“Connected Substitutes and Invertibility of Demand
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We consider the invertibility (injectivity) of a nonparametric nonseparable demand system. Invertibility of demand is important in several contexts, including identiﬁcation 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 suﬀicient for invertibility. The connected substitutes conditions require weak substitution between all goods and suﬀicient strict substitution to necessitate treating them in a single demand system. The connected substitutes conditions have transparent economic interpretation, are easily checked, and are satisﬁed in many standard models. They need only hold undersome 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 suﬀicient conditions are also “nearly necessary” for even local invertibility.
Berry, Steven T.; Gandhi, Amit; and Haile, Philip A., "“Connected Substitutes and Invertibility of Demand" (2011). Cowles Foundation Discussion Papers. 2151.