Consider a market where the price of labor is w = 2, the price of a unit of capital is r = 1. A firm with the following production functionQ=K^1/2 + L^1/2

where K is the amount of capital the firm uses, and L is the amount of labor it uses.

(a) What is the marginal rate of technical substitution for this firm at any combination of labor and capital L and K?

(b) Show that, for this firm, the cost-minimizing combination of labor and capital has 4 times as much capital as it does labor.

(c) For this firm, what is the cost-minimizing combination of labor and capital that produces a quantity of 6?

(d) What will the cost of the combination of labor and capital chosen in (c) be?

(e) For this firm, what is the cost-minimizing combination of labor and capital that produces a quantity of Q?

(f) Write down the firms cost function C(Q); that is, what is the minimum cost for this firm to produce Q units of output?