COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1806R Connected Substitutes and Invertibility of Demand Steven Berry, Amit Gandhi, and Philip Haile June 2011 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 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 sufficient conditions are also "nearly
necessary" for even local invertibility. |
