Question

1. Consider two firms that produce a single output good, y, using two inputs: Capital, K, and labor, L. the prices of each unit of capital and labor are r and w, respectively. The output good y sells for $p per unit.
Firm A’s production function is y = fA(K,L) = K1/4 L1/4. The profit function is thus:
A(K,L) = K1/4 L1/4 – rK -wL

a. Find the profit maximizing levels of K and L as functions of r, w, and p
b. Suppose that r = w = $1 and p = $4. What is the profit maximizing level of output, y?

Answer 1

$$\frac {\delta fA}{\delta K}=\frac {pL^{\frac{1}{4}}}{4K^{\frac{3}{4}}}-r$$

$$\frac {\delta fA}{\delta K}=\frac {pK^{\frac{1}{4}}}{4L^{\frac{3}{4}}}-w$$

$$K=\sqrt [11]{\frac{p^{16}}{4^{16}w^4r^{12}}}$$

$$L=\sqrt[11]{\frac{r^8p^4}{4^4w^{12}}}$$

If r=w=$1, p=$4

$$K=\sqrt [11] {\frac {4^{16}}{4^{16}{1^4 \times1^{12}}}}=1$$

$$L=\sqrt[11]{\frac{1^8 4^4}{4^4 1^{12}}}=1$$