The Restrictiveness of Flexible Functional Forms and the Measurement of Regulatory Constraint

Abstract

The paper by Fare and Logan raises several issues concerning the application of duality theory to models with rate of return regulation. The major point of the F-L paper appears to be that the standard specification of the A-J constraint used in our original paper is consistent with a duality between rate of return regulated cost and production. Thus, according to F-L, if a differentiable cost function is specified for a rate of return regulated firm it would not in general be true that the cost function could be used to recover the essential characteristics of a uniquely corresponding production technology. The lack of such a duality, if it were true, would have serious implications for applied modeling. F-L attempt to resolve this apparent problem by suggesting an alternate model in which duality is preserved. They point to some of the properties of their regulated cost function and further develop a translog example. This paper provides both a response to the F-L arguments and an extension of our previous results. First, we show that on the basis of the arguments of F-L, it is incorrect to argue that no duality can exist in the case of A-J regulated cost functions. Secondly, we show exactly how to construct an A-J regulated cost function which preserves full duality. Finally, we show the relationship between the specification of F-L and the A-J specification. tion is not concave1 in factor prices. They then argue, again correctly, that on the basis of their result, it will not in general be the case that (A-J) regulated cost functions are concave in factor prices. Finally, they argue that this lack of concavity implies that there can be no duality between production and cost functions under A-J constraint. By this they presumably mean that there is not a unique production technology corresponding to a regulated cost function satisfying an A-J constraint. This argument is, of course, false. Before turning to our demonstration of this fact, it is important to clarify two issues. First, the F-L argument that no duality exists is very weak. In essence, they note that, in the case of no regulation, the dual cost function will be concave and that, given their demonstration of nonconcavity in the regulated case, duality cannot be present in the A-J regulated case. This argument is only valid, if one has demonstrated ex ante that concavity is necessary for duality in the regulated case. Such a demonstration was not provided by F-L. Secondly, F-L are correct in pointing out that the derivative properties of the regulated cost function which we refer to as an extended form of Shephard's Lemma were not rigorously established. In order for an extension of Shephard's Lemma to hold, duality must be established. In this paper we demonstrate the