## Formula to Calculate Cross-Price Elasticity of Demand

Cross price elasticity of demand formula is used to measure the percentage change in quantity demanded of a product with respect to the percentage change in the price of a related product and it can be evaluated by dividing the percentage change in quantity demanded of a particular product by the percentage change in the price of its related product.

**E**

_{c}= Percent change in a quantity of good A/Percent change in the price of good Bor

**Ec = [(P**

_{1}^{A}+ P_{2}^{A})/(Q_{1}^{B}+ Q_{2}^{B})] * [(Q_{1}^{B}– Q_{2}^{B})/(P_{1}^{A}– P_{2}^{A})]Where,

- Ec is the cross-price elasticity of the demand
- P1
^{A}is the price of good A at time 1 - P2
^{A}is the price of good A at time 2 - Q1
^{B}is the quantity of good B at time 1 - Q2
^{B}is the quantity of good B at time 2

### Explanation

- If the cross-price elasticity of demand is positive, the two goods are said to be supplementary goods i.e. if the price of one good increases then the demand for other goods will increase.
- However, if the cross-price elasticity is negative, then the two goods are said to be complementary goods i.e. if the price of one good increases the demand for the other good will be decreased.

### Examples

#### Example #1

**The Company producing torches and batteries is analyzing the cross-price elasticity of the two goods. The demand for torches was 10,000 when the price of batteries was $ 10 and the demand rose to 15,000 when the price of batteries was reduced to 8$.**

Solution-

- Percentage change in quantity of torches = (15000 – 10000)/(15000 + 10000)/2 = 5000/12500 = 40%
- Percentage change in price of batteries = (8 – 10)/(10 + 8)/2 = -2/9 = -22.22%
- Thus, cross price elasticity of demand = 40%/-22.22% = -1.8

Since the cross-price elasticity of demand of torches and batteries is negative, thus these two are complementary goods.

#### Example #2

**Calculate the cross-price elasticity of demand. The percentage change in the price of apple juice changed by 18% and the percentage change in the quantity of demand changed of orange juice by 12%.**

The following is the data used for the calculation of Cross price elasticity of demand.

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Therefore, it will be

= 12%/18% = 0.667

**Cross-price elasticity of demand will be –**

The cross-price elasticity of demand formula of apple juice and orange juice is positive hence they are substitute goods.

#### Example #3

**The annual price of cinema tickets sold in the year 2010 was $ 3.5 whereas the number of popcorns sold at cinema halls was 100,000. The ticket price increased from $ 3.5 in 2010 to $ 6 in the year 2015. There was a decrease in the sale of popcorns to 80,000 units.**

The following is the data used for the calculation of Cross Price Elasticity of Demand.

**Percentage Change in Price of Ticket**

- Percentage change in the price of ticket = (6-3.5)/(6+3.5)/2
- =0.131579

**Percentage Change in the Quantity of Popcorn Sold**

- Percentage change in the quantity of popcorn sold = (80000-100000)/(80000+100000)/2
- =-0.05556

Calculation of Cross Price Elasticity of Demand is as follows –

**Cross price elasticity of demand will be –**

=-0.422222

Since the cross elasticity of demand is negative the two products are complementary.

### Relevance and use

- Cross-price elasticity of demand formula measures the demand sensitivity of one product (say A) when the price of an unrelated product (say B) is changed.
- Cross-price Elasticity of Demand is used to classify goods. The goods are classified as a substitute or complementary goods based on cross-price elasticity of demand. If the cross elasticity of demand is positive the two goods are the substitute and if the cross elasticity is negative the two goods are complementary. Further, if the magnitude of cross elasticity is high, the two goods are a closer substitute or closer complementary depending on the sign.
- It also helps in classifying the market structure. If the cross elasticity of demand is infinite the markets are considered as perfectly competitive whereas zero or close to zero-cross elasticity makes the market structure a monopoly. If there is a high cross-elasticity it is called an imperfect market.
- Cross-price elasticity of the demand helps large firms to decide pricing policy. Large firms generally have more variety of similar and related goods. Thus, cross elasticity of demand helps such firms in decision making whether to increase the price of such related products.
- Cross-price elasticity of the demand formula helps in the classification of products between various industries. If the goods are complimentary that is the cross elasticity is negative, they are classified in different industries. If the goods have positive cross-price elasticity i.e. they are substitute goods then they belong to one industry.

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