Exercise 5: A firm's production function is Q(K, L) = [bK^+ (1 b)L^ ]^/ , with 1.

1. Show that the function Q(K, L) is homogeneous of degree .

2. Compute the marginal rate of technical substitution, and show that it is independent of .

3. Show that the marginal products of Q(K, L) satisfies the relation K Q/K + L Q/L = Q(K, L). Now suppose that = 1, b = 1/2, = 1, and = 2. Consider point (K₀, L₀) = (0, 2).

4. Estimate the overall effect on Q when K increases by 0.15 from K₀ and L increases by 0.1 from L0.

5. Now the firm would like to decrease L from L₀ by 0.2, to keep the total output unchanged, the firm wants to know how much K should be increased from K₀. Can we use the implicit function theorem to estimate the change of K in this case? Explain your answer.