Analysis on the forward market equilibrium model

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Abstract

We establish the existence results for the Allaz–Vila [B. Allaz, J.-L. Vila, Cournot competition, forward markets and efficiency, J. Econ. Theory 59 (1993) 1–16] forward market equilibrium model when the M producers have different linear cost functions. We also consider an example with three asymmetric producers. The computational results supplement the conclusion in that the forward trading would increase market efficiency.

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:

Mnumber of producers
fiproducer i's forward sales in the first period
Qfthe total forward sales in the first period
xithe production of producer i in the second period
siproducer i's spot sales in the second period
Qsthe total spot sales in the second period
ci(·)the cost function of producer i
ui(·)the payoff function of producer i from the spot market in the second period
πi(·)the overall profit function of producer i
pf(·)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., ci(xi)=cxi, for i=1,,M. In particular, in the case of Cournot duopolists, and the demand function p(q)=a-q, with 0<c<a, the unique forward market equilibrium outcome isx1=x2=2(a-c)5;f1=f2=a-c5;p=c+a-c5.

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 si(Qf) is piecewise linear concave in Qf on {Qf:si(Qf)>0}; 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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