Optimal Separation of Twin Convex Sets under Externalities

ABSTRACT

This paper studies the outcomes of independent and interdependent pair-wise contests between economic agents subject to an optimal external decision problem for each pair. The external decision maker like the government or regulator is faced with the problem of how to devise rules and regulations regarding contests. In this paper, a decision problem is faced under negative and positive externalities. A pair of entities is represented by disjoint convex sets in a small area in a neighborhood. I assume that each entity imposes an equal externality on the other (and the other only) and thus they can be considered to be twins. Among the group of twins in any neighborhood, there is a set of twin pairs such that, for each pair in the set, each twin can impose a strictly negative externality on the other (and the other only), and this is a potential welfare loss which concerns the decision maker. A separating hyper-plane can block the negative externalities between any pair of twins given convexity. However, this can be costly if positive externality from the neighborhood is also blocked by the separation technology. Thus, this paper compares the pair-wise utility from separation to that of non-separation. A simple representation of the decision problem is developed with respect to a single and isolated neighborhood. A complete characterization of the decision problem is obtained with a large number of pair-wise intersecting neighborhoods.

Cite this paper

Mallick, I. (2014) Optimal Separation of Twin Convex Sets under Externalities.*Advances in Pure Mathematics*, **4**, 381-390. doi: 10.4236/apm.2014.48049.

Mallick, I. (2014) Optimal Separation of Twin Convex Sets under Externalities.

References

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http://dx.doi.org/10.1093/0198288816.001.0001

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[3] Keynes, J.M. (1936) The General Theory of Employment, Interest and Money. Macmillan, London.

[4] Ray, D. (1999) Development Economics. Oxford University Press, New Delhi.

[5] Fudenberg, D. and Jean, T. (2005) Game Theory. Ane Books, India.

[6] Gale, D. (2000) Strategic Foundations of General Equilibrium: Dynamic Matching and Bargaining Games. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9780511492310

[7] Osborne, M.J. and Ariel, R. (1990) Bargaining and Markets. Academic Press, Inc., San Diego.

[8] Coase, R.H. (1988) The Firm, the Market and the Law. The University of Chicago Press, Chicago.

[9] Laffont, J.-J. (1989) The Economics of Uncertainty and Information. The MIT Press, Cambridge.

[10] Mas-Colell, A., Whinston, M.D. and Green, J.R. (1995) Microeconomic Theory. Oxford University Press, Oxford.

[11] Bowles, S. (2005) Microeconomics: Behavior, Institutions and Evolution. Oxford University Press, New Delhi.

[12] Hayami, Y. (2001) Development Economics: From the Poverty to the Wealth of Nations. Oxford University Press, Oxford. http://dx.doi.org/10.1093/0199243972.001.0001

[13] Azariadis, C. (1993) Intertemporal Macroeconomics. Blackwell Publishers, Cambridge.

[14] Blanchard, O.J. and Fischer, S. (1989) Lectures on Macroeconomics. The MIT Press, Cambridge.

[15] Mallick, I. (2002) Strategic Competition in Banking: Theory and Policy. Ph.D. Thesis, Jadavpur University, Kolkata.

[16] Dewatripont, M. and Tirole, J. (1993) The Prudential Regulation of Banks. The MIT Press, Cambridge.

[17] Gale, D. (1983) Money: In Disequilibrium. Cambridge University Press, Cambridge.

[18] Stiglitz, J.E. (1986) Economics of the Public Sector. W. W. Norton & Company, New York.

[19] Viscusi, W.K., Harrington, J.E. and Vernon, J.M. (2005) Economics of Regulation and Antitrust. The MIT Press, Cambridge.

[20] Bernard, M. (2012) Conflict, Cooperation and Coordination: Essays in Game Theory and Experimental Economics. Ph.D. Thesis, Stockholm School of Economics, Stockholm.

[21] Tirole, J. (1988) The Theory of Industrial Organization. The MIT Press, Cambridge.

[22] Hart, O. (1995) Firms, Contracts, and Financial Structure. Oxford University Press, Oxford.

http://dx.doi.org/10.1093/0198288816.001.0001

[23] Smith, J.M. (1982) Evolution and the Theory of Games. Cambridge University Press, Cambridge.

[24] Edgeworth, F.Y. (1881) Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences. Kegan Paul, London.

[25] Duffie, D. (2012) Dark Markets: Asset Pricing and Information Transmission in Over-the-Counter Markets. 3rd Edition, Princeton University Press, Princeton.

[26] Shavell, S. (2004) Foundations of Economic Analysis of Law. Harvard University Press, Cambridge.

[27] Milgrom, P. and Roberts, J. (1992) Economics, Organization and Management. Prentice-Hall, Inc., Upper Saddle River.

[28] Minkowski, H. (1911) Theorie der Konvexen Korper, Insbesondere Begrundung ihres Oberflachenbegriffs. Gessamelte Abhandlungen II, Leipzig.

[29] Rockafellar, R.T. (1970) Convex Analysis. Princeton University Press, Princeton.

[30] Gale, D. (1960) The Theory of Linear Economic Models. McGraw-Hill, Inc., New York.

[31] Luenberger, D.G. (1969) Optimization by Vector Space Methods. Wiley, New York.

[32] Dixit, A.K. (1990) Optimization in Economic Theory. Oxford University Press, Oxford.

[33] Rudin, W. (1976) Principles of Mathematical Analysis. McGraw-Hill, Inc., London.