Stability and instability in oligopoly
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.
scientific article; zbMATH DE number 4020790
scientific article