Lower Hemi-Continuity, Open Sections, and Convexity: Counter Examples in Infinite Dimensional Spaces

Author(s)
Adib Bagh

ABSTRACT

A lower hemi-continuous correspondence with open and convex values in Rn must have open lower sections. This well- known fact has been used to establish the existence of continuous selections, maximal elements, and fixed points of correspondences in various economic applications. Since there is an increasing number of economic models that use correspondences in an infinite-dimensional setting, it is important to know whether or not the above fact remains valid in such applications. The aim of this paper is to show that the above fact no longer holds when Rn is replaced with an infinite-dimensional space. This is accomplished by using the standard orthonormal base in a Hilbert space H to construct two correspondences with values in H equipped with the weak topology. The first correspondence is lower hemi-continuous with open and convex values but does not have open lower sections. The second is a lower hemi-continuous correspondence that fails to have an open graph despite having open and convex upper and lower sections. These counter-examples demonstrate that in an infinite-dimensional setting, it is no longer possible to rely on the geometric properties of a lower hemi-continuous map (the convexity of its sections) to establish the topological properties (open lower sections, open graph) needed in many economic applications.

A lower hemi-continuous correspondence with open and convex values in Rn must have open lower sections. This well- known fact has been used to establish the existence of continuous selections, maximal elements, and fixed points of correspondences in various economic applications. Since there is an increasing number of economic models that use correspondences in an infinite-dimensional setting, it is important to know whether or not the above fact remains valid in such applications. The aim of this paper is to show that the above fact no longer holds when Rn is replaced with an infinite-dimensional space. This is accomplished by using the standard orthonormal base in a Hilbert space H to construct two correspondences with values in H equipped with the weak topology. The first correspondence is lower hemi-continuous with open and convex values but does not have open lower sections. The second is a lower hemi-continuous correspondence that fails to have an open graph despite having open and convex upper and lower sections. These counter-examples demonstrate that in an infinite-dimensional setting, it is no longer possible to rely on the geometric properties of a lower hemi-continuous map (the convexity of its sections) to establish the topological properties (open lower sections, open graph) needed in many economic applications.

Cite this paper

A. Bagh, "Lower Hemi-Continuity, Open Sections, and Convexity: Counter Examples in Infinite Dimensional Spaces,"*Theoretical Economics Letters*, Vol. 2 No. 2, 2012, pp. 121-124. doi: 10.4236/tel.2012.22022.

A. Bagh, "Lower Hemi-Continuity, Open Sections, and Convexity: Counter Examples in Infinite Dimensional Spaces,"

References

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[23] T. C. Bergstrom, R. P. Parks and T. Rader, “Preferences Which Have Open Graphs,” Journal of Mathematical Economics, Vol. 3, 1976, pp. 265-268. doi:10.1016/0304-4068(76)90012-4

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[25] C. D. Aliprantis and R. Tourky, “Equilibria in Incomplete Assets Economics with Infinite Dimensional Spot Markets,” Economic Theory, Vol. 38, No. 2, 2008, pp. 221-262. doi:10.1007/s00199-007-0247-2

[1] J. C. Moore, “Mathematical Methods for Economic Theory, 2,” Studies in Economic Theory, Springer-Verlag, Berlin, Vol. 10, 1999.

[2] R. T. Rockafellar and R. J.-B. Wets, “Variational Analysis,” Grundlehren der mathematischen Wissenchaften, Springer, Berlin, Vol. 317, 2009.

[3] N. Stokey, R. E. Lucas and E. Prescott, “Recursive Methods in Economic Dynamics,” Harvard University Press, Cambridge, 1989.

[4] C. D. Alip-rantis, G. Camera and F. Ruscitti, “Monetary Equilibrium and the Differentiability of the Value Function,” Journal of Economic Dynamics and Control, Vol. 33, No. 2, 2009, pp. 454-462. doi:10.1016/j.jedc.2008.06.010

[5] L. M. Benveniste and J. A. Scheinkman, “On the Differentiability of the Value Function in Dynamic Models of Economics,” Econometrica, Vol. 47, No. 3, 1979, pp. 727-732. doi:10.2307/1910417

[6] T. Kim, “Dif-ferentiability of the Value Function: A New Characterization,” Seoul Journal of Economics, Vol. 6, 1993, pp. 257-265.

[7] D. Gale and A. Mas-Collell, “An Equilibrium Existence Result Theorem for a General Model without Ordered Preference,” Journal of Mathematical Economics, Vol. 2, 1975, pp. 9-15. doi:10.1016/0304-4068(75)90009-9

[8] D. Gale and A. Mas-Collell, “Corrections to an Equilibrium Existence Theorem for a General Model without Ordered Preferences,” Journal of Mathematical Economics, Vol. 6, No. 3, 1979, pp. 297-298. doi:10.1016/0304-4068(79)90015-6

[9] E. Michael, “Continuous Selections I,” Annals of Mathematics, Vol. 64, 2, 1956, 361-382. doi:10.2307/1969615

[10] G. Tian, “On the Existence of Equilibria in Generalized Games,” IJGM, Vol. 20, 1992, pp. 247-254.

[11] N. C. Yannelis and N. B. Prabhakar, “Existence of Maximal Elements and Equilibria in Linear Topological Spaces,” Journal of Mathematical Economics, Vol. 12, 1983, pp. 233-245. doi:10.1016/0304-4068(83)90041-1

[12] N. Sun, “Bewley’s Limiting Approach to Infinite Dimensional Econo-mies with lsc Preferences,” Economics Letters, Vol. 92, No. 1, 2006, pp. 7-13. doi:10.1016/j.econlet.2006.01.006

[13] T. F. Bewley, “Existence of Equilibria in Economies with Infinitely Many Commodities,” Journal of Economic Theory, Vol. 4, No. 3, 1972, pp. 514-540. doi:10.1016/0022-0531(72)90136-6

[14] G. Chichilnisky and P. Kalman, “An Application of Functional Analysis of Model of Optimal Allocation of Resources with an Infinite Horizon,” Journal of Optimization Theory and Applications, Vol. 30, No. 1, 1980, pp. 19-32. doi:10.1007/BF00934586

[15] G. Chichilnisky and Y. Zhou, “Smooth Infinite Economies,” Journal of Mathematical Economics, Vol. 29, 1998, pp. 27-42. doi:10.1016/S0304-4068(97)00009-8

[16] M. Florenzano, “On the Existence of Equilibria Space in Economies with an Infinite Dimensional Commodity Space,” Journal of Mathematical Economics, Vol. 13, 1983, pp. 207-219. doi:10.1016/0304-4068(83)90039-3

[17] A. Mas-Collel and W. R. Zame, “Equilibrium in Infinite Dimensional Spaces,” In: W. Hildenbrand and H. Sonnenschein, Eds., The Handbook of Mathematical Economics, Elsevier, London, Chapter 34, Vol. 4, 1991.

[18] S. Toussaint, “On the Existence of Equilibria in Economies with Infinitely Many Commodities and without Ordered Preferences,” Journal of Economic Theory, Vol. 33, No. 1, 1984, pp. 98-115. doi:10.1016/0022-0531(84)90043-7

[19] N. C. Yannelis and M. Ali-Khan, “Equilibrium Theory in Infinite Dimensional Spaces,” Springer-Verlag, New York, 1991.

[20] W. Zame, “Competitive Equilibria Space in Production Economies with an Infinite Dimensional Commodity Space,” Econometrica, Vol. 33, 1987, pp. 1075-1108. doi:10.2307/1911262

[21] W. Rudin, “Functional Analysis,” 2nd Edition, Interntional Series in Pure and Applied Mathe-matics, McGraw- Hill, New York, 1991.

[22] C. D. Aliprantis and K. C. Border, “Infinite Dimensional Analysis: A Hitch-hiker’s Guide,” 3rd Edition, Springer-Verlag, Berlin, 2006.

[23] T. C. Bergstrom, R. P. Parks and T. Rader, “Preferences Which Have Open Graphs,” Journal of Mathematical Economics, Vol. 3, 1976, pp. 265-268. doi:10.1016/0304-4068(76)90012-4

[24] J. Zhou, “On the Existence of Equilibrium for Abstract Economies,” Journal of Mathematical Analysis and Applications, Vol. 193, No. 3, 1995, pp. 839-858. doi:10.1006/jmaa.1995.1271

[25] C. D. Aliprantis and R. Tourky, “Equilibria in Incomplete Assets Economics with Infinite Dimensional Spot Markets,” Economic Theory, Vol. 38, No. 2, 2008, pp. 221-262. doi:10.1007/s00199-007-0247-2