Question

A local garage uses two types of inputs, capital and labour, to service used cars. Their production function is described by Q = 8K^0.8L^0.2, where Q is the number of cars serviced, K is the amount of capital, equipment and tools used in the process and L denotes the input of human labour. The costs of inputs are £16 per unit of labour and £1 per unit of capital. The garage has received an order from a taxi company to service a fleet of 48 cars. Consider a constrained cost minimisation problem and answer the following questions.

a. What is the cost minimising quantity of capital the garage should use?
K* =
b. What is the cost minimising quantity of labour the garage should use?
L* =
c. What is the lowest cost at which 48 cars can be serviced?

Answer 1

a. The cost minimising quantity of capital the garage should use is at MPK/r = MPL/w, so:

$MPK = Q'(K) = 6.4K^{-0.2}L^{0.2},$

$MPL = Q'(L) = 1.6K^{0.8}L^{-0.8},$

$6.4K^{-0.2}L^{0.2}/1 = 1.6K^{0.8} L^{-0.8}/16,$

K = 64L,

$8×(64L)^{0.8}L^{0.2}= 48,$

L = 0.215,

K = 0.215×64 = 13.78.

K* = 13.78.

b. The cost minimising quantity of labour the garage should use is:

L* = 0.215.

c. The lowest cost at which 48 cars can be serviced is: 0.215×16 + 13.78×1 = 17.22.