On Approximating the Gradient of the Value Function

Alexandre Dmitriev

doi:10.4236/tel.2019.91011

Abstract

The optimality conditions for macroeconomic problems with limited commitment often contain partial derivatives of the optimal value function, corresponding to the outside option. This paper contributes to the literature on recursive contracts by proposing an algorithm for approximating the gradient of the value function using simulation-based methods. Our method combines numerical solution and simulation of the model, Monte-Carlo integration and numerical differentiation. It does not suffer from the curse of dimensionality and is therefore convenient for models involving many state variables. The algorithm inherits the speed and accuracy limitations of the numerical solution method it relies on. Our accuracy analysis is limited to a few classical examples from macroeconomic literature.

Highlights

This paper contributes to the literature on recursive contracts by proposing an algorithm for approximating the gradient of the value function using simulation-based methods
The purpose of this paper is to propose a simple algorithm for computing partial derivatives of the optimal value function
Often the optimality conditions for this class of problems involve partial derivatives with respect to endogenous state variables of the optimal value function corresponding to the dynamic programming formulation of an outside option

Summary

Introduction

The purpose of this paper is to propose a simple algorithm for computing partial derivatives of the optimal value function. To circumvent the problem of finding the values of the derivatives in [3], the authors proposed a method based on the ideas of Benveniste and Scheinkman [4] Their method has limited applicability since it depends on the availability of an analytical solution for the derivatives as conditional expectations of the known functions of the model solution. The initial step of our algorithm involves obtaining numerical solution to a problem using a procedure which satisfies three criteria It approximates some unknown function with flexible functional forms of finite elements.

Dmitriev DOI

The Algorithm

Implementation of the Algorithm

Numerical Accuracy: A Comparison

Concluding Remarks