# American Institute of Mathematical Sciences

2015, 2(1): 65-87. doi: 10.3934/jdg.2015.2.65

## Discrete time dynamic oligopolies with adjustment constraints

 1 Department of Economics, University of Colorado, Boulder, CO 80309-0256, United States 2 Department of Economics, Society, Politics, Università degli Studi di Urbino, 61029 Urbino, Italy 3 Department of Applied Mathematics, University of Pécs, Pécs, 7624, Hungary

Received  October 2014 Revised  March 2015 Published  June 2015

A classical $n$-firm oligopoly is considered first with linear demand and cost functions which has a unique equilibrium. We then assume that the output levels of the firms are bounded in a sense that they are unwilling to make small changes, the output levels are bounded from above, and if the optimal output level is very small then the firms quit producing, which are realistic assumptions in real economies. In the first part of the paper, the best responses of the firms are determined and the existence of infinitely many equilibria is verified. The second part of the paper examines the global dynamics of the duopoly version of the game. In particular we study the stability of the system, the bifurcations which can occur and the basins of attraction of the existing attracting sets, as a function of the speed of adjustment parameter.
Citation: Chrystie Burr, Laura Gardini, Ferenc Szidarovszky. Discrete time dynamic oligopolies with adjustment constraints. Journal of Dynamics & Games, 2015, 2 (1) : 65-87. doi: 10.3934/jdg.2015.2.65
##### References:
 [1] R. Amir, Cournot oligopoly and the theory of supermodular games,, Games and Economic Behavior, 15 (1996), 132. doi: 10.1006/game.1996.0062. [2] R. Amir and V. E. Lambson, On the effects of entry in Cournot markets,, Review of Economic Studies, 67 (2000), 235. doi: 10.1111/1467-937X.00129. [3] G. I. Bischi, C. Chiarella, M. Kopel and F. Szidarovszky, Nonlinear Oligopolies: Stability and Bifurcations,, Springer-Verlag, (2010). doi: 10.1007/978-3-642-02106-0. [4] G. I. Bischi, L. Gardini and U. Merlone, Impulsivity in binary choices and the emergence of periodicity,, Discrete Dynamics in Nature and Society, (2009). doi: 10.1155/2009/407913. [5] A. Dal Forno, L. Gardini and U. Merlone, Ternary choices in repeated games and border collision bifurcations,, Chaos Solitons & Fractals, 45 (2012), 294. doi: 10.1016/j.chaos.2011.12.003. [6] R. Day, Complex Economic Dynamics,, MIT Press, (1994). [7] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications,, Applied Mathematical Sciences 163, (2008). [8] T. Bresnahan and V. Ramey, Output Fluctuations at the Plant Level,, Quaterly Journal of Economics, 109 (1994), 593. doi: 10.2307/2118415. [9] C. Fershtman and M. Kamien, Dynamic duopolistic competition with sticky prices,, Econometrica, 55 (1987), 1151. doi: 10.2307/1911265. [10] L. Gardini, I. Sushko and A. Naimzada, Growing through chaotic intervals,, Journal of Economic Theory, 143 (2008), 541. doi: 10.1016/j.jet.2008.03.005. [11] L. Gardini, U. Merlone and F. Tramontana, Inertia in binary choices: Continuity breaking and big-bang bifurcation points,, Journal of Economic Behavior & Organization, 80 (2011), 153. doi: 10.1016/j.jebo.2011.03.004. [12] W. Huang and R. Day, Chaotically switching bear and bull markets: The derivation of stock price distributions from behavioral rules,, in Nonlinear Dynamics and Evolutionary Economics (eds. R. Day, (1993). [13] W. Novshek, On the existence of Cournot equilibrium,, Review of Economic Studies, 52 (1985), 85. doi: 10.2307/2297471. [14] H. E. Nusse and J. A. Yorke, Border-collision bifurcations including period two to period three for piecewise smooth systems,, Physica D, 57 (1992), 39. doi: 10.1016/0167-2789(92)90087-4. [15] H. E. Nusse and J. A. Yorke, Border-collision bifurcation for piecewise smooth one-dimensional maps,, Int. J. Bifurcation Chaos, 5 (1995), 189. doi: 10.1142/S0218127495000156. [16] K. Okuguchi, Expectations and Stability in Oligopoly Models,, Springer-Verlag, (1976). [17] K. Okuguchi and F. Szidarovszky, The Theory of Oligopoly with Multi-product Firms,, Lecture Notes in Economics and Mathematical Systems, (1990). doi: 10.1007/978-3-662-02622-9. [18] T. Puu and I. Sushko (eds.), Oligopoly Dynamics, Models and Tools,, Springer Verlag, (2002). [19] T. Puu and I. Sushko (eds.), Business Cycle Dynamics, Models and Tools,, Springer Verlag, (2006). [20] D. Radi, L. Gardini and V. Avrutin, The role of constraints in a segregation model: The symmetric case,, Chaos, 66 (2014), 103. doi: 10.1016/j.chaos.2014.05.009. [21] D. Radi, L. Gardini and V. Avrutin, The role of constraints in a segregation model: The asymmetric Case,, Discrete Dynamics in Nature and Society, (2014). doi: 10.1155/2014/569296. [22] M. Simaan and T. Takayama, Game theory applied to dynamic duopoly problems with production constraint,, Automatica, 14 (1978), 161. doi: 10.1016/0005-1098(78)90022-5. [23] I. Sushko and L. Gardini, Degenerate bifurcations and border collisions in piecewise smooth 1D and 2D maps,, Int. J. Bif. and Chaos, 20 (2010), 2045. doi: 10.1142/S0218127410026927. [24] I. Sushko, L. Gardini and K. Matsuyama, Superstable credit cycles and u-sequence,, Chaos Solitons & Fractals, 59 (2014), 13. doi: 10.1016/j.chaos.2013.11.006. [25] F. Tramontana, F. Westerhoff and L. Gardini, On the complicated price dynamics of a simple one-dimensional discontinuous financial market model with heterogeneous interacting traders,, J. Econ. Behav. Organ., 74 (2010), 187. doi: 10.1016/j.jebo.2010.02.008. [26] F. Tramontana, L. Gardini and F. Westerhoff, Heterogeneous speculators and asset price dynamics: Further results from a one-dimensional discontinuous piecewise-linear model,, Computational Economics, 38 (2011), 329. doi: 10.1007/s10614-011-9284-9. [27] Z. T. Zhusubaliyev and E. Mosekilde, Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems,, World Scientific, (2003).

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##### References:
 [1] R. Amir, Cournot oligopoly and the theory of supermodular games,, Games and Economic Behavior, 15 (1996), 132. doi: 10.1006/game.1996.0062. [2] R. Amir and V. E. Lambson, On the effects of entry in Cournot markets,, Review of Economic Studies, 67 (2000), 235. doi: 10.1111/1467-937X.00129. [3] G. I. Bischi, C. Chiarella, M. Kopel and F. Szidarovszky, Nonlinear Oligopolies: Stability and Bifurcations,, Springer-Verlag, (2010). doi: 10.1007/978-3-642-02106-0. [4] G. I. Bischi, L. Gardini and U. Merlone, Impulsivity in binary choices and the emergence of periodicity,, Discrete Dynamics in Nature and Society, (2009). doi: 10.1155/2009/407913. [5] A. Dal Forno, L. Gardini and U. Merlone, Ternary choices in repeated games and border collision bifurcations,, Chaos Solitons & Fractals, 45 (2012), 294. doi: 10.1016/j.chaos.2011.12.003. [6] R. Day, Complex Economic Dynamics,, MIT Press, (1994). [7] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications,, Applied Mathematical Sciences 163, (2008). [8] T. Bresnahan and V. Ramey, Output Fluctuations at the Plant Level,, Quaterly Journal of Economics, 109 (1994), 593. doi: 10.2307/2118415. [9] C. Fershtman and M. Kamien, Dynamic duopolistic competition with sticky prices,, Econometrica, 55 (1987), 1151. doi: 10.2307/1911265. [10] L. Gardini, I. Sushko and A. Naimzada, Growing through chaotic intervals,, Journal of Economic Theory, 143 (2008), 541. doi: 10.1016/j.jet.2008.03.005. [11] L. Gardini, U. Merlone and F. Tramontana, Inertia in binary choices: Continuity breaking and big-bang bifurcation points,, Journal of Economic Behavior & Organization, 80 (2011), 153. doi: 10.1016/j.jebo.2011.03.004. [12] W. Huang and R. Day, Chaotically switching bear and bull markets: The derivation of stock price distributions from behavioral rules,, in Nonlinear Dynamics and Evolutionary Economics (eds. R. Day, (1993). [13] W. Novshek, On the existence of Cournot equilibrium,, Review of Economic Studies, 52 (1985), 85. doi: 10.2307/2297471. [14] H. E. Nusse and J. A. Yorke, Border-collision bifurcations including period two to period three for piecewise smooth systems,, Physica D, 57 (1992), 39. doi: 10.1016/0167-2789(92)90087-4. [15] H. E. Nusse and J. A. Yorke, Border-collision bifurcation for piecewise smooth one-dimensional maps,, Int. J. Bifurcation Chaos, 5 (1995), 189. doi: 10.1142/S0218127495000156. [16] K. Okuguchi, Expectations and Stability in Oligopoly Models,, Springer-Verlag, (1976). [17] K. Okuguchi and F. Szidarovszky, The Theory of Oligopoly with Multi-product Firms,, Lecture Notes in Economics and Mathematical Systems, (1990). doi: 10.1007/978-3-662-02622-9. [18] T. Puu and I. Sushko (eds.), Oligopoly Dynamics, Models and Tools,, Springer Verlag, (2002). [19] T. Puu and I. Sushko (eds.), Business Cycle Dynamics, Models and Tools,, Springer Verlag, (2006). [20] D. Radi, L. Gardini and V. Avrutin, The role of constraints in a segregation model: The symmetric case,, Chaos, 66 (2014), 103. doi: 10.1016/j.chaos.2014.05.009. [21] D. Radi, L. Gardini and V. Avrutin, The role of constraints in a segregation model: The asymmetric Case,, Discrete Dynamics in Nature and Society, (2014). doi: 10.1155/2014/569296. [22] M. Simaan and T. Takayama, Game theory applied to dynamic duopoly problems with production constraint,, Automatica, 14 (1978), 161. doi: 10.1016/0005-1098(78)90022-5. [23] I. Sushko and L. Gardini, Degenerate bifurcations and border collisions in piecewise smooth 1D and 2D maps,, Int. J. Bif. and Chaos, 20 (2010), 2045. doi: 10.1142/S0218127410026927. [24] I. Sushko, L. Gardini and K. Matsuyama, Superstable credit cycles and u-sequence,, Chaos Solitons & Fractals, 59 (2014), 13. doi: 10.1016/j.chaos.2013.11.006. [25] F. Tramontana, F. Westerhoff and L. Gardini, On the complicated price dynamics of a simple one-dimensional discontinuous financial market model with heterogeneous interacting traders,, J. Econ. Behav. Organ., 74 (2010), 187. doi: 10.1016/j.jebo.2010.02.008. [26] F. Tramontana, L. Gardini and F. Westerhoff, Heterogeneous speculators and asset price dynamics: Further results from a one-dimensional discontinuous piecewise-linear model,, Computational Economics, 38 (2011), 329. doi: 10.1007/s10614-011-9284-9. [27] Z. T. Zhusubaliyev and E. Mosekilde, Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems,, World Scientific, (2003).
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