Abstract
During the past several years it has become increasingly common to use mathematical programming methods for deriving economic equilibria of supply and demand. Well-defined approaches exist for the case of a single firm (monopoly) and for the case of many firms (perfect competition). In this paper a certain family of convex programs is formulated to determine equilibria for the case of a few firms (oligopoly). Solutions to this family of convex programs are shown to be Nash equilibria in the formal sense ofN person games. This equivalence leads to a mathematical programming-based algorithm for determining an oligopolistic market equilibrium.
Similar content being viewed by others
References
H.W. Brock and D.M. Nesbitt, “Large-scale energy planning models: A methodological analysis”, Decision Analysis Group, Stanford Research Institute, Menlo Park, CA (May 1977).
A.V. Fiacco, “Continuity of the optimal value function under the Mangasarian—Fromovitz constraint qualification”, Technical Report Serial T-432, George Washington University, Washington, DC (1980).
W.W. Hogan, “Energy policy models for project independence”,Computers and Operations Research 2 (1975) 251–271.
C.E. Lemke and J.T. Howson Jr., “Equilibrium points of bimatrix games”,Journal of the Society for Industrial and Applied Mathematics 12 (1964) 413–423.
J. Nash, “Non-cooperative games”,Annnals of Mathematics 54 (1951) 286–295.
K. Okuguchi, “Expectations and stability in oligopoly models” in:Lecture Notes in Economics and Mathematical Systems 138 (Springer, Berlin, 1976).
J. Rosenmuller, “On a generalization of the Lemke-Howson algorithm to non-cooperativeN-person games”,SIAM Journal on Applied Mathematics 21 (1971) 74–79.
R.B. Rovinsky, C.A. Shoemaker and M.J. Todd, “Determining optimal use of resources among regional producers using different levels of cooperation”,Operations Research 28 (1980) 859–866.
S.W. Salant, “Imperfect competition in the international energy market: A computerized Nash—Cournot model”, paper presented at the ORSA/TIMS National Meeting, Washington, DC (May 1980).
H. Scarf,The computation of economic equilibria (Yale University Press, New Haven, CT, 1973).
M. Sobel, “An algorithm for a game equilibrium point”, Discussion Paper #7035, CORE, Universite Catholique de Louvain, Louvain-la-Neuve (November 1970).
M. Spence, “The implicit maximization of a function of monopolistically competitive markets”, Working Paper, Institute of Economic Research, Harvard University, Cambridge, MA (1976).
F. Szidarovsky and S. Yakowitz, “A new proof of the existence and uniqueness of the Cournot equilibrium”,International Economic Review 18 (1977) 787–789.
R. Wilson, “Computing equilibria ofN-person games”,SIAM Journal on Applied Mathematics 21 (1971) 80–87.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Murphy, F.H., Sherali, H.D. & Soyster, A.L. A mathematical programming approach for determining oligopolistic market equilibrium. Mathematical Programming 24, 92–106 (1982). https://doi.org/10.1007/BF01585096
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01585096