Answer to Question #252508 in Microeconomics for Puku

Question

The estimated production function of a firm is Q = 100K0.7L0.4. The wage rate is Rs. 50 and rate of interest is Rs. 40. 

i. Compute the marginal products of two inputs.

ii. Determine the amount of labor and capital that the firm should use in order to minimize cost of producing 1200 units of output. What is the cost of the firm?

iii. What will be the optimal employment of labor and capital in order to maximize output under given total cost of Rs. 1500?

iv. What is the degree of returns to scale depicted by the production function?

Answer 1

Solution:

i.). Marginal Product of Labor (MP_L) = $\frac{\partial Q} {\partial L}$ = 40K^0.7L^-0.6

Marginal Product of Capital (MP_K) = $\frac{\partial Q} {\partial K}$ = 70K^-0.3L^0.4

 

ii.). Cost minimizing is where MRTS = $\frac{MP_{L} } {MP_{K}} = \frac{w} {r}$

w = 50

r = 40

$\frac{40K^{0.7}L^{-0.6} }{70K^{-0.3}L^{0.4}} = \frac{50}{40}$

$\frac{4K }{7L} = \frac{5}{4}$

K = 2.1875L

Substitute for K in the production function and solve where L yields an output of 1,200 units:

Q = 100K^0.7L^0.4

1,200 = 100(2.1875L^0.7) L^0.4

1,200 = 218.75L^1.1

L = 4.7

K = 2.1875L = 2.1875(4.7) = 10.28125

K = 10.28125

TC = wL + rK

TC = (50)(4.7) + (40)(10.28125)

TC = 235 + 411.25 = 646.25

Total Cost = 646.25

iii.). 1,500 = 100(2.1875L^0.7) L^0.4

1,500 = 218.75L^1.1

L = 5.76

K = 2.1875L = 2.1875(5.76) = 12.6

The optimal employment of labor and capital in order to maximize output under a given total cost of Rs. 1500 = (5.76, 12.6)

 

iv.). The production function exhibits an increasing return to scale. That is when the output increases by a larger proportion than the increase in inputs during the production process.