Question

Suppose the hourly wage is $20 and the price of each unit of capital is $20. The price of output is constant at $50 per unit. The production function is f(E,K) = E1/5 K1/2, How much labor and capital should the firm employ in the long run? How much profit will the firm earn?

Answer 1

Wage rate is 20 and rental price of capital is 20. This makes w/r = 20/20 = 1. From production function , we

have MRTS = MPE/MPK = (1/5)E^-4/5 K^1/2 divided by (1/2)E^1/5 K^-1/2 or 0.4K/E. At the optimum input mix,

we have MRTS = w/r or 0.4K/E = 1. This implies E = 0.4K

Use this in production function Q = (0.4K)^1/5 K^1/2 or K = (Q/0.4^0.2)^(1/0.7)

This is simplified to K = 1.3Q^1.4286 and E = 0.52Q^1.4286

Cost function is C = wE + rK

C = 20* 0.52Q^1.4286 + 20*1.3Q^1.4286

C = 36.4Q^1.4286

Profit = revenue - cost = 50Q - 36.4Q^1.4286

Profit is maximum when marginal profit is 0

50 = 36.4*1.4286 x Q^0.4286

Q = 4.82

Optimum amount of K = 12.3 units and E = 4.9

Profit = 50*4.82 - 12.3*20 - 4.9*20 = -103.