Profit function is a mathematical expression that represents the relationship between a firm’s total profit, total revenue, and total costs. In simple terms, it shows how much profit a business can make based on its revenues and expenses. In mathematics, it is used to model the relationship between a firm’s output and its profit, helping to identify the optimal production level that maximizes profit. There are three components of profit function i.e.,

• Total Revenue (R(Q))
• Total Cost (C(Q))
• Profit (π(Q))

In this article, we will discuss the profit function in detail, including its components, formula, and solved examples.

What is Profit Function?

A profit function is a mathematical representation of a firm’s profit, defined as the difference between total revenue and total cost. It is used to calculate the profit of a business based on its revenue and costs. It is typically expressed as:

Profit (π) = Total Revenue (R) – Total Cost (C),

Where π represents profit, R represents total revenue from selling a certain number of units of a good or service, and C represents the total cost of producing those units.

Profit Function Definition

A profit function is a mathematical representation that measures a firm’s profit by calculating the difference between total revenue and total cost.

Profit Function is used to analyze and determine the level of output that maximizes a firm’s profit.

Components of a Profit Function

There are two components of profit function i.e.,

• Revenue Function
• Cost Function

Let’s discuss these in detail.

Revenue Function

The revenue function represents the total income generated from selling a certain quantity of goods or services at a given price. Mathematically, it is expressed as:

R(q) = p(q) × q

Where:

• R(q) is the total revenue generated from selling q units of the product,
• p(q) is the price per unit as a function of quantity q,
• q is the quantity of goods or services sold.

Cost Function

The cost function represents the total cost incurred by a firm in producing a certain quantity of goods or services. It includes all expenses such as raw materials, labor, overhead costs, etc.

The cost function can be expressed as:

C(q) = C_fixed (q) + C_variable(q)

Where,

• C(q) is the total cost of producing q units of the product,
• C_fixed (q) represents the fixed costs that do not vary with the level of output,
• C_variable(q) is the variable cost that depends on the quantity q produced.

Formula of Profit Function

Profit (π) = Total Revenue (R) – Total Cost (C)

OR

π(q) = R(q) – C(q)

Where,

• π (profit) is the difference between revenue and cost,
• R (total revenue) is the income generated from selling goods or services,
• C (total cost) is the expense incurred in producing those goods or services.

Examples questions on Profit Function

1: Suppose a company manufactures and sells widgets. The revenue function R(q) and cost function C(q) are given by:

R(q) = 20q and C(q) = 5q + 50

Find profit function.

To find the profit function Π(q), we subtract the cost function from the revenue function:

π(q) = R(q) – C(q)

⇒ π(q) = 20q – (5q + 50)

⇒ π(q) = 20q – 5q – 50

⇒ π(q) = 15q – 50

So, the profit function π(q) in this example is Π(q) = 15q – 50.

2: Consider another scenario where the revenue function R(q) and cost function C(q) are quadratic:

R(q) = 10q – 0.5q² and C(q) = 2q² + 5q + 20

Find profit function.

To find the profit function Π(q), we subtract the cost function from the revenue function:

Π(q) = R(q) – C(q)

⇒ π(q) = (10q – 0.5q²) – (2q² + 5q + 20)

⇒ π(q) = 10q – 0.5q² – 2q² – 5q – 20

⇒ π(q) = -2.5q² + 5q – 20

Therefore, the profit function π(q) in this example is π(q) = -2.5q² + 5q – 20.

In both examples, the profit function π(q) shows how profit changes with the quantity q produced and sold. It helps managers and economists analyze the impact of production levels on profitability and make informed decisions regarding pricing, production volumes, and cost management strategies.

How to Use the Profit Function?

We can use the profit function using the following steps:

Step 1: Determine the selling price of each unit of your product or service.

Step 2: Determine the number of units sold.

Step 3: Multiply the price per unit by the quantity sold.

Step 4: Determine all fixed costs associated with production.

Step 5: Determine the cost to produce one unit of your product or service.

Step 6: Multiply the variable cost per unit by the quantity produced.

Step 7: Add fixed costs to total variable costs.

Step 8: Subtract the total costs from the total revenue.

Solved Examples on Profit Function

1: A company sells widgets. The total revenue (R) from selling q widgets is given by R = 50q, and the total cost (C) of producing q widgets is given by C=30q + 200. Determine the profit function and find the profit when 20 widgets are produced and sold.

Given: Revenue Function = 50q and cost function = 30q + 200

As we know, Profit (π) = Total Revenue (R) – Total Cost (C)

⇒ π(q) = 50q – (30q + 200)

⇒ π(q) = 50q – 30q – 200

⇒ π(q) = 20q – 200

Profit for 20 Widgets is π(20)

⇒ π(20) = 20(20) – 200

⇒ π(20) = 400 – 200

⇒ π(20) = 200

So, the profit when 20 widgets are produced and sold is $200.

2: A firm has the total revenue function R = 100q – q² and the total cost function C = 20q + 300. Determine the profit function and find the profit when 10 units are produced and sold.

Given: Revenue Function = 100q – q² and cost function = 20q + 300

As we know, Profit (π) = Total Revenue (R) – Total Cost (C)

π(q) = (100q – q²) – (20q + 300)

⇒ π(q) = 100q – q² – 20q – 300

⇒ π(q) = 80q – q² – 300

Profit for 10 units is π(10)

Thus, π(10) = 80(10) – 10² – 300

⇒ π(10) = 800 – 100 – 300

⇒ π(10) = 400

So, the profit when 10 units are produced and sold is $400.

3: The total revenue for a company is given by R = 75q and the total cost is C = 25q + 150. What is the break-even point, where profit is zero?

Given: Revenue Function = 75q and cost function = 25q + 150

As we know, Profit (π) = Total Revenue (R) – Total Cost (C)

⇒ π(q) = 75q – (25q + 150)

⇒ π(q) = 75q – 25q – 150

⇒ π(q) = 50q – 150

To find break-even point, set profit (π) to zero and solve for q.

Thus, 0 = 50q – 150

⇒ 150 = 50q

⇒ q = 150 / 50

⇒ q = 3

So, the break-even point is when 3 units are produced and sold.

Practice Problems on Profit Function

1: A bakery sells cakes for $15 each. The fixed costs are $200 per month, and the variable cost per cake is $5. Calculate the profit if 100 cakes are sold in a month.

2:A factory produces widgets at a variable cost of $8 per widget. The selling price is $20 per widget, and the fixed costs are $1,000 per month. Determine the profit if the factory produces and sells 150 widgets in a month.

3: A coffee shop has fixed costs of $300 and sells a cup of coffee for $4. The variable cost per cup is $1. Calculate the profit if 250 cups of coffee are sold.

4: An online store sells gadgets at $50 each. The variable cost per gadget is $20, and the fixed costs are $1,500 per month. Find the profit if 100 gadgets are sold in a month.

5: A car wash business has fixed costs of $600 per month and charges $12 per car. The variable cost per car is $3. Determine the profit if 200 cars are washed in a month.

Read More,

• Profit and Loss
• Linear Programming
• Business Studies

Conclusion

In this article, we’ve explored the profit function, a vital tool for any business looking to understand its profitability. By breaking down the profit function into its essential components—total revenue and total costs—we’ve seen how businesses can calculate their profit accurately. Understanding fixed and variable costs and knowing how to use the profit formula can help businesses make informed decisions about pricing, production, and cost management.

Profit Function – FAQs

What is a profit function?

A profit function is a mathematical formula used to calculate the profit of a business.

What are the components of a profit function?

The main components of a profit function are:

• Total Revenue
• Total Costs

How do you calculate total revenue?

Total revenue is calculated by multiplying the price per unit by the quantity sold.

How do you calculate total costs?

Total costs are calculated by adding fixed costs and variable costs.

What is the difference between fixed costs and variable costs?

• Fixed Costs: These are costs that remain constant regardless of the level of production, such as rent, insurance, and salaries.
• Variable Costs: These are costs that vary directly with the level of production, such as costs for raw materials and direct labor