For this question, assume that the firm has a production function given by f(z1, z2) = 7z1^ 1/3z2^2/3

Answer the following:

a) Calculate the TRS1,2 between input 1 and input 2. Evaluate it at z1 = z2 = 1.

b) Using the expression for the TRS you used in a) (not its value), apply FOC to find the conditional input demand functions for input 1 and 2. What is the conditional input demand for input 1 when w1 = w2 = 4 and q = 2?

c) Using the expression for the conditional input demand functions for input 1 and 2 you found in b), find the expression for the cost function c(w1, w2, q). What is the value of the cost function at w1 = w2 = 1 and q = 18?

d) Suppose the firm faces some legal constraints of the total output it can produce. Namely, the firm can only produce q [0, 5169]. But its technology, and therefore its cost function, is not affected by this legal restriction. Using the expression for the cost function you found in c), find the maximal output level legally produced at which marginal cost is at least as high as average cost.