# Optimal Separation of Twin Convex Sets Under Externalities

[1] Allen, F. and Douglas, G. (2007) Understanding Financial Crises. Oxford University Press, Oxford. [2] Hahn, F. (1985) Money, Growth and Stability. The MIT Press, Cambridge. [3] Keynes, J.M. (1936) The General Theory of Employment, Interest and Money. Macmillan, London. [4] Ray, D.

11 Pages Posted: 12 May 2020

See all articles by Indrajit Mallick

## Indrajit Mallick

Centre for Studies in Social Sciences Calcutta

Date Written: April 17, 2020

### 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.

Keywords: Convex Set, Separation, Externality

JEL Classification: E, E0

Suggested Citation

Mallick, Indrajit, Optimal Separation of Twin Convex Sets Under Externalities (April 17, 2020). [1] Allen, F. and Douglas, G. (2007) Understanding Financial Crises. Oxford University Press, Oxford. [2] Hahn, F. (1985) Money, Growth and Stability. The MIT Press, Cambridge. [3] Keynes, J.M. (1936) The General Theory of Employment, Interest and Money. Macmillan, London. [4] Ray, D., Available at SSRN: https://ssrn.com/abstract=3578861