INSEAD Library

Doriot: for 2012-2013 courses

Includes bibliographical references and index

Digitized

Dynamic General Equilibrium Modeling Computational Methods and Applications Table of Contents Preface ........................................................................................... VII List of Figures .......................................................................... XXIII List of Symbols........................................................................ XXVII List of Programs....................................................................... XXIX Part I. Representative Agent Models 1 Basic Models ................................................................................ 3 1.1 The Deterministic Finite-Horizon Ramsey Model and Non-Linear Programming ........................................... 4 1.1.1 The Ramsey Problem ................................................ 4 1.1.2 The Kuhn-Tucker Theorem ...................................... 6 1.2 The Deterministic Infinite-Horizon Ramsey Model and Dynamic Programming ............................................... 9 1.2.1 Recursive Utility ....................................................... 9 1.2.2 Euler Equations........................................................ 11 1.2.3 Dynamic Programming............................................ 13 1.2.4 The Saddle Path ...................................................... 16 1.2.5 Models with Analytic Solution ............................... 21 1.3 The Stochastic Ramsey Model ........................................... 25 1.3.1 Stochastic Output .................................................... 25 1.3.2 Stochastic Euler Equations ..................................... 28 1.3.3 Stochastic Dynamic Programming .......................... 30 1.4 Labor Supply, Growth, and the Decentralized Economy .......................................................................... 33 1.4.1 Substitution of Leisure ............................................ 33 1.4.2 Growth and Restrictions on Technology and Preferences ............................................................ 34 1.4.3 The Decentralized Economy .................................... 40 1.5 Model Calibration and Evaluation ........................................ 44 1.5.1 The Benchmark Model.............................................. 44 1.5.2 Calibration ................................................................ 46 1.5.3 Model Evaluation ...................................................... 51 1.6 Numerical Solution Methods ............................................... 59 1.6.1 Characterization ........................................................ 59 1.6.2 Accuracy of Solutions ................................................ 61 Appendices ................................................................................ 65 A.1 Solution to Example 1.2.1 ................................................... 65 A.2 Restrictions on Technology and Preferences ....................... 67 Problems .................................................................................... 72 2 Perturbation Methods ................................................................. 75 2.1 Linear Solutions for Deterministic Models .......................... 77 2.2 The. Stochastic Linear Quadratic Model .............................. 84 2.3 LQ Approximation................................................................ 89 2.3.1 A Warning ................................................................. 89 2.3.2 An Illustrative Example............................................. 91 2.3.3 The General Method.................................................. 95 2.4 Linear Approximation .......................................................... 98 2.4.1 An Illustrative Example............................................. 99 2.4.2 The General Method................................................ 106 2.5 Quadratic Approximation .................................................. 114 2.5.1 Introduction ............................................................. 114 2.5.2 The Deterministic Growth Model ............................ 116 2.5.3 The Stochastic Growth Model ................................. 118 2.5.4 Generalization ......................................................... 124 2.6 Applications ....................................................................... 131 2:6.1 The Benchmark Model ............................................ 131 2.6.2 Time to Build .......................................................... 138 2.6.3 New Keynesian Phillips Curve ............................... 143 Appendices .............................................................................. 158 A.3 Solution of the Stochastic LQ Problem .............................. 158 A.4 Derivation of the Log-Linear Model of the New Keynesian Phillips Curve ................................................ 160 Problems .................................................................................. 169 3 Deterministic Extended Path .................................................... 175 3.1 Solution of Deterministic Models........................................ 176 3.1.1 Finite-Horizon Models ............................................ 176 3.1.2 Infinite-Horizon Models .......................................... 179 3.2 Solution of Stochastic Models............................................. 181 3.2.1 An Illustrative Example........................................... 182 3.2.2 The Algorithm in General ....................................... 184 3.3 Further Applications ........................................................... 186 3.3.1 The Benchmark Model............................................. 186 3.3.2 A Small Open Economy........................................... 189 Problems .................................................................................. 200 4 Discrete State Space Methods.................................................... 207 4.1 Solution of Deterministic Models........................................ 208 4.2 Solution of Stochastic Models............................................. 221 4.3 Further Applications ........................................................... 232 4.3.1 Non-Negative Investment........................................ 232 4.3.2 The Benchmark Model............................................. 235 Problems ................................................................................... 238 5 Parameterized Expectations......................................................... 243 5.1 Characterization of Approximate Solutions ....................... 244 5.1.1 An Illustrative Example........................................... 244 5.1.2 A General Framework.............................................. 247 5.1.3 Adaptive Learning.................................................... 249 5.2 Computation of the Approximate Solution ........................ 252 5.2.1 Choice of T and W .................................................... 252 5.2.2 Iterative Computation of the Fixed Point ................ 254 5.2.3 Direct Computation of the Fixed Point.................... 255 5.2.4 Starting Points.......................................................... 257 5.3 Applications ........................................................................ 259 5.3.1 Stochastic Growth with Non-Negative Investment ............................................................. 259 5.3.2 The Benchmark Model............................................. 265 5.3.3 Limited Participation Model of Money ................... 268 Problems .................................................................................. 281 6 Projection Methods.................................................................... 285 6.1 Characterization of Projection Methods............................. 286 6.1.1 An Example.............................................................. 286 6.1.2 The General Framework.......................................... 289 6.1.3 Relation to Parameterized Expectations ................. 291 6.2 The Building Blocks of Projection Methods ...................... 293 6.2.1 Approximating Function ......................................... 293 6.2.2 Residual Function ................................................... 294 6.2.3 Projection and Solution ........................................... 295 6.2.4 Accuracy of Solution............................................... 297 6.3 Applications ........................................................................ 297 6.3.1 The Deterministic Growth Model............................ 298 6.3.2 The Stochastic Growth Model with Non-Negative Investment...................................... 302 6.3.3 The Benchmark Model............................................. 309 6.3.4 The Equity Premium Puzzle ................................... 311 Problems .................................................................................. 324 Part II. Heterogeneous Agent Models 7 Computation of Stationary Distributions................................ 329 7.1 A Simple Heterogeneous-Agent Model with Aggregate Certainty.......................................................... 331 7.2 The Stationary Equilibrium of a Heterogeneous-Agent Economy ...................................... 338 7.3 Applications ........................................................................ 359 7.3.1 The Risk-Free Rate in Economies with Heterogeneous Agents and Incomplete Insurance .............................................................. 359 7.3.2 Heterogeneous Productivity and Income Distribution ........................................................... 367 Problems .................................................................................. 384 8 Dynamics of the Distribution Function.................................... 389 8.1 Introduction ........................................................................ 390 8.2 Transition Dynamics .......................................................... 393 8.2.1 Partial Information................................................... 395 8.2.2 Guessing a Finite Time Path for the Factor Prices...................................................................... 406 8.3 Aggregate Uncertainty........................................................ 411 8.4 Applications ........................................................................ 421 8.4.1 Costs of Business Cycles with Liquidity Constraints and Indivisibilities .............................. 422 8.4.2 Business Cycle Dynamics of the Income Distribution ........................................................... 431 8.5 Epilogue............................................................................... 446 Problems .................................................................................. 449 9 Deterministic Overlapping Generations Models ........................ 9.1 The Steady State ................................................................. 9.1.1 An Illustrative Example........................................... 9.1.2 Computation of the Steady State ............................ 9.2 The Transition Path............................................................. 9.2.1 A Stylized 6-Period Model...................................... 9.2.2 Computation of the Transition Path ........................ 9.3 Application: The Demographic Transition ........................ 9.3.1 The Model ............................................................... 9.3.2 Computation ............................................................ 9.3.3 Results ..................................................................... Problems .................................................................................. 451 453 454 458 469 471 473 482 483 489 499 502 10 Stochastic Overlapping Generations Models......................... 507 10.1 Individual Uncertainty....................................................... 507 10.2 Aggregate Uncertainty...................................................... 520 10.2.1 Log-Linearization ................................................. 522 10.2.2 The Algorithm of Krusell and Smith in Overlapping Generations Models ......................... 533 Appendix.................................................................................. 549 A.5 Parameters of the AR(1)-Process with Annual Periods .............................................................................. 549 Problems .................................................................................. 551 Part III. Tools 11 Numerical Methods.................................................................. 555 11.1 A Quick Refresher in Linear Algebra .............................. 555 11.1.1 Complex Numbers................................................. 555 11.1.2 Vectors .................................................................. 557 11.1.3 Norms .................................................................... 558 11.1.4 Linear Independence.............................................. 558 11.1.5 Matrices ................................................................. 558 11.1.6 Linear and Quadratic Forms ................................. 563 11.1.7 Eigenvalues and Eigenvectors .............................. 564 11.1.8 Matrix Factorization.............................................. 565 11.1.9 Givens Rotation..................................................... 570 11.2 Function Approximation .................................................. 570 11.2.1 Taylor's Theorem .................................................. 571 11.2.2 Implicit Function Theorem ................................... 574 11.2.3 Linear Interpolation .............................................. 575 11.2.4 Cubic Splines ........................................................ 577 11.2.5 Families of Polynomials ....................................... 578 11.2.6 Chebyshev Polynomials ........................................ 581 11.2.7 Multidimensional Approximation ......................... 588 11.2.8 Neural Networks ................................................... 591 11.3 Numerical Differentiation and Integration ....................... 593 11.3.1 Differentiation ....................................................... 593 11.3.2 Numerical Integration............................................ 598 11.4 Stopping Criteria for Iterative Algorithms ....................... 603 11.5 Non-Linear Equations ...................................................... 606 11.5.1 Single Equations ................................................... 607 11.5.2 Multiple Equations ................................................ 610 11.6 Numerical Optimization ................................................... 622 11.6.1 Golden Section Search .......................................... 623 11.6.2 Gauss-Newton Met ho d........................................ 626 11.6.3 Quasi-Newton ....................................................... 630 11.6.4 Genetic Algorithms................................................ 633 12 Various Other Tools ................................................................ 645 12.1 Difference Equations ........................................................ 645 12.1.1 Linear Difference Equations.................................. 645 12.1.2 Non-Linear Difference Equations ......................... 648 12.2 Markov Processes ............................................................. 651 12.3 DM-Statistic ...................................................................... 658 12.4 The HP-Filter .................................................................... 663 References ...................................................................................... 667 Name Index .................................................................................... 685 Subject Index.................................................................................. 691