Abstract
We present higher order necessary conditions for a model of welfare economics, where the preference mapping has a star-shape property. We assume that the preferences can be satiable and can be described by an arbitrary preference set, without the use of utility functions. These conditions are formulated in terms of higher-order directional derivatives of multivalued mappings, and the variable domination structure is not given by cones.
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Allouch, N., Le Van, C., Page, F.H.: Arbitrage and equilibrium in unbounded exchange economies with satiation. J. Math. Econ. 42(06), 661–674 (2006)
Allouch, N., Le Van, C.: Erratum to “Walras and dividends equilibrium with possibly satiated consumers”. J. Math. Econ. 45, 320–328 (2009)
Aubin, J.-P.: Optima and Equilibria: an Introduction to Nonlinear Analysis. Springer, Berlin (1998)
Bao, T.Q., Mordukhovich, B.S.: Set-valued optimization in welfare economics. Adv. Math. Econ. 13, 113–153 (2010)
Bao, T.Q., Mordukhovich, B.S.: Necessary nondomination conditions in set and vector optimization with variable ordering structures. J. Optim. Theory. Appl. 162, 350–370 (2014)
Durea, M., Strugariu, R., Tammer, C: On set-valued optimization problems with variable ordering structure. J. Glob. Optim. 61, 745–767 (2015)
Flores-Bazán, F., Jiménez, B.: Strict efficiency in set-valued optimization. SIAM J. Control Optim. 48(2), 881–908 (2009)
Gale, D., Mas-Colell, A.: On the role of complete, transitive preferences in equilibrium theory. In: Schwödiauer, G. (ed.) Equilibrium and Disequilibrium in Economic Theory, pp. 7–14. D. Reidel Publishing Company, Dordrecht (1977)
Isac, G., Khan, A.A.: Second-order optimality conditions in set-valued optimization by a new tangential derivative. Acta Mathematica Vietnamica 34, 81–90 (2009)
Jahn, J.: Vector Optimization. Springer, Berlin (2004)
Jahn, J., Khan, A.A., Zeilinger, P.: Second order optimality conditions in set-valued optimization. J. Optim. Theory Appl. 125, 331–347 (2005)
Khan, A.A., Soleimani, B., Tammer, C.: Second-order optimality conditions in set-valued optimization with variable ordering structure. Pure and Applied Functional Analysis 2, 305–316 (2017)
Khan, A.A., Tammer, C.: Generalized Dubovitskii-Milyutin approach in set-valued optimization. Vietnam J. Math. 40, 285–304 (2012)
Khan, A.A., Tammer, C.: Second-order optimality conditions in set valued optimization via asymptotic derivatives. Optimization 62, 743–758 (2013)
Khan, A.A., Tammer, C., Zălinescu, C.: Set-Valued Optimization. An Introduction with Applications. Springer, Berlin (2015)
Li, S.J., Sun, X.K., Zhu, S.K.: Higher-order optimality conditions for strict minimality in set-valued optimization. Journal of Nonlinear and Convex Analysis 13(2), 281–291 (2012)
Mas-Collel, A.: Equilibrium theory with possibly satiated preferences. In: Majumdar, M. (ed.) Equilibrium and Dynamics: Proceedings of the Essays in Honour of David Gale, pp. 201–213, MacMillan, London (1992)
Sato, N.: Satiation and existence of competitive equilibrium. J. Math. Econ. 46, 534–551 (2000)
Studniarski, M.: Optimization with respect to general preference mappings. Folia Math. 19(1), 50–54 (2017)
Werner, J.: Arbitrage and the existence of competitive equilibrium. Econometrica 55, 1403–1418 (1987)
Won, D.C., Yannelis, N.C.: Equilibrium theory with satiable and non-ordered preferences. J. Math. Econ. 47, 245–250 (2011)
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The authors would like to thank two anonymous referees for their valuable comments which have greatly helped to improve the previous version of this paper.
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Michalak, A., Studniarski, M. Higher-Order Optimality Conditions in Set-Valued Optimization with Respect to General Preference Mappings. Set-Valued Var. Anal 30, 975–993 (2022). https://doi.org/10.1007/s11228-022-00627-2
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DOI: https://doi.org/10.1007/s11228-022-00627-2