On the dynamics of a durable commodity market

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