The Open Graph Theorem for Correspondences: A New Proof and Some Applications

Affiliation(s)

Department of Economics and Law, University La Sapienza, Rome, Italy.

Department of Political and Social Sciences, John Cabot University, Rome, Italy.

Department of Economics and Law, University La Sapienza, Rome, Italy.

Department of Political and Social Sciences, John Cabot University, Rome, Italy.

ABSTRACT

It is known that a correspondence from a topological space to a Euclidean space, with open and convex upper sections, has an open graph if and only if it is lower hemicontinuous. We refer to this result as the open graph theorem. We provide a new and simple proof of the open graph theorem. We also show that the open graph theorem leads to novel results on the existence of constant selections and fixed points for correspondences with non-compact and non-convex domain. Finally, we present an economic application of our results to a principal-agent model.

It is known that a correspondence from a topological space to a Euclidean space, with open and convex upper sections, has an open graph if and only if it is lower hemicontinuous. We refer to this result as the open graph theorem. We provide a new and simple proof of the open graph theorem. We also show that the open graph theorem leads to novel results on the existence of constant selections and fixed points for correspondences with non-compact and non-convex domain. Finally, we present an economic application of our results to a principal-agent model.

KEYWORDS

Open Graph Theorem; Lower Hemicontinuity; Constant Selections; Fixed Points; Support of a Measure

Open Graph Theorem; Lower Hemicontinuity; Constant Selections; Fixed Points; Support of a Measure

Cite this paper

G. Impicciatore and F. Ruscitti, "The Open Graph Theorem for Correspondences: A New Proof and Some Applications,"*Theoretical Economics Letters*, Vol. 2 No. 3, 2012, pp. 270-273. doi: 10.4236/tel.2012.23049.

G. Impicciatore and F. Ruscitti, "The Open Graph Theorem for Correspondences: A New Proof and Some Applications,"

References

[1] 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

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

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

[4] K. Border, “Fixed Point Theorems with Applications to Economics and Game Theory,” Cambridge University Press, Cambridge, 1985. doi:10.1017/CBO9780511625756

[5] R. T. Rockafellar and R. Wets, “Variational Analysis,” 3rd Edition, A Series of Comprehensive Studies in Mathematics, Springer, Berlin, Vol. 317, 2009.

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

[7] T. Kim, “Differentiability of the Value Function: A New Characterization,” Seoul Journal of Economics, Vol. 6, No. 3,1993, pp. 257-265.

[8] 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

[9] C. D. Aliprantis, 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

[10] C. D. Horvath, “On the Existence of Constant Selections,” Topology and Its Applications, Vol. 104, No. 1-3, 2000, pp. 119-139. doi:10.1016/S0166-8641(99)00022-X

[11] S. J. Grossman and O. D. Hart, “An Analysis of the Principal-Agent Problem,” Econometrica, Vol. 51, No. 1, 1983, pp. 7-45. doi:10.2307/1912246

[12] B. Caillaud and B. E. Hermalin, “Hidden Action and Incentives,” 2000. http://faculty.haas.berkeley.edu/hermalin/agencyread.pdf

[1] 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

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

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

[4] K. Border, “Fixed Point Theorems with Applications to Economics and Game Theory,” Cambridge University Press, Cambridge, 1985. doi:10.1017/CBO9780511625756

[5] R. T. Rockafellar and R. Wets, “Variational Analysis,” 3rd Edition, A Series of Comprehensive Studies in Mathematics, Springer, Berlin, Vol. 317, 2009.

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

[7] T. Kim, “Differentiability of the Value Function: A New Characterization,” Seoul Journal of Economics, Vol. 6, No. 3,1993, pp. 257-265.

[8] 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

[9] C. D. Aliprantis, 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

[10] C. D. Horvath, “On the Existence of Constant Selections,” Topology and Its Applications, Vol. 104, No. 1-3, 2000, pp. 119-139. doi:10.1016/S0166-8641(99)00022-X

[11] S. J. Grossman and O. D. Hart, “An Analysis of the Principal-Agent Problem,” Econometrica, Vol. 51, No. 1, 1983, pp. 7-45. doi:10.2307/1912246

[12] B. Caillaud and B. E. Hermalin, “Hidden Action and Incentives,” 2000. http://faculty.haas.berkeley.edu/hermalin/agencyread.pdf