This question is meant to give you a better understanding of the optimality condition for profit maximization. It will also give you practice in doing unconstrained optimiza- tion with two variables. It is slightly different from what you've seen in class, but the translation of words into math is easy. Remember profit is total revenue minus total cost. Once you have done the translation, its very easy optimization math.

(a) Consider a firm that has two plants (or factories), each with a different cost function. Factory A can produce output at cost

C_A(y_A) =y²_A

and factory B can produce output at cost

C_B(y_B) = exp(y_B)

This firm can sell the total output from both plants (in only one market) at the price p.

How should the firm optimally allocate production between the two plants? Interpret your answer in terms of marginal costs and marginal revenue.

[Note: SOC satisfied for the optimization problem.]

(b) Consider a firm that can sell its output in the US as well as Canada. The output price in US isp_Uand in Canada isp_C. Supposep_U> p_C. This firm only has one plant that produces at cost

C(y) =y².

How should this firm allocate production from its only plant between the two markets? Think intuitively if you are stuck - what would you do if you managed this firm but had never attended intermediate micro? [Math note:(a+b)² a²+b²]