To determine the firm's short-run average cost function (SRAC), we need to follow these steps:

1. Understanding the Production Function

The production function is given as:

$y = \frac{1}{1000} \cdot K \cdot \sqrt{L}$

Here, $K$ (capital) is fixed in the short run. Let $K = K_0$ (a constant).

2. Solving for Labor ($L$) in Terms of Output ($y$)

Rearranging the production function to isolate $L$:

$y = \frac{1}{1000} \cdot K_0 \cdot \sqrt{L}$ $1000y = K_0 \cdot \sqrt{L}$ $\sqrt{L} = \frac{1000y}{K_0}$ $L = \left(\frac{1000y}{K_0}\right)^2$

3. Cost Function

The total cost ($TC$) in the short run includes:

• Labor Cost: $w \cdot L$, where $w = 10$
• Fixed Capital Cost: $r \cdot K_0$, where $r = 20$

$TC = w \cdot L + r \cdot K_0$

Substitute $L$ into the total cost equation: $TC = 10 \cdot \left(\frac{1000y}{K_0}\right)^2 + 20 \cdot K_0$

$TC = 10 \cdot \frac{(1000^2 \cdot y^2)}{K_0^2} + 20 \cdot K_0$

$TC = \frac{10 \cdot 1000^2 \cdot y^2}{K_0^2} + 20 \cdot K_0$

4. Short-Run Average Cost Function (SRAC)

The short-run average cost (SRAC) is: $SRAC = \frac{TC}{y}$

Substitute $TC$: $SRAC = \frac{\frac{10 \cdot 1000^2 \cdot y^2}{K_0^2} + 20 \cdot K_0}{y}$

Simplify: $SRAC = \frac{10 \cdot 1000^2 \cdot y}{K_0^2} + \frac{20 \cdot K_0}{y}$

$SRAC = \frac{10,000,000 \cdot y}{K_0^2} + \frac{20 \cdot K_0}{y}$

This is the short-run average cost function in terms of $y$ and $K_0$.

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Clarification or Questions?

Would you like a specific example or a graph for visualization?

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5 Related Questions

1. How does the SRAC change as output $y$ increases?
2. How would the SRAC differ if the wage rate or rental rate changed?
3. What is the marginal cost function for this production technology?
4. How does the fixed $K_0$ affect cost efficiency in the short run?
5. What happens to the SRAC if $K_0$ is doubled?

Tip:

Always examine how input prices (e.g., $w$ and $r$) affect both variable and