The Economics of Inaction: Stochastic Control Models with Fixed Costs

by Nancy L. Stokey

Overview

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.


In The 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.

Product Details

ISBN-13:	9780691135052
Publisher:	Princeton University Press
Publication date:	11/09/2008
Edition description:	New Edition
Pages:	320
Product dimensions:	6.20(w) x 9.30(h) x 1.10(d)
Age Range:	18 Years

About the Author

Nancy L. Stokey is the Frederick Henry Prince Professor of Economics at the University of Chicago. She is the author, with Robert Lucas and Edward Prescott, of Recursive Methods in Economic Dynamics.

Read an Excerpt

The Economics of Inaction

Stochastic Control Models with Fixed Costs

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By Nancy L. Stokey

Princeton University Press

Copyright © 2009 Princeton University Press
All right reserved.
ISBN: 978-0-691-13505-2

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Chapter One

Introduction

In situations where action entails a fixed cost, optimal policies involve doing nothing most of the time and exercising control only occasionally. Interest in economic models that exhibit this type of behavior has exploded in recent years, spurred by growing evidence that "lumpy" adjustment is typical in a number of important economic settings.

For example, the short-run effects of monetary policy are connected with the degree of price stickiness. Data from the U.S. Bureau of Labor Statistics on price changes at retail establishments for the period 1988-2003 suggest that price adjustment is sluggish, at least for some products. The average duration of a price varies greatly across different types of products, with some-gasoline, airfares, and produce-displaying frequent price changes, and others-taxi fares and many types of personal services-displaying much longer durations. The average duration across all products is about 5 months, but the range is wide, from 1.5 months at the 10th percentile to 15 months at the 90th percentile. The size distribution of adjustments is perhaps even more informative about price stickiness. Many adjustments are large, even after short-term sale prices have been removed. The average size is more than 8%. Moreover, 1988-2003 was a period of low inflation, so in the broad sample 45% of price changes are negative, and price cuts are almost as large as price increases. More detailed data for the largest three urban areas (New York, Los Angeles, and Chicago), show that about 30% of price changes are greater than 10% in magnitude, and the figure is about the same for increases and decreases.

The fact that many price changes are large suggests that there may be substantial fixed costs associated with changing a price. Otherwise it is difficult to explain why the changes are not carried out in series of smaller, more frequent increments.

Investment behavior provides a second example. Establishment-level data from the U.S. Census Bureau on 13,700 manufacturing plants for the period 1972-1988 show lumpy adjustment in two ways. First, more than half of the plants in the sample display an episode of large adjustment, at least 37%. In addition, a substantial fraction of aggregate investment, 25%, is concentrated in plants that are increasing their capital stock by more than 30%. At the other end of the distribution, over half of plants increase their capital stock by less than 2.5%, accounting for about 20% of total investment. Thus, for individual plants, changes in size come in substantial part from large one-year increases. From the aggregate point of view, a sizable share of total investment is devoted to these episodes of large increase. As with the price data, the evidence on investment suggests that fixed costs are important.

Evidence on job creation and destruction displays patterns similar to those in the investment data. The opening of a new establishment, the expansion of an old one, or the addition of a shift leads to concentrated job creation, while plant closings and mergers lead to concentrated destruction. Data from the U.S. Census Bureau on 300,000-400,000 manufacturing plants for the period 1972-88 show that two-thirds of total job creation and destruction occurs at plants where employment expands or contracts by more than 25% within a twelve-month period. One-quarter of job destruction arises from plant closings. Again, these adjustment patterns suggest that fixed costs are important.

For consumers, fixed costs are clearly important for durable goods such as housing and automobiles. U.S. Census Bureau data for 1996 show that among individuals aged 15 and older living in owner-occupied housing, the median tenure at the current residence is eight years, and about 40% have been at their current residence for more than eleven years. Although individuals have many motives for staying in their current residences, the substantial transaction costs-including time costs-involved in buying, selling, and moving clearly make frequent moves unattractive.

Evidence suggests that fixed costs are also important for explaining individual portfolio behavior. Absent fixed costs, it is hard to understand why so many households fail to participate in the stock market. The fact that lagged stock market participation increases the likelihood of current participation very strongly-by 32 percentage points, even after controlling for age, education, race, income, and other factors-strongly suggests that fixed entry costs are important. In addition, the fact that wealthier households are more likely both to own stocks and to trade suggests that fixed costs per period or per transaction are also important. In this context and others the costs of gathering and processing information are likely to be important components of the fixed cost.

These examples suggest that models using a representative firm and representative household may be inadequate for studying some questions. For example, the economic effects of an aggregate productivity shock or aggregate demand shock depend on the investment and hiring/firing response of firms. If fixed costs are important, describing the aggregate response requires averaging over a group of firms that are doing little or nothing and a group making substantial adjustments. Although it may happen that movements in the aggregate can be generated by a representative agent, it is difficult to confirm this-or to determine what the representative agent should look like-without explicitly carrying out the aggregation. Moreover, a representative agent that serves as an adequate proxy during periods of calm may be misleading during episodes when the economic environment becomes more turbulent. And on the household side, explicitly taking heterogeneity into account may have a substantial impact on conclusions about welfare.

In situations where fixed costs are important, continuous-time models in which the stochastic shocks follow a Brownian motion or some other diffusion have strong theoretical appeal. An optimal policy in this type of setting involves taking action when the state variable reaches or exceeds appropriately chosen upper and/or lower thresholds and doing nothing when the state lies inside the region defined by those thresholds. Continuous-time models permit a very sharp characterization of the thresholds that trigger an adjustment and the level(s) to which adjustment is made. Indeed, the thresholds and return points can often be characterized as the solution to a system of three or four equations in as many unknowns. The goal in this book is to develop the mathematical apparatus for analyzing models of this type. In the rest of this introduction the structure of typical models is briefly described and a few examples are discussed.

Suppose that in the absence of control the increments to a state variable X(t) are those of a Brownian motion. The (flow) return to the decision maker at any date, g(X(t)), depends on the current state, where the function g is continuous and single peaked. Suppose the peak is at x = a, so that g is increasing on (-[infinity], a) and decreasing on (a, +[infinity]). The decision maker can adjust the state by discrete amounts, and there is a fixed cost c associated with making any adjustment. For now suppose that the fixed cost is the only cost of adjustment. The decision maker's objective is to maximize the expected discounted value of returns net of adjustment costs, where future returns and costs are discounted at a constant interest rate r. The standard menu cost model has this structure, with X(t) interpreted as the firm's price (in log form) relative to an industrywide or economywide price index that fluctuates stochastically.

The problem for the decision maker is to balance two conflicting goals: maintaining a high return flow by keeping the state in the region around a and avoiding frequent payment of adjustment costs. An optimal policy in this setting has the following form: the decision maker chooses threshold values b, B for the points where control will be exercised, with b < a < B, and a return point S [member of] (b, B) to which the state is adjusted when control is exercised. As long as the state remains inside the open interval (b, B)-called the inaction region-the decision maker exercises no control: adjustment is not required. When the state falls to b or rises to B the fixed cost is paid and the state variable is adjusted to the return point S. If the initial state lies outside (b, B) an adjustment to S is made immediately. If the fixed cost c is sufficiently large relative to the range of the return function g, it is possible that b = -[infinity], B = +[infinity], or both.

Notice that in general an optimal policy does not involve returning to the point a where instantaneous returns are at a maximum. That is, in general S [not equal to] a. For example, if the drift is positive, ? > 0, the decision maker might choose S < a, anticipating that (on average) the state will rise. Or, if the return function g is asymmetric around a, he might skew the return point in the direction of higher returns.

Let v(x) denote the expected discounted return from following an optimal policy, given the initial state X(0) = x. The first step in finding the optimum is to formulate a Bellman equation involving the function v. That equation can then be used to characterize the optimal policy. Indeed, it can be used in two different ways.

Suppose that thresholds b and B have been selected and that the initial state X(0) = x lies between them, b < x < B. Let [E.sub.x] [?] and [Pr.sub.x] [?] denote expectations and probabilities conditional on the initial state x, and define the random variable T = T (b) [and] T (B) as the first time the process X(t) reaches b or B. T is an example of a stopping time. It is useful to think of v(x) as the sum of three terms:

v(x) = expected returns over [0, T) + expected returns over [T, +[infinity]) if b is reached before B + expected returns over [T, +[infinity]) if B is reached before b.

Let w(x, b, B) denote the first of these terms, the expected returns up to the stopping time T. The key to making this problem tractable is the fact that w(x, b, B), which is the expected value of an integral over time up to the stopping time T = T (b) [and] T(B), can be written as an integral over states in the interval [b, B]. Specifically,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the expected discounted local time function. It is a weighting function-like a density-for each state [xi] up to the stopping time T(b) [and] T(B), conditional on the initial state x. For the second and third terms it is useful to define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, [psi](x, b, B) is the expected discounted value, conditional on the initial state x, of an indicator function for the event of reaching the lower threshold b before the upper threshold B is reached. The value [psi](x, b, B) has a similar interpretation, with the roles of the thresholds reversed. For any r [greater than or equal to] 0, clearly [psi] and [psi] are bounded, taking values on the interval [0, 1]. With w, [psi], and [psi] so defined, the Principle of Optimality implies that v satisfies the Bellman equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

where the optimization is over the choice of the threshold values b and B and return point S.

If X is a Brownian motion or geometric Brownian motion, closed-form expressions can be derived for the functions [??], [pai], and [pai], and (1.1) provides a direct method for characterizing the optimal policy. If X is a more general diffusion, closed forms for [??], [psi], and [psi] are not available, but fairly sharp characterizations can often be obtained. In either case, several properties of the solution are worth noting. First, it is immediate from (1.1) that the optimal return point [S.sup.*] maximizes v(S) and does not depend on x. Second, the Principal of Optimality implies that the thresholds [b.sup.*] and [B.sup.*] do not depend on x. That is, if [b.sup.*], [B.sup.*] attain the maximum in (1.1) for any x [member of] ([b.sup.*], [B.sup.*]), then they attain the maximum for all x [member of] [b.sup.*], [B.sup.*]. The rational decision maker does not alter his choice of thresholds as the state variable evolves. Finally, it is immediate from (1.1) that the value function is a constant function outside the inaction region, v(x) = v([S.sup.*]) - c, for all x [not member of] ([b.sup.*], [B.sup.*]).

An alternative method for characterizing the optimum involves an indirect approach based on (1.1). Notice that if x lies inside the open interval ([b.sup.*], [B.sup.*]), then for [DELTA]t sufficiently small the probability of reaching either threshold can be made arbitrarily small. Hence it follows from (1.1) and the definition of v that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [DELTA]X is the (random) increment to the state over [DELTA]t. As will be shown more formally in Chapter 3, if X is a Brownian motion with parameters (?, [[sigma].sup.2]), then a second-order Taylor series approximation gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using this fact, rearranging terms, and letting [DELTA]t -> 0 gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

This equation, a second-order ordinary differential equation (ODE), is called the Hamilton-Jacobi-Bellman (HJB) equation. The optimal value function v satisfies this equation on the inaction region, the interval ([b.sup.*], [B.sup.*]). To complete the solution of a second-order ODE two boundary conditions are needed. In addition, for this problem the thresholds [b.sup.*] and [B.sup.*] must be determined. Recall that v is known outside the inaction region, v(x) = v(S) - c, all x [not member of] ([b.sup.*], [B.sup.*]). The two boundary conditions for (1.2) and the thresholds [b.sup.*] and [B.sup.*] are determined by requiring that v and v' be continuous at [b.sup.*] and [B.sup.*]. These conditions, called value matching and smooth pasting, reproduce the solution obtained by maximizing in (1.1).

This approach can be applied to a variety of problems. For inventory or investment models it is natural to assume that there are proportional costs of adjustment as well as fixed costs, and that both types of costs can be different for upward and downward adjustments. Even with these changes, however, the overall structure of the solution is similar to the one for the menu cost model. The main difference is that there are two return points, [s.sup.*] < [S.sup.*], where the former applies for upward adjustments, from [b.sup.*], and the latter for downward adjustments, from [B.sup.*].

(Continues...)

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Table of Contents

Preface ix

Chapter 1: Introduction 1
Notes 12

Part I: Mathematical Preliminaries 15

Chapter 2: Stochastic Processes, Brownian Motions, and Diffusions 17
2.1. Random Variables and Stochastic Processes 17
2.2. Independence 18
2.3. Wiener Processes and Brownian Motions 19
2.4. Random Walk Approximation of a Brownian Motion 20
2.5. Stopping Times 24
2.6. Strong Markov Property 24
2.7. Diffusions 25
2.8. Discrete Approximation of an Ornstein-Uhlenbeck Process 27
Notes 28

Chapter 3: Stochastic Integrals and Ito's Lemma 30
3.1. The Hamilton-Jacobi-Bellman Equation 31
3.2. Stochastic Integrals 34
3.3. Ito's Lemma 37
3.4. Geometric Brownian Motion 38
3.5. Occupancy Measure and Local Time 41
3.6. Tanaka's Formula 43
3.7. The Kolmogorov Backward Equation 47
3.8. The Kolmogorov Forward Equation 50
Notes 51

Chapter 4: Martingales 53
4.1. Definition and Examples 53
4.2. Martingales Based on Eigenvalues 57
4.3. The Wald Martingale 58
4.4. Sub- and Supermartingales 60
4.5. Optional Stopping Theorem 63
4.6. Optional Stopping Theorem, Extended 67
4.7. Martingale Convergence Theorem 70
Notes 74

Chapter 5: Useful Formulas for Brownian Motions 75
5.1. Stopping Times Defined by Thresholds 78
5.2. Expected Values for Wald Martingales 79
5.3. The Functions ψ and Ψ 82
5.4. ODEs for Brownian Motions 87
5.5. Solutions for Brownian Motions When r = 0 88
5.6. Solutions for Brownian Motions When r > 0 93
5.7. ODEs for Diffusions 98
5.8. Solutions for Diffusions When r = 0 98
5.9. Solutions for Diffusions When r > 0 102
Notes 106

Part II: Impulse Control Models 107

Chapter 6: Exercising an Option 109
6.1. The Deterministic Problem 110
6.2. The Stochastic Problem: A Direct Approach 116
6.3. Using the Hamilton-Jacobi-Bellman Equation 119
6.4. An Example 125
Notes 128

Chapter 7: Models with Fixed Costs 129
7.1. A Menu Cost Model 130
7.2. Preliminary Results 133
7.3. Optimizing: A Direct Approach 136
7.4. Using the Hamilton-Jacobi-Bellman Equation 140
7.5. Random Opportunities for Costless Adjustment 145
7.6. An Example 146
Notes 152

Chapter 8: Models with Fixed and Variable Costs 153
8.1. An Inventory Model 154
8.2. Preliminary Results 157
8.3. Optimizing: A Direct Approach 160
8.4. Using the Hamilton-Jacobi-Bellman Equation 162
8.5. Long-Run Averages 164
8.6. Examples 166
8.7. Strictly Convex Adjustment Costs 174
Notes 175

Chapter 9: Models with Continuous Control Variables 176
9.1. Housing and Portfolio Choice with No Transaction Cost 178
9.2. The Model with Transaction Costs 182
9.3. Using the Hamilton-Jacobi-Bellman Equation 184
9.4. Extensions 191
Notes 196

Part III: Instantaneous Control Models 197

Chapter 10: Regulated Brownian Motion 199
10.1. One- and Two-Sided Regulators 201
10.2. Discounted Values 205
10.3. The Stationary Distribution 212
10.4. An Inventory Example 218
Notes 224

Chapter 11: Investment: Linear and Convex Adjustment Costs 225
11.1. Investment with Linear Costs 227
11.2. Investment with Convex Adjustment Costs 232
11.3. Some Special Cases 236
11.4. Irreversible Investment 239
11.5. Irreversible Investment with Two Shocks 243
11.6. A Two-Sector Economy 247
Notes 248

Part IV: Aggregation 251

Chapter 12: An Aggregate Model with Fixed Costs 253
12.1. The Economic Environment 256
12.2. An Economy with Monetary Neutrality 259
12.3. An Economy with a Phillips Curve 261
12.4. Optimizing Behavior and the Phillips Curve 265
12.5. Motivating the Loss Function 278
Notes 280

A Continuous Stochastic Processes 283
A.1. Modes of Convergence 283
A.2. Continuous Stochastic Processes 285
A.3. Wiener Measure 287
A.4. Nondifferentiability of Sample Paths 288
Notes 289

B Optional Stopping Theorem 290
B.1. Stopping with a Uniform Bound, T ≤ N 290
B.2. Stopping with Pr {T < ∞} = 1 292
Notes 294

References 295
Part Index 303

What People are Saying About This

Prescott

Fixed adjustment costs are pervasive in economic modeling and until this book there was no place where the needed tools were developed in a way that was accessible to a broad group of economists. Now there is. This brilliantly lucid book is self-contained, first developing the mathematical preliminaries and then using the tools in a number of illustrative economic applications. I advise economists to add this book to their bookshelf.
— Edward C. Prescott, Nobel Laureate in Economics

Darrell Duffie

Nancy Stokey has given us a clear, elegant, and rigorous distillation of why and how we should delay action until the status of a decision problem changes enough. By combining the mathematical apparatus with a wealth of applications to production, macroeconomics, and other fields, this book immediately becomes the definitive treatment. It will be the stepping stone into the subject for almost every interested researcher.
— Darrell Duffie, Graduate School of Business, Stanford University

From the Publisher

"Fixed adjustment costs are pervasive in economic modeling and until this book there was no place where the needed tools were developed in a way that was accessible to a broad group of economists. Now there is. This brilliantly lucid book is self-contained, first developing the mathematical preliminaries and then using the tools in a number of illustrative economic applications. I advise economists to add this book to their bookshelf."—Edward C. Prescott, Nobel Laureate in Economics

"Stochastic control problems arise everywhere in modern economics. The Economics of Inaction gives a wonderful treatment for students and practitioners alike. It is rigorous yet clear, concise yet thorough. Inaction would not be the optimal decision about this book: read it now!"—Avinash Dixit, Princeton University

"Nancy Stokey has given us a clear, elegant, and rigorous distillation of why and how we should delay action until the status of a decision problem changes enough. By combining the mathematical apparatus with a wealth of applications to production, macroeconomics, and other fields, this book immediately becomes the definitive treatment. It will be the stepping stone into the subject for almost every interested researcher."—Darrell Duffie, Graduate School of Business, Stanford University

Avinash Dixit

Stochastic control problems arise everywhere in modern economics. The Economics of Inaction gives a wonderful treatment for students and practitioners alike. It is rigorous yet clear, concise yet thorough. Inaction would not be the optimal decision about this book: read it now!
— Avinash Dixit, Princeton University

Editorial Reviews

"The presentation of all these problems and solutions is impeccably precise, perfectly appropriate for textbook use in a taught course, and suitable for independent reading by readers with standard mathematical and economic background."—-Giuseppe Bertola, Journal of Economic Literature

From the Publisher

The presentation of all these problems and solutions is impeccably precise, perfectly appropriate for textbook use in a taught course, and suitable for independent reading by readers with standard mathematical and economic background.

Journal of Economic Literature - Giuseppe Bertola

The presentation of all these problems and solutions is impeccably precise, perfectly appropriate for textbook use in a taught course, and suitable for independent reading by readers with standard mathematical and economic background.
— Giuseppe Bertola

Journal of Economic Literature