11. optimal plan without cost Ralph has a one product firm that uses two factors of production to make some good or service. Call the two factors capital, denoted K ≥ 0, and labor, denoted L ≥ 0. Also denote the output quantity by q. To make q ≥ 0 units of output, Ralph must supply labor and capital in such quantities that the following constraint is satisfied:
q ≤
√KL Output sells for 15 per unit, while capital costs 9 per unit and labor costs 4 per unit. Ralph’s profit, of course, is how much he receives from customers less how much he pays for capital and labor, or 15q − 9K − 4L. The catch is Ralph cannot use capital in excess of 7 units. Putting everything together, your task is to solve the following maximization exercise:
max q≥0,K≥0,L≥0 15q − 9K − 4L s.t. q ≤
√KL;K ≤ 7