# American Institute of Mathematical Sciences

April 2018, 5(2): 165-185. doi: 10.3934/jdg.2018010

## Stable manifold market sequences

 National and Kapodistrian University of Athens, Department of Economics, Sofokleous 1, 10559, Athens, Greece

* Corresponding author: E. Camouzis

Received  October 2017 Revised  February 2018 Published  March 2018

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, $\{m_t\}$.

Citation: E. Camouzis, H. Kollias, I. Leventides. Stable manifold market sequences. Journal of Dynamics & Games, 2018, 5 (2) : 165-185. doi: 10.3934/jdg.2018010
##### References:
 [1] L. Brand, Advanced Calculus, An Introduction to Classical Analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1955. [2] T. Bresnahan, Duopoly models with consistent conjectures, The American Economic Review, (1981), 934-945. [3] M. H. Busse, When Supply and Demand Just Won't Do: Using the Equilibrium Locus to Think about Comparative Statics, mimeo, Northwestern University, 2012. [4] L. Cabral, Experience advantages and entry dynamics, Journal of Economic Theory, 59 (1993), 403-416. doi: 10.1006/jeth.1993.1025. [5] E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations: With Open Problems and Conjectures, Chapman & Hall/CRC Press, November 2007. [6] A. Cournot, Recherches sur les Principes Mathematiques de la Theorie des Richesses, Hachette, Paris, 1838. [7] P. S. Dasgupta and Y. Ushio, On the rate of convergence of oligopoly equilibria in large markets: An example, Econom. Lett., 8 (1981), 13-17. doi: 10.1016/0165-1765(81)90086-0. [8] J. Fraysse and M. Moreaux, Cournot equilibrium in large markets under increasing returns, Econom. Lett., 8 (1981), 217-220. doi: 10.1016/0165-1765(81)90069-0. [9] S. Grossmman, Nash equilibrium and the industrial organization of markets with large fixed costs, Econometrica, 49 (1981), 1149-1172. doi: 10.2307/1912748. [10] O. Hart, Imperfect Competition in General Equilibrium: An Overview of Recent Work, in Frontier of Economics Oxford, 1985. [11] P. D. Klemperer and M. A. Meyer, Supply Function Equilibria in Oligopoly under Uncerainty, Econometrica, 57 (1989), 1243-1277. doi: 10.2307/1913707. [12] H. Kollias, E. Camouzis and Y. Leventides, Global analysis of solutions on the cournot-theoharis duopoly with variable marginal costs, Journal of Dynamics and Games, 4 (2017), 25-39. [13] A. Mas-Colell, Walrasian equilibria as limits of noncooperative equilibria. Ⅰ. Mixed strategies, J. Econom. Theory, 30 (1983), 153-170. doi: 10.1016/0022-0531(83)90098-4. [14] F. Menezes and J. Quiggin, Inferring the strategy space from market outcomes, Econometrica, RSMG Working Paper Series, 2013. [15] W. Novshek, Cournot equilibrium with free entry, Review of Economic Studies, 47 (1980), 473-486. doi: 10.2307/2297299. [16] W. Novshek, Perfectly competitive markets as the limits of cournot markets, Journal Of Economic Theory, 35 (1985), 72-82. doi: 10.1016/0022-0531(85)90062-6. [17] W. Novshek and H. Sonnenschein, Walrasian equilibria as limits of noncooperative equilibria. Ⅱ. Pure strategies, J. Econom. Theory, 30 (1983), 171-187. doi: 10.1016/0022-0531(83)90099-6. [18] M. Okuno, A. Postlewaite and J. Roberts, Oligopoly and competition in large markets, The American Economic Review, 70 (1980), 22-31. [19] T. E. Pallander, Konkurrens och marknadsjämvikt vid duopolo och oligopol, Ekonomisk Tidskrift, 41 (1939), 124-145, 222-250. [20] R. D. Theocharis, On the Stability of the Cournot Solution on the oligopoly problem, Review of Economic Studies, 27 (1959), 133-134. doi: 10.2307/2296135. [21] Y. Ushio, Cournot equilibrium with free entry: The case of decreasing average cost functions, Rev. Econom. Stud., 50 (1983), 347-354. doi: 10.2307/2297420. [22] X. Vives, Cournot and the oligopoly problem, European Economic Review, 33 (1989), 503-514. doi: 10.1016/0014-2921(89)90129-3.

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##### References:
 [1] L. Brand, Advanced Calculus, An Introduction to Classical Analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1955. [2] T. Bresnahan, Duopoly models with consistent conjectures, The American Economic Review, (1981), 934-945. [3] M. H. Busse, When Supply and Demand Just Won't Do: Using the Equilibrium Locus to Think about Comparative Statics, mimeo, Northwestern University, 2012. [4] L. Cabral, Experience advantages and entry dynamics, Journal of Economic Theory, 59 (1993), 403-416. doi: 10.1006/jeth.1993.1025. [5] E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations: With Open Problems and Conjectures, Chapman & Hall/CRC Press, November 2007. [6] A. Cournot, Recherches sur les Principes Mathematiques de la Theorie des Richesses, Hachette, Paris, 1838. [7] P. S. Dasgupta and Y. Ushio, On the rate of convergence of oligopoly equilibria in large markets: An example, Econom. Lett., 8 (1981), 13-17. doi: 10.1016/0165-1765(81)90086-0. [8] J. Fraysse and M. Moreaux, Cournot equilibrium in large markets under increasing returns, Econom. Lett., 8 (1981), 217-220. doi: 10.1016/0165-1765(81)90069-0. [9] S. Grossmman, Nash equilibrium and the industrial organization of markets with large fixed costs, Econometrica, 49 (1981), 1149-1172. doi: 10.2307/1912748. [10] O. Hart, Imperfect Competition in General Equilibrium: An Overview of Recent Work, in Frontier of Economics Oxford, 1985. [11] P. D. Klemperer and M. A. Meyer, Supply Function Equilibria in Oligopoly under Uncerainty, Econometrica, 57 (1989), 1243-1277. doi: 10.2307/1913707. [12] H. Kollias, E. Camouzis and Y. Leventides, Global analysis of solutions on the cournot-theoharis duopoly with variable marginal costs, Journal of Dynamics and Games, 4 (2017), 25-39. [13] A. Mas-Colell, Walrasian equilibria as limits of noncooperative equilibria. Ⅰ. Mixed strategies, J. Econom. Theory, 30 (1983), 153-170. doi: 10.1016/0022-0531(83)90098-4. [14] F. Menezes and J. Quiggin, Inferring the strategy space from market outcomes, Econometrica, RSMG Working Paper Series, 2013. [15] W. Novshek, Cournot equilibrium with free entry, Review of Economic Studies, 47 (1980), 473-486. doi: 10.2307/2297299. [16] W. Novshek, Perfectly competitive markets as the limits of cournot markets, Journal Of Economic Theory, 35 (1985), 72-82. doi: 10.1016/0022-0531(85)90062-6. [17] W. Novshek and H. Sonnenschein, Walrasian equilibria as limits of noncooperative equilibria. Ⅱ. Pure strategies, J. Econom. Theory, 30 (1983), 171-187. doi: 10.1016/0022-0531(83)90099-6. [18] M. Okuno, A. Postlewaite and J. Roberts, Oligopoly and competition in large markets, The American Economic Review, 70 (1980), 22-31. [19] T. E. Pallander, Konkurrens och marknadsjämvikt vid duopolo och oligopol, Ekonomisk Tidskrift, 41 (1939), 124-145, 222-250. [20] R. D. Theocharis, On the Stability of the Cournot Solution on the oligopoly problem, Review of Economic Studies, 27 (1959), 133-134. doi: 10.2307/2296135. [21] Y. Ushio, Cournot equilibrium with free entry: The case of decreasing average cost functions, Rev. Econom. Stud., 50 (1983), 347-354. doi: 10.2307/2297420. [22] X. Vives, Cournot and the oligopoly problem, European Economic Review, 33 (1989), 503-514. doi: 10.1016/0014-2921(89)90129-3.
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: $A = 3$, $b = 1$, $c_j = 1-\frac{1}{(j+1)^2}, j\geq 1$
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: $A = 3$, $b = 1$, $c_j = 1-\frac{1}{(j+1)^2}, j\geq 1$
Market Supply and Price related to the numerical solution presented in Figure 2
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