Question

1. A microbrewer desires to produce 120 cases of beer while incurring the least possible cost. The microbrewer’s production function is Q = 8K^.5L^.5 MPL = 4K^.5L^-.5 MPK = 4L^.5K^-.5 PK = $27 PL = $3 • Write the expression for the firm’s expansion path • What information does the expansion path show? • What is the optimal combination of K and L that the firm should use to produce 120 cases? • What is the cost that the firm incurs.

Answer 1

1. Firm'sm expansion path is given by MRTS = PL/PK = 3/27 = 1/9
MRTS = MPL/MPK = (4K^.5L^-.5)/4L^.5K^-.5) = K^.5+.5/L^.5+.5 = K/L
So, K/L = 1/9
So, L = 9K
Or, L - 9K = 0

2. It shows that as the firm expands production then both K and L will increase. If 1 unit of K is used, 9 units of L will be used.

3. Q = 8K^.5L^.5 = 120
So, K^.5(9K)^.5 = 120/8
So, 3K^(.5+.5) = 120/8
So, K = 120/(8*3) = 5
Thus, K = 5
L = 9K = 9*5 = 45
So, L = 45

4, Cost = PL*L + PK*K = 3(45) + (27)*(5) = 135 + 135 = 270
Thus, cost = 270