Application: Pricing and Cost Changes

3.   Application: Pricing and Cost Changes

The preceding theorem permits us to make a number of predictions about the behavior of the profit-maximizing firm and to set up some normative "operations research" rules for its operation. We can determine not only the optimal output, but also the profit-maximizing price with the aid of the demand curve for the product of the firm. For, given the opti­mal output, we can find out from the demand curve what price will permit the company to sell this quantity, and that is necessarily the opti­mal price. In Figure 1, where the optimal output is OQ_m we see that the corresponding price is Q_mP_m where point P_m is the point on the demand curve above Q_m (note that P_m is not the point of intersection of the marginal cost and the marginal revenue curves).

It was shown in the last section of Chapter 4 how our theorem can also enable us to predict the effect of a change in tax rates or some other change in cost on the firm's output and pricing. We need merely determine how this change shifts the marginal cost curve to find the new profit-maximizing price-output combination by finding the new point of intersection of the marginal cost and marginal revenue curves. Let us recall one particular result for use later in this chapter—the theorem about the effects of a change in fixed costs. It will be remembered that a change in fixed costs never has any effect on the firm's marginal cost curve because marginal fixed cost is always zero (by definition, an additional unit of output adds nothing to fixed costs). Hence, if the profit-maximizing firm's rents, its total assessed taxes, or some other fixed cost increases, there will be no change in the output-price level at which its marginal cost equals its marginal revenue. In other words, the profit-maximizing firm will make no price or output changes in response to any increase or decrease in its fixed costs! This rather unexpected result is certainly not in accord with common business practice and requires some further comment which will be supplied presently.

4.   Extension: Multiple Products and Inputs

The firm's output decisions- are normally more complicated, even in principle, than the preceding decisions suggest. Almost all companies pro­duce a variety of products and these various commodities typically compete for the firm's investment funds and its productive capacity. At any given time there are limits to what the company can produce, and often, if it decides to increase its production of product x, this must be done at the expense of product y. In other words, such a company cannot simply expand the output of x to its optimum level without taking into account the effects of this decision on the output of y.

For a profit-maximizing decision which takes both commodities into account we have a marginal rule which is a special case of Rule 2 of Chapter 3:

Any limited input (including investment funds) should be allocated between the two outputs x and у in such a way that the marginal profit yield of the input, i, in the production of x equals the marginal profit yield of the input in the production of y.

 If the condition is violated the firm cannot be maximizing its profits, because the firm can add to its earnings simply by shifting some of г out of the product where it obtains the lower return and into the manufacture of the other.

Stated another way, this last theorem asserts that if the firm is maxi­mizing its profits, a reduction in its output of x by an amount which is worth, say, $5, should release just exactly enough productive capacity, C, to permit the output of у to be increased $5 worth. For this means that the marginal return of the released capacity is exactly the same in the produc­tion of either x or y, which is what the previous version of this rule asserted.³

Still another version of this result is worth describing: Suppose the price of each product is fixed and independent of output levels. Then we require that the marginal cost of each output be proportionate to its price, i.e., that where P_x and MC_X are, respectively, the price and the marginal cost of x, etc.

In this discussion we have considered only the output decisions of a profit-maximizing firm. Of course, the firm has other decisions to make. In particular, it must decide on the amounts of its inputs including its marketing inputs (advertising, sales force, etc.). There are similar rules for these decisions, as discussed in Chapter 11 and in Chapter 17, Section 6. The main result here is that profit maximization requires for any inputs г and j

where MP_t represents the marginal profit contribution of input г and Pi is its price, etc.

Having discussed the consequences of profit maximization, let us see now what difference it makes if the firm adopts an alternative objective, one to which we have already alluded — the maximization of the value of its sales (total revenue) under the requirement that the firm's profits not fall short of some given minimum level.