## Cross Price Elasticity of Demand Definition

Cross Price Elasticity of Demandmeasures the relationship between price a demand i.e., change in quantity demanded by one product with a change in price of the second product, where if both products are substitutes, it will show a positive cross elasticity of demand and if both are complementary goods, it would show an indirect or a negative cross elasticity of demand. In simple terms, it measures the sensitivity of demand for one quantity X when the price of related good Y is changed.

### Cross Price Elasticity of Demand formula

It is calculated by dividing the percentage change in the quantity of good X by percentage change in the price of good Y which is represented mathematically as

**Cross Price Elasticity of Demand = (∆Q**

_{X}/Q_{X}) ÷ (∆P_{Y}/P_{Y})Further, the formula for cross-price elasticity of demand can be elaborated into

**Cross Price Elasticity of Demand = (Q**

_{1X}– Q_{0X}) / (Q_{1X}+ Q_{0X}) ÷ (P_{1Y}– P_{0Y}) / (P_{1Y}+ P_{0Y}),where

- Q
_{0X}= Initial demanded quantity of good X, - Q
_{1X}= Final demanded quantity of good X, - P
_{0Y}= Initial price of good Y and - P
_{1Y}= Final price of good Y

### Step by Step Calculation of the Cross Price Elasticity of Demand

This can be determined in the following five steps:

**Step #1:**Firstly, identify P_{0Y}and Q_{0X}which is the initial price of good Y an initially demanded quantity of good X respectively.**Step #2:**Now, determine the final demanded quantity of good X and the final price of good Y which are termed as Q_{1X}and P_{1Y}respectively.**Step #3:**Now work out the numerator of the formula which represents the percentage change in quantity. It is arrived at by dividing the difference of final and initial quantities (Q_{1X}– Q_{0X}) by summation of the final and initial quantities (Q_{1X}+ Q_{0X}) i.e. (Q_{1X}– Q_{0X}) / (Q_{1X}+ Q_{0X}).**Step #4:**Now work out the denominator of the formula which represents the percentage change in price. It is arrived at by dividing the difference of final and initial prices (P_{1Y}– P_{0Y}) by summation of the final and initial prices (P_{1Y}+ P_{0Y}) i.e. (P_{1Y}– P_{0Y}) / (P_{1Y}+ P_{0Y}).**Step #5:**Finally, the cross-price elasticity of demand is calculated by dividing the expression in Step 3 by expression in Step 4 as shown below.

**Cross price elasticity of demand Formula** = (Q_{1X} – Q_{0X}) / (Q_{1X} + Q_{0X}) ÷ (P_{1Y} – P_{0Y}) / (P_{1Y} + P_{0Y})

### Examples

#### Example #1

**Let us take the simple example of gasoline and passenger vehicles. Now let us assume that a surge of 50% in gasoline price resulted in a decline in the purchase of passenger vehicles by 10%. Calculate the cross-price elasticity of demand in this case.**

Using the above-mentioned formula the cross-price elasticity of demand can be calculated as:

Percentage change then the number of passenger vehicles ÷ Percentage change the price of gasoline

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Since we can see a negative value for cross elasticity of demand, it vindicates the complementary relationship between gasoline and passenger vehicles.

#### Example #2

**Let us assume that there two companies in the business of selling soft drinks. At present, company 2 sells soft drinks Y at $3.50 per bottle, while company 1 is able to sell 4,000 bottles of soft drinks Y per week. In order to impact the sales of company 1, company 2 has been decided to decrease the price to $2.50 which resulted in decreased sales of 3,000 bottles of soft drinks Y per week. Calculate the cross-price elasticity of demand in the case.**

Given, Q_{0X} = 4,000 bottles, Q_{1X} = 3,000 bottles, P_{0Y} = $3.50 and P_{1Y} = $2.50

Therefore, the cross price elasticity of demand can be calculated using above formula as,

- Cross price elasticity of demand = (3,000 – 4,000) / (3,000 + 4,000) ÷ ($2.50 – $3.50) / ($2.50 + $3.50)
- = (-1 / 7) ÷ (-1 / 6)
- = 6/7 or 0.857

Since, we can see a positive value for cross elasticity of demand, it vindicates the competitive relationship between soft drink X and soft drink Y.

### Relevance and Use

It is of paramount importance for a business to understand the concept and relevance of cross-price elasticity of demand to understand the relationship between the price of a good and the quantity demanded of another good at that price. It can be used to decide the pricing policy for different markets and for various products or services. The cross-price elasticity behaves differently based on the type of relationship between the goods which are discussed below.

**#1 – Substitute products**

In case both goods which are perfect substitutes to each other resulting in perfect competition, then an increase in the price of one goodwill lead to an increase in demand for the rival product. For example, various brands of cereal are examples of substitute goods. It is to be noted that the cross-price elasticity for two substitutes will be positive.

**#2 – Complementary products**

If in case one good is complementary to the other good, then a decrease in the price of one goodwill leads to an increase in demand for the complementary good. The stronger the relationship between the two products, the higher will be the coefficient of cross-price elasticity of demand. For example, game consoles and software games are examples of complementary goods. It is to be noted that the cross elasticity will be negative for complementary goods.

**#3 – Unrelated products**

In case there is no relationship between the goods, then an increase in the price of one good will not affect the demand for the other product. As such, unrelated products have a zero cross elasticity. For example, the effect of changes in taxi fares on the market demand for milk.

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