The stability of generalised stackelberg equilibria in heterogeneous oligopoly

Abstract

In this paper we have derived sufficient conditions for the stability of the G. S. E. These conditions follow from the condition on the stability of the G. C. E., given by Okuguchi (1978). When the conditions for asymptotic stability are fulfilled, the existence of the G. S. E. may be demonstrated in the same way as the existence of the G. C. E. has been demonstrated by Okuguchi (1978), that is: the G. S. E. is the fixpoint of a contracting mapping. In the case of linear demand functions and quadratic cost functions we have found the rather strong result that the sufficient conditions for (asymptotic) stability of the G. C. E. imply the (asymptotic) stability of the G. S. E. as well. This conclusion may be surprising to followers of Stackelberg. Mostly the outcome of a Stackelberg oligopoly is considered as adisequilibrium. In this paper we have made clear that a firm maximizing expected profit and using a generalised Stackelberg strategy, like a firm using a generalised Cournot strategy, only correctly predicts inequilibrium the price of the other firms. (See also on this point Heertje and Furth (1979)). When the outcome of a G. S. E. gives one or more firms less profit than in a G. C. E. then they expect, any price change, under-taken by them, will lower their expected profit even more.