Equilibrium Multiplicity: A Systematic Approach using Homotopies, with an Application to Chicago

Discrete choice models with social interactions or spillovers may exhibit multiple equilibria. This paper provides a systematic approach to enumerating them for a quantitative spatial model with discrete locations, social interactions, and elastic housing supply. The approach relies on two homotopies. A homotopy is a smooth function that transforms the solutions of a simpler city where solutions are known, to a city with heterogeneous locations and finite supply elasticity. The first homotopy is that, in the set of cities with perfectly elastic floor surface supply, an economy with heterogeneous locations is homotopic to an economy with homogeneous locations, whose solutions can be comprehensively enumerated. Such an economy is epsilon close to an economy whose equilibria are the zeros of a system of polynomials. This is a well-studied area of mathematics where the enumeration of equilibria can be guaranteed. The second homotopy is that a city with perfectly elastic housing supply is homotopic to a city with an arbitrary supply elasticity. In a small number of cases, the path may bifurcate and a single path yields two or more equilibria. By running the method on thousands of cities, we obtain a large number of equilibria. Each equilibrium has different population distributions. We provide a method that is computationally feasible for economies with a large number of locations choices, with an empirical application to the City of Chicago. There exist multiple ``counterfactual Chicagos'' consistent with the estimated parameters. Population distribution, prices, and welfare are not uniquely pinned down by amenities. The paper's method can be applied to models in trade and IO. Further applications of algebraic geometry are suggested.