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Stability and instability in oligopoly. (English) Zbl 0627.90011
This paper analyzes the stability properties of the equilibria of two types of models of oligopoly. The first is a homogeneous good quantity setting model with inverse demand given by \(p=f(x_ 1,...,x_ n)\) where \(x_ i\) is the output of firm i. The second is a heterogeneous good price setting model with demand \(x_ i=f_ i(p_ 1,...,p_ n)\) for \(i=1,...,n\), where \(p_ i\) is the price of firm i. The cost function for firm in i both cases is \(C_ i=C_ i(x_ i).\)
In the homogeneous good market there is an upper bound M, on the output of a given firm and \(V=\{x\in R^ n|\) for all \(i=1,..,n:\) \(x_ i\in [0,M]\}\). Let J be an open interval containing [0,nM] and U an open neighborhood in \(R^ n\) of \([0,N]^ n\), where N is a positive number whose importance is indicated below. Define \(p_{-1}=(p_ 1,...,p_{i- 1},p_{i+1},...,p_ n)\). The following assumptions are made: (i) \(f\in C^{\infty}(J,R)\) and for all \(i=1,...,n\), \(f_ i\in C^{\infty}(U,R)\); (ii) \(f'<0\) and for all \(i=1,...,n\), \(\partial f_ i/\partial p_ i<0\); (iii) \(f(n,M)<0<f(0)\) and for all \(i=1,...,n\) and all \(p_{-1}\in [0,N]^{n-1}\), \(f_ i(N;p_{-1})<0<f_ i(0;p_{- 1})\); (iv) for all \(i=1,...,n\); \(C_ i\in C^{\infty}(R,R)\), \(C_ i(x_ i)>0\) whenever \(x_ i>0\); (v) \(C_ i'(x_ i)>0.\)
For the homogeneous market V is the feasible region. For the heterogeneous market define \(V(f_ 1,...,f_ n)\) to be the feasible region. The profit of firm i in the homogeneous (heterogeneous) oligopoly is \[ \pi_ i(x_ 1,...,x_ n)=x_ if(x_ i+x_{-i})-C_ i(x_ i),\quad (\pi_ i(p_ 1,...,p_ n)=p_ if_ i(p_ i,p_{-i})-C_ i(f_ i(p_ i;p_{-1})). \] Define for all \(i: \phi_ i(x_ 1,...,x_ n)=\partial \pi_ i/\partial x_ i\) and \(\psi_ i(p_ 1,...,p_ n)=\partial \pi_ i/\partial p_ i\). This generates vector fields \(\phi =(\phi_ 1,...,\phi_ n):\) \(V\to R^ n\) and \(\psi =(\psi_ 1,...,\psi_ n):\) \(V(f_ 1,...,f_ n)\to R^ n\). These vector fields define dynamical systems through the differential equations \(\dot x_ i=\phi_ i(x_ i;x_{-i})\) and \(\dot p_ i=\psi_ i(p_ i;p_{-i})\), \(i=1,...,n\). With the further assumption, (vi) the vector fields \(\phi\) and \(\psi\) have no cycles; the author proves the following main results for regular oligopolies.
Theorem: When \(\phi\) (\(\psi)\) points inwards at the boundary of V \((V(f_ 1,...,f_ n))\), then (i) there is at least one stable equilibrium (a nondegenerate critical point of the vector field); (ii) the number of equilibria is odd; (iii) when for all \(x\in V\) \((p\in V(f_ 1,...,f_ n))\) \[ sign(\frac{\partial (\phi_ 1,...,\phi_ n)}{\partial (x_ 1,...,x_ n)})=(-1)^ n,\quad (sign(\frac{\partial (\psi_ 1,...,\psi_ n)}{\partial (p_ 1,...,p_ n)})=(-1)^ n) \] there is a unique stable equilibrium.
Reviewer: D.Kovenock

91B24 Microeconomic theory (price theory and economic markets)
91A12 Cooperative games
91A40 Other game-theoretic models
91B50 General equilibrium theory
91A80 Applications of game theory
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[1] ()
[2] Bonnanno, G; Zeeman, E.C, Limited knowledge of demand and oligopoly equilibria, J. econ. theory, 35, 276-283, (1985) · Zbl 0597.90014
[3] Braess, D, Morse theorie für berandette mannigfaltigkeiten, Math. ann., 208, 133-148, (1974) · Zbl 0263.58005
[4] Debreu, G, Economies with a finite set of equilibria, Econometrica, 387-392, (1970) · Zbl 0253.90009
[5] Fisher, F.M, The stability of the Cournot oligopoly solution: the effects of speeds of adjustments and increasing marginal costs, Rev. econ. stud., 28, 125-135, (1961)
[6] Furth, D, The oligopoly game, Dissertation, (January 1982), Amsterdam
[7] Golubitsky, M; Guillemin, V, Stable mappings and their singularities, (1973), Springer-Verlag New York/Heidelberg/Berlin · Zbl 0294.58004
[8] Hahn, F.H, The stability of the Cournot oligopoly solution, Rev. econ. stud., 29, 329-331, (1962)
[9] Hirsh, M.W, A proof of the non-contractibility of a cell onto its boundary, (), 364-365 · Zbl 0113.16704
[10] Hosomatsu, Y, A note on the stability conditions in Cournot’s dynamic market solution, when neither the actual market demand functions nor the production level of of the rivals are known, Rev. econ. stud., 36, 117-121, (1969) · Zbl 0181.23108
[11] Jongen, H.Th, On non-convex optimization, Dissertation, (1977), Enschede · Zbl 0393.90078
[12] Jongen, H.Th; Jonker, P; Twilt, F, ()
[13] Milnor, J.W, Topology from the differentiable viewpoint, (1965), Virginia Univ. Press Charlottesville, Va., · Zbl 0136.20402
[14] Neudecker, H, Cournot’s dynamic market solution and Hosomatsu’s lemma: an alternative proof, Rev. econ. stud., 37, 447-448, (1970)
[15] Al-Nowaihi, A; Levine, P.L, The stability of the Cournot oligopoly model, a reassessment, J. econ. theory, 35, 307-321, (1985) · Zbl 0598.90019
[16] Okuguchi, K, Expectation and stability in oligopoly models, () · Zbl 0718.90016
[17] Quandt, R.E, On the stability of price adjusting oligopoly, Southern econ. J., 33, 332-336, (1967)
[18] Rand, D, Thresholds in Pareto sets, J. math. econ., 3, 139-154, (1976) · Zbl 0357.90016
[19] Seade, J, The stability of Cournot revisited, J. econ. theory, 23, 15-27, (1980) · Zbl 0453.90018
[20] Smale, S, Morse inequalities for a dynamical system, Bull. amer. math. soc., 66, 43-49, (1960) · Zbl 0100.29701
[21] Smale, S, On gradient dynamical systems, Ann. of math., 74, 199-206, (1961) · Zbl 0136.43702
[22] Smale, S, Pareto theory with constrains, J. math. econ., 1, 213-221, (1974) · Zbl 0357.90010
[23] Smale, S, An approach to the analysis of dynamic processes in economic systems, (), 363-367
[24] Theocharis, R.D, On the stability of the Cournot solution on the oligopoly problem, Rev. econ. stud., 27, 133-134, (1959)
[25] Wan, Y.H, On local Pareto optima, J. math. econ., 2, 35-42, (1975) · Zbl 0309.90049
[26] \scE. C. Zeeman, “Applications of Catastrophe Theory,” in Proc. Tokyo Manifold Conference, pp. 11-23. · Zbl 0311.58004
[27] Rouche, N; Habets, P; Laloy, M, Stability theory by Liapunov’s direct method, (1977), Springer-Verlag New York/Berlin · Zbl 0364.34022
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