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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

##### MSC:
 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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