Patrick can manufacture toasters using the production technology q = K^2 + 4KL + L^ 2 10, where as usual, K denotes capital and L denotes hours of labor. If K^2 + 4KL + L^ 2 10 would be less than zero, q=0. He rents capital at a rate r and hires workers at an hourly wage of w. Patrick is good at his job, and always minimizes his costs of production.

(a) (3) Solve for Patrick's Marginal rate of Technical Substitution, MRT SL,K.

(b) (5) In the short run, Patrick has capital K = 5. What is his short-run cost total cost function?

(c) (3) Set up (but do not solve yet) Patrick's long-run cost minimization problem, using the Lagrangian

(d) (6) Now assume that w = 10 and r = 15, and solve (using whatever method you prefer) for the long-run total cost function.

(e) (5) Suppose that Patrick wants to produce Q = 815 units. Assuming that w = 10 and r = 15, and that in the short-run his K = K = 5 (as in the previous parts) how much cheaper is it to produce those units in the long run than in the short run? What does that tell you about the right amount of capital to use to produce 815 units?

(f) (4) Does this production function exhibit economies of scale? diseconomies of scale? constant economies of scale? To earn marks, you must explain your answer.