Article Contents
Article Contents

# On the dynamics of a durable commodity market

• * Corresponding author: Jose.Canovas@upct.es

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.

Mathematics Subject Classification: Primary: 37N40; Secondary: 26A18, 37B20.

 Citation:

• Figure 1.  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

Figure 2.  Ricker map $f(p) = p e^{\delta(1-\alpha)}\, e^{-\delta\,\alpha\,p}$ for $\alpha = 0.3$ and $\delta = 4.7$

Figure 3.  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

Figure 4.  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 $l = 1.5$

Figure 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

Figure 6.  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

Figure 7.  The set $A_{0.3,1}$ is shadowed light. No isolated lines give us the set, which is achieved by a limit

Figure 8.  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 $A_{\alpha,p_0}$

Figure 9.  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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