Analysis on the forward market equilibrium model
Introduction
Allaz and Vila [1] presented a forward market model with identical Cournot duopolists. They showed that even with certainty and perfect foresight, forward trading can improve market efficiency. Each of the producers will sell forward so as to make them worse off and make consumers better off than would be the case if the forward market did not exist. This phenomenon is similar to that of the prisoners’ dilemma.
In the forward market model mentioned above, the inverse demand function is affine and the producers have the same linear cost function. Hence, one can solve for a Nash equilibrium of the forward market in closed form; see [1]. However, it is not clear that a Nash equilibrium would exist if the producers had nonidentical cost functions. Indeed, one can construct a simple example of Cournot duopolists with nonidentical linear cost functions for which the Allaz–Vila approach is not valid. If fact, the two-period forward market model belongs to a new class of mathematical programs called Equilibrium Problems with Equilibrium Constraints (EPECs), where each player solves a nonconvex mathematical program with equilibrium constraints (MPEC) [2], and a Nash equilibrium for an EPEC may not exist because of the nonconvexity in each player's problem. Pang and Fukushima [4] give a simple numerical example of such a case.
We observe that the mathematical structure of the two-period forward market model is similar to that of the multiple leader Stackelberg model analyzed by Sherali [6]. The similarity becomes evident when new variables are introduced for spot market sales. An immediate result is that we are able to adapt the analysis in [6] to establish the existence of a forward market equilibrium for M producers with nonidentical linear cost functions.
The remainder of this paper is organized as follows. In the next section, we give a general formulation for the two-period forward market model with M producers. In Section 3, we reformulate the forward market equilibrium model by introducing new variables on spot market sales. Assuming that the inverse demand function is affine and allowing the producers to have nonidentical linear cost functions, we establish the existence of a forward market equilibrium. In Section 4, we use the sequential nonlinear complementarity (SNCP) algorithm proposed in [8] to compute a forward market Nash equilibrium for a three-producer example.
Section snippets
The two-period forward market model
We use the following notation throughout this paper:number of producers producer i's forward sales in the first period the total forward sales in the first period the production of producer in the second period producer i's spot sales in the second period the total spot sales in the second period the cost function of producer i the payoff function of producer i from the spot market in the second period the overall profit function of producer i the forward price (or
Existence of a forward market equilibrium
In [1], Allaz and Vila showed that one can solve for the forward market Nash equilibrium in closed form when demand and cost functions are affine and the producers have the same cost function, i.e., , for . In particular, in the case of Cournot duopolists, and the demand function , with , the unique forward market equilibrium outcome is
The purpose of this paper is to establish an existence theorem for the forward market equilibrium
An EPEC approach for computing a forward market equilibrium
The computation of a forward market equilibrium involves solving a family of concave maximization problems, as defined in (7). However, the objective functions are nonsmooth in these problems because is piecewise linear concave in on ; see Theorem 4. One might encounter difficulties in solving these nonsmooth problems with nonlinear programming solvers. An alternative to avoid the nonsmooth objective functions is to formulate the forward market equilibrium problem as an
Acknowledgments
I am grateful to Professor Richard W. Cottle for his invaluable guidance and support. I thank Dr. Kenneth L. Judd, Professor Michael A. Saunders and Professor Hanif D. Sherali for their comments. This research was conducted when the author was at Stanford University and was supported by the Electric Power Research Institute (EPRI) and the Jerome Kaseberg Doolan Fellowship.
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