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Stable manifold market sequences

  • * Corresponding author: E. Camouzis

    * Corresponding author: E. Camouzis 
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  • In this article, we construct examples of discrete-time, dynamic, partial equilibrium, single product, competition market sequences, namely, $\{m_{t}\}_{t = 0}^{∞}$, in which, potentially active firms, are countably infinite, the inverse demand function is linear, and the initial market $m_0$ is null. For Cournot markets, in which, the number of firms is defined exogenously, as a finite positive integer, namely $n: n>3$, the long term behavior of the quantity supplied, into the market, by Cournot firms is not well explored and is unknown. In this article, we conjecture, that in all such cases, the Cournot equilibrium, provided that it exists, is unreachable. We construct Cournot market sequences, which might be viewed, as appropriate resource tools, through which, the "unreachability" of Cournot equilibrium points is being resolved. Our construction guidelines are, the stable manifolds of Cournot equilibrium points. Moreover, if the number of active firms, increases to infinity and the marginal costs of all active firms are identical, the aggregate market supply, increases to a competitive limit and each firm, at infinity, faces a market price equal to its marginal cost. Hence, the market sequence approaches a perfectly competitive equilibrium. In the case, where marginal costs are not identical, we show, that there exists a market sequence, $\{m_{t}\}_{t = 0}^{∞}$, which approaches an infinite dimensional Cournot equilibrium point. In addition, we construct a sequence of Cournot market sequences, namely, $\{m_{it}\}_{t = 0}^{∞}, i ≥ 1$, which, for each, $i$, approaches an imperfectly competitive equilibrium. The sequence of the equilibrium points and the double sequence, $\{m_{it}\}$, both approach, the equilibrium, at infinity, of the market sequence, $\{m_t\}$.

    Mathematics Subject Classification: Primary: 91A06; Secondary: 91A50.

    Citation:

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  • Figure 1.  The variable on the horizontal axis is time. Numerical solution of System (4.0.3), representing a sustainable market, within a time frame from 1 to 20. The lower part of the graph shows the distributions of the amounts supplied into the market by 20 firms, which become active sequentially from stage 1 to stage 20. The values of the parameters are: $A = 3$, $b = 1$, $c_j = 1-\frac{1}{(j+1)^2}, j\geq 1$

    Figure 2.  The variable on the horizontal axis is time. Numerical Solution of System (2.0.3), which represents a non-sustainable market (The Stable Manifold Hypothesis is not incorporated). Time frame varies from 1 to 9. Firms, 1-6, become active sequentially. Firms 7 and 8 are not visible. They become active only at stage 8. At stage 9, all firms exit the market. The values of the parameters are: $A = 3$, $b = 1$, $c_j = 1-\frac{1}{(j+1)^2}, j\geq 1$

    Figure 3.  Market Supply and Price related to the numerical solution presented in Figure 2

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