The Informational Efficiency of Finite Price Mechanisms

Abstract

This paper obtains finite counterparts of previous results that showed the informational efficiency of the Walrasian mechanism among all mechanisms yielding Pareto-optimal individually rational trades in an exchange economy while using a continuum of possible messages. Such mechanisms lack realism, since it is not possible to transmit or announce all points of a continuum, and it generally takes infinite time to find an equilibrium message, among all the messages in a continuum. Accordingly, the paper studies approximations of the continuum Walrasian mechanism, in which the number of messages is finite. It applies general results from a companion paper, which considered finite approximations of continuum mechanisms in general organizations, with exchange economies as a particular example. For classic exchange economies, we compare the continuum Walrasian mechanism with alternative continuum mechanisms that also find a Pareto-optimal and individually rational allocation. There are many of them, and some of them, like the continuum Direct Revelation mechanism, do not use prices at all. A finite approximation to a continuum mechanism will have an error. Its overall error for a given class of economies is the worst distance (over all members of the class) between the continuum mechanism's final allocation and the approximation's final allocation. We measure a finite mechanism's cost by the number of its (equilibrium) messages. We consider exchange economies in which traders' utility functions are quasi-linear and strictly concave. We find that the overall error of a sufficiently fine finite approximation of the Walrasian mechanism is arbitrarily close to the overall error of a not more costly approximation of an alternative continuum mechanism that has the same number of message variables. The former overall error is smaller than the latter if the alternative continuum mechanism has a larger number of message variables. A continuum Direct Revelation mechanism is an example of an alternative mechanism with a larger number of message variables than the Walrasian mechanism. Thus the informational superiority of the Walrasian mechanism emerges again when we approximate it and take the finite number of messages as our cost measure.