Question

Suppose that the price of labor (w) is $ 6 and the price of capital (r) is $ 18. Assume the firm has a budget of $ 20,000 to spend on labor and capital. Also assume that the firm’s production function is Q = 4L^.5 6K^.5 .

A. Write the equation for this firm’s isocost line.

B. What is the optimal combination of inputs for this firm? Show your work.

C.        At that input combination, how much output can this firm produce?

Answer 1

A. Equation of isocost line is wL + rK = Budget
So, 6L + 18K = 20,000

B. Optimal combination is determined where MRTS = w/r = 6/18 = 1/3
So, MRTS = MPL/MPK = (∂Q/∂L)/(∂Q/∂K) = .5(4L^.5-16K^.5)/.5(4L^.56K^.5-1) = L^-.5K^.5/L^.5K^-.5 = K^.5+.5/L^.5+.5 = K/L
So, K/L = 1/3
So, L = 3K

Now, 6L + 18K = 20,000
So, 6(3K) + 18K = 20,000
So, 18K + 18K = 36K = 20,000
So, K = 20,000/36 = 555.56

L = 3K = 3(555.56) = 1,666.68

Thus, L = 1,666.68; K = 555.56

C. Q = (4L^.5 )(6K^.5) = 4(1666.68^.5)6(555.56)^.5 = 4(40.82)*(6)(23.57)
So, Q = 23,091.06