Question

2.   Bridgestone Company has the following production function for tires: Q = 20 K ^0.2 L ^0.8, where K represents machine hours and L represents labor hours. They pay $ 15 per hour to rent their machines and $ 10 per hour to their workers. They have $ 12,000 to spend on capital and labor.

A. Does this production function exhibit constant, increasing, or decreasing returns to scale?

B. Does this production function exhibit diminishing marginal returns to capital and labor?

C. Derive the optimal bundle of capital and labor to achieve the desired level of output. What is the maximum amount of output that they can produce with this $ 12,000 in expenditures?

Answer 1

A. Production function exhibits constant returns to scale.

Since the production function is a Cobb Douglas production function.

a+b=1 as 0.2+0.8=1, So the production function is a constant returns to scale.

B. Marginal product of capital= 0.2*20* k^-0.8 * L^0.8

Marginal product of labor= 0.8*20* K^0.2* L^-0.2

Marginal product diminishes as Capital increases and marginal product diminishes as labor increases.

So this function exhibits diminishing Returns in Capital and labor.

c. MRTS= 4k/L

At equilibrium , MRTS= r/w=15/10=1.5

L=6K

12000= wk+rL

12000=15k+10*6k

12000=75k

k*= 160

L*= 960

Q=20* (160)^0.2 (960)^0.8

Q= 13419.1