The Economics of Inaction: Stochastic Control Models With Fixed Costs

by Stokey, Nancy L.

• ISBN13: 9781400829811
• ISBN10: 140082981X
• Copyright: 2009-03-01
• Publisher: Princeton Univ Pr

Summary

In economic situations where action entails a fixed cost, inaction is the norm. Action is taken infrequently, and adjustments are large when they occur. Interest in economic models that exhibit ''lumpy'' behavior of this kind has exploded in recent years, spurred by growing evidence that it is typical in many important economic decisions, including price setting, investment, hiring, durable goods purchases, and portfolio management. InThe Economics of Inaction, leading economist Nancy Stokey shows how the tools of stochastic control can be applied to dynamic problems of decision making under uncertainty when fixed costs are present. Stokey provides a self-contained, rigorous, and clear treatment of two types of models, impulse and instantaneous control. She presents the relevant results about Brownian motion and other diffusion processes, develops methods for analyzing each type of problem, and discusses applications to price setting, investment, and durable goods purchases. This authoritative book will be essential reading for graduate students and researchers in macroeconomics.

Table of Contents

Preface	p. ix
Introduction	p. 1
Notes	p. 12
Mathematical Preliminaries	p. 15
Stochastic Processes, Brownian Motions, and Diffusions	p. 17
Random Variables and Stochastic Processes	p. 17
Independence	p. 18
Wiener Processes and Brownian Motions	p. 19
Random Walk Approximation of a Brownian Motion	p. 20
Stopping Times	p. 24
Strong Markov Property	p. 24
Diffusions	p. 25
Discrete Approximation of an Ornstein-Uhlenbeck Process	p. 27
Notes	p. 28
Stochastic Integrals and Ito's Lemma	p. 30
The Hamilton-Jacobi-Bellman Equation	p. 31
Stochastic Integrals	p. 34
Ito's Lemma	p. 37
Geometric Brownian Motion	p. 38
Occupancy Measure and Local Time	p. 41
Tanaka's Formula	p. 43
The Kolmogorov Backward Equation	p. 47
The Kolmogorov Forward Equation	p. 50
Notes	p. 51
Martingales	p. 53
Definition and Examples	p. 53
Martingales Based on Eigenvalues	p. 57
The Wald Martingale	p. 58
Sub- and Supermartingales	p. 60
Optional Stopping Theorem	p. 63
Optional Stopping Theorem, Extended	p. 67
Martingale Convergence Theorem	p. 70
Notes	p. 74
Useful Formulas for Brownian Motions	p. 75
Stopping Times Defined by Thresholds	p. 78
Expected Values for Wald Martingales	p. 79
The Functions ¿ and ¿	p. 82
ODEs for Brownian Motions	p. 87
Solutions for Brownian Motions When r = 0	p. 88
Solutions for Brownian Motions When r > 0	p. 93
ODEs for Diffusions	p. 98
Solutions for Diffusions When r = 0	p. 98
Solutions for Diffusions When r > 0	p. 102
Notes	p. 106
Impulse Control Models	p. 107
Exercising an Option	p. 109
The Deterministic Problem	p. 110
The Stochastic Problem: A Direct Approach	p. 116
Using the Hamilton-Jacobi-Bellman Equation	p. 119
An Example	p. 125
Notes	p. 128
Models with Fixed Costs	p. 129
A Menu Cost Model	p. 130
Preliminary Results	p. 133
Optimizing: A Direct Approach	p. 136
Using the Hamilton-Jacobi-Bellman Equation	p. 140
Random Opportunities for Costless Adjustment	p. 145
An Example	p. 146
Notes	p. 152
Models with Fixed and Variable Costs	p. 153
An Inventory Model	p. 154
Preliminary Results	p. 157
Optimizing: A Direct Approach	p. 160
Using the Hamilton-Jacobi-Bellman Equation	p. 162
Long-Run Averages	p. 164
Examples	p. 166
Strictly Convex Adjustment Costs	p. 174
Notes	p. 175
Models with Continuous Control Variables	p. 176
Housing and Portfolio Choice with No Transaction Cost	p. 178
The Model with Transaction Costs	p. 182
Using the Hamilton-Jacobi-Bellman Equation	p. 184
Extensions	p. 191
Notes	p. 196
Instantaneous Control Models	p. 197
Regulated Brownian Motion	p. 199
One- and Two-Sided Regulators	p. 201
Discounted Values	p. 205
The Stationary Distribution	p. 212
An Inventory Example	p. 218
Notes	p. 224
Investment: Linear and Convex Adjustment Costs	p. 225
Investment with Linear Costs	p. 227
Investment with Convex Adjustment Costs	p. 232
Some Special Cases	p. 236
Irreversible Investment	p. 239
Irreversible Investment with Two Shocks	p. 243
A Two-Sector Economy	p. 247
Notes	p. 248
Aggregation	p. 251
An Aggregate Model with Fixed Costs	p. 253
The Economic Environment	p. 256
An Economy with Monetary Neutrality	p. 259
An Economy with a Phillips Curve	p. 261
Optimizing Behavior and the Phillips Curve	p. 265
Motivating the Loss Function	p. 278
Notes	p. 280
Continuous Stochastic Processes	p. 283
Modes of Convergence	p. 283
Continuous Stochastic Processes	p. 285
Wiener Measure	p. 287
Nondifferentiability of Sample Paths	p. 288
Notes	p. 289
Optional Stopping Theorem	p. 290
Stopping with a Uniform Bound, T ≤ N	p. 290
Stopping with Pr {T < ∞} = 1	p. 292
Notes	p. 294
References	p. 295
Part Index	p. 303
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