Connected Substitutes and Invertibility of Demand

We consider the invertibility (injectivity) of a nonparametric nonseparable demand system. Invertibility of demand is important in several contexts, including identi…cation of demand, estimation of demand, testing of revealed preference, and economic theory exploiting existence of an inverse demand function or (in an exchange economy) uniqueness of Walrasian equilibrium prices. We introduce the notion of “connected substitutes”and show that this structure is su¢ cient for invertibility. The connected substitutes conditions require weak substitution between all goods and su¢ cient strict substitution to necessitate treating them in a single demand system. The connected substitutes conditions have transparent economic interpretation, are easily checked, and are satis…ed in many standard models. They need only hold under some transformation of demand and can accommodate many models in which goods are complements. They allow one to show invertibility without strict gross substitutes, functional form restrictions, smoothness assumptions, or strong domain restrictions. When the restriction to weak substitutes is maintained, our su¢ cient conditions are also “nearly necessary”for even local invertibility. Keywords: univalence, injectivity, weak substitutes, complements This paper combines and expands on selected material …rst explored in Gandhi (2008) and Berry and Haile (2009). We have bene…tted from helpful comments from Jean-Marc Robin, Don Brown, the referees, and participants in seminars at LSE, UCL, Wisconsin, Yale, the 2011 UCL workshop on “Consumer Behavior and Welfare Measurement,” and the 2011 “Econometrics of Demand” conference at MIT. Adam Kapor provided capable research assistance. Financial support from the National Science Foundation is gratefully acknowledged.