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A Computational Analysis of the Core of a Trading Economy with Three Competitive Equilibria and a Finite Number of Traders

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  • Alok Kumar
  • Martin Shubik

Abstract

In this paper we examine the structure of the core of a trading economy with three competitive equilibria as the number of traders (N) is varied. We also examine the sensitivity of the multiplicity of equilibria and of the core to variations in individual initial endowments. Computational results show that the core first splits into two pieces at N = 5 and then splits a second time into three pieces at N = 12. Both of these splits occur not at a point but as a contiguous gap. As N is increased further, the core shrinks by N = 600 with essentially only the 3 competitive equilibria remaining. We find that the speed of convergence of the core toward the three competitive equilibria is not uniform. Initially, for small N, it is not of the order 1/N but when N is large, the convergence rate is approximately of the order 1/N. Small variations in the initial individual endowments along the price rays to the competitive equilibria make the respective competitive equilibrium (CE) unique and once a CE becomes unique, it remains so for all allocations on the price ray. Sensitivity analysis of the core reveals that in the large part of the endowment space where the competitive equilibrium is unique, the core either converges to the single CE or it splits into two segments, one of which converges to the CE and the other disappears.

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Bibliographic Info

Paper provided by Yale School of Management in its series Yale School of Management Working Papers with number ysm223.

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Date of creation: 01 Sep 2001
Date of revision: 01 Nov 2003
Handle: RePEc:ysm:somwrk:ysm223

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Web page: http://icf.som.yale.edu/
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Keywords: Core; Multiple competitive equilibria; Speed of convergence; Sensitivity Analysis.;

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  1. Debreu, Gerard, 1975. "The rate of convergence of the core of an economy," Journal of Mathematical Economics, Elsevier, vol. 2(1), pages 1-7, March.
  2. Shapley, Lloyd S & Shubik, Martin, 1977. "An Example of a Trading Economy with Three Competitive Equilibria," Journal of Political Economy, University of Chicago Press, vol. 85(4), pages 873-75, August.
  3. Shapley, L S, 1975. "An Example of a Slow-Converging Core," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 16(2), pages 345-51, June.
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