The problem gives a production function $q = K^{1/2}L^{1/3}$, where $q$ is the quantity of output, $K$ is capital, and $L$ is labor. In the short run, capital is fixed at $K = 100$. The wage rate is $w = \$10$ and the rental rate on capital is $r = \$20$. The goal is to find the short-run average cost (AC).

1. Problem Description

The problem gives a production function $q = K^{1/2}L^{1/3}$, where $q$ is the quantity of output, $K$ is capital, and $L$ is labor. In the short run, capital is fixed at $K = 100$. The wage rate is w = \10$and the rental rate on capital is$r = \$20$. The goal is to find the short-run average cost (AC).

2. Solution Steps

First, we plug in the fixed value of capital $K = 100$ into the production function:

$q = (100)^{1/2}L^{1/3} = 10L^{1/3}$

Next, solve for $L$ in terms of $q$:

$q = 10L^{1/3}$

$L^{1/3} = \frac{q}{10}$

$L = (\frac{q}{10})^3 = \frac{q^3}{1000}$

The total cost (TC) is the sum of the cost of labor and the cost of capital:

$TC = wL + rK$

Substitute the given values for $w$, $r$, and $K$:

$TC = 10L + 20(100) = 10L + 2000$

Now substitute the expression for $L$ in terms of $q$:

$TC = 10(\frac{q^3}{1000}) + 2000 = \frac{q^3}{100} + 2000$

The average cost (AC) is the total cost divided by the quantity of output:

$AC = \frac{TC}{q} = \frac{\frac{q^3}{100} + 2000}{q} = \frac{q^3}{100q} + \frac{2000}{q} = \frac{q^2}{100} + \frac{2000}{q}$

3. Final Answer

The short-run average cost (AC) is:

$AC = \frac{2000}{q} + \frac{q^2}{100}$

Therefore, the answer is d.