Question 11.12: A firm faces the production function Q = 20K^0.4L^0.6 . It c......
A firm faces the production function Q = 20K^{0.4}L^{0.6} . It can buy inputs K and L for £400 a unit and £200 a unit respectively. What combination of L and K should be used to maximize output if its input budget is constrained to £6,000?
MP_L = \frac{\partial Q}{\partial L} = 12K^{0.4}L^{-0.4} MP_K = \frac{\partial Q}{\partial K} = 8K^{-0.6}L^{0.6}
Optimal input mix requires
\frac{MP_L}{P_L} = \frac{MP_K}{P_K}
Therefore
\frac{12K^{0.4}L^{-0.4}}{200} = \frac{8K^{-0.6}L^{0.6}}{400}
Cross multiplying gives
4,800K = 1,600L
3K = L
Substituting this result into the budget constraint
200L + 400K = 6,000
gives
200(3K) + 400K = 6,000
600K + 400K = 6,000
1,000K = 6,000
K = 6
Therefore
L = 3K = 18