Question

A firm has a supply function given by \(S(p)=4 p .\) Its fixed costs are 100 . If the price changes from 10 to \(20,\) what is the change in its profits?

Short answer

Profit increases by 1200 units.

Step-by-step solution

Understand the Supply Function

The supply function gives the quantity supplied as a function of price, represented as \(S(p) = 4p\). This means for every unit increase in price \(p\), the quantity supplied increases by \(4\) units.

Calculate Initial Profit at Price 10

First, find quantity supplied at \(p = 10\). Substitute into the supply function: \(S(10) = 4 \times 10 = 40\). Profit at this price is revenue minus fixed costs. Revenue \(R = p \times S(p) = 10 \times 40 = 400\). Subtract fixed costs: Profit \(= 400 - 100 = 300\).

Calculate New Profit at Price 20

Next, find quantity supplied at \(p = 20\). Substitute into the supply function: \(S(20) = 4 \times 20 = 80\). Calculate revenue: \(R = 20 \times 80 = 1600\). Subtract fixed costs: Profit \(= 1600 - 100 = 1500\).

Calculate Change in Profit

Subtract initial profit from new profit to find the change: Change in Profit \(= 1500 - 300 = 1200\). This is the increase in profit when the price changes from 10 to 20.

Key concepts

Profit Calculation

Profit calculation is a fundamental aspect of understanding the financial health of a business. It involves determining the profitability by subtracting total costs from total revenue. In the exercise example, the supply function describes how many units are available based on the price.
To calculate profit, first find the revenue, which is simply the product of price and quantity supplied. Then, subtract the fixed costs, which do not change with the level of production. In our exercise, the initial profit at a price of 10 was \(400 - 100 = 300\). After recalculating for a price increase to 20, the profit became \(1600 - 100 = 1500\).

• Revenue is calculated by multiplying the selling price by the quantity supplied.
• Subtract fixed costs to get profit.

Understanding these steps can help grasp how businesses make decisions based on profitability.

Fixed Costs

Fixed costs are expenses that remain unchanged regardless of the amount produced or sold. They are often called "sunk costs" since they must be paid even if the business is not active or producing goods. Examples of fixed costs include rent, salaries, and insurance.
In the exercise, the firm's fixed costs are 100. These costs need to be subtracted from the total revenue to calculate the profit, thereby showing how even consistent expenses like fixed costs can impact profitability.

• Fixed costs do not vary with production levels or sales.
• They must be subtracted from revenue to calculate profit.

Recognizing fixed costs is crucial for budgeting and financial forecasting.

Quantity Supplied

Quantity supplied refers to how much of a product a firm is willing to produce and sell at a given price. It is directly influenced by the market price. In the context of the supply function \(S(p) = 4p\), it indicates that for every increase in price, the quantity supplied increases by 4 units.
In the given exercise:

• At a price of 10, the supply was calculated as \(4 \times 10 = 40\) units.
• When the price increased to 20, the supply rose to \(4 \times 20 = 80\) units.

Understanding this relationship between price and quantity is essential for grasping how market dynamics influence supply. The supply function provides a simple yet powerful way to predict a firm's production behavior based on prices.