Elasticity

Lecture

Watch first: Marginal Revolution University on the elasticity of demand ^[1], and on calculating it ^[2].

The Crimson Barnacle shuddered as it dropped out of the time-stream and settled into the dim glow of a parking lot somewhere in suburban Earth, March 2020. Captain Pirate Pete had wanted gasoline and diesel. What he found instead was chaos. Two shoppers were fighting over the last shrink-wrapped bale of toilet paper, and a shelf that should have held a hundred packages held exactly none. "Blisterin' barnacles," Pete muttered, "it's the Great Toilet Paper Shortage. We've sailed right into the whirlpool."

Pete planted his boots between the empty shelves and the last bale. "We burned the diesel to get here," he said. "We're buying the paper. Whatever it costs."

Socks called down from the gangplank, hooves ticking on the ramp. "The diesel is burned, Captain. You will not see it again. Spent, and unrecoverable — that is a sunk cost, and it does not get a vote. The next pack is the only decision left. If the price is a panic price, we walk. Coming all this way is not a reason to pay it."

Pete's jaw worked. He did not reach for the bale.

First Mate Orpheus poked his head out from the galley, wiping his hands on his apron. "Captain, I've seen this before. Not the paper — the panic. People will pay anything when they think they can't get a thing. But raise the price of a cup of coffee by a dime and they march off to the competitor down the street. Why does the price of one thing barely move buyers while the price of another sends them stampeding?" Stella Starlight, still half-dreaming of the North Atlantic from which the crew had plucked her, blinked. "You mean some things we just... can't stop buying? That seems terribly unfair."

Socks the GOAT trotted down the gangplank, hooves clicking, navigational instruments slung around his neck. "It isn't fairness, Stella. It's elasticity — the single most useful number in all of economics, and the one the Hypno Hippo most wants you never to learn." At the mention of the villain, a flicker of purple light pulsed from a nearby billboard: a TokPod ad, feeding, always feeding. Petunia Pareto adjusted her spectacles and sighed happily. "Elasticity is just a derivative in disguise. A ratio of percentage changes. Oh, it's beautiful."

Socks fixed the crew with a steady stare. "Today we learn to measure responsiveness. We'll learn why some sellers can double their prices and grow rich, while others try the same and go bankrupt. We'll learn why a tax on cigarettes lands on smokers but a tax on luxury yachts lands on yacht-builders. Pay attention — this is the knowledge that turns a victim of the market into a navigator of it. X marks the spot, but elasticity tells you how hard the current is pulling." And so, in a ransacked big-box store at the dawn of a pandemic, the lecture began.

1. What Elasticity Measures: Responsiveness, Not Slope

Socks the GOAT planted his hooves and began. "The first thing you must unlearn, Stella, is the temptation to say 'how much does quantity change when price changes' in raw units. Units lie. A dollar means one thing for a pack of gum and nothing for a yacht. So economists do not measure responsiveness in units. We measure it in percentages."

1.1 The core definition

The price elasticity of demand answers one question: when the price of a good rises by one percent, by what percentage does the quantity demanded fall? Formally, it is the ratio of the percentage change in quantity demanded to the percentage change in price ^[3]:

Because the law of demand tells us that price and quantity demanded move in opposite directions, this ratio is almost always negative. By long convention economists drop the minus sign and talk about the absolute value of elasticity, so that a bigger number always means "more responsive." When Socks says demand is "more elastic," he means buyers run away faster when the price climbs.

1.2 Why percentages, not units

Petunia floated over, delighted. "Consider, Stella. Suppose the toilet paper price rises from four dollars to five dollars — that's a twenty-five percent increase. And suppose a yacht's price rises from one million to one million and one dollars. Both went up by roughly the same raw... no wait, they didn't." She caught herself. "The point is that only percentages let us compare a gum market to a yacht market on the same ruler." Because elasticity is a ratio of two percentages, it is a pure number with no units — it is the same whether we price in dollars, doubloons, or the Hippo's dopamine-tokens.

1.3 The three buckets

Socks drew three lines in the dust of the parking lot. Economists sort every good into three buckets based on the absolute value of its elasticity. When \(|E_d| > 1\), demand is elastic: quantity is very responsive, and buyers flee a price hike. When \(|E_d| < 1\), demand is inelastic: quantity barely budges, and buyers grit their teeth and pay. And at the knife's edge, when \(|E_d| = 1\), demand is unit elastic: the percentage change in quantity exactly matches the percentage change in price.

2. The Midpoint Method: Doing the Arithmetic Honestly

"Now," said Socks, "there is a trap in the arithmetic, and the Hypno Hippo loves traps. If you compute percentage change the naive way, you get two different answers depending on whether the price went up or came down. That is nonsense. The responsiveness of a market segment should not depend on which direction you happened to walk along it."

2.1 The problem with ordinary percentage change

Suppose quantity goes from 100 to 150. Using the starting value as the base, that's a 50% increase. But going from 150 back to 100 is only a 33% decrease over the same stretch. Same two points, two different percentages. Petunia frowned. "The derivative doesn't care which direction you approach from. Neither should we." The fix is to divide the change not by the starting value but by the average of the two values — the midpoint.

2.2 The midpoint formula

The midpoint method (also called the arc-elasticity method) computes each percentage change relative to the average of the endpoints:

Because both the numerator and denominator use the midpoint as their base, you get the same elasticity whether you move from point 1 to point 2 or back again. This is the standard the course will use for every calculation.

2.3 A worked calculation

Stella raised her hand. "Could we try a real one? With the toilet paper?" Socks nodded. Imagine that before the panic, a store sold 500 packages a week at $4. During a modest price increase to $5, weekly sales fell to 450. Let's measure.

Walking through it: the quantity change is \(450-500=-50\), and the average quantity is \((450+500)/2=475\), so the percentage change in quantity is \(-50/475 = -0.1053\), or about −10.5%. The price change is \(5-4=1\), and the average price is \((5+4)/2=4.5\), so the percentage change in price is \(1/4.5 = 0.2222\), or about +22.2%. Dividing, \(-0.1053 / 0.2222 = -0.474\). The absolute value, 0.47, is less than one — toilet paper demand in this range is inelastic. "Of course it is," muttered Orpheus. "Nobody shops around for a cheaper substitute when nature calls."

3. Elastic, Inelastic, and Unit Elastic: Three Illustrative Goods

Socks produced a small chalkboard from the Crimson Barnacle's hold. "Let me show you three goods side by side, so you can see responsiveness with your own eyes." These are teaching numbers, chosen to make the three cases crisp.

3.1 Three demand schedules

Simplified example: the table below shows three illustrative goods, each facing the same price rise from $10 to $12. Insulin is a near-necessity with no substitute; a mid-range restaurant meal has some substitutes; and a particular brand of soda has many close substitutes. The quantities are pedagogical, not empirical.

For insulin, quantity falls from 100 to 97 — a tiny response to a 20% price hike, giving \(|E_d|\approx 0.17\). For the restaurant meal, quantity falls from 100 to 82, giving an elasticity of almost exactly 1. For Brand-X soda — which buyers will happily abandon for Brand-Y — quantity collapses from 100 to 60, giving \(|E_d|\approx 2.75\).

3.2 Visualizing the three curves

The chart below plots the three demand curves from the table above. Notice that the flatter curve (soda) is the more elastic one over this range, and the steeper curve (insulin) is the more inelastic.

3.3 A caution about slope versus elasticity

4. The Determinants of Elasticity: Why Some Goods Resist Price

Stella was puzzled. "But Socks, how does a buyer decide to be responsive? What makes insulin stubborn and soda flighty?" The goat smiled. "Excellent question. Elasticity is not random. Four forces drive it."

4.1 Substitutes: the master key

The single most important determinant is the availability of close substitutes. If a good has many close substitutes, buyers can switch the moment its price rises, and demand is elastic. If it has none, buyers are trapped and demand is inelastic. This is why a particular brand of soda is far more elastic than soda in general, which is in turn more elastic than all beverages. The narrower you draw the category, the more substitutes exist outside it, and the more elastic demand becomes.

4.2 Necessities versus luxuries

Goods that feel like necessities — insulin, basic food, home heating — tend to be inelastic, because buyers cut other spending before they cut these. Luxuries — exotic vacations, designer handbags, a second yacht — tend to be elastic, because they are the first things abandoned when the price climbs. "The Hypno Hippo," Socks added darkly, "works very hard to make his TokPods feel like necessities. An addiction is just a demand curve he has bent to vertical."

4.3 Share of the budget

A good that eats a large share of your budget is one you watch closely; a price rise there forces you to respond. A trivial purchase — a pack of salt, a box of toothpicks — can double in price without most buyers noticing, so demand is inelastic. The larger the budget share, the more elastic the demand tends to be.

4.4 Time horizon

The longer the time horizon, the more ways buyers find to substitute away, and the more elastic demand becomes. A short-run elasticity and a long-run elasticity for the very same good can differ by a factor of three or more.

5. Elasticity Along a Straight-Line Demand Curve

"Now I will show you the thing that trips up half of all students," said Socks, "and I will show it with a picture." Petunia was already vibrating with excitement at the prospect of a changing derivative.

5.1 The paradox of the straight line

A straight-line (linear) demand curve has a constant slope. One might therefore assume it has a constant elasticity. It does not. Elasticity depends not only on the slope \(\Delta P / \Delta Q\) but on the ratio of price to quantity at the point you're measuring. The relationship between the slope and elasticity runs through the demand slope:

At the top of the demand curve, price is high and quantity is low, so the price-to-quantity ratio is enormous and elasticity is large (elastic). At the bottom, price is low and quantity is high, so the ratio is tiny and elasticity is small (inelastic). Somewhere in the middle, elasticity passes through exactly 1 — unit elastic.

5.2 A worked midpoint at two ends

Consider a simple linear demand curve. Simplified example: let the demand be \(Q = 100 - 10P\). Up near the top, move price from $8 to $9 (quantity from 20 to 10). Down near the bottom, move price from $2 to $3 (quantity from 80 to 70).

Near the top: \(\%\Delta Q = -10/15 = -0.667\); \(\%\Delta P = 1/8.5 = 0.1176\); ratio \(= -5.67\). Strongly elastic. Now the bottom:

Near the bottom: \(\%\Delta Q = -10/75 = -0.1333\); \(\%\Delta P = 1/2.5 = 0.40\); ratio \(= -0.33\). Strongly inelastic. Same curve, same slope, utterly different elasticity.

5.3 The linear demand and supply set

To anchor this, here is a simple linear demand-and-supply pair we'll reuse. Simplified example: demand \(Q_d = 100 - 10P\) and supply \(Q_s = -20 + 20P\). Setting them equal gives equilibrium at \(P=4\), \(Q=60\). The table below traces the demand curve and marks where elasticity transitions.

6. Elasticity and Total Revenue: The Seller's Compass

Orpheus leaned in. "This is the part that matters for a working sailor. If I'm selling fish on the dock, should I raise my price or lower it to make more coin?" Socks grinned. "Elasticity answers exactly that. It is a seller's compass."

6.1 Total revenue defined

Total revenue is simply price times quantity sold: \(TR = P \times Q\). When a seller raises the price, two forces fight. The price effect pushes revenue up (each unit sells for more). The quantity effect pushes revenue down (fewer units sell). Which wins depends entirely on elasticity.

6.2 The total revenue rule

When demand is inelastic (\(|E_d|<1\)), quantity barely falls when price rises, so the price effect dominates: raising price raises total revenue. When demand is elastic (\(|E_d|>1\)), quantity falls sharply, so the quantity effect dominates: raising price lowers total revenue. And at unit elasticity, the two effects exactly cancel and total revenue is at its maximum.

6.3 A worked revenue table

Using our demand curve \(Q = 100 - 10P\), let's compute total revenue at each price and watch it rise, peak, then fall.

6.4 The fish connection

Petunia clapped. "It's the fish again! Revenue humped like a well-fed fish, fattest in the middle where elasticity equals one." Socks nodded — the crew had seen this shape before, hunting the elusive maximum where marginal revenue meets marginal cost. "At the top of this revenue hump, raising the price one more cent costs you exactly as much in lost quantity as it gains you in margin. That is unit elasticity, and it is where a revenue-maximizing seller plants her flag."

Socks glanced at Stella. "That next cent is the margin: judge the unit still in front of you, and keep a substitute in mind so you can still refuse it."

7. Cross-Price and Income Elasticity: Mapping the Web of Goods

"Price elasticity is only the first of a family," said Socks. "Two cousins tell you how a good relates to other goods and to income."

7.1 Cross-price elasticity

The cross-price elasticity of demand measures how the quantity demanded of one good responds to a price change in another good:

The sign tells you the relationship. If \(E_{XY}>0\), the goods are substitutes: when coffee's price rises, people buy more tea (tea quantity up, coffee price up — same direction, positive). If \(E_{XY}<0\), the goods are complements: when printers get more expensive, people buy fewer ink cartridges (cartridge quantity down, printer price up — opposite direction, negative). If near zero, the goods are unrelated.

A cross-price elasticity of +0.5 means tea and coffee are substitutes, but mild ones — a 30% jump in coffee's price only nudges tea purchases up 15%.

7.2 Income elasticity

The income elasticity of demand measures how quantity demanded responds to a change in income. If positive, the good is normal (you buy more as you get richer). If large and positive (greater than 1), it's a luxury. If negative, it's an inferior good (you buy less as you get richer — think instant ramen replaced by restaurant meals).

An income elasticity of 2.0 means restaurant meals are a luxury: a 12% rise in income produces a 24% rise in restaurant dining. Note we reuse the same ratio-of-percentages machinery as Eq. 1 — only the denominator changes from price to income.

8. Tax Incidence: Who Really Pays Depends on Elasticity

Vilfredo Vitriol, lurking at the edge of the parking lot, sneered at the mention of taxes — he had made fortunes by knowing who would truly bear them. "You children," he drawled, "think a tax falls on whoever the law names. How quaint." Socks did not take the bait; he simply taught.

8.1 Statutory versus economic incidence

Tax incidence is the study of who actually bears the burden of a tax, which is often very different from who legally writes the check. The law can require sellers to remit a tax, but sellers raise their prices to pass part of it to buyers. The economic incidence — the real burden — is decided not by the legislature but by relative elasticity.

8.2 The incidence rule

The core result: the more inelastic side of the market bears more of the tax. If buyers can't easily leave (inelastic demand), they get stuck with most of the burden. If sellers can't easily exit (inelastic supply), they eat most of it. Intuitively, whoever can most easily walk away from the transaction escapes the tax; whoever is trapped pays.

8.3 Why cigarette and yacht taxes land so differently

8.4 A worked incidence split

Simplified example: suppose a $2 per-unit tax is imposed in the §5.3 market, where demand is \(Q_d = 100 - 10P\) and supply is \(Q_s = -20 + 20P\), so the pre-tax equilibrium is \(P = 4\), \(Q = 60\). After the tax, the price buyers pay rises from $4.00 to $5.33, while the price sellers keep falls from $4.00 to $3.33, and quantity falls from 60 to about 46.67.

The buyer price rose by $1.33 of the $2.00 tax, so buyers bear two-thirds of the burden and sellers bear the remaining one-third (the $0.67 drop in what they keep). This split tells us demand here is more inelastic than supply — buyers were the more trapped side. It matches the incidence rule: at \(P = 4\), \(Q = 60\), \(|E_d| = 10 \times 4/60 \approx 0.67\) and \(E_s = 20 \times 4/60 \approx 1.33\), so the buyers' share is \(E_s/(E_s + |E_d|) = 1.33/2 = 2/3\).

9. Real Inflation Data: Reading the CPI Through an Elasticity Lens

Socks pulled a dusty ledger from the Barnacle. "Before we leave 2020, let me connect elasticity to a number you'll hear every month: the price level itself. The Bureau of Labor Statistics ^[5] tracks the Consumer Price Index — the average price of a basket of goods. Watching it rise tells us how much the cost of living climbs; watching how buyers respond is where elasticity lives."

9.1 The CPI series

The table below shows the U.S. Consumer Price Index for All Urban Consumers (series CUUR0000SA0), December values, drawn from the Bureau of Labor Statistics ^[5]. These are index numbers, not dollars; they measure the price level relative to the 1982–84 base of 100.

9.2 The inflation connection

The year-over-year change in this index is the inflation rate, computed as the percentage change from one December to the next:

From December 2020 to December 2021 the index rose from 260.474 to 278.802, an increase of about 7.0%. "Here is the elasticity lesson," said Socks. "When the price level jumps like this, households respond differently for different goods. For inelastic necessities — food staples, rent, medicine — they keep buying and simply feel poorer. For elastic goods, they substitute or cut back. The pain of inflation is distributed exactly according to elasticity. The Hypno Hippo prospers in such confusion, because a population that can't read the CPI can't tell real pain from engineered panic."

10. Elasticity Across the Centuries: From 1625 to 2425

As the lesson wound down, the Crimson Barnacle's time-engine thrummed, and the crew glimpsed how elasticity echoes across the ages. "This is why we sail through time," Pete said. "The principle is eternal; only the goods change."

10.1 1625: the mercantilist salt trade

In 1625, salt was the great inelastic good — the only way to preserve food through winter, with no substitute. Rulers taxed it mercilessly (the infamous salt taxes of the era) precisely because its inelastic demand made the revenue reliable and the burden fell squarely on the common buyer. A monopolist or a monarch who understands inelasticity can extract enormous rents; a population that doesn't understand it simply pays.

10.2 2025: digital goods and the Hippo's engineered inelasticity

In the present, the most valuable business skill is manufacturing inelasticity. Lock-in, network effects, and addiction all do the same thing: they strip away substitutes and bend the demand curve toward vertical. The Hypno Hippo's TokPods are the purest case — a good engineered so that users cannot walk away, which lets him raise the "price" (in time and attention) without losing the audience.

10.3 2425: the only defense

Stella Starlight looked up at the purple TokPod billboard, then back at Socks, and for the first time she did not feel dreamy or clueless. She felt responsive. "So if I always keep a substitute in mind," she said slowly, "no one can ever trap me." Socks nodded. "Now you're navigating, lass. Now you read the scoreboard." And the Crimson Barnacle lifted off, bound for the next century, its goat-skull sail catching the light.

Key Takeaways

• Price elasticity of demand measures responsiveness as a ratio of percentage changes (%ΔQ over %ΔP), making it a unitless number that compares any two markets on the same ruler.
• Demand is elastic when |E_d| > 1 (buyers flee price hikes), inelastic when |E_d| < 1 (buyers stay), and unit elastic when |E_d| = 1 (the two percentage changes match).
• Always use the midpoint method so the elasticity is the same whether price rises or falls; it divides each change by the average of the endpoints.
• Four determinants drive elasticity: availability of close substitutes (the master key), necessity versus luxury, share of the budget, and the time horizon.
• The total revenue rule: raise price when demand is inelastic, cut price when demand is elastic, and revenue is maximized at unit elasticity.
• Cross-price elasticity reveals substitutes (positive) and complements (negative); income elasticity reveals normal, luxury, and inferior goods.
• Tax incidence falls on the more inelastic side of the market — whoever can least easily walk away bears the heaviest burden, regardless of who the law names.

Check Yourself

Discussion Questions

1. The Hypno Hippo's TokPods are described as "engineered inelasticity." Using the four determinants of elasticity, explain precisely which levers a company pulls to make its product's demand more inelastic — and what a critical-thinking defense against each lever would look like.
2. In 1625 rulers taxed salt heavily because its demand was inelastic. Compare the salt tax to the modern cigarette tax and the failed luxury yacht tax from the lecture. Why did two of these raise reliable revenue while the third collapsed its own tax base? What does this teach about choosing what to tax?
3. Socks tells Stella that keeping a substitute in mind means "no one can ever trap you." Translate this piece of pirate wisdom into the formal language of elasticity and tax incidence. In what sense is an economically literate consumer always on the "elastic" side of every market, and why does that protect them?

Glossary

price elasticity of demand

A measure of how responsive the quantity demanded of a good is to a change in its price, expressed as the ratio of the percentage change in quantity to the percentage change in price.

law of demand

The principle that, all else equal, the quantity demanded of a good falls as its price rises.

absolute value

A number's magnitude without regard to sign; used so that larger elasticity values always indicate greater responsiveness.

elastic

Demand for which the absolute value of elasticity exceeds one, meaning quantity responds more than proportionally to a price change.

inelastic

Demand for which the absolute value of elasticity is less than one, meaning quantity responds less than proportionally to a price change.

unit elastic

Demand for which the absolute value of elasticity equals one, so the percentage change in quantity exactly matches the percentage change in price.

percentages

Relative changes expressed per hundred, used in elasticity so that goods measured in different units can be compared on a common scale.

midpoint method

A technique for computing elasticity that divides each change by the average of the two endpoint values, so the result is the same in either direction.

availability of close substitutes

The degree to which other goods can replace a given good; the primary determinant of how elastic its demand is.

time horizon

The length of time buyers have to adjust to a price change; longer horizons generally make demand more elastic.

demand slope

The change in price divided by the change in quantity along a demand curve, measured in raw units rather than percentages.

total revenue

The total receipts of a seller, equal to price multiplied by quantity sold.

total revenue rule

The principle that raising price increases total revenue when demand is inelastic and decreases it when demand is elastic, with revenue maximized at unit elasticity.

cross-price elasticity of demand

A measure of how the quantity demanded of one good responds to a change in the price of another good; its sign indicates whether the goods are substitutes or complements.

substitutes

Goods that can replace one another, so that a rise in the price of one increases demand for the other, giving a positive cross-price elasticity.

complements

Goods consumed together, so that a rise in the price of one decreases demand for the other, giving a negative cross-price elasticity.

income elasticity of demand

A measure of how quantity demanded responds to a change in consumer income, used to classify goods as normal, luxury, or inferior.

normal

A good whose quantity demanded rises as consumer income rises, giving a positive income elasticity.

luxury

A normal good with income elasticity greater than one, so demand rises more than proportionally with income.

inferior good

A good whose quantity demanded falls as consumer income rises, giving a negative income elasticity.

tax incidence

The analysis of who actually bears the economic burden of a tax, which depends on relative elasticities rather than on who legally pays.

inflation rate

The percentage change in the general price level from one period to the next.