Original article was published by Moeedlodhi on Artificial Intelligence on Medium

# A detailed introduction to Price Elasticity- with example.

Significance of Price Elasticity and how it is used to find the optimum price point.

As a Data Science internee, I have come to the realization that **the amount of value you provide is directly proportional to the price tag you get.**

The number of skills you possess only matters if those skills can translate to added value to your customer in the form of increased sales or decreases costs. Simple. **The more value you provide, the more valuable you are as a Data Scientist.**

In today’s article, I will be going over a significant topic that I have come across repeatedly during my time as an internee and that topic is **Price Elasticity.**

**What is “Price Elasticity”?**

Price Elasticity tells us **how sensitive sales of a particular product are to a unit change in its price.**

It is nothing more than** the ratio of the percentage change in sales to the percentage change in price. for example:**

Let’s say, On a daily basis, A particular store sells 20 units of handbags.

The store has set a price tag of let’s to say $200 on the bag and they are making a profit on it.

One day, the store manager decides to** increase the price of the bag to $220 which brings the sales down to 15.**

The percentage change in Price is:

And the percentage change in Sales is:

So if I were to calculate the Price Elasticity for the bag, I would take the ratio of **change in sales to change in price. Which turns out to be:**

So what does this 2.5 tell us? To put it in simple terms, a unit change in price will have a **2.5 times effect on the volume of sales.** This value is also known as the** Elasticity coefficient which is just a measure of how sensitive sales are to a unit change in price.**

In this example, We got a value of 2.5 for the Elasticity, But what if our Elasticity is less than 1? Or what if it’s 0? Let’s take a look at these cases.

## 1)Perfectly inelastic(Coefficient = 0)

If we have a “Perfectly Inelastic” case where our **Elasticity coefficient is 0, **Our Sales will not be affected whatsoever by the change in Price. Essential goods such as fuel are perfectly Inelastic meaning, even if the price goes up considerably, There won’t be a drop in Sales because of the high dependency people have on it.

**2)Perfectly elastic( Coefficient = **∞)

If we have a “Perfectly elastic” case where our **Elasticity coefficient is almost infinite, **A unit price change would have a drastic effect on the Sales of the product. Products that have **substitutes **and “don’t affect our everyday lives” tend to have a high Elasticity coefficient. for example: If the price of apples goes up, People will look for substitutes or even not buy apples at all because they can “survive” without them.

## 3)Unit elastic (Coefficient = 1)

In the case of Unit elasticity, a unit change in price will have a unit change in the quantity sold. for example, a 15 percent increase in price leads to a 15 percent contraction in Sales. So basically the Elasticity coefficient is 1.

## 4)Elasticity between 0 and 1

If Elasticity is between 0 and 1, It is also an Inelastic case. Obviously, not perfectly inelastic but a change in price will have a very small effect on the change in sales but will **result in an increase in Revenue.**

# Log-Log model

Now we have gone over the concept of Elasticity and the different types of Elasticities. But the question remains, **Why is it important? As a business owner, why do I care?**

Well, the thing is, Business owners want to know how much their Sales volume will change with a change in price. And they are also interested in the Selling point which would **maximize their profits**. And, This is where the concept of Elasticity helps and is used extensively.

To find Elasticity, we turn to the** Log-Log model**.

I have Sales data for a particular product for different prices over the course of 2 years.

Looking at this graphically, I get

For months where price has shown a decrease, Average Daily Sales have shown a considerable increase. There are a few points where Sales rise and drop unexpectedly but this can be attributed to special events that happen throughout the year such as Christmas, etc.

Let’s visualize this further

The Scatterplot gives us the same finding, Lower prices have increased Sales.

Moving onto Elasticity.

## The Log-Log Model

Now that we have visualized the data, We will move towards taking the logarithm of both the dependent and independent variable

`data['Log(price)'] = np.log(data['Price'])`

data['Log(Avg Sales)'] = np.log(data['AvgDailySales'])

data['constant'] = data['Price']/data['Price']

We now fit a Linear Regression model to our variables and find the Elasticity coefficient.

from sklearn.linear_model import LinearRegression

LR = LinearRegression()

X= data[['Log(price)','constant']]

y = data['Log(Avg Sales)']LR.fit(X,y)

coef = LR.coef_

coefintercept = LR.intercept_

intercept

The coefficient is -2.934 and the intercept is 25.672

**The -2.934 here is the Elasticity coefficient for our variables**.

The log-log model equation can be written as:

where **B1 is -2.934** and **Bo is 25.672.**

The same equation can be simplified and written as:

Where “Alpha” is the **exponential of the intercept**, X is the **price**, “Beta” is our **Elasticity coefficient **and Y is our** Expected Demand. Let’s look at this further.**

We will add another column to our dataset by the name of “**Expected Demand**” and visualize it.

So far, So good.

The minimum Price the handbag is sold for is 1917, Let’s say, The cost of “producing one bag” is 1700. Can we find an **optimum price point**? Let’s take a look

What I have done in the above visual is set a **base price of 1700 which is the unit price of one bag.**

The cost is the **product of Expected Demand and Base Price **and Rev is the** product of Expected Demand and Price.**

The profit is nothing but the difference between the Rev(Revenue) and the Cost.

Plotting the profit curve gives us:

A steady increase in profit for a decrease in price but a further decrease in price (2795 to 1917) results in a drop. So for this case, The Price for the products should not go down from 2795 as that will **minimize the profits and is not the optimum point.**

Note: We had to make an assumption of the base price of 1700 so that does play a role in determining the optimum sales point.

# Conclusion

In this article, I went over the basics of Elasticity and how Elasticity coefficients can be used to better understand the behavior of Data and find the “optimum Selling price”.

Retail is an extremely interesting domain and Data Scientists working on Retail projects play a very important role in the business.

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