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. Google Scholar

[2]

T. Bresnahan, Duopoly models with consistent conjectures, The American Economic Review, (1981), 934-945. Google Scholar

[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.Google Scholar

[4]

L. Cabral, Experience advantages and entry dynamics, Journal of Economic Theory, 59 (1993), 403-416. doi: 10.1006/jeth.1993.1025. Google Scholar

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E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations: With Open Problems and Conjectures, Chapman & Hall/CRC Press, November 2007.Google Scholar

[6]

A. Cournot, Recherches sur les Principes Mathematiques de la Theorie des Richesses, Hachette, Paris, 1838.Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

S. Grossmman, Nash equilibrium and the industrial organization of markets with large fixed costs, Econometrica, 49 (1981), 1149-1172. doi: 10.2307/1912748. Google Scholar

[10]

O. Hart, Imperfect Competition in General Equilibrium: An Overview of Recent Work, in Frontier of Economics Oxford, 1985. Google Scholar

[11]

P. D. Klemperer and M. A. Meyer, Supply Function Equilibria in Oligopoly under Uncerainty, Econometrica, 57 (1989), 1243-1277. doi: 10.2307/1913707. Google Scholar

[12]

H. KolliasE. 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. Google Scholar

[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. Google Scholar

[14]

F. Menezes and J. Quiggin, Inferring the strategy space from market outcomes, Econometrica, RSMG Working Paper Series, 2013.Google Scholar

[15]

W. Novshek, Cournot equilibrium with free entry, Review of Economic Studies, 47 (1980), 473-486. doi: 10.2307/2297299. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

M. OkunoA. Postlewaite and J. Roberts, Oligopoly and competition in large markets, The American Economic Review, 70 (1980), 22-31. Google Scholar

[19]

T. E. Pallander, Konkurrens och marknadsjämvikt vid duopolo och oligopol, Ekonomisk Tidskrift, 41 (1939), 124-145, 222-250. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

X. Vives, Cournot and the oligopoly problem, European Economic Review, 33 (1989), 503-514. doi: 10.1016/0014-2921(89)90129-3. Google Scholar

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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. Google Scholar

[2]

T. Bresnahan, Duopoly models with consistent conjectures, The American Economic Review, (1981), 934-945. Google Scholar

[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.Google Scholar

[4]

L. Cabral, Experience advantages and entry dynamics, Journal of Economic Theory, 59 (1993), 403-416. doi: 10.1006/jeth.1993.1025. Google Scholar

[5]

E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations: With Open Problems and Conjectures, Chapman & Hall/CRC Press, November 2007.Google Scholar

[6]

A. Cournot, Recherches sur les Principes Mathematiques de la Theorie des Richesses, Hachette, Paris, 1838.Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

S. Grossmman, Nash equilibrium and the industrial organization of markets with large fixed costs, Econometrica, 49 (1981), 1149-1172. doi: 10.2307/1912748. Google Scholar

[10]

O. Hart, Imperfect Competition in General Equilibrium: An Overview of Recent Work, in Frontier of Economics Oxford, 1985. Google Scholar

[11]

P. D. Klemperer and M. A. Meyer, Supply Function Equilibria in Oligopoly under Uncerainty, Econometrica, 57 (1989), 1243-1277. doi: 10.2307/1913707. Google Scholar

[12]

H. KolliasE. 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. Google Scholar

[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. Google Scholar

[14]

F. Menezes and J. Quiggin, Inferring the strategy space from market outcomes, Econometrica, RSMG Working Paper Series, 2013.Google Scholar

[15]

W. Novshek, Cournot equilibrium with free entry, Review of Economic Studies, 47 (1980), 473-486. doi: 10.2307/2297299. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

M. OkunoA. Postlewaite and J. Roberts, Oligopoly and competition in large markets, The American Economic Review, 70 (1980), 22-31. Google Scholar

[19]

T. E. Pallander, Konkurrens och marknadsjämvikt vid duopolo och oligopol, Ekonomisk Tidskrift, 41 (1939), 124-145, 222-250. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

X. Vives, Cournot and the oligopoly problem, European Economic Review, 33 (1989), 503-514. doi: 10.1016/0014-2921(89)90129-3. Google Scholar

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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