An Approximate Law of One Price in Random Assignment Games

Abstract

The "law of one price" asserts that homogeneous goods must sell for the same price across locations and vendors. While many deviations from this 'law' have been observed in the real world, it remains a useful building block in economic theory, and serves as a benchmark for empirical studies. A crucial underlying assumption used in arguing for the validity of the law is the homogeneity of goods and buyers: buyers do not care which of the goods they buy, or which seller they are buying it from, nor do sellers care about the identity of the buyers. In other words, any two instances of the good are perfect substitutes for the buyers, as are any two buyers from any seller's point of view. This paper makes the formal claim that even in the presence of heterogeneous preferences, an approximate version of the law remains valid, and the approximation improves as the market grows large. To prove this result we use the assignment game model of Shapley and Shubik (1971) in which there is a finite set of firms and a finite set of workers, and each firm is looking to hire exactly one worker in exchange for a negotiable salary. Each firm has a (possibly different) value for hiring each of the workers, and each worker has a (possibly different) reservation value for working for each of the firms, and utilities are assumed to be linear in money. We study three different random models which generate the value: In the independent and bounded model, we assume that this productivity is separable in the firm's quality, the worker's human capital level, and an idiosyncratic component that is independently and identically distributed according to some bounded distribution.