# 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:

show all references

##### References:
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
 [1] Rabah Amir, Igor V. Evstigneev. A new perspective on the classical Cournot duopoly. Journal of Dynamics & Games, 2017, 4 (4) : 361-367. doi: 10.3934/jdg.2017019 [2] Hongming Yang, C. Y. Chung, Xiaojiao Tong, Pingping Bing. Research on dynamic equilibrium of power market with complex network constraints based on nonlinear complementarity function. Journal of Industrial & Management Optimization, 2008, 4 (3) : 617-630. doi: 10.3934/jimo.2008.4.617 [3] Shimin Li, Jaume Llibre. On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-17. doi: 10.3934/dcdsb.2019111 [4] Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001 [5] Nickolas J. Michelacakis. Strategic delegation effects on Cournot and Stackelberg competition. Journal of Dynamics & Games, 2018, 5 (3) : 231-242. doi: 10.3934/jdg.2018015 [6] P. Daniele, S. Giuffrè, S. Pia. Competitive financial equilibrium problems with policy interventions. Journal of Industrial & Management Optimization, 2005, 1 (1) : 39-52. doi: 10.3934/jimo.2005.1.39 [7] Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091 [8] De-Jun Feng, Antti Käenmäki. Equilibrium states of the pressure function for products of matrices. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 699-708. doi: 10.3934/dcds.2011.30.699 [9] Iraklis Kollias, Elias Camouzis, John Leventides. Global analysis of solutions on the Cournot-Theocharis duopoly with variable marginal costs. Journal of Dynamics & Games, 2017, 4 (1) : 25-39. doi: 10.3934/jdg.2017002 [10] Lukas Neumann, Christian Schmeiser. A kinetic reaction model: Decay to equilibrium and macroscopic limit. Kinetic & Related Models, 2016, 9 (3) : 571-585. doi: 10.3934/krm.2016007 [11] Elvio Accinelli, Bruno Bazzano, Franco Robledo, Pablo Romero. Nash Equilibrium in evolutionary competitive models of firms and workers under external regulation. Journal of Dynamics & Games, 2015, 2 (1) : 1-32. doi: 10.3934/jdg.2015.2.1 [12] Michal Kočvara, Jiří V. Outrata. Inverse truss design as a conic mathematical program with equilibrium constraints. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1329-1350. doi: 10.3934/dcdss.2017071 [13] Jin Ma, Xinyang Wang, Jianfeng Zhang. Dynamic equilibrium limit order book model and optimal execution problem. Mathematical Control & Related Fields, 2015, 5 (3) : 557-583. doi: 10.3934/mcrf.2015.5.557 [14] Yunan Wu, T. C. Edwin Cheng. Classical duality and existence results for a multi-criteria supply-demand network equilibrium model. Journal of Industrial & Management Optimization, 2009, 5 (3) : 615-628. doi: 10.3934/jimo.2009.5.615 [15] T.C. Edwin Cheng, Yunan Wu. Henig efficiency of a multi-criterion supply-demand network equilibrium model. Journal of Industrial & Management Optimization, 2006, 2 (3) : 269-286. doi: 10.3934/jimo.2006.2.269 [16] Sin-Man Choi, Ximin Huang, Wai-Ki Ching. Minimizing equilibrium expected sojourn time via performance-based mixed threshold demand allocation in a multiple-server queueing environment. Journal of Industrial & Management Optimization, 2012, 8 (2) : 299-323. doi: 10.3934/jimo.2012.8.299 [17] Liping Zhang, Soon-Yi Wu, Shu-Cherng Fang. Convergence and error bound of a D-gap function based Newton-type algorithm for equilibrium problems. Journal of Industrial & Management Optimization, 2010, 6 (2) : 333-346. doi: 10.3934/jimo.2010.6.333 [18] Hongyu He, Naohiro Kato. Equilibrium submanifold for a biological system. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1429-1441. doi: 10.3934/dcdss.2011.4.1429 [19] Alain Chenciner. The angular momentum of a relative equilibrium. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1033-1047. doi: 10.3934/dcds.2013.33.1033 [20] Luis C. Corchón. Imperfectly competitive markets, trade unions and inflation: Do imperfectly competitive markets transmit more inflation than perfectly competitive ones? A theoretical appraisal. Journal of Dynamics & Games, 2018, 5 (3) : 189-201. doi: 10.3934/jdg.2018012

Impact Factor: