Question

14. Suppose the production function for cars is ? = 4??^(1/4)?^(3/4)?. The cost of capital is $10 per hour, and the

wage for labor is $15 per hour. If the company wants to produce 1000 cars, what combination of capital and labor should it use:
a. 50 units of capital, 25 units of labor
b. 25 units of capital, 50 units of labor

c. 12.9 units of capital, 58 .1units of labor

d. 148.7 units of capita, 297.3 units of labor

Answer 1

As per the question the production function of car is Q=4K^1/4L^3/4

Where L=labour and K=Capital

Cost of labour (w) per unit=$15

Cost of capital (r) per unit =$10

Company want to produce Q = 1000 cars

Marginal Product (MP) of Labour =dQ/dL = 3K^1/4L^-1/4=3 K^1/4/ L^1/4

Marginal Product (MP) of capital =dQ/dK = L^3/4K^-3/4=L^3/4/ K^3/4

Marginal rate of technical substitution (MRTS) = MP of Labour / MP of capital

Marginal rate of technical substitution (MRTS) = (3 K^1/4/ L^1/4) / (L^3/4/ K^3/4) =3K/L

At equilibrium level of output, MRTS = cost of labour / cost of capital

At equilibrium level of output,    3K/L = 15 / 10     

At equilibrium level of output,     3K/L = 3/2   So L =2K   and    K=L/2

As per the question the production function Q=4K^1/4L^3/4

At cost minimising level of output or equilibrium level of output where, MRTS = Price of labour (w) / Price of capital (r)  

So we got

L =2K   and    K=L/2

For producing Q = 1000 level of output

1000=4K^1/4L^3/4    (replacing the value of L=2K)

1000=4K^1/4(2K)^3/4

1000 = 4(2)^3/4K

1000 =6.727K

K= 1000/6.727=148.65   or Approx 148.7

For producing Q = 1000 level of output

1000=4K^1/4L^3/4    (replacing the value of K=L/2=0.5L)

1000=4(0.5L)^1/4L^3/4

1000=4(0.5)^1/4L

1000 =3.3635L

L= 297.30  

To produce 1000 units of cars the company should use 148.7 units of capital and 297.30 units of labour

Answer Option (D) 148.7 units of capital, 297.3 units of labour