A firm produces output according to the production function 𝑌 = 𝐹(𝐾, 𝐿) = 𝐾 0.5 𝐿 0.5 . Suppose it sells its output in a perfectly competitive market at a price of 10, and the factors of production are priced at 𝑟 per unit of K and 𝑤 per unit of L. In the short run, K is fixed at 4. c. What is this firm’s short-run demand for labour? [20] d. How much output should it produce to maximize profits in the short run? [20] e. How much output should it produce to maximize profits in the long run? [30]

Let's break down the firm's production function and address each question step by step. The production function is given as:

\[
Y = F(K, L) = K^{0.5} L^{0.5}
\]

In the short run, capital \( K \) is fixed at \( K = 4 \), and the firm sells output in a perfectly competitive market at a price \( P = 10 \). The cost of labor \( L \) is represented by \( w \) (the wage rate), and the cost of capital is represented by \( r \) (the rental rate of capital).

### (c) Short-run demand for labor

In the short run, \( K \) is fixed at 4. Therefore, the production function simplifies to:

\[
Y = F(4, L) = 4^{0.5} L^{0.5} = 2L^{0.5}
\]

The firm's revenue is given by:

\[
\text{Revenue} = P \cdot Y = 10 \cdot 2L^{0.5} = 20L^{0.5}
\]

The firm's cost in the short run consists only of labor costs, since \( K \) is fixed:

\[
\text{Cost} = w \cdot L
\]

To find the firm's short-run demand for labor, we need to maximize profit. Profit is defined as:

\[
\text{Profit} = \text{Revenue} - \text{Cost} = 20L^{0.5} - wL
\]

To maximize profit, we take the derivative of the profit function with respect to \( L \), and set it equal to 0:

\[
\frac{d}{dL}\left( 20L^{0.5} - wL \right) = 0
\]

First, compute the derivative:

\[
\frac{d}{dL}\left( 20L^{0.5} \right) = 10L^{-0.5}
\]
\[
\frac{d}{dL}\left( wL \right) = w
\]

Setting the derivative equal to 0 gives:

\[
10L^{-0.5} = w
\]

Solve for \( L \):

\[
L^{-0.5} = \frac{w}{10}
\]
\[
L^{0.5} = \frac{10}{w}
\]
\[
L = \left( \frac{10}{w} \right)^2
\]
\[
L = \frac{100}{w^2}
\]

Thus, the firm's **short-run demand for labor** is:

\[
L = \frac{100}{w^2}
\]

### (d) Short-run profit-maximizing output

From part (c), we know the labor demand \( L = \frac{100}{w^2} \). Now we substitute this into the production function to find the short-run output \( Y \):

\[
Y = 2L^{0.5}
\]

Substitute \( L = \frac{100}{w^2} \):

\[
Y = 2 \left( \frac{100}{w^2} \right)^{0.5}
\]
\[
Y = 2 \times \frac{10}{w}
\]
\[
Y = \frac{20}{w}
\]

Therefore, the firm's **short-run profit-maximizing output** is:

\[
Y = \frac{20}{w}
\]

### (e) Long-run profit-maximizing output

In the long run, both capital \( K \) and labor \( L \) are variable. The firm will choose the optimal combination of \( K \) and \( L

A firm produces output according to the production function 𝑌 = 𝐹(𝐾, 𝐿) = 𝐾^0.5 𝐿^0.5. Suppose it sells its output in a perfectly competitive market at a price of 10, and the factors of production are priced at 𝑟 per unit of K and 𝑤 per unit of L. In the short run, K is fixed at 4. c. What is this firm’s short-run demand for labour? d. How much output should it produce to maximize profits in the short run? e. How much output should it produce to maximize profits in the long run?

Explore how a firm maximizes profits using a Cobb-Douglas production function with fixed capital in the short run. Learn how to determine labor demand and calculate profit-maximizing output both in the short run and long run. Suitable for undergraduate economics students.

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