# American Institute of Mathematical Sciences

December  2019, 24(12): 6621-6631. doi: 10.3934/dcdsb.2019159

## On the dynamics of a durable commodity market

* Corresponding author: Jose.Canovas@upct.es

Received  April 2018 Revised  April 2019 Published  July 2019

Fund Project: Authors have been partially supported by the Grant MTM2017-84079-P from Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER).

Disequilibria phenomenon appears in the economic model of durable stocks proposed by A. Panchuck and T. Puu in [7]. In this paper, assuming that agents have the same utility functions, we give not only bounds of the disequilibrium but also prove the existence of a compact set of no-trade points such that it does not depend on the initial stock distribution. We also give a description of the nature of $\omega$–limit sets in the general case proving that disequilibrium points can be attained as limit points of orbits.

Citation: Jose S. Cánovas, María Muñoz-Guillermo. On the dynamics of a durable commodity market. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6621-6631. doi: 10.3934/dcdsb.2019159
##### References:
 [1] L. Block, J. Keesling, S. H. Li and K. Peterson, An improved algorithm for computing topological entropy, Journal of Statistical Physics, 55 (1989), 929-939.  doi: 10.1007/BF01041072.  Google Scholar [2] J. S. Cánovas, On $\omega$-limit sets of non-autonomous discrete systems, Journal of Difference Equations and Applications, 12 (2006), 95-100.  doi: 10.1080/10236190500424274.  Google Scholar [3] J. S. Cánovas and M. Muñoz-Guillermo, On the complexity of economic dynamics: An approach through topological entropy, Chaos, Solitons Fractals, 103 (2017), 163-176.  doi: 10.1016/j.chaos.2017.05.030.  Google Scholar [4] W. de Melo and S. van Strien, One-dimensional Dynamics, Springer-Verlag, 1993. doi: 10.1007/978-3-642-78043-1.  Google Scholar [5] F. Y. Edgeworth, The pure theory of international values, Econ. J., (1894), 35–50. Google Scholar [6] J. Graczyk, D. Sands and G. Światec, Metric attractors for smooth unimodal maps, Ann. Math., 159 (2004), 725-740.  doi: 10.4007/annals.2004.159.725.  Google Scholar [7] A. Panchuck and T. Puu, Dynamics of a durable commodity market involving trade at disequilibrium, Commun. Nonlinear Sci. Numer. Simulat., 58 (2018), 2-14.  doi: 10.1016/j.cnsns.2017.08.003.  Google Scholar [8] T. Puu, Disequilibrium trade and the dynamics of stock markets, in M. Faggini, A. Parziale eds. Complexity in Economics: Cutting Edge Research, Springer, (2014), 225–245.  Google Scholar [9] W. E. Ricker, Stock and recruitment, Journal of The Fisheries Research Board of Canada, 11 (1954), 559-623.  doi: 10.1139/f54-039.  Google Scholar [10] D. Singer, Stable orbits and bifurcations of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267.  doi: 10.1137/0135020.  Google Scholar

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##### References:
 [1] L. Block, J. Keesling, S. H. Li and K. Peterson, An improved algorithm for computing topological entropy, Journal of Statistical Physics, 55 (1989), 929-939.  doi: 10.1007/BF01041072.  Google Scholar [2] J. S. Cánovas, On $\omega$-limit sets of non-autonomous discrete systems, Journal of Difference Equations and Applications, 12 (2006), 95-100.  doi: 10.1080/10236190500424274.  Google Scholar [3] J. S. Cánovas and M. Muñoz-Guillermo, On the complexity of economic dynamics: An approach through topological entropy, Chaos, Solitons Fractals, 103 (2017), 163-176.  doi: 10.1016/j.chaos.2017.05.030.  Google Scholar [4] W. de Melo and S. van Strien, One-dimensional Dynamics, Springer-Verlag, 1993. doi: 10.1007/978-3-642-78043-1.  Google Scholar [5] F. Y. Edgeworth, The pure theory of international values, Econ. J., (1894), 35–50. Google Scholar [6] J. Graczyk, D. Sands and G. Światec, Metric attractors for smooth unimodal maps, Ann. Math., 159 (2004), 725-740.  doi: 10.4007/annals.2004.159.725.  Google Scholar [7] A. Panchuck and T. Puu, Dynamics of a durable commodity market involving trade at disequilibrium, Commun. Nonlinear Sci. Numer. Simulat., 58 (2018), 2-14.  doi: 10.1016/j.cnsns.2017.08.003.  Google Scholar [8] T. Puu, Disequilibrium trade and the dynamics of stock markets, in M. Faggini, A. Parziale eds. Complexity in Economics: Cutting Edge Research, Springer, (2014), 225–245.  Google Scholar [9] W. E. Ricker, Stock and recruitment, Journal of The Fisheries Research Board of Canada, 11 (1954), 559-623.  doi: 10.1139/f54-039.  Google Scholar [10] D. Singer, Stable orbits and bifurcations of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267.  doi: 10.1137/0135020.  Google Scholar
Edgeworth box in which the preferences of both agents are shown, where $\alpha = 0.4$, $\beta = 0.6$, the price $p = 2$ and the budget line is $x+p y = 1.5$. The points $(x_1,y_1)$ and $(x_2,y_2)$ represent the preferences of both agents for that static situation, since in those points they maximize their own level of satisfaction
Ricker map $f(p) = p e^{\delta(1-\alpha)}\, e^{-\delta\,\alpha\,p}$ for $\alpha = 0.3$ and $\delta = 4.7$
On the left, bifurcation diagram of price for $\alpha = 0.3$. We have computed orbits of length equal to $50000$ and we have drawn the last $250$ points. On the center and right, topological entropy and estimation of Lyapunov exponents, respectively
The shaded area represents $A_{\alpha}(p)$ for $\alpha = 0.6$ and $p = 2$ whereas the line represents the points $(x,y)$ such that for $p = 2$ the budget is $l = 1.5$
The set $A_{\alpha ,p_0}$ is constructed for $\alpha = \beta = 0.6$, $\delta = 7$ and $p_0 = \frac{1}{\alpha \delta}$ as the interior white set limited by the lines and containing the diagonal of the square
The set $A_{0.3,2.4}$ is shadowed light. Isolated lines defines the set, indicating that it cannot be approached by as a limit, and orbits stick on it
The set $A_{0.3,1}$ is shadowed light. No isolated lines give us the set, which is achieved by a limit
The region limited by the lines defined $A_{\alpha,p_0}$ for $\alpha = 0.6$ and $\delta = 5.4$, $p_0 = \frac{1}{\alpha\delta}$. We have considered initial conditions $(x_0,y_0) = (\frac{j}{100},0.1)$ for $j = 1,\ldots 100$, we have computed orbits of length $1000$ and we have drawn the last $200$ points. We can see as the projection of $\omega$–limit sets on $[0,1]^2$ are points that belong to the set $A_{\alpha,p_0}$
Fix $\delta \in [0,5]$. From left to right, in dark, stability regions are showed for parameter values $\alpha = 0.1$ and $\beta = 0.9$ (left), $\alpha = 0.3$ and $\beta = 0.7$ (middle) and $\alpha = 0.4$ and $\beta = 0.5$ (right). $\delta$ is in Y–axis and $x$ on the X–axis
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