How to Calculate Maximum Revenue

Co-authored by wikiHow Staff

Last Updated: March 24, 2023

Business statisticians know how to use sales data to determine mathematical functions for sales and demand. Using these functions and some basic calculus, it is possible to calculate the maximum revenue that the company can expect. If you know the revenue function, you can find the first derivative of that function and then determine the maximum point of the function.

Part 1

Part 1 of 3:

Using the Revenue Function

1
Understand the relation between price and demand. Economics study shows that, for most traditional businesses, as the demand for any item increases, the price for that item should decrease. Conversely, as the price decreases, demand should increase. Using data from actual sales, a company can determine a supply and demand graph. That data can be used to calculate a price function.
- For more information on graphing supply and demand data, see Find and Analyze Demand Function Curve.
2
Create a price function. The price function consists of two primary pieces of information. The first is the intercept. This is the theoretical price if no items are sold. The second detail is a decreasing slope. The slope of the graph represents the drop in price for each item. A sample price function might look like:
- $p=500-{\frac {1}{50}}q$ $p=500-{\frac {1}{50}}q$
  - p = price
  - q = demand, in number of units
- This function sets the “zero price” at $500. For each unit sold, the price decreases by 1/50th of a dollar (two cents).
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3
Determine the revenue function. Revenue is the product of price times the number of units sold. Since the price function includes the number of units, this will result in a squared variable. Using the price function from above, the revenue function becomes:^{[1]
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- $R(q)=p*q$
- $R(q)=[500-{\frac {1}{50}}q]*q$
- $R(q)=500q-{\frac {1}{50}}q^{2}$

Part 2

Part 2 of 3:

Finding the Maximum Revenue Value

1
Find the first derivative of the revenue function. In calculus, the derivative of any function is used to find the rate of change of that function. The maximum value of a given function occurs when the derivative equals zero. So, to maximize the revenue, find the first derivative of the revenue function.^{[2]
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- Suppose the revenue function, in terms of number of units sold, is $R(q)=500q-{\frac {1}{50}}q^{2}$ $R(q)=500q-{\frac {1}{50}}q^{2}$ . The first derivative, therefore, is:
  - $R^{\prime }(q)=500-{\frac {2}{50}}q$
- For a review of derivatives, see the wikiHow article on how to Take Derivatives.
2
Set the derivative equal to 0. When the derivative is zero, the graph of the original function is at either a peak or a trough. This will be either the maximum or minimum value. For some higher level functions, there may be more than one solution to the zero derivative, but not a basic price-demand function.^{[3]
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- $R^{\prime }(q)=500-{\frac {2}{50}}q$
- $0=500-{\frac {2}{50}}q$
3
Solve for the number of items at the 0 value. Use basic algebra to solve the derivative for the number of items to sell where the derivative is equal to zero. This will give you the number of items that will maximize the revenue.^{[4]
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- $0=500-{\frac {2}{50}}q$
- ${\frac {2}{50}}q=500$
- ${\frac {1}{50}}q=250$
- $q=50*250$
- $q=12,500$
4
Calculate the maximum price. Using the optimal number of sales from the derivative calculation, you can enter that value into the original price formula to find the optimal price.^{[5]
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- $p=500-{\frac {1}{50}}q$
- $p=500-{\frac {1}{50}}12,500$
- $p=500-250$
- $p=250$
5
Combine the results to calculate maximum revenue. After you have found the optimal number of sales and the optimal price, multiply them to find the maximum revenue. Recall that $R=p*q$ $R=p*q$ . The maximum revenue for this example, therefore, is:^{[6]
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- $R=p*q$
- $R=(250)(12,500)$
- $R=3,125,000$
6
Summarize the results. Based on these calculations, the optimal number of units to sell is 12,500, at the optimal price of $250 each. This will result in a maximum revenue, for this sample problem, of $3,125,000.^{[7]
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Part 3

Part 3 of 3:

Solving Another Sample Problem

1
Begin with the price function. Suppose that another business has collected price and sales data. Using that data, the company has determined that the initial price is $100, and each additional unit sold will cut the price by one cent. Using this data, the following price function is:
- $p=100-0.01q$
2
Determine the revenue function. Recall that revenue is equal to price times quantity. Using the price function above, the revenue function is:
- $R(q)=[100-0.01q]*q$
- $R(q)=100q-0.01q^{2}$
3
Find the derivative of the revenue function. Using basic calculus, find the derivative of the revenue function:
- $R(q)=100q-0.01q^{2}$
- $R^{\prime }(q)=100-(2)0.01q$
- $R^{\prime }(q)=100-0.02q$
4
Find the maximum value. Set the derivative equal to zero and solve for $q$ $q$ to find the optimal number of sales. This calculation is as follows:
- $R^{\prime }(q)=100-0.02q$
- $0=100-0.02q$
- $0.02q=100$
- $q=100/0.02$
- $q=5,000$
5
Calculate the optimal price. Use the optimal sales value in the original price formula to find the optimal sales price. For this example, this works as follows:
- $p=100-0.01q$
- $p=100-0.01(5,000)$
- $p=100-50$
- $p=50$
6
Combine the maximum sales and optimal price to find maximum revenue. Using the relationship that revenue equals price times quantity, you can find the maximum revenue as follows:
- $R(q)=p*q$
- $R(q)=50*5,000$
- $R(q)=250,000$
7
Interpret the results. Using this data and based on the price function $p=100-0.01q$ , the company’s maximum revenue is $250,000. This assumes a unit price of $50 and a sale of 5,000 units.

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Community Q&A

Question

How do I find demand function from price function only?

Maximilian Pagsaligan

Community Answer

Derive the demand function, which sets the price equal to the slope times the number of units plus the price at which no product will sell, which is called the y-intercept, or "b." The demand function has the form y = mx + b, where "y" is the price, "m" is the slope and "x" is the quantity sold.