12. optimal plan with cost This is a continuation of the immediately above problem. Now, however, we decompose Ralph’s choice into a cost exercise followed by a quantity exercise.

(a) Determine Ralph’s cost of producing q ≥ 0 units, C(q; P).

(b) Now that we have Ralph’s cost curve fully specified, we turn to his revenue and profit. Revenue, of course, is 15q, as each unit of output sells for 15. So Ralph’s profit is 15q−C(q; P). Determine Ralph’s profit maximizing choice of output and corresponding maximal profit.

(c) What is the connection between this choice and the shadow prices that emerge in constructing Ralph’s cost curve?