Global invertibility of excess demand functions

Covarrubias, Enrique (2013): Global invertibility of excess demand functions.

Abstract

In this paper we provide necessary and sufficient conditions for the excess demand function of a pure exchange economy to be globally invertible so that there is a unique equilibrium. Indeed, we show that an excess demand function is globally invertible if and only if its Jacobian never vanishes and it is a proper map. Our result includes as special cases many partial results found in the literature that imply global uniqueness including Gale-Nikaido conditions and properties related to stability of equilibria. Furthermore, by showing that the condition is necessary, we are implicitly finding the weakest possible condition.

• Item Type: MPRA Paper
• Language: English
• Keywords: Uniqueness Equilibrium
• Subjects: D - Microeconomics > D5 - General Equilibrium and Disequilibrium; D - Microeconomics > D5 - General Equilibrium and Disequilibrium > D50 - General; D - Microeconomics > D5 - General Equilibrium and Disequilibrium > D51 - Exchange and Production Economies

References

Arrow, K. and Debreu, G. (1954) Existence of an equilibrium for a competitive economy. \emph{Econometrica} 22, 265-290.

Arrow, K. and Hahn, F. (1971) General competitive equilibrium. North-Holland, Amsterdam, New York and Tokyo.

Benhabib, J. and Nishimura, K. (1979) On the uniqueness of steady states in an economy with heterogeneous capital goods. \emph{International Economic Review} 20-1, 59-82.

Chichilnisky, G. (1998) Topology and invertible maps. \emph{Advances in Applied Mathematics} 21, 113-123.

Covarrubias, E. (2013) The number of equilibria of smooth infinite economies. \emph{Journal of Mathematical Economics}, in press.

Debreu, G. (1983) Economic theory in mathematical mode. Nobel Memorial Lecture.

Dierker, E. (1972) Two remarks on the number of equilibria of an economy. \emph{Econometrica} 40-5, 951-953.

Gale, D. Univalence theorems for differentiable mappings. Discussion Paper No. 29, Institute of Social and Economic Research, Osaka University (November, 1962).

Gale, D. and Nikaido, H. (1965) The Jacobian matrix and global univalent mappings. \emph{Mathematische Annalen} 159, 81-93.

Hahn, F. (1982) Stability. in Handbook of Mathematical Economics, ch. 16, vol. II, edited by K.J. Arrow and M.D. Intrilligator. North-Holland.

Hatcher, A. (2002) Algebraic topology. Cambridge University Press, Cambridge.

Ho, C.W. (1975) A note on proper maps. \emph{Proc. Amer. Math. Soc.} 51, 237-241.

Mas-Colell, A. (1979) Homeomorphisms of compact, convex sets and the Jacobian matrix. \emph{SIAM Journal on Mathematical Analysis} 10-6, 1105-1109.

Mas-Colell, A. (1985) The Theory of General Economic Equilibrium. A Differentiable Approach. Econometric Society Monographs, Cambridge: Cambridge University Press, 1985.

Mukherji, A. (1995) A Locally Stable Adjustment Process. \emph{Econometrica} 63 (1995) 441-448.

Mukherji, A. (1997) On the uniqueness of competitive equilibrium. \emph{Economic Theory} 10, 509-520.

Nikaido, H. (1962) Uniqueness of Solution of Certain Equations, I, II. Discussion Papers Nos. 28, 30, Institute of Social and Economic Research, Osaka University.

Nishimura, K. (1978) A further remark on the number of equilibria of an economy. \emph{International Economic Review} 19-3, 679-685.

Nishimura, K. (1979) On the uniqueness theorems by Arrow and Hahn. \emph{Journal of Economic Theory} 21-2, 348-352.

Palais, R.S. (1970) When proper maps are closed. \emph{Proc. Amer. Math. Soc.} 24, 835-836.

Pearce, I.F. and Wise, J. (1973) On the uniqueness of competitive equilibrium: part I, unbounded demand. \emph{Econometrica} 41, 817-828.

Pearce, I.F. and Wise, J. (1974) On the uniqueness of competitive equilibrium: part II, bounded demand. \emph{Econometrica} 42, 921-932.

Shafer, W. and H. Sonnenschein (1982) Market demand and excess demand functions, in Handbook of Mathematical Economics, ch. 14, vol. II, edited by K.J. Arrow and M.D. Intrilligator. North-Holland.

Varian, H. R. (1975) A third remark on the number of equilibria of an economy. \emph{Econometrica} 43-5/6, 985-986.

Wagstaff, P. (1975) A uniqueness theorem. \emph{ International Economic Review} 16-2, 521-524.

Yun, K. K. (1981) A note of Nishimura's uniqueness theorems of general equilibrium. \emph{International Economic Review} 22-2, 471-473.