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Example 2 Let us now transplant the problem of Example 1 into the setting of a monopolistic market. By virtue of this new market-structure assumption, the revenue function must be modified to reflect the fact that the prices of the two products will now vary with their output levels (which are assumed to be identical with their sales levels, no inventory accumulation being contemplated in the model). The exact manner in which prices will vary with output levels is, of course, to be found in the demand functions for the firm's two products.

Suppose that the demands facing the monopolist firm are as follows:

These equations reveal that the two commodities are related in consumption', specifically, they are substitute goods, because an increase in the price of one will raise the demand for the other. As given. (11.30) expresses the quantities demanded 0, and 02 as functions of prices, but for our present purposes it will be more convenient to have prices P, and P, expressed in terms of the sales volumes 0, and 02, that is' t0 have average-revenue functions for the two products. Since

(11.30) can be rewritten as

we may (considering 0, and 02 as parameters) apply Cramer's rule to solve for P, and p-, as follows:

P2 = 70 " 0, " 20, These constitute the desired average-revenue functions, since P, = AR, and

Consequently, the firm's total-revenue function can be written as

R = PxQx + P2Q2 = (55 - 0, - 02)Qi + (70 - 0, - 2 0 2)02 [by (11.30')] = 550, + 7002 - 20,02 - 0? - 202 If we again assume the total-cost function to be

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