Abstract
The paper consists of two parts. The first is devoted to a general subdifferential theory based on an axiomatic approach. Along with the list of “axioms” which summarizes properties shared by all major subdifferentials studied in variational analysis, we also consider four “optional” properties which specific subdifferentials may or may not have, such as trustworthiness, robustness, tightness (validity of a certain fuzzy subdifferential inequality) and geometric compatibility (connection between subdifferentials and normal cones). The concluding result says that the approximate G-subdifferential is the only subdifferential that has the four properties on all Banach space.
The second part is devoted to application of the general subdifferential theory to a model of welfare economics with a Banach commodity space. Here we begin with subdifferential characterization of nonconvex separation property in general and also for a special case of one of the sets being a shifted kernel of a linear epimorphism, and then apply the results to characterize Pareto and weak Pareto optimal allocations in welfare economics. The final result is a strengthening of earlier versions of the second welfare theorem due to Khan-Vohra, Cornet, Joffre and Mordukhovich. In particular, a weaker and more symmetric version of Cornet's qualification condition appears in the characterization of Pareto optimality.
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References
Aussel, D., Corvellec, J.-N., Lassonde, M.: Mean value property and subdifferential criteria for lower semicontinuous functions. Trans. Am. Math. Soc. 347, 4147–4161 (1995)
Aussel, D., Corvellec, J.-N., Lassonde, M.: Nonsmooth constrained optimization and multidirectional mean value inequalities. SIAM J. Optimization 9, 690–706 (1999)
Bonnisseau, J.-M., Cornet, B.: Valuation equilibrium and Pareto optimum in non-convex economies. J. Math. Econ. 17, 293–308 (1988)
Borwein, D., Borwein, J.M., Wang, X.: Approximate subgradient and coderivatives in IR n, Set-valued Anal. 4, 375–398 (1996)
Borwein, J.M., Joffre, A.: Nonconvex separation theorem in Banach spaces. J. Oper. Res. Appl. Math. 48, 169–180 (1998)
Borwein, J.M., Lucet, Y., Mordukhovich, B.S.: Compactly epi-Lipschitzian convex sets and functions in normed spaces. J. Convex Anal. 7, 375–393 (2000)
Borwein, J.M., Strojwas, H.: Tangential approximations. Nonlinear Anal. Theory Methods Appl. 9, 1347–1366 (1985)
Borwein, J.M., Wang, X.: Lipschitz functions with maximal Clarke subdifferentials are generic. Proc. Am. Math. Soc. 128, 3221–3229 (2000)
Borwein, J.M., Zhu, J.: Techniques of variational analysis. CMS Books in Mathematics, vol. 20. Berlin: Springer 2006
Cornet, B.: The second welfare theorem in non-convex economies. CORE discussion paper 8630 (1986)
Correa, R., Joffré, A., Thibault, L.: Subdifferential monotonicity as characterization of convex functions. Numer. Funct. Ana. Optimization 15, 1167–1183 (1994)
Fabian, M.: Subdifferentiability and trustworthiness in light of the new variational principle of Borwein and Preiss. Acta Univ. Carolinae 30, 51–56 (1989)
Hamano, T.: On the non-existence of the marginal cost price equilibrium and the Ioffe normal cone. Z. Nationalökonomie 50, 47–53 (1989)
Ioffe, A.D.: Sous-différentielles approchées des fonctions numériques. C. R. Acad. Sci. Paris 292 (1981), 675–678
Ioffe, A.D.: On subdifferentiability spaces. Ann. N.Y. Acad. Sci. 410, 107–121 (1983)
Ioffe, A.D.: Approximate subdifferentials and applications 2. Mathematika 33, 111–128 (1986)
Ioffe, A.D.: On the local surjection property. Nonlinear Anal. Theory Methods Appl. 11, 565–592 (1987)
Ioffe, A.D.: Fuzzy principles and characterization of trustworthiness. Set-valued Anal. 6, 265–276 (1998)
Ioffe, A.D.: Codirectional compactness, metric regularity and subdifferential calculus. In: Thera, M. (ed.): Constructive, experimental and nonlinear analysis, CMS Conference Proceeding, vol. 27, pp. 123–165 (2000)
Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspehi Mat. Nauk 55(3), 103–162 (2000) (in Russian), English translation: Russian Math. Surveys 55(3), 501–558 (2000)
Joffre, A.: A second welfare theorem in nonconvex economies. In: Thera, M. (ed.): Constructive, experimental and nonlinear analysis, CMS Conference Proceeding, vol. 27, pp. 175–184 (2000)
Katriel, G.: Are the approximate and the Clarke subgradients generically equal? J. Math. Anal. Appl. 193, 588–593 (1995)
Khan, M.A.: Ioffe's normal cone and foundations of welfare economics. The infinite dimensional theory. J. Math. Anal. Appl. 161, 284–298 (1991)
Khan, M.A.: The Mordukhovich normal cone and foundations of welfare economics. J. Public Econ. Theory 1, 309–338 (1999)
Khan, M.A., Vohra, R.: An extension of the second welfare theorem to economies with non-convexities and public goods. Q. J. Econ. 102, 223–245 (1987)
Lassonde, M.: First order rules for nonsmooth constrained optimization. Nonlinear Anal. Theory Methods Appl. 44 (2001), 1031–1056
Mordukhovich, B.S.: Metric approximations and necessary conditions for optimality for general classes of nonsmooth optimization problems. Dokl. Acad. Nauk SSSR 254 (1980), 1072–1076
Mordukhovich, B.S.: Abstract extremal principle with applications to wellfare economics. J. Math. Anal. Appl. 251, 187–216 (2000)
Mordukhovich, B.S.: Variational analysis and generalized differentiation. Berlin: Springer 2005
Penot, J.-P.: The compatibility with order of some subdifferentials. Positivity 6, 413–432 (2002)
Rockafellar, R.T.: Maximal monotone relations and the second derivatives of nonsmooth functions. Ann. Inst. H. Poincaré. Analyse Non-Linéaire 2, 167–184 (1985)
Zhu, Q.J.: The equivalence of several basic theorems for subdifferentials. Setvalued Anal. 6, 171–185 (1998)
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Ioffe, A.D. (2009). Variational analysis and mathematical economics 1: Subdifferential calculus and the second theorem of welfare economics. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics. Advance in Mathematical Economics, vol 12. Springer, Tokyo. https://doi.org/10.1007/978-4-431-92935-2_3
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DOI: https://doi.org/10.1007/978-4-431-92935-2_3
Publisher Name: Springer, Tokyo
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