Question

Please show all work.

Cost Minimization:

a) If input prices are w = 4, and r = 1, and q = 4K^0.5L^0.5, what is the least-cost input combination (L*, K*) required to produce 40 units of output?

b) What is the total cost (C) associated with 40 units?

c) Suppose instead that capital was fixed at 16 units. What would be the implications for labor usage (L*) and total cost (C)?

Answer 1

q = 4K^0.5L^0.5

Total cost (C) = wL + rK = 4L + K

(a) q = 4K^0.5L^0.5 = 40, therefore K^0.5L^0.5 = 10

Total cost is minimized when MPL / MPK = w/r = 4/1 = 4

MPL = q / L = 4 x 0.5 x (K / L)^0.5

MPK = q / K = 4 x 0.5 x (L / K)^0.5

MPL / MPK = K / L = 4

K = 4L

Substituting in production function,

(4L)^0.5L^0.5 = 10

2 x L^0.5L^0.5 = 10

2 x L = 10

L = 5

K = 4 x 5 = 20

(b) Total cost = 4 x 5 + 1 x 20 = 20 + 20 = 40

(c) q = 4K^0.5L^0.5 and K = 16

q = 4(16)^0.5L^0.5 = 4 x 4 x L^0.5 = 16 x L^0.5

MPL / MPK = K / L = 4

4L = K

4L = 16

L = 4

Therefore, labor usage decreased by 1 unit (= 5 - 4).

Total cost = 4 x 4 + 16 x 1 = 16 + 16 = 32

Therefore, total cost decreased by 8 units (= 40 - 32).