Equilibrium sorting of heterogeneous consumers across locations: theory and empirical implications

Abstract

This paper develops a model in which a continuum of consumers choose froma continuum of locations indexed by school quality. It computes equilibria that are sustained by a price function that matches consumers to different locations based on their willingness to pay for school quality. In equilibrium each location is inhabited by a set of people with varying levels of education, ability, intensity of preference for education, and income. The distributions of characteristics within each location are determined by the structural elements of the model. The paper also develops a set of computational algorithms that solve several complex numerical problems. These problems include the calculation of a number of diffcult integrals, the calculation of asymptotic approximations to those integrals, the solution of an implicitly defined differential equation that depends on the integrals previously calculated, and the maximization of a likelihood function that depends on the solution of the differential equation. Finally, this paper demonstrates how the equilibrium implications of a structural economic matching model can be used to solve two important econometric identification problems. First, it is likely that regressions that seek to estimate the effects of school quality on educational outcomes produce biased and inconsistent estimates because people choose where their children go to school. The model in the paper solves this problem by using a consumer location choice equation and an equilibrium pricing relation to create a valid instrument for the school quality variable. Second, hedonic estimation problems in a single market are unidentified because the marginal price function is unknown or collinear with the level of the product demanded. This paper solves this problem by exploiting the restrictions that equilibrium in the sorting economy imposes on the equilibrium price function. The equilibrium price equation introduces a non-linearity into the system that is suffcient for identification.