Economies of Scale and the Profitability of Marginal-Cost Pricing: A Note

Abstract

We are grateful to John Scott [1979] for detecting and noting the computational error in our polynomial example. However, it requires only accurate algebra to construct a correct example. In particular, C (y) = y [2 (y 1)3], for 0 < y < 1.5, a well-defined cost function for which average cost strictly decreasing and yet for which average and marginal cost somewhere coincide. Here, AC(y) = 2 (y -1)3, MC(y) = 2 (4y 1) (y 1)2, and at y = 1, both average and marginal costs are 2. Despite decreasing average costs, a marginal-costpricing firm would exactly break even at that scale of operation. The other purpose of Scott's note is ... to point up problems of using a narrowly defined production technology to define the economies of scale relevant for understanding industrial concentration [Scott, 1979, p. 741]. Of course, we agree that a measure of technological scale economies would not reflect other effects of a firm's size on its costs. However, we maintain that any analysis of the relationships between costs and output levels must include consideration of the technological returns to scale, and that the measures S and S developed in our paper [Panzar and Willig, 1977] are appropriate tools for such studies. The proof of Theorem 2 in Panzar and Willig [1977, p. 490] shows that S equals the ratio of cost to the revenue from marginal cost pricing, no matter what complex of considerations determines costs. In contrast, S defined with reference to only the technology. When costs are determined by efficient choices from the technology set, at parametric input prices, then S = S, given the regularity conditions. However, if other considerations were to influence costs, S would deviate from S, the deviation between them would be an indicator of the scale effects of the other considerations, and still, S would be an important determinant of S. For an example reflecting some of Scott's concerns, suppose that the unit price faced by a firm for each of its factors depends on the quantity of the factor that it purchases: i.e., w= w (xi). Extending a result of Fergueson [1969] to our multi-output framework, we obtain, under our regularity conditions,