Question

Suppose that the production function of the firm is:

Q = 100L^1/2.K^1/2

 K= 100, P = $1, w = $50 and r = $40. Determine the quantity of labor that the firm should hire in order to maximize the profits. What is the maximum profit of this firm?

Answer 1

Maximizing profit:

$Profit (Pr) = P\times Q - Cost=P\times Q - (L+K)=1\times100\times L^{0.5}\times K^{0.5}-(L+100)=100\times L^{0.5}\times 100^{0.5}-(L+100)=1000\times L^{0.5}-L-100$

equilibrium condition

$\frac{MPL}{MPK} =\frac{ w}{ r}$

$\frac{MPL}{MPK} =\frac{ 50}{ 40}=1.25$

$\frac{50(\frac{K}{L})^{0.5}}{50(\frac{L}{K})^{0.5}}=1.25$

$\frac{K}{L}=1.25$

$L=\frac{100}{1.25}=80$

$Profit (Pr) = 1000\times L^{0.5}-L-100=1000\times 80^{0.5}-80-100=8764.27$