A Bayesian Approach to the Production of Information with a Linear Utility Function

Abstract

The paper considers the problem facing a consumer deciding whether to purchase a good whose effect on utility is unknown. The consumer is allowed to learn about the effect over time according to a Bayesian updating procedure. Of interest is the quantity of the good the consumer will purchase in the first period. The paper allows the consumer's budget constraint to be spread over more than one period and consequently generates results which contradicts earlier work. This paper is an example of the general approach to the production of information outlined in Grossman, Kihlstrom and Mirman (1977), hereafter referred to as GKM. They consider the decision problem facing a consumer operating in a two period environment when uncertainty is present in each period. The consumer's utility function contains a random variable and the consumer must choose a quantity of the good y, in the first period to maximize the discounted present value of expected utility. The consumer has a fixed level of income which must be exhausted in each period; any remaining income after purchasing the optimal y is spent on an alternative good x. The distribution function describing the random variable is adaptive in the sense that the objective values of the parameters in the distribution function are unknown, but the consumer's subjective beliefs about these parameters are updated over time according to Bayesian rules. The consumer makes a decision in the first period Yl, and consequently observes the realisation of a number of random variables. That is, the consumer observes a random sample and can infer from this sample the likelihood that it came from a particular distribution. The size of the sample is given by Yl. In making a decision Yl, not only