Separability Testing in Production Economics: Comment

Abstract

Pope and Hallam (PH) proposed a test for indirect separability based on the rank of an appropriately defined matrix involving the bordered Hessian of the production function. test is interesting and important, but its presentation obscures the main idea because the discussion, in the context of flexible functional forms, is marred by a crucial error. Blackorby, Primont, and showed that a test for weak separability of flexible functional forms imposes strong and unwanted restrictions on a model's structure. After the publication of their paper, the consensus has been that a test for weak separability of flexible functional forms cannot be performed with the available functions. PH refer to this issue by stating: This (test) moderates in some cases the criticism by Blackorby, Primont, and Russell (p. 143). The reader is tantalized by this statement, but the rest of the paper does not shed any further light upon it. PH discuss necessary and sufficient rank conditions for indirect separability but present also a lengthy elaboration of a necessary condition. The former set of conditions obviously provides a stronger test, and thus we focus our discussion upon it. Furthermore, PH's necessary rank condition as applied to the empirical example is wrongly stated and is inferior with respect to the stronger test. We think that the contribution of PH can be stated in the following terms: The principal interest in the rank condition rests upon the possibility of testing the necessary and sufficient conditions for indirect separability. When not rejected, PH's rank condition, rank [fxfl](N,, Nf) = 1, implies indirect separability and, at least in two cases (generalized Leontief and quadratic functions), does not imply direct separability. Conversely, (in flexible functional forms), direct separability implies the rank condition for indirect separability, and this is the reason why this second test is subject to the criticism of Blackorby, Primont, and Russell. Thus, the rank condition applied to dual functions is noteworthy. However, its application in the context of flexible functional forms suggests that, for the translog function, its test is equivalent to the separability test originally proposed by Berndt and Christensen while, in the case of the generalized Leontief and quadratic functions, the two tests are substantially different. The numerical example chosen by PH is not suitable for illustrating the working of the necessary and sufficient rank condition which, furthermore, is erroneously implemented. The interesting question, therefore, is, are there other flexible functional forms (aside from the generalized Leontief and quadratic functions) for which the rank condition for indirect separability does not coincide with the conditions for direct separability? An answer to this question is conjectured in the conclusion.