Question

A firm is producing output using only capital. Its production function is \(Q=10 \mathrm{~K}\) - K2. The firm sells its product in a competitive market at a price of \(£ 2\) and it rents capital from a competitive market at a rental rate of \(r\) per unit of capital. Write down the profit function of the firm and find its capital demand function

Short answer

Profit function: \( \pi = (20 - r)K - 2K^2 \); Capital demand: \( K = \frac{20 - r}{4} \).

Step-by-step solution

Define the Profit Function

To find the profit function, we need to know how the firm's revenue and costs are determined. The revenue is calculated as the product of the price per unit (\( £2 \)) and the quantity produced (\( Q \)). Therefore, the revenue function is given by \( R = 2Q \). The cost is the product of the rental rate per unit of capital (\( r \)) and the amount of capital used (\( K \)). Thus, the cost function is \( C = rK \). The profit function \( \pi \) is then defined as revenue minus cost: \( \pi = R - C = 2Q - rK \).

Substitute the Production Function into the Profit Function

Substitute the production function \( Q = 10K - K^2 \) into the profit function. \[\pi = 2(10K - K^2) - rK = 20K - 2K^2 - rK\] Simplify the expression to get the profit function in terms of capital \( K \). \[\pi = 20K - 2K^2 - rK = (20 - r)K - 2K^2\]

Find the First Derivative of the Profit Function

To find the capital demand function, take the derivative of the profit function with respect to \( K \) and set it equal to zero to find the critical points. Differentiate: \[\frac{d\pi}{dK} = 20 - r - 4K\] Set the first derivative equal to zero and solve for \( K \): \[20 - r - 4K = 0\]

Solve for Capital Demand Function

Solve the equation from the previous step for \( K \) to find the capital demand function. \[4K = 20 - r\] \[K = \frac{20 - r}{4}\] The capital demand function is \( K = \frac{20 - r}{4} \).

Key concepts

Revenue Function

In economics, the revenue function is a crucial component for understanding how a firm generates income from its products or services. The revenue function is calculated by multiplying the price per unit by the quantity of the product sold. In this particular exercise, we are given that the firm sells its product at a price of £2 per unit. Since the production function is represented by \( Q = 10K - K^2 \), the revenue function can be expressed as:

• Revenue (R): \( R = 2Q \)
• Substitute Production: \( R = 2(10K - K^2) \)
• Simplified Revenue: \( R = 20K - 2K^2 \)

A solid grasp of the revenue function is essential because it aids in determining how changes in production and pricing affect the firm's total income. Understanding how revenue varies with different levels of output allows a firm to strategize effectively on pricing and production quantities.

Cost Function

The cost function is another fundamental concept in understanding a firm's financial health. It represents the total costs incurred by a firm while producing its goods or services. In this exercise, the cost is associated with the rental rate of capital, which is designated as \( r \). The firm rents units of capital to produce its goods, and the cost function is expressed as:

• Cost (C): \( C = rK \)

This formula shows that the cost is dependent on both the rental rate and the amount of capital used. By analyzing the cost function, firms can understand how changes in input costs, like an increase in the rental rate \( r \), will affect their total production costs. Accurately evaluating costs is key in identifying measures to improve efficiency and maximize profits.

Capital Demand Function

The capital demand function is derived from the firm's objective to maximize its profit. It indicates the optimal amount of capital that the firm should utilize to achieve maximum profitability. We derive this function from the profit function by first substituting the production function into it, and then through calculus by finding the critical points of the profit function. Here's how it unfolds in our exercise:

• Profit Function: \( \pi = (20 - r)K - 2K^2 \)
• Derivative Analysis: Initially differentiate the profit function concerning \( K \).

After differentiating, we solve for \( K \) when the derivative is set to zero, leading to the capital demand function:

• Capital Demand: \( K = \frac{20 - r}{4} \)

This expression gives us the amount of capital the firm should employ for optimal profitability. The capital demand function shows the relationship between the desired level of capital and variables such as the rental rate \( r \), helping businesses make strategic decisions on capital investments.

Derivative

In mathematics and economics, the derivative is a powerful tool that helps to determine the rate of change of a function with respect to one of its variables. When analyzing the firm's profit function, taking the derivative with respect to capital \( K \) helps find the level of capital that maximizes profit. In our exercise, the process is as follows:

• Profit Derivative: We compute \( \frac{d\pi}{dK} = 20 - r - 4K \)
• Finding Critical Point: Set \( \frac{d\pi}{dK} = 0 \) to find critical values.
• Solving for \( K \): We solve \( 20 - r - 4K = 0 \) resulting in \( K = \frac{20 - r}{4} \)

Applying derivatives guides us to optimal solutions, like finding minimum costs or maximum profits. It's a foundational concept that empowers economic analysis and decision-making by providing insight into how variables impact outcomes. Understanding derivatives enables a firm to navigate real-world financial scenarios efficiently.