Connected Substitutes and Invertibility of Demand

34 Pages Posted: 17 Jul 2012

See all articles by Steven Berry

Steven Berry

Yale University - Department of Economics; National Bureau of Economic Research (NBER); Yale University - Cowles Foundation

Amit Gandhi

University of Wisconsin - Madison

Philip A. Haile

Yale University - Department of Economics; National Bureau of Economic Research (NBER); Yale University - Cowles Foundation

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Date Written: July 17, 2012

Abstract

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.

Keywords: Univalence, Injectivity, Weak substitutes, Complements

JEL Classification: D01, C3

Suggested Citation

Berry, Steven T. and Gandhi, Amit and Haile, Philip A., Connected Substitutes and Invertibility of Demand (July 17, 2012). Cowles Foundation Discussion Paper No. 1806R. Available at SSRN: https://ssrn.com/abstract=2111382 or http://dx.doi.org/10.2139/ssrn.2111382

Steven T. Berry

Yale University - Department of Economics ( email )

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National Bureau of Economic Research (NBER)

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Yale University - Cowles Foundation

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Amit Gandhi

University of Wisconsin - Madison ( email )

716 Langdon Street
Madison, WI 53706-1481
United States

Philip A. Haile (Contact Author)

Yale University - Department of Economics ( email )

28 Hillhouse Ave
New Haven, CT 06520-8264
United States
203-432-3568 (Phone)
203-432-6323 (Fax)

National Bureau of Economic Research (NBER)

1050 Massachusetts Avenue
Cambridge, MA 02138
United States

Yale University - Cowles Foundation

Box 208281
New Haven, CT 06520-8281
United States

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