Question

A firm sells its output in a perfectly competitive market at a fixed price of R200 per unit. It
Buys the two inputs K and L at prices of R42 per unit and R5 per unit respectively, and faces the production function
q = 3.1K0.3L0.25
What combination of K and L should it use to maximize profit?

Answer 1

The firm will hire labor and capital up to the point where:

$$\dfrac{MPL}{MPK} = \dfrac{W}{r}$$

From the given production function, the marginal product of labor is:

$$MPL = \dfrac{\delta q}{\delta L} = 0.775K^{0.3}L^{-0.75}$$

And the marginal product of capital is:

$$MPK = \dfrac{\delta q}{\delta K} = 0.93K^{-0.7}L^{0.25}$$

The price of labor is W = R5. and the price of capital is r = R42 Therefore:

$$\dfrac{0.775K^{0.3}L^{-0.75}}{0.93K^{-0.7}L^{0.25} } = \dfrac{5}{42}$$

$$\dfrac{K}{L} = \dfrac{1}{7}$$

Solving for L and K each at a time, we get:

$$K = \dfrac{L}{7}......(i)$$ $$L = 7K......(ii)$$

Substituting equations (i) and (ii) into the production function each at a time:

$$q = 3.1\left(\dfrac{L}{7}\right)^{0.3}L^{0.25}$$

$$q = 1.729L^{0.55}$$ $$\color{red}{L^* \approx 0.37q^{20/11}}$$

$$q = 3.1K^{0.3}(7K)^{0.25}$$

$$q = 3.1K^{0.55}$$

$$\color{red}{K^* \approx 0.13q^{20/11}}$$