# American Institute of Mathematical Sciences

doi: 10.3934/jdg.2021002

## A dynamic for production economies with multiple equilibria

 1 Universidad Autónoma de San Luis Potosí, Facultad de Economía, San Luis Potosí, 78213, México 2 Instituto Potosino de Investigación Científica y Tecnológica, División de Control y Sistemas Dinámicos, San Luis Potosí, 78216, México

* Corresponding author: Humberto Muñiz

Received  September 2020 Revised  December 2020 Published  January 2021

In this article, we extend to private ownership production economies, the results presented by Bergstrom, Shimomura, and Yamato (2009) on the multiplicity of equilibria for the special kind of pure-exchanges economies called Shapley-Shubik economies. Furthermore, a dynamic system that represents the changes in the distribution of the firms on the production branches is introduced. For the first purpose, we introduce a particular, but large enough, production sector to the Shapley-Shubik economies, for which a simple technique to build private-ownership economies with a multiplicity of equilibria is developed. In this context, we analyze the repercussions on the behavior of the economy when the number of possible equilibria changes due to rational decisions on the production side. For the second purpose, we assume that the rational decisions on the production side provoke a change in the distribution of the firms over the set of branches of production.

Citation: Elvio Accinelli, Humberto Muñiz. A dynamic for production economies with multiple equilibria. Journal of Dynamics & Games, doi: 10.3934/jdg.2021002
##### References:
 [1] E. Accinelli and E. Covarrubias, Evolution and jump in a Walrasian framework, J. Dyn. Games, 3 (2016), 279-301.  doi: 10.3934/jdg.2016015.  Google Scholar [2] T. C. Bergstrom, K.-I. Shimomura and T. Yamato, Simple economies with multiple equilibria, B. E. J. Theor. Econ., 9 (2009), 31pp. doi: 10.2202/1935-1704.1609.  Google Scholar [3] E. Dierker, Two remarks on the number of equilibria of an economy, Econometrica, 40 (1972), 951-953.  doi: 10.2307/1912091.  Google Scholar [4] T. Hens and B. Pilgrim, The index-theorem, in General Equilibrium Foundations of Finance, Theory and Decision Library, 33, Springer, Boston, MA, 2002. doi: 10.1007/978-1-4757-5317-2_4.  Google Scholar [5] T. J. Kehoe, An index theorem for general equilibrium models with production, Econometrica, 48 (1980), 1211-1232.  doi: 10.2307/1912179.  Google Scholar [6] T. J. Kehoe, Multiplicty of equilbria and compartive statics, Quart. J. Econom., 100 (1985), 119-147.  doi: 10.2307/1885738.  Google Scholar [7] A. Mas-Colell, The Theory of General Economic Equilbrium. A Differential Approach, Econometric Society Monographs, 9, Cambridge University Press, Cambridge, 1989.   Google Scholar [8] P. A. Samuelson, Foundations of Economic Analysis, Harvard University Press, Cambridge, Mass., 1947.   Google Scholar [9] L. Shapley and M. Shubik, An example of a trading economy with three competitive equilibria, J. Political Economy, 85 (1997), 873-875.  doi: 10.1086/260607.  Google Scholar

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##### References:
 [1] E. Accinelli and E. Covarrubias, Evolution and jump in a Walrasian framework, J. Dyn. Games, 3 (2016), 279-301.  doi: 10.3934/jdg.2016015.  Google Scholar [2] T. C. Bergstrom, K.-I. Shimomura and T. Yamato, Simple economies with multiple equilibria, B. E. J. Theor. Econ., 9 (2009), 31pp. doi: 10.2202/1935-1704.1609.  Google Scholar [3] E. Dierker, Two remarks on the number of equilibria of an economy, Econometrica, 40 (1972), 951-953.  doi: 10.2307/1912091.  Google Scholar [4] T. Hens and B. Pilgrim, The index-theorem, in General Equilibrium Foundations of Finance, Theory and Decision Library, 33, Springer, Boston, MA, 2002. doi: 10.1007/978-1-4757-5317-2_4.  Google Scholar [5] T. J. Kehoe, An index theorem for general equilibrium models with production, Econometrica, 48 (1980), 1211-1232.  doi: 10.2307/1912179.  Google Scholar [6] T. J. Kehoe, Multiplicty of equilbria and compartive statics, Quart. J. Econom., 100 (1985), 119-147.  doi: 10.2307/1885738.  Google Scholar [7] A. Mas-Colell, The Theory of General Economic Equilbrium. A Differential Approach, Econometric Society Monographs, 9, Cambridge University Press, Cambridge, 1989.   Google Scholar [8] P. A. Samuelson, Foundations of Economic Analysis, Harvard University Press, Cambridge, Mass., 1947.   Google Scholar [9] L. Shapley and M. Shubik, An example of a trading economy with three competitive equilibria, J. Political Economy, 85 (1997), 873-875.  doi: 10.1086/260607.  Google Scholar
Example 1: $\omega=\left((5,2),(2,3)\right)$, $f(x)=5.5x-\dfrac{1}{2}x^2$, equilibria prices at $p_1=1/2$, $p_2=1$, $p_3=2$
Example 2: $r = 7/9$ equilibrium price at $p_1 = 0.5$ is singular
Example 4: equilibria prices at $p\approx 0.154648,$ $p=1$ and $p=6.194385$
Example 5: equilibria prices at $p_1\approx 0.1792472498915,$ $p_2=2$ and $p_3\approx 2.6915745984313$
Example 6: equilibria prices at $p_1\approx 0.125623594624,$ $p_2=1$ and $p_3\approx 1.0747180810635$. Distribution $(10,40)$
Example 6: equilibria prices at $p_1\approx 0.17025278062395,$ $p_2 \approx 0.6155470462368$ and $p_3\approx 1.5223658675626$. Distribution $(11,39)$
Example 7: Blue line represents the demand function $x_2^1(p) = \phi_1(p^{-1})$, red line represent right hand side of equation 26. Distribution $(12,38)$
Example 7: Blue line represents the demand function $x_2^1(p)=\phi_1(p^{-1})$, red line represent right hand side of equation 26. Distribution $(14,36)$
Profit analysis example 7: Blue line represents $\pi_1(p)$, while red line represent $\pi_2(p)$
Profit table for example 7
Utility table for example 7
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