A firm uses two inputs, X and Y and its production function is Q =root(xy), where here we are using x and y to represent the quantities of the two inputs.
Assume that the demand for the firm’s product is QD = 24 -4P, where P is measured in dollars. The firm has no control over the exogenous prices of its inputs, px>0 and py>0. The firm’s problem is to choose x and y to maximize its profit.
(g) Calculate the first-order condition for the firm’s problem.
(h) What are the boundary points of this problem? Under what conditions would the firm choose a boundary point? Be as specific as possible about which boundary point the firm would choose.
(i) Assuming that the solution occurs at a point that satisfies the first-order condition, find four equations that determine the solution to the firm’s problem. The four unknowns are x, y, Q, and P.