Stability and instability in oligopoly
type: Item
instance of: scholarly article
title: Stability and instability in oligopoly
zbMATH Open document ID: 0627.90011
DOI: 10.1016/0022-0531(86)90072-4
author: Dave Furth
published in: Journal of Economic Theory
publication date: 1986-01-01T00:00:00Z
review text: 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_ …
Mathematics Subject Classification ID: 91B24
Mathematics Subject Classification ID: 91A12
Mathematics Subject Classification ID: 91A40
Mathematics Subject Classification ID: 91B50
Mathematics Subject Classification ID: 91A80
zbMATH DE Number: 4020790
zbMATH Keywords: stability properties of the equilibria
zbMATH Keywords: oligopoly
zbMATH Keywords: dynamical systems
reviewed by: Dan Kovenock
MaRDI profile type: Publication
full work available at URL: 0022-0531(86)90072-4
OpenAlex ID: W2033378983
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description: scientific article; zbMATH DE number 4020790