A Note on the Equilibrium Properties of Locational Sorting Models

A central feature of many models of location choice -- whether of firms or households, within or across cities -- is the role of local interactions or spillovers, whereby the payoffs from choosing a location depend in part on the number or attributes of other individuals or firms that choose the same or nearby locations in equilibrium. The main goal of this paper is to develop the equilibrium properties of a broadly applicable and readily estimable class of sorting models that allow the location decision to depend on both fixed local attributes (including unobserved attributes) and such local interactions. In particular, we prove uniqueness in the case of congestion effects and use a series of simulations to demonstrate that a unique equilibrium is more likely to obtain (i) the smaller are any agglomeration effects, (ii) the larger are the set of choices available to the agents, (iii) the more "meaningful variation" there is in those choices, and (iv) the more heterogeneous are the agents themselves. This is encouraging for the use of our model to describe the sorting of individuals and firms over geographic space, where the number of choices is usually large and variation in exogenous fixed attributes can be important. Moreover, these results conveniently coincide with the conditions required for econometric identification of our model.