Comparative Statics in Markets for Indivisible Goods

We complete the study of comparative statics initiated in Caplin and Leahy [2010], which introduced a new mathematical apparatus for understanding NTU allocation markets, as such covering the housing market and other markets for large indivisible goods. We introduce homotopy methods to characterize how equilibrium changes in response to arbitrary parameter changes. Generically, we show that there can be five and only five qualitatively distinct forms of market transition: Graft; Prune and Plant; Prune and Graft; Cycle and Reverse; and Shift and Replant. Our path-following methods identify new algorithms for computing market equilibria.