Experimentation And Learning In Rational Addiction Models With Multiple Addictive Goods

The purpose of this paper is to explore and evaluate smooth approximation methods for value functions. These approximation methods are increasingly important in numerical dynamic programming since they allow researchers to solve models with a multitude of continuous state variables. In this paper we focus on a new approximation method which has been recently developed in the context of semi-nonparametric estimation by Coppejans (1999). The basic idea of this approach is to represent a function of several variables as superpositions of functions of one variable. The one-dimensional functions as well as the superpositions are represented as B-splines which have nice computational properties. This approach has two distinct advantages. First, it allows us to impose useful properties on the value function such as monotonicity and concavity. Second, and more importantly, it allows us to parameterize the value function by a fairly low dimensional object which alleviates the curse of dimensionality typically encountered in these type of problems. In order to evaluate this new method we compare it with more commonly used methods like Chebychev Polynomials. The comparison of the two methods is based on dynamic model of rational addiction under uncertainty. Orphanides Zervos (1995) argue that uncertainty and learning through experimentation need to be incorporated into the rational addiction framework in order to account for `involuntary'' addiction. We extend their simple model to allow for wealth accumulation as well as uncertainty in income and asset returns. This gives rise to rich dynamic model with five continuous state variables and hence provides a good model to test the two approximation algorithms of interest.