Question 11.6: A firm can buy two inputs K and L at £18 per unit and £8 per......
A firm can buy two inputs K and L at £18 per unit and £8 per unit respectively and faces the production function Q = 24 K^{0.6}L^{0.3} . What is the maximum output it can produce for a budget of £50,000? (Work to nearest whole units of K, L and Q.)
The budget constraint is 50,000 − 18K − 8L = 0 and the function to be maximized is Q=24K^{0.6}L^{0.3} . The Lagrangian for this problem is therefore
G = 24 K^{0.6}L^{0.3} + λ(50,000 − 18K − 8L)
Partially differentiating to find the stationary points of G gives
\frac{\partial G}{\partial K } = 14.4 K^{-0.4}L^{0.3} − 18λ = 0
\frac{14.4L^{0.3}}{18K^{0.4} } = λ (1)
\frac{\partial G}{\partial L } = 7.2 K^{0.6}L^{-0.7} − 8λ = 0
\frac{7.2K^{0.6}}{8L^{0.7} } = λ (2)
\frac{\partial G}{\partial \lambda } = 50,000 − 18K − 8L = 0 (3)
Setting (1) equal to (2) to eliminate λ
\frac{14.4L^{0.3}}{18K^{0.4} } = \frac{7.2K^{0.6}}{8L^{0.7} }
115.2L = 129.6K
L = 1.125K (4)
Substituting (4) into (3)
50,000 − 18K − 8(1.125K) = 0
50,000 − 18K − 9K = 0
50,000 = 27K
1,851.8519 = K (5)
Substituting (5) into (4)
L = 1.125(1,851.8519) = 2,083.3334
Thus, to the nearest whole unit, optimum values of K and L are 1,852 and 2,083 respectively.
We can check that when these whole values of K and L are used the total cost will be
TC = 18K + 8L = 18(1,852) + 8(2,083) = 33,336 + 16,664 = £50,000
and so the budget constraint is satisfied. The actual maximum output level will be
Q = 24 K^{0.6}L^{0.3} = 24 (1,852)^{0.6}(2,083)^{0.3} = 21,697 units