Question

given production function q=12L^0.15K. a firm has 20000 to spend on cars. wage is $8 and capital cost is $30. assuming the firm wants to produce as many cars with 20000, how many employees and capital should it employ? how many cars should the firm produce

Answer 1

Given,

q = 12L^0.15 K

Hence,

MP(L) = derivative of q with respect to L

            = (d/dL) [12L^0.15 K]

            = (12 × 0.15)KL^(0.15 -1)

            = 1.8K / L^0.85

MP(K) = derivative of q with respect to K

            = (d/dK) [12L^0.15 K]

            = 12L^0.15

Now the equality,

[MP(L) / w] = [MP(K) / r]

[(1.8K / L^0.85) / 8] = [(12L^0.15) / 30]

Or, 1.8K / 8L^0.85 = 12L^0.15 / 30

Or, 1.8K × 30 = 12L^0.15 × 8L^0.85

Or, 54K = 96L^(0.15 + 0.85)

Or, 54K = 96L^1

Or, 54K = 96L

Or, L = 54K/96

Or, L = 0.5625K ……. (eq 1)

Budget constraint,

L × w + K × r = 20,000

8L + 30K = 20,000

Now by putting the (eq 1)

8 × 0.5625K + 30K = 20,000

Or, 4.5K + 30K = 20,000

Or, 34.5K = 20,000

Or, K = 20,000/34.5 = 580 (rounded to whole number)

Now by putting it in (eq 1)

L = 0.5625 × 580 = 326

Therefore,

q = 12L^0.15 K

   = 12 × 326^0.15 × 580

   = 12 × 2.38222 × 580

   = 16,580

Answers:

Employees = 326

Capital = 580

Cars = 16,580