The firm and its objectives
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 Px and MCX 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 MPt represents the marginal profit contribution of input г and Pi is its price, etc.
