pg. 396)
13.1.3 Weighting Matrix and Covariance Matrix Estimation (pg. 398)
13.1.4 Overidentifying Restrictions (pg. 399)
13.1.5 Implementation (pg. 400)
13.1.6 Relation to Other Estimation Methods (pg. 401)
13.2 Maximum Likelihood (pg. 401)
13.2.1 Estimation (pg. 401)
13.2.2 Asymptotic Properties (pg. 402)
13.2.3 Hypothesis Testing (pg. 403)
13.3 Simulation-Based Methods (pg. 404)
13.3.1 Simulated Method of Moments (pg. 405)
13.3.2 Simulated Maximum Likelihood (pg. 407)
13.3.3 Indirect Inference (pg. 408)
13.4 Exercises (pg. 411)
III. Equilibrium Analysis (pg. 413)
14. Complete Markets Exchange Economies (pg. 415)
14.1 Uncertainty, Preferences, and Endowments (pg. 415)
14.2 Pareto Optimum (pg. 416)
14.3 Time 0 Trading (pg. 417)
14.4 Sequential Trading (pg. 423)
14.5 Equivalence of Equilibria (pg. 431)
14.6 Asset Price Bubbles (pg. 434)
14.7 Recursive Formulation (pg. 439)
14.8 Asset Pricing (pg. 440)
14.9 Exercises (pg. 445)
15. Neoclassical Growth Models (pg. 449)
15.1 Deterministic Models (pg. 449)
15.1.1 A Basic Ramsey Model (pg. 449)
15.1.2 Incorporating Fiscal Policy (pg. 458)
15.2 A Basic RBC Model (pg. 461)
15.2.1 Steady State (pg. 463)
15.2.2 Calibration (pg. 463)
15.2.3 Log-Linearized System (pg. 464)
15.2.4 Business Cycle Statistics and Model Results (pg. 469)
15.2.5 Impact of a Permanent TFP Shock (pg. 471)
15.2.6 Impact of a Temporary TFP Shock (pg. 472)
15.2.7 Effects of Persistence and Critiques of the RBC Model (pg. 473)
15.3 Extensions of the Basic RBC Model (pg. 474)
15.3.1 Various Utility Functions (pg. 474)
15.3.2 Capacity Utilization (pg. 479)
15.3.3 Capital or Investment Adjustment Costs (pg. 480)
15.3.4 Stochastic Trends (pg. 485)
15.3.5 Other Sources of Shocks (pg. 487)
15.4 Exercises (pg. 491)
16. Bayesian Estimation of DSGE Models Using Dynare (pg. 493)
16.1 Principles of Bayesian Estimation (pg. 494)
16.2 Bayesian Estimation of DSGE Models (pg. 495)
16.2.1 Numerical Solution and State-Space Representation (pg. 496)
16.2.2 Evaluating the Likelihood Function (pg. 497)
16.2.3 Computing the Posterior (pg. 499)
16.2.4 Identification (pg. 501)
16.2.5 Model Comparison (pg. 502)
16.2.6 Model Diagnosis: Predictive Checks (pg. 502)
16.3 An Example (pg. 503)
16.3.1 Dynare Codes (pg. 503)
16.3.2 Dynare Output (pg. 507)
16.3.3 Stochastic Trends (pg. 508)
16.4 Exercises (pg. 509)
17. Overlapping Generations Models (pg. 511)
17.1 Exchange Economies (pg. 511)
17.1.1 A Special Case and Multiple Equilibria (pg. 513)
17.1.2 Existence and Efficiency (pg. 518)
17.2 Production Economies (pg. 524)
17.2.1 Multiple Equilibria (pg. 526)
17.2.2 Dynamic Efficiency (pg. 529)
17.2.3 Altruism, Bequests, and Infinite Horizons (pg. 540)
17.3 Asset Price Bubbles (pg. 542)
17.4 Sunspots and Self-Fulfilling Prophecies (pg. 546)
17.5 Exercises (pg. 548)
18. Incomplete Markets Models (pg. 551)
18.1 Production Economies (pg. 551)
18.1.1 Income Fluctuation Problem (pg. 552)
18.1.2 Production (pg. 553)
18.1.3 Stationary Recursive Equilibrium (pg. 554)
18.1.4 Computation and Implications (pg. 555)
18.2 Endowment Economies (pg. 559)
18.2.1 Risk-Free Rate (pg. 559)
18.2.2 Fiat Money (pg. 561)
18.2.3 Interest on Currency (pg. 561)
18.2.4 Seigniorage (pg. 564)
18.3 Aggregate Shocks (pg. 566)
18.3.1 Recursive Equilibrium (pg. 566)
18.3.2 Krusell–Smith Method (pg. 567)
18.4 Uninsured Idiosyncratic Investment Risk (pg. 569)
18.5 Exercises (pg. 570)
19. Search and Matching Models of Unemployment (pg. 573)
19.1 A Basic DMP Model (pg. 574)
19.1.1 Steady State (pg. 577)
19.1.2 Transitional Dynamics (pg. 579)
19.1.3 Large Firms (pg. 581)
19.1.4 Efficiency (pg. 583)
19.2 Cyclical Volatilities of Unemployment and Vacancies (pg. 584)
19.3 Endogenous Job Destruction (pg. 587)
19.3.1 Steady State (pg. 590)
19.3.2 Transitional Dynamics (pg. 593)
19.4 Unemployment and Business Cycles (pg. 593)
19.4.1 Households (pg. 593)
19.4.2 Firms (pg. 595)
19.4.3 Nash Bargained Wages (pg. 597)
19.4.4 Equilibrium (pg. 598)
19.5 Exercises (pg. 598)
20. Dynamic New Keynesian Models (pg. 601)
20.1 A Basic DNK Model (pg. 601)
20.1.1 Households (pg. 602)
20.1.2 Final Goods Firms (pg. 603)
20.1.3 Intermediate Goods Firms (pg. 604)
20.1.4 Central Bank (pg. 606)
20.1.5 Sticky-Price Equilibrium (pg. 607)
20.1.6 Flexible-Price Equilibrium (pg. 607)
20.1.7 Log-Linearized System (pg. 608)
20.2 Monetary Policy Design (pg. 614)
20.2.1 Efficient Allocation (pg. 614)
20.2.2 Quadratic Approximation to Utility (pg. 616)
20.2.3 Commitment versus Discretion (pg. 620)
20.3 Fiscal Stimulus (pg. 624)
20.3.1 A Neoclassical Model (pg. 624)
20.3.2 Monopolistic Competition (pg. 625)
20.3.3 A DNK Model (pg. 627)
20.3.4 Zero-Interest-Rate Lower Bound (pg. 630)
20.3.5 Duration of Fiscal Stimulus (pg. 635)
20.3.6 Government Purchases and Welfare (pg. 636)
20.4 A Medium-Scale DSGE Model (pg. 640)
20.4.1 Households (pg. 641)
20.4.2 Firms (pg. 644)
20.4.3 Monetary and Fiscal Policies (pg. 646)
20.4.4 Aggregation and Equilibrium (pg. 646)
20.5 Exercises (pg. 647)
IV. Further Topics (pg. 651)
21. Recursive Utility (pg. 653)
21.1 Deterministic Case (pg. 654)
21.2 Stochastic Case (pg. 659)
21.3 Properties of Recursive Utility (pg. 675)
21.4 Portfolio Choice and Asset Pricing (pg. 680)
21.5 Pareto Optimality (pg. 698)
21.6 Exercises (pg. 703)
22. Dynamic Games (pg. 705)
22.1 Repeated Games (pg. 706)
22.1.1 Perfect Monitoring (pg. 706)
22.1.2 Equilibrium Payoff Set (pg. 708)
22.1.3 Computation (pg. 711)
22.1.4 Simple Strategies (pg. 712)
22.1.5 Imperfect Public Monitoring (pg. 713)
22.2 Dynamic Stochastic Games (pg. 716)
22.3 Application: The Great Fish War (pg. 718)
22.4 Credible Government Policies (pg. 720)
22.4.1 One-Period Economy (pg. 721)
22.4.2 Infinitely Repeated Economy (pg. 723)
22.4.3 Equilibrium Value Set (pg. 725)
22.4.4 Best and Worst SPE Values (pg. 727)
22.4.5 Recursive Strategies (pg. 729)
22.5 Exercises (pg. 731)
23. Recursive Contracts (pg. 733)
23.1 Limited Commitment (pg. 734)
23.1.1 A Dynamic Programming Method (pg. 735)
23.1.2 A Lagrangian Method (pg. 737)
23.1.3 An Alternative Characterization (pg. 738)
23.2 Hidden Action (pg. 739)
23.3 Hidden Information (pg. 745)
23.3.1 Characterizations (pg. 746)
23.3.2 Long-Run Poverty (pg. 751)
23.4 Exercises (pg. 752)
Mathematical Appendixes (pg. 755)
A: Linear Algebra (pg. 757)
B: Real and Functional Analysis (pg. 763)
C: Convex Analysis (pg. 771)
D: Measure and Probability Theory (pg. 779)
References (pg. 787)
Matlab Index (pg. 811)
Name Index (pg. 813)
Subject Index (pg. 819)

Jianjun Miao

Jianjun Miao is Professor of Economics at Boston University.

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