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# A dynamic for production economies with multiple equilibria

• * Corresponding author: Humberto Muñiz
• 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.

Mathematics Subject Classification: Primary:91B50, 91B55;Secondary:91B38.

 Citation: • • Figure 1.  Example 1: $\omega=\left((5,2),(2,3)\right)$, $f(x)=5.5x-\dfrac{1}{2}x^2$, equilibria prices at $p_1=1/2$, $p_2=1$, $p_3=2$

Figure 2.  Example 2: $r = 7/9$ equilibrium price at $p_1 = 0.5$ is singular

Figure 3.  Example 4: equilibria prices at $p\approx 0.154648,$ $p=1$ and $p=6.194385$

Figure 4.  Example 5: equilibria prices at $p_1\approx 0.1792472498915,$ $p_2=2$ and $p_3\approx 2.6915745984313$

Figure 5.  Example 6: equilibria prices at $p_1\approx 0.125623594624,$ $p_2=1$ and $p_3\approx 1.0747180810635$. Distribution $(10,40)$

Figure 6.  Example 6: equilibria prices at $p_1\approx 0.17025278062395,$ $p_2 \approx 0.6155470462368$ and $p_3\approx 1.5223658675626$. Distribution $(11,39)$

Figure 7.  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 $(12,38)$

Figure 8.  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 $(14,36)$

Figure 9.  Profit analysis example 7: Blue line represents $\pi_1(p)$, while red line represent $\pi_2(p)$

Figure 10.  Profit table for example 7

Figure 11.  Utility table for example 7

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