This document outlines a 5-step process for determining the optimal price for a product to maximize profit or revenue. The steps are: 1) find critical points of the revenue and profit functions, 2) analyze historical price and demand data, 3) calculate the estimated optimal prices, 4) determine estimated revenue and profit using these prices, and 5) compare estimated to actual revenue and profit curves. The optimal price to maximize profit is higher than for revenue. The example calculates optimal prices of $8.35 for profit and $6.35 for revenue using a linear demand curve estimated from historical data.

1. Mary Suma|tmarysuma@gmail.com|http://www.linkedin.com/in/sumayyappan| /tmarysuma
PRICE OPTIMIZTION
Determining the optimal selling price using Demand, Revenue
and Profit Function
-Mary Suma
Contents
Background...................................................................................................................................................................2
Abstract..........................................................................................................................................................................2
Step1 : Find Critical Points ......................................................................................................................................3
Revenue Function ..................................................................................................................................................3
Profit Function.........................................................................................................................................................3
Step 2: Analyze price variation in Historical data...........................................................................................3
Step3: Find Optimal price........................................................................................................................................5
Step 4: Find Profit and Revenue using Optimal Price...................................................................................5
Step5 : Pricing for maximum profit or revenue...............................................................................................5
Conclusion.....................................................................................................................................................................7
Reference.......................................................................................................................................................................7

2. Mary Suma|tmarysuma@gmail.com|http://www.linkedin.com/in/sumayyappan| /tmarysuma
Background
Price optimization is the process of finding the sweet spot and the perfect balance between
profit and value. In this presentation, ‘Pricing to maximize profit’ is explained in 5 simple steps.
• to estimate the right price of a product
• to determine how customers, react to different pricing strategies.
Machine Learning uses historical behaviour of customer’s demand to find the most optimal price
to maximize the profit and revenue.
Abstract
Demand curve represents the relationship between price of the product and the quantity sold at
a certain point in time.
Linear Demand Curve is given by
Demand Curve D(x) = ax+b
Total Revenue is given by
Revenue R(x) = x * D(x) = ax^2+bx
Total Profit is given by
Total Profit P(x) = (x-c) * D(x) = ax^2-axc+bx-bc
Imagine a company that has been selling a product, that follows linear demand Curve
D(x) = ax+b , where alpha(a) = -40 and beta(b) = 500.
Demand function, Quantity vs Price

3. Mary Suma|tmarysuma@gmail.com|http://www.linkedin.com/in/sumayyappan| /tmarysuma
Step1 : Find Critical Points
Now, to optimize the profit, find the critical points and check if these critical points of profit
function are maximum points or minimum points. If the second derivative is negative, it is the
maximum point. If one of critical point is maximum, that is the optimal price.
Revenue Function
First Derivative of Revenue function, R’(x) = 2ax+b
• Critical point is when R’(x) = 0
• 2ax+b = 0 ; x = -b/2a, which is the optimal price for Profit
Optimal Price to maximize profit is given by P_MaxProfit = -b/2a
Profit Function
First Derivative of Profit function P’(x) = 2ax – ac+b
• Critical point is when P’(x) = 0
• 2ax – ac+b = 0; x = (-b+ac)/2a
Optimal Price to maximize revenue is given by P_MaxRevenue = (-b+ac)/2a
The price that maximizes profit is always bigger than the one that maximizes total revenue since c is
always positive.
Step 2: Analyze price variation in Historical data
The graph below shows the historical variation in prices and demand for the last 365 days.
Create Demand Curve using historical data.

4. Mary Suma|tmarysuma@gmail.com|http://www.linkedin.com/in/sumayyappan| /tmarysuma
Historical Performance
Demand function of Historical data.
The model may be significant at p<=0.05

5. Mary Suma|tmarysuma@gmail.com|http://www.linkedin.com/in/sumayyappan| /tmarysuma
Step3: Find Optimal price
Calculate the estimate optimal price for Profit and Revenue function using the formula derived
in step1.
P_MaxProfit = -b/2a
P_MaxRevenue = (-b+ac)/2a
From historical demand function,
Alpha, a = -38.85226
Beta, b = 493.6576 (Intercept)
Hence
P_MaxProfit = 8.35301
P_MaxRevenue = 6.35301
Step 4: Find Profit and Revenue using Optimal Price
Using the estimated price for Revenue and Profit, we now calculate the Revenue and total profit.
Replacing alpha = -40 and beta = 500 in the demand function, we get
• Revenue R(x) = xD(x) = ax^2+bx
Revenue with estimated optimal price, 6.35301 = 1562.076
• Total Profit P(x) = (x-c)D(x) = ax^2-axc+bx-bc
Profit with estimated optimal price, 8.35301 = 722.0756
Step5 : Pricing for maximum profit or revenue
Now lets plot the relationships between estimated profit , estimated revenue , true profit and
true revenue. stat_function can abe used to plot graph of a quadratic function. An alternate

6. Mary Suma|tmarysuma@gmail.com|http://www.linkedin.com/in/sumayyappan| /tmarysuma
approach is to define a new dataframe containing x and y coordinates for the function then
use geom_line to draw the curve.
Estimated Revenue vs Actual Revenue
As represented in the revenue curve, the optimal price is 6.35 and about 248 units will be sold in
order to maximize the revenue to 1,562.
Estimated Profit vs Actual Profit
As represented in the profit curve, the optimal price is 8.35 and about 148 units will be sold in
order to maximize the profit to 722.
(8.35, 722)
(6.35, 1562)

7. Mary Suma|tmarysuma@gmail.com|http://www.linkedin.com/in/sumayyappan| /tmarysuma
Inference:
• When the primary objective is maximizing the profit, the price is set as close as
possible to the peak of revenue curve or the profit curve.
• In the example, estimated revenue and estimated profit curves are similar to the
actual revenue and actual profit
Conclusion
Using Price optimization, businesses predict to what degree demand is altered with the change
of price. It is used to measure how sensitive customers can be to price changes. Pricing for
maximum profit, is most effective with established products in stable markets. New products aim
to increase the market share than increasing revenue and hence different pricing strategy
should be used.
Reference
Pricing and revenue optimization - Robert Lewis
Price Optimization - Yuri Fonseca