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Anything goes with heterogeneous, but not always with homogeneous oligopoly. (English) Zbl 1170.91436
Summary: L. C. Corchón and A. Mas-Colell [Econ. Lett. 51, No. 1, 59–65 (1996; Zbl 0875.90004)] showed that in heterogeneous oligopoly (almost) everything is possible. In order to obtain a similar result for homogeneous oligopoly, either one needs an externality in the cost function, or the reaction correspondences should fulfill a special condition.

##### MSC:
 91B50 General equilibrium theory
##### Keywords:
Cournot oligopoly; (In)Stability; (Limit)Cycles; Morse theory
Full Text:
##### References:
 [1] Amir, R.; Jin, J.Y., Cournot and bertrand equilibria compared: substitutability, complementarity, and concavity, International journal of industrial organization, 19, 303-317, (2001) [2] Arrow, K.J.; Debreu, G., Existence of an equilibrium for a competitive economy, Econometrica, 22, 265-290, (1954) · Zbl 0055.38007 [3] Bischi, G.-I.; Lamantia, F., Nonlinear duopoly games with positive cost externalities due to spillover effects, Chaos, solitons & fractals, 13, 701-721, (2002) · Zbl 1052.91008 [4] Bonanno, G., Oligopoly equilibria when firms have local knowledge of demand, International economic review, 29, 45-55, (1988) · Zbl 0689.90008 [5] Cheng, L., Comparing bertrand and Cournot equilibria: a geometric approach, The rand journal of economics, 16, 146-152, (1985) [6] Corchón, L.C., Theories of imperfectly competitive markets, (2001), Springer Berlin, (2nd revised and enlarged edition) · Zbl 0986.91010 [7] Corchón, L.C.; Mas-Colell, A., On the stability of best reply and gradient systems with applications to imperfectly competitive models, Economics letters, 51, 59-65, (1996) · Zbl 0875.90004 [8] Dastidar, K.G., Comparing Cournot and bertrand in a homogeneous product market, Journal of economic theory, 75, 205-212, (1997) · Zbl 0888.90027 [9] Debreu, G., Economies with a finite set of equilibria, Econometrica, 38, 387-392, (1970) · Zbl 0253.90009 [10] Debreu, G., Excess demand functions, Journal of mathematical economics, 1, 15-23, (1974) · Zbl 0283.90005 [11] Friedman, J.W., Oligopoly and the theory of games, (1977), North-Holland Amsterdam · Zbl 0385.90001 [12] Furth, D., Stability and instability in oligopoly, Journal of economic theory, 40, 197-228, (1986) · Zbl 0627.90011 [13] Furth, D.; Sierksma, G., The rank and eigenvalues of main diagonal perturbed matrices, Linear and multilinear algebra, 25, 191-204, (1989) · Zbl 0682.15007 [14] Goresky, M., McPherson, R., 1983. Stratified Morse Theory. Ergebnisse der Mathematik und ihre Grenzgebiete, 3. Folge, Band 14. Springer, Berlin. [15] Hartman, P., Ordinary differential equations, (1964), Wiley New York · Zbl 0125.32102 [16] Hathaway, N.J.; Rickard, J.A., Equilibria of price-setting and quantity-setting duopolies, Economics letters, 3, 133-137, (1979) [17] Hirsch, M.W.; Smale, S., Differential equations, dynamical systems and linear algebra, (1974), Academic Press New York · Zbl 0309.34001 [18] Jongen, H.Th.; Jonker, P.; Twilt, F., Nonlinear optimization in $$\mathbb{R}^n$$, (1983), Peter Lang Verlag Bern · Zbl 0527.90064 [19] Kopel, M., Simple and complex adjustment dynamics in Cournot duopoly models, Chaos, solitons & fractals, 7, 2031-2048, (1996) · Zbl 1080.91541 [20] Lorenz, H.-W., Nonlinear dynamical economics and chaotic motion, (1993), Springer Berlin, (2nd revised and enlarged edition) · Zbl 0841.90036 [21] Mantel, R., On the characterization of aggregate excess demand functions, Journal of economic theory, 7, 348-353, (1974) [22] Matsumoto, Y., 2002. An Introduction to Morse Theory. Translation of Mathematical Monographs, vol. 208. American Mathematical Society, Providence, RI. · Zbl 0990.57001 [23] McKenzie, L.W., On equilibrium in Graham’s model of world trade and other competitive systems, Econometrica, 22, 147-160, (1954) · Zbl 0055.13702 [24] McManus, M., 1964. Equilibrium, numbers and size in Cournot oligopoly. Yorkshire Bulletin of Social Science and Economic Research 16. [25] Milnor, J., Morse theory. annals of mathematics studies, vol. 51, (1963), Princeton University Press Princeton, NJ [26] Morse, M., The calculus of variation in the large, (1934), American Mathematical Society Providence, RI · JFM 60.0450.01 [27] Morse, M.; Cairns, S.S., Critical point theory in global analysis and differential topology, (1969), Academic Press New York · Zbl 0177.52102 [28] Okuguchi, K., Equilibrium prices in the bertrand and Cournot oligopolies, Journal of economic theory, 42, 128-139, (1987) · Zbl 0631.90009 [29] Palander, T.F., Konkurrens och marknadsjämvikt vid duopol och oligopol, The Scandinavian journal of economics (then ekonomisk tidskrift), 41, 124-145, (1939), 222-250 [30] Palis, J.; de Melo, W., Geometric theory of dynamical systems, (1982), Springer Berlin [31] Puu, T., Chaos in oligopoly pricing, Chaos, solitons & fractals, 1, 573-581, (1991) · Zbl 0754.90015 [32] Puu, T., Sushko, I. (Eds.), 2002. Oligopoly Dynamics. Springer, Berlin. · Zbl 1001.00069 [33] Rand, D., Exotic phenomena in games and duopoly models, Journal of mathematical economics, 5, 173-184, (1978) · Zbl 0393.90014 [34] Roberts, J.; Sonnenschein, H., On the existence of Cournot equilibrium without concave profit functions, Journal of economic theory, 13, 112-117, (1976) · Zbl 0341.90011 [35] Scarf, H., Some examples of global instability of competitive equilibrium, International economic review, 1, 157-172, (1960) · Zbl 0096.14204 [36] Singh, N.; Vives, X., Price and quantity competition in a differentiated duopoly, The rand journal of economics, 16, 546-554, (1984) [37] Smale, S., Morse inequalities for dynamical systems, Bulletin of the American mathematical society, 66, 43-49, (1960) · Zbl 0100.29701 [38] Smale, S., On gradient dynamical systems, Annals of mathematics, 74, 199-206, (1961) · Zbl 0136.43702 [39] Sonnenschein, H., Do Walras identity and continuity characterize the class of community excess demand functions?, Journal of economic theory, 6, 345-354, (1973) [40] Theocharis, R.D., On the stability of the Cournot solution on the oligopoly problem, Review of economic studies, 27, 133-134, (1960) [41] Tirole, J., The theory of industrial organization, (1989), MIT Press Cambridge [42] Vives, X., On the efficiency of bertrand and Cournot equilibria with product differentiation, Journal of economic theory, 36, 166-175, (1985) · Zbl 0596.90017
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