Accuracy of policy function approximations for strongly concave recursive problems

• Published date: 01 September 2019
• Author: Osvaldo Candido,Wilfredo L. Maldonado,Luis Felipe V. N. Pereira
• DOI: http://doi.org/10.1111/ijet.12171
• Date: 01 September 2019

doi: 10.1111/ijet.12171

Accuracy of policy function approximations for strongly

concave recursive problems

Wilfredo L. Maldonado,∗Osvaldo Candido∗and Luis Felipe V. N. Pereira∗

Underthe hypotheses of strong concavity of the aggregator function and concavity of the stochas-

tic operator which define the objectivefunction of the stochastic dynamic programming problem

(SDPP), we prove that the policy function approximationof the problem is a H ¨

older continuous

function with respect to the value function approximation. From this, explicit error bounds for

computation of the solution of such problems are provided. To illustrate the results we apply the

error control formula to the solution of two SDPPs with aggregator functions: the neoclassical

Ramsey economic growth model and the Lucas asset pricing model.

Key wor ds stochastic dynamic programming problem, strongly concave aggregator function,

error bounds of the policy function

JEL classification C61, D90

Accepted 23 October 2017

1 Introduction

The rejection of expenditure on consumption as a measure of satisfaction levels reachedin particular

periods motivated Koopmans(1960) to find other ways of modeling the agent’s utility.In a framework

where the bundle of commodities consumedin the first period should have an effect on the preference

between alternative sequences of bundles in the remainingfuture, Koopmans suggested the following

deterministic aggregate utility defined on infinite horizon consumption paths:

U(1x)=W(u1(x1),U

2(2x)),

where W(u1,U

2) is a continuous and increasing function called the aggregator function,u1(x1)is

the current utility or one-period utility (at time t=1) and U2(2x) is called prospective utility (as

from time t=2 on).1

These aggregator functions were studied by Lucas and Stokey (1984) for the bounded case,

and later by Boyd (1990) for the unbounded case.2The use of aggregator functions in stochastic

frameworks requires the definition of the stochastic aggregator Mthat might be different from the

classical mathematical expectation operator E. The inclusion of a concave stochastic aggregator in

∗Graduate School of Economics, Catholic Universityof Bras ´

ılia, Bras´

ılia, Brazil. Email: wilfredo@pos.ucb.br

This author would like to thank the CNPq of Brazil for financial support 303420/2012-0 and Edital Universal

470923/2012-1. The authors greatly appreciate the valuable comments and review of an anonymous referee. All

remaining errors are our responsibility.

1As in Koopmans, we denote 1x=(x1,x

2,x

3,...,x

t,...)=(x1,2x)=....

2Results regarding unbounded aggregators can be found in Duran (2000), Le Vanand Vailakis (2005), Rinc´

on-Zapatero

and Rodr´

ıguez-Palmero (2007) and Martins-da-Rocha and Vailakis (2010). Martins-da-Rocha and Vailakis(2013) dealt

with aggregators bounded from below.

International Journal of Economic Theory (2018) 1–19 © IAET 1

International Journal of Economic Theory

International Journal of Economic Theory 15 (2019) 249–267 © IAET 249

Accuracy of policy function approximations Wilfredo L. Maldonado et al.

the problem where Wis an additive function amplifies the risk aversion above the intertemporal

inelasticity. On the other hand, the convexity of Mimplies the opposite effect (see Epstein and

Zin 1989; Farmer 1990; Weil 1990). Marinacci and Montrucchio (2010) studied the existence and

uniqueness of solutions for general nonlinear stochastic equations defined by an aggregator function

Wand a stochastic aggregator M, deriving conditions for the classes of Thompson and Blackwell

aggregator functions.

Epstein and Zin (1991) used that kind of aggregator functions when dealing with consumption-

based capital asset pricing models. These utility functions are commonly used to obtain an explana-

tion for the equity premium puzzle.3Another use of recursive utilities with stochastic aggregators

was proposed by Epstein and Wang (1994) as a way of modeling Knightian uncertainty.

Despite the relevance and flexibility of working with aggregator functions, little attention is

given to the accuracy of the calculation of the optimal policy function in dynamic programming

problems with recursive aggregators. Ozaki and Streufert (1996) derived conditions for existence

of this optimum and provided a broad family of examples, illustrating with several applications in

economics. However, once the existence of this optimal policy function is proved, one still has to

deal with the problem of finding accurate approximations of it.

Considering the case of additiveagg regatorsand mathematical expectations as stochastic aggrega-

tors, it results in the classical stochastic dynamic programming problem(SDPP). In that case, asymp-

totic convergence results for the sequence of policy functions were obtained by Christiano (1990)

and Tauchen (1990), without numerical error bounds. Maldonado and Svaiter (2007) showed that

strong concavity of the one-period return function implies H¨

older continuity of the policy function

approximation on the value function approximation. Santos and Vigo-Aguiar (1998) demonstrated

analogous results under stronger conditions (interiority of solutions and twice differentiability of

the one-period return function). More recently, Li (2015) derived and applied the error bounds for

some examples of strongly concave problems with non-interior solutions. In the latter three works,

the error bounds in the optimal policy function depend on the error bound of the value function of

the problem, which can be calculated using the methodologies of Christiano (1990), Tauchen(1990)

or more recently those introduced by Stachurski (2009).

In this work we extend the results of Maldonado and Svaiter (2007) to the case of dynamic

programming problems defined from aggregator functions and stochastic aggregators. Namely, it is

proved that if the aggregator function is strongly concave and the stochastic aggregator is concave,

then the policy function approximation is H¨

older continuous in the value function approximation.

That result allows us to obtain accurate approximations to the optimal policy function of the prob-

lem. We illustrate the usage of the error bounds methodology through two examples where the

optimal policy function is accurately found: the Ramsey economic growth model and the Lucas asset

pricing model.

The paper is organized into five sections. Section 2 describes the framework and the hypothe-

ses that we will consider. For the sake of completeness, we provide a simple proof of the existence

and uniqueness of a fixed-point utility function for a bounded aggregator function and stochas-

tic aggregator as well as the validity of the Bellman equation for this case. Section 3 states the

main result of the paper, the H¨

older continuity of the policy function approximation on the value

function approximation. In Section 4 we provide two numerical examples in order to illustrate

the method and the usage of the error bounds. Section 5 concludes, and proofs are given in

the Appendix.

3See Mehra and Prescott (1985) for seminal work on the equity premium puzzle.

2International Journal of Economic Theory (2018) 1–19 © IAET

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250