Forward-Looking Decision Making: Dynamic Programming Models Applied to Health, Risk, Employment, and Financial Stability: 3 (The Gorman Lectures in Economics) - Hardcover

Hall, Robert E.

Hardcover

ISBN 10: 0691142424 ISBN 13: 9780691142425

Verlag: Princeton University Press, 2010

Inhaltsangabe

Individuals and families make key decisions that impact many aspects of financial stability and determine the future of the economy. These decisions involve balancing current sacrifice against future benefits. People have to decide how much to invest in health care, exercise, their diet, and insurance. They must decide how much debt to take on, and how much to save. And they make choices about jobs that determine employment and unemployment levels. Forward-Looking Decision Making is about modeling this individual or family-based decision making using an optimizing dynamic programming model.


Robert Hall first reviews ideas about dynamic programs and introduces new ideas about numerical solutions and the representation of solved models as Markov processes. He surveys recent research on the parameters of preferences--the intertemporal elasticity of substitution, the Frisch elasticity of labor supply, and the Frisch cross-elasticity. He then examines dynamic programming models applied to health spending, long-term care insurance, employment, entrepreneurial risk-taking, and consumer debt.


Linking theory with data and applying them to real-world problems, Forward-Looking Decision Making uses dynamic optimization programming models to shed light on individual behaviors and their economic implications.

Über diesen Titel

Über die Autorin bzw. den Autor

Robert E. Hall is the Robert and Carole McNeil Professor of Economics and Hoover Senior Fellow at Stanford University. He is the author of "The Rational Consumer, Booms and Recessions in a Noisy Economy", and "Digital Dealing".

Von der hinteren Coverseite

"Forward-Looking Decision Making provides interesting applications of the dynamic programming approach for analyzing individual decisions that balance current and future welfare. The subjects are timely and the book contains a good selection of topics, united by a common analytical theme."--John Ermisch, University of Essex

Aus dem Klappentext

"Forward-Looking Decision Making provides interesting applications of the dynamic programming approach for analyzing individual decisions that balance current and future welfare. The subjects are timely and the book contains a good selection of topics, united by a common analytical theme."--John Ermisch, University of Essex

Auszug. © Genehmigter Nachdruck. Alle Rechte vorbehalten.

Forward-Looking Decision Making

Dynamic-Programming Models Applied to Health, Risk, Employment, and Financial StabilityBy Robert E. Hall

Princeton University Press

Copyright © 2010 Princeton University Press
All right reserved.
ISBN: 978-0-691-14242-5

Contents

Foreword..................................................................viiPreface...................................................................ix1 Basic Analysis of Forward-Looking Decision Making.......................12 Research on Properties of Preferences...................................103 Health..................................................................234 Insurance...............................................................425 Employment..............................................................506 Idiosyncratic Risk......................................................707 Financial Stability with Government-Guaranteed Debt.....................87References................................................................119Index.....................................................................123

Chapter One

Basic Analysis of Forward-Looking Decision Making

Individuals and families make the key decisions that determine the future of the economy. The decisions involve balancing current sacrifice against future benefits. People decide how much to invest in health care, exercise, and good diet, and so determine their longevity and future satisfaction. They make choices about jobs that determine employment and unemployment levels. Their investment decisions are at the heart of some issues of financial stability.

1.1 The Dynamic Program

Economists have gravitated to the dynamic program as the workhorse model of the way that people balance the present against the future. The dynamic program is one of the two tools economists tend to reach for when solving problems of optimization over time. The other is the Euler equation. The two approaches are perfectly equivalent: if a problem is susceptible to solution by a dynamic program, it is susceptible to an Euler equation solution, and vice versa. The class of models suited to either method have the property that the trade-off between this year and next year— the marginal rate of substitution—depends only on consumption this year and next year. Utility functions with this property are additively separable. They have the form,

[summation over (t)][β^tu(x_t), (1.1)

where x_t is a vector of things the family cares about.

In words, the idea of a dynamic program is to summarize the entire future in a value function, which shows how much lifetime utility the family will enjoy starting next year based on the resources for future spending left after this year. People makes choices about this year by balancing the immediate marginal benefits from using resources this year against the marginal value of the remaining resources as of next year, according to the value function. A dynamic program uses backward recursion. Start with a known or assumed value function for some distant year. Find the value function in the previous year by solving the year-to-year balancing problem for all possible levels of resources in that year. Keep moving backward in time until you reach a year from now. Finally, solve this year's optimization by taking actual resources and solving the balancing problem for this year's utility and next year's value function. The dynamic-programming approach is conceptually simple and numerically robust. A famous book by Stokey and Lucas (1989) helped persuade economists of the virtues of dynamic programming for recursive problems.

The Euler equation approach considers a possible choice this year and then uses the marginal rate of substitution to map it into a choice next year. Apply this as a forward recursion, to see where the immediate choice leads to at some distant future date. Again, as in the dynamic program, you need to have some concept of a distant target—for example, the family cannot be deeply in debt at that time. Keep trying out initial choices to find the one where the family meets its distant target. Though this approach has some appeal in explaining dynamic models, it fails completely as a way to solve models numerically. The Euler equation is numerically unstable. Good methods exist for dealing with the instability, but are rarely used in modern dynamic economics because the dynamic program approach works so well. The instability of the Euler equation creates conceptual confusion as well, because one might fall into the trap of thinking that the aberrant paths that do not reach the target might describe actual behavior.

All the models in this book rest on dynamic programming. At the beginning of the year, the family inherits some state variables as a result of choices made last year and events during that year. These could be wealth, health status, employment status, or debt. The family picks values of choice variables: consumption, health spending, job search, or borrowing. A law of motion shows how the inherited values of the state variables, the values of the choice variables, and events during the year map into next year's values of the state variables. The events include financial returns, health outcomes, job loss, and fluctuations in earnings.

The Bellman equation describes the condition for recursive optimization:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

Here V(s) is the value function, which assigns lifetime expected utility based on the state variables, s. The choice variables are in the vector x_t. The function u(s, x) is the period utility that describes the flow of satisfaction that the family receives when choosing x given s. The general law of motion is

s_t+1 = f_t(s_t, x_t, ε_t) (1.3)

Here ε_t describes the random events that occur during year t. Anything about year t that is known in advance can be built into the function f_t.

As a simple example, consider the standard lifecycle saving model. The single state variable is wealth, W_t. The only choice variable is current consumption, c_t. The Bellman equation is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

and the law of motion for wealth is

W_t+1 = (1 + r_t)(W_t - c_t + y_t + ε_t) (1.5)

Here r_t is the known, time-varying return to savings, y_t is the known part of income, and ε_t is the random part of income. Although it would be easy to write down the first-order condition for the maximization in the Bellman equation, the condition usually does not add much to understanding. The best approach is often just to leave the maximization to software (Matlab).

As I noted earlier, to solve for the family's optimal choice this year, x_t, start with a known value function in some distant year T, V_T (s_T), iterate backwards to find V_t+1, and then solve the Bellman equation for the optimal x_t. Sometimes (but never in this book), the value functions have known functional forms. For the great majority of interesting problems, the functions need to be represented as well-chosen approximations. Judd (1998) discusses this topic at an advanced level. The state variables may be discrete or continuous. Discrete variables might tell whether a person was employed or unemployed, or well or sick. Discrete variables may also describe random outside events, even some, such as financial returns, that might be considered continuous. Endogenous state variables such as wealth, that result from choices that are continuous, need to be treated as continuous. One should avoid the temptation to convert them to discrete variables, as the resulting approximation is hard to manage and gives unreliable results.

Handling discrete state variables is straightforward: just subscript the value function by the discrete state. Thus the Bellman equation when s_t is discrete is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

1.2 Approximation

Many interesting models have only a single continuous state variable, including all the models in this book. A useful family of approximations in that case is a weighted sum of known functions of s. Let φ_i(s) denote these known functions—typically there are several hundred of them. It is convenient to normalize them so that they equal 1 at a give value s_i and 0 at the points corresponding to other of the functions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

The approximation is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)

Under the normalization of the φs, the value V_i,t is the value of V_t(s) at s = s_i. The approximation interpolates between the V_i,t points for intermediate values of s. The point defined by an s_i and V_i pair is called a knot.

A convenient choice for the interpolation functions is a tent:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)

The function V_t(s) is then the linear interpolation between the knots.

Given a set of interpolation functions, the backward recursion to find the current knot values is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.10)

For the models in this book, with twenty years of backward recursion and 500 knots in the approximation to the value function, a standard PC, vintage 2008, takes around half a minute to calculate the value functions.

When calculating the value functions, it is usually a good idea to store away the choice functions, represented as values x_i,t of the optimal choice at time t given state variable value s_i (these are also called policy functions).

1.3 Stationary Case

Sometimes the stationary value function is interesting. Suppose that the decision maker is embedded in an unchanging environment with the random s drawn from an unchanging distribution. Suppose further that the horizon is infinite. Then the value function becomes stationary, in the sense that it loses its time subscript. The stationary approximating Bellman equation is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.11)

To find the values V_i that solve this equation, we can treat the Bellman equation as a big system of nonlinear equations to be solved for the unknowns, V_i. This method is called projection and can be tricky, but when it works it is usually fast. An alternative, foolproof method is value function iteration. Start with arbitrary Vi, put them on the right-hand side of the Bellman equation, and calculate a new set. Iterate this operation to convergence, which is guaranteed (see Judd 1998, p. 412). This approach can be slow for bigger problems (Judd discusses a number of ways to speed it up).

1.4 Markov Representation

The solution to the dynamic program describes the way a decision maker responds to an uncertain environment. From the stochastic driving force , the model generates the decision maker's stochastic response, in the sense of the joint distribution of the endogenous variables of the model. Most researchers describe the joint distribution by simulation. They start at given values of the state variables s, evaluate the choice functions x(s), draw a random from the appropriate distribution, compute the new state vector from the law of motion, and continue for many simulated years. But simulation is extremely inefficient: to drive down the sampling errors from simulation, which cause the joint distribution of the simulated data to differ from the true joint distribution generated by the model, to acceptable low values, you may need to simulate for days. Some (but not all) of the aspects of the joint distribution are available with essentially perfect accuracy by direct calculation rather than simulation.

A recursive model is a Markov process. For given current values of the state variables s, the choice functions and the law of motion generate a probability distribution across states in the coming year. If the model is stationary, the Markov process has constant probabilities; otherwise, they vary with time. A Markov process is fully defined by its transition matrix. Interesting aspects of the joint distribution can be calculated by standard matrix operations applied to the matrix. For example, transition probabilities over more than one year are powers of the transition matrix and the stationary probabilities of a stationary model can be calculated in no time by matrix inversion.

For a continuous state variable, the true transition matrix is infinitely big, so again we need to use an approximation. I treat the model as assuming that the state variables originate from only the grid of points used in the earlier approximation, s_i. Then I calculate the transition probability from state s_i this year to s'_i next year as the probability that a person starting from the exact point s_i this year winds up in an interval containing s_i' next year. The interval runs from halfway between s_i'-1 and s_i' to halfway between s_i' and s_i'+1. I denote the transition probability as Π_i,i' and calculate it as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.12)

Solve the linear system πΠ = π and [summation]_iπ_i = 1 to find stationary probabilities π_i.

1.5 Distribution of the Stochastic Driving Force

The calculation of the Bellman equation requires the evaluation of an expectation over the distribution of the random ε. One could imagine assuming a continuous distribution of the disturbance with a known functional form and performing an analytic or numerical integration to form the expectation. But it is rare to know that the disturbance has a particular functional form and often impossible to do the integration analytically and challenging to do it numerically. It is usually better to use a purely empirical distribution. For example, if the disturbance is productivity, one can take 50 realizations of actual productivity. The integration is replaced by adding up the value function at 50 values and dividing by 50. Chapter 6 takes this approach with 14,000 values of startup companies.

Chapter Two

Research on Properties of Preferences

The studies in this book use information about preferences from research on individual behavior. Consider the standard intertemporal consumption–hours problem without unemployment,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

subject to the budget constraint,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

Here R_tτ, is the price at time t of a unit of goods delivered at time t + τ.

I let c(λ, λw) be the Frisch consumption demand and h(λ, λw) be the Frisch supply of hours per worker. See Browning et al. (1985) for a complete discussion of Frisch systems in general. The functions satisfy

U_c(c(λ, λw), h(λ, λw)) = λ (2.3)

and

U_h(c(λ, λw), h(λ, λw)) = λw. (2.4)

Here λ is the Lagrange multiplier for the budget constraint.

The Frisch functions have symmetric cross-price responses: c₂ = -h₁. They have three basic first-order or slope properties:

• Intertemporal substitution in consumption, c₁(λ, λw), the response of consumption to changes in its price.

• Frisch labor-supply response, h₂(λ, λw), the response of hours to changes in the wage.

• Consumption–hours cross-effect, c2(λ, λw), the response of consumption to changes in the wage (and the negative of the response of hours to the consumption price). The expected property is that the cross-effect is positive, implying substitutability between consumption and hours of nonwork or complementarity between consumption and hours of work.

(Continues...)

Excerpted from Forward-Looking Decision Makingby Robert E. Hall Copyright © 2010 by Princeton University Press. Excerpted by permission of Princeton University Press. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 2010
• Sprache: Englisch
• ISBN 10: 0691142424
• ISBN 13: 9780691142425
• Einband: Tapa dura
• Anzahl der Seiten: 146
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Nicht verfügbar