Information-Gathering and Producing: Complements or Substitutes?

Abstract

We aim at some simple theoretical underpinnings for the study of a complex empirical question studied by labor economists and others: does Information-technology improvement lead to occupational shifts -- toward workers and away from other occupations -- and to changes in the productivity of non-information workers? In our simple model there is a Producer, whose payoff depends on a production quantity and an unknown state of the world, and an Information-gatherer (IG) who expends effort to learn more about the unknown state. The IG's effort yields a signal which is conveyed to the Producer. The Producer uses the signal to revise prior beliefs about the state and uses the posterior to make an expected-payoff-maximizing quantity choice. Our central aim is to find conditions on the IG and the Producer under which more IG effort leads to a larger average production quantity (Complements) and conditions under which it leads to a smaller average quantity (Substitutes). For each of the IG's possible efforts there is an information structure, which specifies a signal distribution for every state and (for a given prior) a posterior state distribution for every signal. We start by considering a Blackwell IG. For such an IG, the possible structures can be ranked so that a higher-ranking structure is more useful to every Producer, no matter what the prior and the payoff function may be. For the Blackwell IGs whom we consider, a higher-ranking structure is reasonably interpreted as a higher-effort structure. The Blackwell theorems state that one structure ranks above another if and only if the expected value (over the possible signals) of any convex function on the posteriors is not less for the higher-ranked structure. So we have Complements (Substitutes) if the Producer's best quantity is indeed a convex (concave) function of the posteriors. That gives us Complements/Substitutes results for a variety of Producers. We then turn to a non-Blackwell IG who partitions the state set into n equal-probability intervals. The IG can choose any positive integer n and n is the effort measure. We recapture some of the results from the Blackwell-IG case, but far different techniques are needed, since the Blackwell theorems cannot be used.