Question 11.3: A firm faces the production function Q = 120L + 200K − L² − ......
A firm faces the production function
Q = 120L + 200K − L² − 2K²
for positive values of Q. It can buy L at £5 a unit and K at £8 a unit and has a budget of £70. What is the maximum output it can produce?
MP_L = \frac{\partial Q}{\partial L} = 120 − 2L MP_K = \frac{\partial Q}{\partial K} = 200 − 4K
For optimal input combination
\frac{MP_L}{P_L} = \frac{MP_K}{P_K}
Therefore, substituting MP_K and MP_L and the given input prices
\frac{120 – 2L}{5} = \frac{200 – 4K}{8}
8(120 − 2L) = 5(200 − 4K)
960 − 16L = 1,000 − 20K
20K = 40 + 16L
K = 2 + 0.8L (1)
Substituting (1) into the budget constraint
5L + 8K = 70
gives
5L + 8(2 + 0.8L) = 70
5L + 16 + 6.4L = 70
11.4L = 54
L = 4.74 (to 2 dp)
Substituting this result into (1)
K = 2 + 0.8(4.74) = 5.79
Therefore maximum output is
Q = 120L + 200K − L² − 2K²
= 120(4.74) + 200(5.79) − (4.74)² − 2(5.79)²
= 1,637.28