The Process Analysis Alternative to Statistical Cost Functions: Comment
Since my recent article, which cast considerable doubt on the statistical estimation approach to the derivation of cost functions, I have been expecting a comment from proponents of that approach. Instead, I am pleasantly surprised to be replying to disciples of the process analysis approach. J. R. Marsden, D. E. Pingry, and A. Whinston (MPW) feel that a linear programming application of process analysis to petroleum refining has basically the same disadvantages as the statistical cost function technique, but that these could be overcome with the adoption of their particular approach. They propose a more general formulation utilizing non-linear programming techniques and allowing for nonconvex production technologies. Let us begin by considering their four objections to my application of process analysis to petroleunm refining. First, MPW assert that the test of the classical cost function assumptions (i.e., marginal costs slope upward and average costs are U-shaped) was not really a test at all but proceeded directly from the convexity assumptions of the linear programming model of the refinery. Certainly, the fixed capital process constraints imply a finite output and a rising marginal cost curve, but the relevant question is over what output range do marginal costs rise. MPW apparently feel that because of the convexity assumption marginal costs must necessarily rise over a broad output range. To demonstrate the error in their assertion, one need only examine some output range from b74) to b(n+l) over which the basis x* does not change. Since the basis is unchanged, the dual solution vector y* will similarly not change, thereby proving that short-run marginal costs (given by the jth element of y*) are constant over the given output range. As an example, Figure 1 of my paper illustrates a case where a basis change did not occur over the output range 8.4 to 8.9 MMB/D and marginal costs are constant. Furthermore, the dots in Figure 1, indicating basis changes, suggest that even after basis changes, marginal costs need not necessarily increase as the basis changes. Therefore, under this standard linear programming problem where the production processes are convex, short-run marginal costs can either rise in a step-wise manner or remain horizontal over the output range up to the full utilization of the capital stock at which point marginal costs become vertical. Either rising short-run marginal costs or an inverted L-shaped short-run marginal cost can be obtained assuming a standard convex production technology. Since the same result may be found in statistical cost studies (i.e., constant short-run marginal costs over the observed output range), the results in both Figures 1 and 2 indicating a rising marginal cost function over a broad output range certainly do not follow from the convexitv assumptions as MPW assert. Secondly, MPW are apparently disturbed because the short-run marginal cost function as drawn in Figure 1 does not change in a step function manner. They argue that the use of parametric programming would have revealed these steps and other useful information regarding capacity limitations. Contrary to MPW's assertion, parametric programming with UNIVAC's Omega package was utilized which reports the activities entering and exiting the basis at each basis change. As indicated in footnote 10, page 49, the particular parametrics option chosen does not report the complete solution vector at each basis change within the 6 increment to the output constraint bj, but rather reports the solution values for the first basis change * Department of economics, University of Pennsylvania and the University of Houston.
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