Question

c) Davy metal company produces brass fittings. Davy's engineers estimate the production function represented below as relevant for their long run capital-labor decisions. where O-annual output measured in kilograms L- labor measured in person hours Kcapital measured in machine hours Davy's employees are relatively highly skilled and earn S15 per hour. The firm estimates a rental charge of $50 per hour on capital. Davy forecasts annual costs of S500,000 per year, measured in dollars. i Derive the firm's marginal product of labor (MP) and marginal product of capital (MPk) equations (4 marks)

Please explain in details with step-by-step solution, Thank you so much

Answer 1

Q = 500L^0.6K^0.8

(i)

MPL = $\partial$ Q/ $\partial$ L = (500 x 0.6 x K^0.8) / L^0.4 = (300 x K^0.8) / L^0.4

MPK = $\partial$ Q/ $\partial$ K = (500 x 0.8 x L^0.6) / K^0.2 = (400 x L^0.6) / K^0.2

(ii)

Total cost (C) = wL + rK

$500,000 = $15L + $50K

100,000 = 3L + 10K

Cost is minimized when (MPL/MPK) = w/r = $15/$50 = 3/10

MPL/MPK = [(300 x K^0.8) / L^0.4] / [(400 x L^0.6) / K^0.2] = (3K) / (4L) = 3/10

30K = 12L

10K = 4L [(K/L) = 4/10 = 0.4]

Substituting in cost function:

100,000 = 3L + 4L = 7L

L = 14,286

K = 4L/10 = (4 x 14,286)/10 = 57,144/10 = 5,714

(iii)

With new prices,

MPL/MPK = 3K/4L = $22.5/$50 = 4.5/10

18L = 30K

K/L = 18/30 = 3/5 = 0.6 (or K = 0.6L)

Therefore, capital-labor ratio will increase from 0.4 to 0.6.

New cost function: 500,000 = 22.5L + 50K

500,000 = 22.5L + 50 x 0.6L

500,000 = 22.5L + 30L = 52.5L

L = 9,524 (Employment will fall by (14,286 - 9,524) = 4,762)

K = 9,524 x 0.6 = 5,714 (Capital will remain unchanged)

(d)

Total cost of production in single plant = $780,000

Total cost of production in different plants = $(540,000 + 180,000) = $720,000

Since Total cost of production in single plant is higher than Total cost of production in different plants, there are diseconomies of scope.