Consumer theory for cheap information

Abstract

In many settings, a decision-maker facing uncertainty must decide not only how much information to purchase, but also from which sources. It is already known that when uncertainty is about a binary variable—there are two possible “states of the world”—a decision-maker with a sufficiently large budget will spend their entire budget on signals from a single source: the most precise one. When there are more than two possible states of the world, however, the story is different: different information sources may differ in the pair of states they are worst at distinguishing, and the decision-maker may therefore purchase information from different sources which “cover for each others’ weaknesses.” I characterize tradeoffs between samples from different sources in a setting with low costs (or large budgets), where the probability of a mistake is small and well-described by large deviations theory. I show that, at large samples, the marginal rate of substitution samples from distinct information sources is approximately given by the ratio of precision-like indices for each source, and that the asymptotically optimal bundle satisfies a maxi-min rule: it maximizes the precision per dollar for the worst case pair of states. Since the precision of each signal does not depend on the characteristics of the decision-maker (their prior beliefs or the payoffs of the decision problem they face), all decision-makers agree on the fraction of signals from each source in an asymptotically optimal bundle. Different information sources with the same worst-case pair of states—the two states the signal is least precisely able to differentiate—are redundant, and are never demanded together; the asymptotically optimal bundle occurs either at a corner, or at one of a finite number of interior “kink points” where the worst-case pair of states switches. To illustrate these results, I consider a number of basic consumer theory exercises and discuss implications for information demand.

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Gary Baker

I’m an economic theory grad student at UW–Madison. I am currently (Spring 2021) on the job market.