Question

If input prices are w = 4, and r = 1, and q = 4K^0.5L^0.5

a. What is the least-cost input combination required to produce 40 units of output?

b. Suppose instead that capital was fixed at 16 units. What would be the implications for labor usage and total cost?

Answer 1

q = 4K0.5L0.5

Total cost, C = wL + rK

C = 4L + K

(a)

Cost is minimized when MPL / MPK = w / r = 4 / 1 = 4

MPL = \partial q / \partial L = 4 x 0.5 x (K / L)0.5 = 2 x (K / L)0.5

MPK = \partial q / \partial K = 4 x 0.5 x (L / K)0.5 = 2 x (L / K)0.5

MPL / MPK = K / L = 4

K = 4L

When q = 40, we get

4 x (4L)0.5 x L0.5 = 40

(4)0.5(L)0.5 x L0.5 = 10

2 x L = 10

L = 5

K = 4 x 5 = 20

[C = 4 x 5 + 20 = 20 + 20 = 40]

(b) If K = 16, we get

4 x (16)0.5 x L0.5 = 40

4 x 4 x L0.5 = 40

L0.5 = 2.5

L = 12.5

TC = 4 x 12.5 + 16 = 50 + 16 = 56

So, when capital is fixed, more labor is required and total cost is higher.