# Cross Elasticity of Demand

Cross elasticity of demand is the ratio of percentage change in quantity demanded of a product to percentage change in price of a related product.

One of the determinants of demand for a good is the price of its related goods. For example, if two goods A and B are consumed together i.e. they are complements, an increase in the price of B will increase the price of the bundle (A + B) which in turn will decrease the demand for A and vice versa. Similarly, if A and B are substitutes, an increase in price of A will increase demand for B because B becomes relatively cheaper and consumer will switch from A to B thereby increasing its demand. Cross elasticity of demand is a quantitative tool which measures the sensitivity of demand of one product, say A, to price changes of the other product, say B.

## Formula

Cross elasticity of demand can be calculated using the following formula:

Cross Elasticity of Demand E_{A, B} | |

= | % increase in quantity demanded of A |

% increase in price of product B |

Percentage changes in the above formula are calculated using the mid-point formula which divides actual change by average of initial and final values.

The formula to calculate cross elasticity thus becomes:

E_{A, B} = | Q_{f} − Q_{i} | ÷ | P_{f} − P_{i} |

(Q_{f} + Q_{i}) ÷ 2 | (P_{f} + P_{i}) ÷ 2 |

Where,

Q_{f} and Q_{i} are the final and initial quantities demanded of product A, respectively; and

P_{f} and P_{i} are the final and initial prices of product B.

## Cross elasticity, substitutes, and complements

Cross elasticity of demand indicates whether any two products are substitutes or complements or independent goods.

### Substitute goods

When the cross elasticity of demand for product A relative to a change in price of product B is positive, it means that in response to an increase (decrease) in price of product B, the quantity demanded of product A has increased (decreased). Since A, say Coke, and B, say Sprite, are substitutes, an increase in price of product B means that more people will consume A instead of B, and this will increase the quantity demanded of product A. Increase in quantity demanded of product A relative to increase in price of product B gives us a positive cross elasticity of demand.

### Complimentary Goods

When the cross elasticity of demand for product A relative to change in price of product B is negative, it means that the quantity demanded of A has decreased (increased) relative to an increase (decrease) in price of product B. As A, say car, and B, say gas, are complimentary goods, and an increase in price of B will reduce the quantity demanded of A. This is because people consume both A and B as a bundle and an increase in price reduces their purchasing power and decreases quantity demanded.

## Rule

A negative (positive) cross elasticity of demand means that the products are substitutes (complements).

## Example 1: cross elasticity and substitutes

The quantity demanded or product A has increased by 12% in response to a 15% increase in price of product B. Calculate the cross elasticity of demand and tell whether the product pair is (a) apples and oranges, or (b) cars and gas.

Cross elasticity of demand

= % change in quantity demanded of A ÷ % change in price of B

= 12% ÷ 15%

= 0.67

Since the cross elasticity of demand is positive, product A and B are substitute goods. They are apples and oranges.

## Example 2: cross elasticity and complements

The government of Selgina is serious about drugs. Possession of drugs is illegal and is severely penalized. However, a black market exists which the government has failed to dismantle despite serious attempts. Khusenichho Chamling, the health minister, is worried about the situation. In early 20X9, a consultant working with health ministry suggested that the government should increase the price of a pack of cigarettes from 200 Selgina dollars (S$) to S$600. A survey conducted in December 20X9 suggested that over the year, the quantity demanded of marijuana decreased from 2,000 kgs per day to just 800 kgs. Calculate the cross elasticity of demand and tell why the policy has proved so effective.

Percentage increase in price of cigarettes

= (600 − 200) ÷ [(600+200) ÷ 2]

= 100%

Percentage increase in quantity demanded of marijuana

= (800 − 2,000) ÷ [(800+2000) ÷ 2]

= -85.71%

Cross elasticity of demand

= % change in quantity demanded ÷ % change in price

= -85.71% ÷ 100%

= -0.86

Cigarettes and marijuana have negative cross elasticity of demand which tells that they are complimentary goods.

The policy has proved effective because cigarettes and marijuana are consumed together. Increase in price of cigarettes increased the price of the whole bundle and reduced the purchasing power of people and resulted in a decrease in consumption of marijuana.

by Obaidullah Jan, ACA, CFA and last modified on

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