Answer to Question #215736 in Macroeconomics for Pee

Question

A firm sells its output in a perfectly competitive market at a fixed price of R800 per unit. It buys the two inputs K and L at prices of R40 per unit and R5 per unit respectively, and faces the production function:
q = 3.1K0.3 L0.25
a. Calculate the total cost curve. (5)
b. Calculate the total revenue curve for this firm. (5)
c. What will be the profit function?

Answer 1

The firm will hire labour 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:

$q = 3.1\left(\dfrac{L}{7}\right)^{0.3}L^{0.25}$ $q = 1.729L^{0.55}$ $\color{red}{L^* \approx 0.37q^{20/11}}$

$q = 3.1K^{0.3}(7K)^{0.25}$ $q = 3.1K^{0.55}$ $\color{red}{K^* \approx 0.13q^{20/11}}$