Question

A firm faces the production function Q = 12K^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? Show graphically as well.

Answer 1

The problem is given to to maximize the function $Q = 12K^{0.4}L^{0.4}$ subject to the budget constraint

$40K + 5L = 800$

The theory of the firm tells us that a firm is optimally allocating a fixed budget if the last £1 spent on each input adds the same amount to output, i.e. marginal product over price should be equal for all inputs. This optimization condition can be written as

$\frac{MP_K}{P_K} = \frac{MP_L}{P_L}$

The marginal products can be determined by partial differentiation:

$MP_K = \frac{dQ}{dK} = 4.8K^{-0.6}L^{0.4} \\ MP_L = \frac{dQ}{dL} = 4.8K^{0.4}L^{-0.6} \\ 4.8K^{-0.6}L^{0.4} = 4.8K^{0.4}L^{-0.6}$

Dividing both sides by 4.8 and multiplying by 40 gives:

$K^{-0.6}L^{0.4}=8K^{0.4}L^{-0.6}$

Multiplying both sides by $K^{0.6}L^{0.6}$ gives:

$L = 8K \\ 40K + 5 \times 8K = 800 \\ 40K + 40K = 800 \\ 80K=800 \\ K = 10 \\ L = 8 \times 10 = 80$