Abstract
We consider an exchange economy where the consumers face linear inequality constraints on consumption. We parametrize the economy with the initial endowments and constraints. We exhibit sufficient conditions on the constraints implying that the demand is locally Lipschitzian and continuously differentiable on an open dense subset of full Lebesgue measure. Using this property, we show that the equilibrium manifold is lipeomorphic to an open, connected subset of an Euclidean space and that the lipeomorphism is almost everywhere continuously differentiable. We prove that regular economies are generic and that they have a finite odd number of equilibrium prices and local differentiable selections of the equilibrium prices.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balasko, Y., Foundations of the General Equilibrium Theory, Academic Press, New York, NY, 1988.
Debreu, G., Economies with a Finite Set of Equilibria, Econometrica, Vol. 38, pp. 387–392, 1970.
Mas-Colell, A., The Theory of General Economic Equilibrium: A Differentiable Approach, Cambridge University Press, Cambridge, UK, 1985.
Smale, S., Global Analysis and Economics, Handbook of Mathematical Economics, Edited by K. Arrow and M. Intriligator, North-Holland, Amsterdam, Netherlands, Vol. 1, Chapter 8, pp. 331–370, 1981.
Deaton, A., and Muellbauer, J., Economics and Consumer Behavior, Cambridge University Press, Cambridge, UK, 1992.
Villanacci, A., A Proof of Existence and Regularity for an Exchange Economy with Nonnegativity Constraints, DIMEFAS Discussion Paper, University of Florence, Florence, Italy, 1993.
Debreu, G., Theory of Value, John Wiley and Sons, New York, NY, 1959.
Cornet, B. and Vial, J. P., Lipschitzian Solutions of Perturbed Nonlinear Programming Problems, SIAM Journal on Control and Optimization, Vol. 24, pp. 1123–1137, 1986.
Fiacco, A., and McCormick, G., Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Classics in Applied Mathematics, SIAM, Philadelphia, Pennsylvania, Vol. 4, 1990.
Rockafellar, R. T., and Wets, R. J. B., Variational Analysis, Springer Verlag, Berlin, Germany, 1998.
Klatte, D., Upper Lipschitz Behavior of Solutions to Perturbed C1,1 Programs, Mathematical Programming, Vol. 88B, pp. 285–311, 2000.
Levy, A. B., Lipschitzian Multifunctions and a Lischitzian Inverse Mapping Theorem, Mathematics of Operations Research, Vol. 26, pp. 105–118, 2001.
Shapiro, A., Sensitivity Analysis of Parametrized Variational Inequalities, Mathematics of Operations Research, Vol. 30, pp. 109–126, 2005.
Rader, T., Nice Demand Functions, Econometrica, Vol. 41, pp. 913–935, 1973.
Shannon, C., Regular Nonsmooth Equations, Journal of Mathematical Economics, Vol. 23, pp. 147–166, 1994.
Bonnisseau, J. M., and Rivera-Cayupi, J., Constrained Consumptions and Regular Economies, Cahier de la Maison des Sciences Economiques 2003-51, Université Paris 1 Panthéon-Sorbonne, 2003; see also ftp://mse.univ-paris1.fr/pub/mse/ cahiers2003/B03051.pdf.
Deimling, K., Nonlinear Functional Analysis, Springer Verlage, Berlin, Germany, 1985.
Federer, H., Geometric Measure Theory, Springer Verlag, Berlin, Germany, 1969.
Author information
Authors and Affiliations
Additional information
Communicated by J. P. Crouzeix
This work was partially supported by CCE, ECOS, and ICM Sistemas Complejos de Ingeniería.
Rights and permissions
About this article
Cite this article
Bonnisseau, J.M., Rivera-Cayupi, J. Constrained Consumptions, Lipschitzian Demands, and Regular Economies. J Optim Theory Appl 131, 179–193 (2006). https://doi.org/10.1007/s10957-006-9147-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-006-9147-z