ARE 201 · Unit 6

Elasticity
Review Session

Measure it · Explain it · Revenue · Sellers & related goods · Taxes & farm income

Today's route

Twenty questions, five big ideas

1

How big is the response?

Compute and classify |Ed| from words, a table, and a graph.

Q1–Q4
2

What do real estimates say?

Read published elasticities and explain why goods differ.

Q5–Q8
3

Does revenue rise or fall?

The total-revenue rule and the unit-elastic peak.

Q9–Q11
4

What about sellers and related goods?

Supply elasticity, time, and the signs of income and cross-price elasticities.

Q12–Q14
5

Who pays, and why do farmers care?

Tax incidence, tax revenue, and farm revenue after a supply shock.

Q15–Q20
Reference card · keep it in view

Six tools cover every question today

1 · Midpoint method

%ΔQ = ΔQ ÷ average Q

%ΔP = ΔP ÷ average P

|Ed| = %ΔQ ÷ %ΔP

2 · Compare to 1

> 1 elastic · < 1 inelastic

= 1 unit elastic

0 perfectly inelastic (vertical)

∞ perfectly elastic (horizontal)

3 · Total revenue

Elastic: price ↑ → TR ↓

Inelastic: price ↑ → TR ↑

Unit elastic: TR unchanged

4 · Supply

Es = %ΔQs ÷ %ΔP

Always positive—no absolute value needed.

More elastic the more time sellers have.

5 · Read the sign

Eincome > 0 normal · < 0 inferior

Ecross > 0 substitutes · < 0 complements

6 · Taxes

The more inelastic side of the market bears more of the tax—no matter who writes the check.

Class convention: report |Ed|—write 0.45, not −0.45. Published tables often keep the minus sign: read the size first, then the sign.
Q1 · Measure it · Warm-up

Price falls 4%. Quantity demanded rises 8%.

What is the price elasticity of demand?

Q2 · Measure it · From words

Wheat: the price rises from $10 to $12

The market

The price of a bushel of wheat rises from $10 to $12. Quantity demanded falls from 1,000 to 800 bushels.

Use the midpoint method to find |Ed|. Is demand elastic or inelastic?
%ΔQ = (Q2 − Q1) ÷ [(Q1 + Q2) ÷ 2]
%ΔP = (P2 − P1) ÷ [(P1 + P2) ÷ 2]
1
Averages: Q = (1,000 + 800) ÷ 2 = 900 · P = ($10 + $12) ÷ 2 = $11Use the same two bases whichever way the price moves.
2
%ΔQ = 200 ÷ 900 = 22.2% · %ΔP = $2 ÷ $11 = 18.2%Drop the signs: we report the absolute value.
3
|Ed| = 22.2% ÷ 18.2% = 1.22 → elasticQuantity responds more than price in percentage terms.
Q3 · Measure it · From a table

A demand schedule: price falls to $6

PriceQuantity demanded
$8120
$6150

Using the midpoint method, what is |Ed| between these two prices?

Q4 · Measure it · From a graph

Find the elasticity between A and B

D B A $3 $2 80 100 |E| = 0.56 $6 $5 20 40 |E| = 3.67 Price Quantity
Use the midpoint method. Is demand elastic or inelastic between A and B?
1
Read the graph: A = (100, $2) · B = (80, $3)
2
Averages: Q = 90 · P = $2.50
3
%ΔQ = 20 ÷ 90 = 22.2% · %ΔP = $1 ÷ $2.50 = 40%
4
|Ed| = 22.2% ÷ 40% = 0.56 → inelastic
5
Higher up the same line: $5 → $6 (Q 40 → 20) gives |Ed| = 3.67Same slope, very different elasticity: elasticity is not slope.
Q5 · Explain it · Reading published estimates

Published estimates: own-price elasticities

Retail-level estimates for U.S. foods
FoodOwn-priceIncome
Grapes−1.37800.4407
Oranges−0.99960.4866
Pork−0.72970.4427
Canned peas−0.69260.3295
Beef and veal−0.61660.4549
Fruit juice−0.56121.1254
Chicken−0.53080.3645
Apples−0.2015−0.3514
Rice−0.1467−0.3664
Cabbage−0.0385−0.3767
Sweeteners−0.0045−0.0928

Published estimates keep the minus sign on own-price elasticities.

(a) Which food in the table has the most elastic demand?

Grapes: |Ed| = 1.38, the only elastic good here. Oranges (0.9996) are essentially unit elastic.

(b) Which has the most inelastic demand?

Sweeteners: |Ed| = 0.0045—close to perfectly inelastic.

(c) The price of canned peas rises 5%. Predict the change in quantity demanded.

−0.6926 × 5% = −3.46%. Inelastic does not mean “no response”: quantity demanded still falls.
Q6 · Explain it · Income elasticity from the same table

Same table, income column: normal or inferior?

Retail-level estimates for U.S. foods
FoodOwn-priceIncome
Grapes−1.37800.4407
Oranges−0.99960.4866
Pork−0.72970.4427
Canned peas−0.69260.3295
Beef and veal−0.61660.4549
Fruit juice−0.56121.1254
Chicken−0.53080.3645
Apples−0.2015−0.3514
Rice−0.1467−0.3664
Cabbage−0.0385−0.3767
Sweeteners−0.0045−0.0928

For income elasticity, the sign is the whole story: + normal, − inferior.

(a) Which food is the best example of a normal good?

Fruit juice: Eincome = 1.13 > 0. Above 1, so technically a luxury.

(b) Which is the best example of an inferior good?

Cabbage: Eincome = −0.38, the most negative. Rice (−0.37) and apples (−0.35) are close behind.

(c) Incomes fall 7%. What happens to the demand for apples?

−0.3514 × (−7%) = +2.46%. Demand rises: apples are inferior here, and an income change shifts the whole demand curve.
Q7 · Explain it · Determinants

Locally grown organic vegetables

Which change would make the demand for locally grown organic vegetables more elastic?

Q8 · Explain it · Rapid round

Elastic or inelastic? Four quick calls

Substitutes

A good with many close substitutes

Elastic. When its price rises, buyers can switch easily.
Necessity or luxury

A luxury good

Elastic. A luxury is easy to skip or postpone.
Share of income

An item that takes 67% of your income

Elastic. A purchase that big is worth shopping around over.
Time horizon

Two estimates for gasoline, 0.4 and 1.8. Which is the short-run one?

0.4. In the short run, drivers cannot change cars, commutes, or homes.
When the determinants disagree—housing is a large budget share and a necessity—the measured number settles it.
Q9 · Revenue · Read the revenue signal

Corn: price and revenue in two different years

Year A

The price of corn rises. Farm revenue rises. Consumer demand for corn must be…

Price and revenue moved together, so the price effect won: demand is inelastic.

Year B

The price of corn falls. Farm revenue rises. Consumer demand for corn must be…

Price and revenue moved in opposite directions, so the quantity effect won: demand is elastic.

Shortcut: P and TR move together → inelastic · opposite directions → elastic · TR unchanged → unit elastic.
Q10 · Revenue · Predict, then check

A farm stand raises its apple price 10%

The manager estimates |Ed| = 1.5 for organic apples. After a 10% price increase, quantity sold falls by ____ and total revenue ____.

Q11 · Revenue · True or false?

Four statements that trip people up

aAlong a straight-line demand curve, the price elasticity of demand is the same at every point.

False. The slope is constant; elasticity is not. Elastic on the upper half, unit elastic at the midpoint, inelastic below.

bTotal revenue is highest where |Ed| = 1.

True. Above that point a price cut raises revenue; below it a price rise does. Revenue peaks where demand is unit elastic.

cRaising the price always increases total revenue.

False. Only when demand is inelastic. With elastic demand, revenue falls.

d“I would pay any amount for my morning coffee.” For this buyer, |Ed| = 0.

True. Quantity does not respond to price at all: perfectly inelastic, a vertical demand curve.
Q12 · Sellers · Supply elasticity

Milk: price and quantity supplied both rise

The market

The price of milk rises from $3.90 to $4.10 per gallon. Quantity supplied rises from 10,000 to 14,000 gallons.

Use the midpoint method to find Es. Is supply elastic or inelastic?
Es = %ΔQs ÷ %ΔP
same midpoint bases as on the demand side
1
Averages: Q = 12,000 gallons · P = $4.00
2
%ΔQs = 4,000 ÷ 12,000 = 33.3% · %ΔP = $0.20 ÷ $4.00 = 5.0%
3
Es = 33.3% ÷ 5.0% = 6.67 → elasticPositive, because price and quantity supplied move together—no absolute value needed.
Q13 · Sellers · Time and supply

Christmas trees: why supply can’t keep up

The market

Real Christmas tree prices have risen because of supply shortages.

(a) Which is the most likely short-run price elasticity of supply for real Christmas trees?

0.1: very inelastic supply. This season’s trees were planted years ago, so quantity supplied can barely respond to the higher price.

(b) Why is short-run supply so inelastic?

A tree takes 8–12 years to grow to market size. Shipping costs or shoppers switching to artificial trees do not explain the sellers’ side.

(c) If high prices last for years, what happens to Es?

It rises. Growers plant more, and as those trees mature, quantity supplied can respond: long-run supply is more elastic.
Q14 · Related goods · Read the headline

Two headlines, two signs

“In Florida, rising menu prices for UberEats and GrubHub have pumped up demand for DoorDash.”

Which elasticity concept is most relevant?

One good’s price changed the quantity demanded of another, in the same direction: Ecross > 0, so UberEats/GrubHub and DoorDash are substitutes.

“In the Midwest, rising gas prices dent demand for tractors.”

The cross-price elasticity here is negative because…

Gas up → tractor demand down: opposite directions, Ecross < 0. Goods used together are complements.

Q15 · Taxes · Who bears the burden?

Apply the incidence rule

Alcohol

Demand for alcohol is highly inelastic. The burden of a higher alcohol tax falls mainly on…

Buyers have the fewest escape routes, so they absorb most of the tax.

Fuel, at the extremes

A fuel tax is imposed where the price elasticity of supply is 0 and demand is perfectly elastic. Who pays the tax?

Buyers leave at any price above the market price, and sellers cannot change quantity at all—so sellers bear the entire tax.

The law decides who writes the check. Elasticity decides who bears the burden.
Q16 · Taxes · Put numbers on it

A $2.00 tax per bottle: who pays how much?

S D $11.60 $10.00 $9.60 96 100 buyers: $1.60 sellers: $0.40 tax $2.00 DWL Price per bottle Quantity (thousand bottles)

Numbers chosen for illustration.

Before: $10.00 and 100,000 bottles. After a $2.00 tax: buyers pay $11.60, sellers keep $9.60, and 96,000 bottles sell.

1
Buyers: $11.60 − $10.00 = $1.60 (80%)Measure each side from the original price.
2
Sellers: $10.00 − $9.60 = $0.40 (20%)|Ed| ≈ 0.28 < Es ≈ 1.0: buyers are more inelastic.
3
Tax revenue: $2.00 × 96,000 = $192,000Both shaded rectangles.
4
Deadweight loss: ½ × $2.00 × 4,000 = $4,000Small: inelastic demand loses few trades.
Q17 · Taxes · Revenue or behavior?

Same tax, two different goals

Goal: raise revenue

A city wants the most revenue from one tax, and a tax works like a price increase. Using the published |Ed| values, which good should it tax?

The most inelastic good: quantity barely falls, so the tax keeps its base (holding market size equal).

Goal: change behavior

A state taxes junk food so that people eat less of it. For the tax to actually deter consumption, demand for junk food should be…

Only elastic demand makes quantity fall a lot. The catch: that same response shrinks the revenue the tax collects.

Raise revenue → tax inelastic goods. Change behavior → the tax works only if demand is elastic.
Q18 · Farm income · Explain it in words

“A drought could raise farm revenue.” How?

Talk to a neighbor for one minute

Using supply, demand, and elasticity, explain how a drought could lead to a substantial increase in farm total revenue.

1
The drought shifts supply left (S₀ → S₁).
2
Price $12 → $15 per cwt; quantity 110 → 100 million cwtWednesday’s sweet potato numbers, run in reverse.
3
Food demand is inelastic: |Ed| = 9.5% ÷ 22.2% = 0.43
4
Revenue rises: $1,320M → $1,500MThe higher price on every unit still sold outweighs the units lost.
S₀ D (inelastic) $12 110 S₁ $15 100 $ / cwt million cwt
S₀ before S₁ after the drought revenue gained revenue lost
Q19 · Farm income · Select all that apply

Fertilizer gets cheaper and corn revenue rises

Fertilizer costs fall significantly, and corn farmers’ total revenue rises. What must have happened?

ASupply shifted to an inelastic part of the demand curve.

False. On an inelastic part, a lower price would have cut revenue.

BSupply shifted to an elastic part of the demand curve.

True. Revenue rose even though price fell, so the quantity effect won.

CIn percentage terms, the change in price exceeded the change in quantity demanded.

False. That is what inelastic demand looks like.

DIn percentage terms, the change in price was less than the change in quantity demanded.

True. |%ΔQ| > |%ΔP| is the definition of elastic.
Cheaper fertilizer shifts supply right: price falls, quantity rises. Compare Q18: there demand was inelastic, so revenue moved opposite to the harvest. Here it is elastic, so revenue moves with it.
Q20 · Put it all together · Select all that apply

One good, four elasticities

Good X: |Ed| = 2 · Ecross (X with Y) = −3 · Es = 1 · Eincome = 4

AGood X is a normal good.

True. Eincome = 4 > 0 (above 1, a luxury).

BGood X is an inferior good.

False. An inferior good has Eincome < 0.

CIf the price of good Y rises 5%, the quantity demanded of X falls 15%.

True. −3 × 5% = −15%: X and Y are complements.

DIf the price of X rises 5%, the quantity supplied of X rises 5%.

True. Es = 1: quantity supplied moves in proportion.

EIf the price of X rises 5%, total revenue from X rises.

False. |Ed| = 2 is elastic: quantity falls 10%, so revenue falls.

FThis market sits at the midpoint of a straight-line demand curve.

False. The midpoint is unit elastic; |Ed| = 2 is on the upper, elastic part.
Before you leave

Six habits prevent most elasticity mistakes

Use the midpoint, every time

Average P and average Q are the bases, so the answer is the same in both directions.

MEASURE

Compare |E| to 1

That one comparison gives both the label and the total-revenue rule.

CLASSIFY

Say the number as a sentence

|Ed| = 0.7: a 10% price rise cuts quantity demanded 7%. Inelastic is not zero.

INTERPRET

Read the sign

Income: + normal, − inferior. Cross: + substitutes, − complements. Supply: always +.

SIGNS

Find the trapped side

The more inelastic side bears more of a tax, no matter who writes the check.

TAXES

Supply shocks meet inelastic food demand

Price moves more than quantity, so farm revenue moves against the size of the harvest.

FARM INCOME
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