The Policy Function of a Discrete-Choice Problem is a Random Number Generator

This paper studies a life-cycle model in which the consumption good is assumed to be indivisible. This assumption requires the number of units of the good purchased in each period to be an integer. It is shown that, if the discount factor is sufficiently small, the policy function takes the form of a pseudo-random number generator or, more precisely, a linear congruential generator. It is also shown that optimal plans are almost always asymptotically non-periodic regardless of the discount factor. Various numerical examples of the value function and the policy function are provided