Question 11.RW.1: A firm faces the production function Q = 12K^0.4L^0.4 and ca......
A firm faces the production function Q = 12 K^{0.4}L^{0.4} and can buy the inputs K and L at prices per unit of £40 and £5 respectively. If it has a budget of £800 what combination of K and L should it use in order to produce the maximum possible output?
From the budget constraint
40K + 5L = 800
5L = 800 − 40K (1)
L = 160 − 8K (2)
Substituting (2) into the objective function Q = 12K^{0.4}L^{0.4} gives
Q = 12K^{0.4}(160 – 8K)^{0.4} (3)
We are now faced with the unconstrained optimization problem of finding the value of K that maximizes the function (3) which has the budget constraint (1) ‘built in’ to it by substitution. This requires us to set d Q/dK = 0. However, it is not straightforward to differentiate the function in (3), and we must wait until further topics in calculus have been covered before proceeding with this solution (see Chapter 12, Example 12.9).