Thanks in advance.
1. If the competitive firm of Example 1 has the cost function C = 207 + 2Q; instead, then: (a) Will the production of the two goods still be technically related? (b) What will be the new optimal levels of Q1 and Qz? (c) What is the value of 12? What does this imply economically?le 1 Let us first postulate a two-product firm under circumstances of pure competition. Since with pure competition the prices of both commodities must be taken as exogenous, will be these will be denoted by Pio and Pzo, respectively. Accordingly, the firm's revenue function R1 = P10 Q1 + P20 Q2 where Q; represents the output level of the ith product per unit of time. The firm's cost function is assumed to be C = 2Q1 + Q1Q2+203 Note that AC/#Q1 = 401 + Q2 (the marginal cost of the first product) is a function not only of Q1 but also of Q2. Similarly, the marginal cost of the second product also depends, in part, on the output level of the first product. Thus, according to the assumed cost func- tion, the two commodities are seen to be technically related in production. The profit function of this hypothetical firm can now be written readily as IT = R - C = P1D Q1 + P2o Q2 - 2Q? - Q1 Q2 - 203 a function of two choice variables (Q1 and Q2) and two price parameters. It is our task to find the levels of Q and Q2 which, in combination, will maximize >. For this purpose, we first find the first-order partial derivatives of the profit function: aQ1. = P10 - 401 - Q2 (11.29) aQ2 = P20 - Q1 - 4Q2 Setting both equal to zero, to satisfy the necessary condition for a maximum, we get the two simultaneous equations 401 + Q2 = P10 Q1 + 4Q2 = P20 which yield the unique solution Q1 = - 4P10 - P20 4 P20 - P10 15 and 15 Thus, if P10 = 12 and Pzo = 18, for example, we have Q; = 2 and Q2 = 4, implying an optimal profit * * = 48 per unit of time. To be sure that this does represent a maximum profit, let us check the second-order con- dition. The second partial derivatives, obtainable by partial differentiation of (11.29), give us the following Hessian: 1H| = 211 #12 Since |Hil= -4 0, the Hessian matrix (or d'z) is negative definite, and the solution does maximize the profit. In fact, since the signs of the leading principal minors do not depend on where they are evaluated, d' z is in this case everywhere negative definite. Thus, according to (11.25), the objective function must be strictly concave, and the maximum profit just found is actually a unique absolute maximum
