On the equal marginal value principle: a comment.

JEL A22 * D24

The equal marginal value principle for allocating resources among multiple processes is a powerful feature in the economist's toolkit. A. J. Yates (The Equal Marginal Value Principle: A Graphical Analysis with Environmental Applications, The Journal of Economic Education,

Quasi-fixed costs feature in all production processes are independent of the output level but are only paid out by the firm if it produces a positive output. For example, electricity used for lighting: if the firm's output is zero, it does not have to provide any lighting, while, if the output is positive, a fixed amount of electricity is used for lighting.

As shown by the following example, the equal marginal value principle does not necessarily apply even if the marginal cost increases with output. Assume that the cost functions of factories 1 and 2 in a firm are: [C.sub.1]([q.sub.1])=[q.sup.2.sub.1]+10, and [C.sub.2]([q.sub.2]) [q.sup.2.sub.2] + 50, where 10 and 50 are the quasi-fixed costs of factories 1 and 2, respectively. The associated marginal cost functions, M[C.sub.i]([q.sub.i])=2[q.sub.i], i=12, are increasing with output and do not reflect the quasi-fixed costs. Let the desired production level be [q.sub.t]=8. According to the equal marginal value principle the optimal division of production, minimizing the cost of production of [q.sub.t], is [q.sub.1]=[q.sub.2]=4, and the associated cost is 92. However, there is a comer solution with [q.sub.1]=8 and [q.sub.2]= 0, in which the related cost is only 74. Moreover, as long as [q.sub.t]<10, the cost of operating factory 1 alone, [q.sup.2.sub.1] + 10, is lower than the cost of operating both factories efficiently, 0.5[q.sup.2.sub.t] + 60, and, vice versa, if [q.sub.t]> 10. Only in the latter case, it is optimal to equalize the marginal costs across the two factories.

The profit-maximizing behavior of a price-taking firm depends on the price range. It might be best for the firm to have zero output in one or both of the factories. In the above example, the minimum average costs of each of the factories are 2 [square root of 10] and 2 [square root of 50], respectively. Thus, for the output price p, p < 2 [square root of 10], the best solution for the firm is to go out of business. If 2 [square root of 10] [less than or equal to ] p < 2 [square root of 50], it is optimal to operate factory 1 alone with a production level of [q.sub.1] = 0.5 p, whereas for p [greater than or equal to] 2 [square root of 50], it is optimal to operate both factories and equalize the marginal costs. In any case, profit maximization implies cost minimization.

When the marginal costs in both factories are constant, the cost is minimized by carrying out all the production in one of the factories, but not necessarily the one with the lower marginal cost. The following example sheds light on this possible case. Let the cost functions be [C.sub.1] ([q.sub.1]) = 10 + 4[q.sub.1], and [C.sub.2]([q.sub.2]) = 20 + 2[q.sub.2], where the quasi-fixed costs are 10 and 20, respectively. Thus, [q.sub.t]=5 is the breakeven level of output, such that for [q.sub.t]<5, the cheapest mode of production is to operate factory 1 alone, and vice versa for [q.sub.t]>5.

Published online: 6 January 2007

N. Kahana

Deparment of Economics, Bar-Ilan University, 52900 Ramat-Gan, Israel

e-mail: kahanan@mail.biu.ac.il