Handout 2: Elasticity

Handout 2: Elasticity

ECON 320

Handout 2: Elasticity

Elasticity in general refers to the responsiveness of one variable with respect another. For example, if suppose changes in X produce changes in Y. Then one can ask what is the elasticity of Y with respect to X. In particular, the change is expressed in percentages. That is, what is the percentage change in Y when X changes by 1%?

The Price Elasticity of Demand

One of the most important applications of elasticity in economics is the price elasticity of demand.

First think why this matters. Suppose I told you that quantity demand falls by 10 units when price rises by LE1. Is this responsive or not responsive? Hard to say. Depends on starting values.

Case 1: Suppose initially quantity demanded was 20 units and price was LE100. Then a 1% price change creates a 50% fall in quantity demanded. Seems very responsive. That is, it is relatively more elastic.

Case 2: Suppose initially quantity demanded was 1000 units and price was LE10. Then a 10% price change creates a 1% fall in quantity demanded. Seems relatively unresponsive. That is, it is relatively less elastic.

Hence putting it into percentage changes makes it unit free, so that comparison is more meaningful. Hence the formula for elasticity is

Or to be more explicit

Note that because price and quantity demanded always move in opposite directions, the elasticity measurement will always be negative. The elasticity measurement, ε, is to be understood as follows:

A 1% increase in price will cause quantity demanded to fall by ε %.

To understand this better let us consider each case. For each case suppose that the initial price and quantity is given by P0 = RO10 and Q0 = 20. Also for each case suppose that the price rises to P1 = RO11. Now let us consider for each case a different response in terms of the quantity demanded after the price increase.

Case 1: Quantity demand falls to Q1=18

In this case the elasticity is given by

In this case it means that a 1% increase in price will cause quantity demanded to fall by 1%.

Case 2: Quantity demand falls to Q1=15

In this case the elasticity is given by

In this case it means that a 1% increase in price will cause quantity demanded to fall by 2.5%.

Case 3: Quantity demand falls to Q1=19

In this case the elasticity is given by

In this case it means that a 1% increase in price will cause quantity demanded to fall by 0.5%.

Terminology:

If ε 1 then we say demand is elastic.

Ifε1 then we say demand is inelastic.

Ifε=1 then we say demand is unit elastic.

Determinants of Price Elasticity

The primary thing that determines price elasticity is availability of substitutes. The more available are subs, the more people switch to other things when the prices rises, hence the greater the elasticity.

Two important things determine the amount of subs:

  1. The definition of the market: Narrow definition implies many subs (like Coke). Broad definition has fewer subs (like beverages).
  2. Time Horizon: The longer the time frame under consideration the more substitutes a person has available. For example the demand for soft drinks per week or soft drinks per year. If the price of soft drink rises over a period of week you have less time to find substitutes. Over a period of a year you have time to seek out more alternatives. Or instead of soft drinks, the demand for housing.

Other Types of Elasticity

For any factor that affects demand one can measure its elasticity; that is, how responsive is demand to a change in that factor. The main types of elasticity are the cross-price elasticity and the income elasticity

Cross Price Elasticity

The cross price elasticity measures the response of demand to a change in the price of a related good; that is, a substitute or complement. Hence what it measures is by what percentage will demand for the good change when the price of a related good changes by one percent.

Letting z represent the price of a related good the cross price elasticity is given by

Substitutes vs Complements

Because changes in the prices of substitutes and complements have different affects on demand, the sign of the cross price elasticity is different.

Recall that when the price of a substitute good rises, the demand for the good in question will rise. Hence in the elasticity formula, both the numerator and denominator are of the same sign, which means the cross price elasticity will be positive.

However, when the price of a complement good rises, the demand for the good in question will fall. Hence in the elasticity formula, both the numerator and denominator are of different signs, which means the cross price elasticity will be negative.

So if the cross price elasticity is εz = 2, this means the related good is a substitute. In particular, we would interpret the number as follows: If the price of the substitute good rises by 1%, demand for the good will rise by 2%.

If the cross price elasticity is εz = -3, this means the related good is a complement. In particular, we would interpret the number as follows: If the price of the complement good rises by 1%, demand for the good will fall by 3%.

Income Elasticity

The income elasticity measures the response of demand to a change in consumer income. Hence what it measures is by what percentage will demand for the good change when consumer income changes by one percent.

Letting I represent consumer income, income elasticity is given by

Because demand rises when income rises, or falls when income falls, the numerator and denominator in the elasticity measure will be of the same sign, which means the elasticity measure will be positive.

As an example, if the elasticity of income is 0.75, we would interpret this as follows: If income rises by 1%, then demand will rise by 0.75%.

Using Other Elasticity Measurements to Forecast Demand

Let us consider how to use these other elasticities to forecaset demand. Suppose you manage a hotel and the income elasticity of demand is 0.5. This means for every 1% change in income, demand will change by 0.5%. Now suppose you then find out foreign income is expected to fall by 10%. Since demand falls by 0.5% for every1% fall in income, when income changes by 10%, demand will fall by 0.5*10% = 5%.

Let us consider another case. Suppose air travel is a complement to hotels. Again, suppose you are a manager of a hotel and the cross price elasticity between airfare and hotel demand is -2. This means for every 1% change in the price of airfare, demand will change by 2%. Now suppose you then find out airfare is expected to rise by 7%. Since demand falls by 2% for every1% rise in airfare, when airfare rises by 7%, demand for hotels will fall by -2*7% = -14%.

In general, since for some factor x the elasticity measurement is defined as

If we know the elasticity measurement, εx, and we know the percentage change in x, %∆x, then we can solve for the percentage change in demand, %∆QD. That is,

.