Elasticity and Its ApplicationsPRINCIPLES OF ECONOMICS (ECON 210)

BEN VAN KAMMEN, PHD

IntroductionThis is the first of 4 chapters that comprise the “middle” of this course.

• These chapters are “extensions” to the Supply and Demand model.

•Elasticity: quantifying how sensitive quantity demanded and quantity supplied are.• To price.• To incomes.• To prices of related goods.

•The Laws of Supply and Demand only say the direction of the relationship between Q and P.• Nothing about the magnitude. Elasticity is about measuring this. How “sensitive” are they?

Elasticity defined•Elasticity: a numerical measure of how responsive 𝑄𝑄𝐷𝐷 or 𝑄𝑄𝑆𝑆 is to one of the factors that determine it.

•Examples,

1. Price Elasticity of Demand

2. Price Elasticity of Supply

3. Income Elasticity of Demand

4. Cross Price Elasticity of Demand

Symbolic definitions of elasticity 1. Price Elasticity of Demand (good x)

𝐸𝐸𝑃𝑃𝑥𝑥,𝑄𝑄𝐷𝐷 =%∆𝑄𝑄𝐷𝐷%∆𝑃𝑃𝑥𝑥

2. Price Elasticity of Supply

𝐸𝐸𝑃𝑃𝑥𝑥,𝑄𝑄𝑆𝑆 =%∆𝑄𝑄𝑆𝑆%∆𝑃𝑃𝑥𝑥

3. Income Elasticity of Demand

𝐸𝐸𝐼𝐼,𝑄𝑄𝐷𝐷 =%∆𝑄𝑄𝐷𝐷%∆𝐼𝐼

4. Cross Price Elasticity of Demand

𝐸𝐸𝑃𝑃𝑦𝑦,𝑄𝑄𝐷𝐷 =%∆𝑄𝑄𝐷𝐷

%∆𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜𝑜𝑃𝑃𝑃𝑃 𝑔𝑔𝑜𝑜𝑜𝑜𝑔𝑔

Illustration• Initially the price is $40/unit. The quantity

demanded is 100.• How does quantity demanded change in

each case when the price rises to $50/unit?• In both cases, we are raising price by the

same amount.

Go to Hotseat, buddy.

Which curve is more “sensitive”?•Both elasticity calculations have the same % change in price.

%∆𝑃𝑃 =50 − 40

40∗ 100 = 25%

•The flatter curve decreases quantity demanded by:20 − 100

100∗ 100 = −80 𝑝𝑝𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑝𝑝𝑜𝑜.

•The steeper curve decreases quantity demanded by:95 − 100

100∗ 100 = −5 𝑝𝑝𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑝𝑝𝑜𝑜.

•The two elasticity calculations are:−8025

= −3.2 and−525

= −0.2,

which reveals that the flatter curve is more elastic.• The larger negative effect means it is more sensitive.

Really cool mathematical note•The terms in the Elasticity definition can be cleverly re-written,

𝐸𝐸𝑃𝑃𝑥𝑥,𝑄𝑄𝐷𝐷 =𝑄𝑄1 − 𝑄𝑄0𝑃𝑃1 − 𝑃𝑃0

∗𝑃𝑃0𝑄𝑄0

.

• As the price change becomes infinitely small, so does the quantity response.• But the ratio of the two changes approaches the (inverse of the) slope of the demand curve:

∆𝑄𝑄∆𝑃𝑃 .

•So you can write elasticity:

𝐸𝐸𝑃𝑃𝑥𝑥,𝑄𝑄𝐷𝐷 =∆𝑄𝑄∆𝑃𝑃

∗𝑃𝑃0𝑄𝑄0

,

where ∆𝑄𝑄∆𝑃𝑃

is the inverse of the slope of the demand curve at the point, (𝑄𝑄0,𝑃𝑃0).

The midpoint method for calculating elasticity

•Maybe you didn’t think that mathematical note was as cool as I did.

•But here’s why it’s useful. Calculating elasticity at one point is less confusing that calculating it based on two endpoints.• Even when you are given two endpoints, you can calculate the elasticity based on the midpoint

between them.

•It’s less confusing because you don’t have to choose which endpoint to use as the “base” when you’re calculating % change. According to the midpoint method, the effect of changing the price (on a linear demand curve) could be calculated as:

𝐸𝐸𝑃𝑃𝑥𝑥,𝑄𝑄𝐷𝐷 =𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑄𝑄𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

∗1

𝑠𝑠𝑠𝑠𝑜𝑜𝑝𝑝𝑃𝑃 𝑜𝑜𝑜𝑜 𝑔𝑔𝑃𝑃𝑑𝑑𝑑𝑑𝑝𝑝𝑔𝑔 𝑃𝑃𝑐𝑐𝑃𝑃𝑐𝑐𝑃𝑃.

The midpoint method (continued)•Using the midpoint eliminates the following confusion. Say you have a linear demand curve that goes through the points: Price $7, Quantity 3 and Price $8 and Quantity 2.

•Elasticity would be sensitive to the “base” and would depend on which end point was the starting point.

𝐸𝐸𝑃𝑃𝑥𝑥,𝑄𝑄𝐷𝐷 =3 − 2

37 − 8

7= −

73

or 𝐸𝐸𝑃𝑃𝑥𝑥,𝑄𝑄𝐷𝐷 =2 − 3

28 − 7

8= −4?

•Using the midpoint is independent of the starting point, though.

𝐸𝐸𝑃𝑃𝑥𝑥,𝑄𝑄𝐷𝐷 =7.52.5

∗1−1

= −3.

Sensitivity to price•To illustrate sensitivity to price, consider two extreme examples.

• This example uses supply, but the concepts of perfectly elastic and perfectly inelastic apply equally to demand.

• Perfectly inelastic means not at all sensitive.

𝐸𝐸𝑃𝑃𝑥𝑥,𝑄𝑄𝑆𝑆 =𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

𝑄𝑄𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∗

1∞ = 0

• Perfectly elastic means infinitely sensitive.

𝐸𝐸𝑃𝑃𝑥𝑥,𝑄𝑄𝑆𝑆 =𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

𝑄𝑄𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∗

10 = ∞

Perfect elasticity and perfect inelasticity•When supply or demand is perfectly elastic, quantity is so sensitive to price that even a small change would make the quantity response infinitely large.

•When it is perfectly inelastic, quantity is completely insensitive to price changes and doesn’t respond at all.

•These are the two extreme ends of the elasticity spectrum.• What determines where a particular demand (supply) curve will fall on this spectrum?

Determinants of elasticity•Demand is more elastic under the following circumstances.

• Availability of close substitutes.• When the good is defined specifically rather than broadly.• When the good is a luxury rather than a necessity.• In the long run rather than the short run.

•Supply is more elastic under the following circumstances.• When producers are able to adjust their output easily.• In the long run rather than the short run.

• Producers can change the scale of their production.

What the other elasticity measures tell you about demand

•Income elasticity can tell you whether a good is:• Normal,• A luxury,• Inferior.

•Cross price elasticity can tell you whether two goods are• substitutes or complements.

Normal, luxury, inferior•A 1% increase in consumers’ incomes can generate one of the following changes in 𝑄𝑄𝐷𝐷.

• An increase of 0-1%• An increase of >1%• A decrease in 𝑄𝑄𝐷𝐷

•The first is a case of a normal good; when consumers’ incomes rise they spend more money on the good in question and the quantity demanded increases.• The increase is roughly that good’s share in the consumers’ budget, so 0 ≤ 𝐸𝐸𝑃𝑃𝑥𝑥,𝑄𝑄𝐷𝐷 ≤ 1.

•The second is a case of a luxury good; consumers make proportionally larger demands than the size of the income increase.• The share of this good in the budget increases as well as the budget, itself.

•Last is an inferior good; when consumers get more income, they substitute away from it and demand less.

Substitutes and complementsWhen the price of another good changes and the demand for the good in question changes, the goods are related.

•Quantity demanded increases when:• The price of a complement decreases, or• the price of a substitute increases.

•Quantity demand decreases when:• The price of a complement increases, or• the price of a substitute decreases.

•So what is the sign on cross price elasticity?

Substitutes and complements (continued)

•Cross price elasticity for a complement?

𝐸𝐸𝑃𝑃𝑦𝑦,𝑄𝑄𝐷𝐷 =%∆𝑄𝑄𝐷𝐷

%∆𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜𝑜𝑃𝑃𝑃𝑃 𝑔𝑔𝑜𝑜𝑜𝑜𝑔𝑔=

−+

< 0

• The price of the other good goes up and you demand less of both (because they go together).

•Cross price elasticity for a substitute?

𝐸𝐸𝑃𝑃𝑦𝑦,𝑄𝑄𝐷𝐷 =%∆𝑄𝑄𝐷𝐷

%∆𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜𝑜𝑃𝑃𝑃𝑃 𝑔𝑔𝑜𝑜𝑜𝑜𝑔𝑔=

++

> 0

• The price of the other good goes up and you demand less of it (and more of this).

Substitutes and complements (concluded)

•Complements are goods that are consumed together, so prices of complements are inversely related to demand.• A negative cross price elasticity indicates complementary goods.

•Substitutes are goods that are perceived as alternatives to each other. The prices of substitutes are positively related to demand.• A positive cross price elasticity indicates that the two goods are substitutes.

Applications of ElasticityECON 210: PRINCIPLES OF ECONOMICS

Pricing a good to maximize revenue•Imagine you are managing a firm that produces some good that has very low marginal cost.• “Scalable” products like software

downloads, subscriptions to periodicals, or streaming media services are good examples.

•You are tasked with finding the price that will make your company the most revenue.• And with near zero variable costs, the most

profit, too!

•Revenue can be shown with ease on a graph of a demand curve.• Revenue is just the price multiplied by the

quantity demanded.

Pricing a good to maximize revenue (continued)

•As it turns out, the price shown on the previous slide is the revenue-maximizing one.• You can confirm this with “trial and error” by trying other prices.• If your current price is $80/unit, you could raise revenue by decreasing price.• If your current price is $10/unit, you could raise revenue by increasing price.

Revenue and elasticity•Revenue changes as the price changes.

• Because of two opposing forces. Cutting (raising) the price tends to decrease (increase) revenue. Butcutting price increases sales (quantity), and the quantity change goes in the opposite direction because of the Law of Demand.

• The effect on revenue depends simply on which force is bigger.%∆ Revenue = %∆𝑃𝑃𝑥𝑥 + %∆𝑄𝑄𝐷𝐷, and

demand elasticity is 𝐸𝐸𝑃𝑃𝑥𝑥,𝑄𝑄𝐷𝐷 =%∆𝑄𝑄𝐷𝐷%∆𝑃𝑃𝑥𝑥

⇔ 𝐸𝐸𝑃𝑃𝑥𝑥,𝑄𝑄𝐷𝐷 ∗ %∆𝑃𝑃𝑥𝑥 = %∆𝑄𝑄𝐷𝐷.

•So, combining these two yields:%∆ Revenue = %∆𝑃𝑃𝑥𝑥 1 + 𝐸𝐸𝑃𝑃𝑥𝑥,𝑄𝑄𝐷𝐷 .

•The effect of a price change depends crucially on the demand elasticity.

An elasticity-based pricing strategy•To be concrete, consider 10% price changes. We know that when the price is “too high”, e.g., above $60/unit, a -10% price change increases revenue. So,

0 < %∆ Revenue. %∆Revenue = −10 ∗ 1 + 𝐸𝐸𝑃𝑃𝑥𝑥,𝑄𝑄𝐷𝐷 .

•For this to hold, the parentheses must be negative. This is true only if the elasticity is greater then 1 in absolute value.• I.e., if demand is elastic, a price cut increases revenue.

An elasticity-based pricing strategy•Conversely when price is “too low” (below $60/unit), a +10% price change increases revenue.

0 < %∆ Revenue. %∆Revenue = 10 ∗ 1 + 𝐸𝐸𝑃𝑃𝑥𝑥,𝑄𝑄𝐷𝐷 .

•Now the number in parentheses must be positive, which holds when elasticity is less than 1 in absolute value.• If demand is inelastic, a price increase increases revenue.

Pricing a good to maximize revenue (concluded)

•To sum up: if demand for your product is elastic, you could raise revenue by decreasing price.

•If demand is inelastic, you could raise revenue by increasing price.

Technological progress when demand is inelastic

•Demand elasticity also determines how producers fare when technology shifts supply outward.

•If demand is elastic, the total revenue in that industry increases.• Price goes down, but quantity goes way up.

•If demand is inelastic, revenue decreases with technological progress.• Price goes down, and quantity only goes up a little.

Technological progress and revenues

Raising transaction costs when demand is inelastic

•Society wants some markets to produce less (none?) of their output.• Drugs.• Prostitution.• Inputs that pollute, e.g., coal.

•On paper it could do this by decreasing supply. The “War on Drugs” is one strategy to decrease the supply of drugs.• By adding additional costs of avoiding law enforcement to the costs of producing and selling drugs.• The equilibrium moves up the demand curve: higher prices of drugs mean less quantity.

•But how elastic is the demand for drugs? How much less drugs do we have?

Research suggests: “not very” elastic•When demand is inelastic and you

increase the price, revenue goes up.• Yes, the revenue of drug dealers.• Probably not the objective of the policy.• Makes the “War” exceedingly difficult

to win.• More money to:

• bribe enforcement agents, • escalate real drug wars, and • invent new drugs.

Raising transaction costs with a tax•The silver lining in this disaster is that the additional revenue spent on drugs does not have to go to drug cartels.

•Recall that taxing a good also decreases its supply.• Something that the states of Washington and Colorado recently remembered. At least with respect to

marijuana laws.

•We will explore the economics of taxation more in a later lecture, but suffice it for now to say that the government could get most of the revenue gains that would otherwise go to sellers in the drugs market.• The light green rectangle goes to the government instead of to sellers.• Inelastic demand means we still wouldn’t discourage much drug use with taxes, but at least we

wouldn’t be enriching the drug cartels.

Cap and trade environmental policies•The right to discharge pollution can be viewed as an economic good.

•In the sense that some production relies on the emission of pollutants.• Or at least is made cheaper by polluting.

•Producers would be willing to pay to emit pollutants if the alternative was shutting down or paying to process pollutants into a “clean” form.

•There is a demand curve for the right to pollute.

Cap and trade environmental policies (continued)

•Pollution is a problem if there is no supply curve, i.e., when no one “owns” the air and can function as seller. Consequently there is no price that firms pay, so they pollute too much.

•“Cap and Trade” policies correct this problem by making the right to pollute scarce (the “cap”).

•Then polluters are allowed to “trade” pollution allowances, allocating them to their most valued uses. • This happy outcome is discussed further in chapter 10.3 of the textbook, but I will not elaborate further

here.

Cap and trade environmental policies (continued)

Equilibrium (less pollution)

Cap and trade and elasticity•This is all very cool, but how does it relate to elasticity?

•The supply of pollution allowances is perfectly inelastic.• If demand increases, the price goes up but quantity does not.

•Environmentalists can reduce pollution by buying pollution permits and not using (“retiring”) them.• This prices some polluters out of the market and reduces pollution even further.• And wouldn’t be possible without tradable pollution permits.

Retiring pollution allowances

New equilibrium

Retired permitsRemaining pollution

“Put a bounty on snakes, you get more snakes”

•What happens when you do the same thing for a good that has elastic supply?

•Snake breeding apparently fits this criterion. And it is not very expensive either.

•We know this because when the governor of Delhi (in India) tried to combat the city’s problem with cobras by offering a bounty for the snakes’ skins, enterprising citizens began farming cobrasfor the purpose of collecting the bounty.

The “cobra effect”: the bounty creates a market for cobras

Equilibrium with more snakesEquilibrium without bounty: snakes are not valuable enough to pay for

“Put a bounty on snakes, you get more snakes” (concluded)

•Similar incarnations of the cobra effect have been observed in Vietnam (with rats), the U.S. (feral pigs and coyotes).• In a non-animal realm, it underlies the

failures of gun buy-back programs and the conjecture that public protesters’ burning of Beatles merchandise (in 1966) actually increased the sales of the band’s records.

Regarding the cobras, economists generally agree that the best solution would be to let this guy handle the problem.

Conclusion•Price elasticity tells you how sensitive quantity demanded (or supplied) is to the good’s price.

•Cross price elasticity measures tell you about relationships (substitutes/complements) between goods.

•Income elasticity measures tell you whether a good is normal or inferior.

•Elasticity matters a great deal for how firms price their products and for designing policies to increase or decrease the production of goods.