Lecture: Elasticity
Lecture script · about 15–20 minutes spoken · Week 5
Opening
Picture a transit planner looking at a bus fare, and a café owner looking at the price of a latte. Both are thinking about a price rise. Both already know which way their buyers will move: a higher price means fewer rides and fewer lattes. What they don't know is how many fewer. Will riders barely notice, or walk away? That one question decides whether the price rise brings in more money or less.
Here's where we are. In Week 1 you met opportunity cost and the habit of holding everything else fixed. In Week 2 you saw the gains from trade. In Week 3 you built demand, and you learned that a change in a good's own price slides buyers along the demand curve. In Week 4 sellers joined in and the market found its price.
This week puts a number on that slide along the curve. I'll start with how to measure a response in percentages. Then we'll read a schedule, sort its pieces into elastic and inelastic, and see why steepness is not the same thing. After that comes the payoff for sellers, the total-revenue test. We'll stretch the same tool to income and to other goods' prices, and finish with what makes buyers more or less responsive.
1. Percentages at the midpoint
Here's the problem with raw numbers. 20 fewer units sounds like a lot. But 20 fewer out of 40 is half the market, and 20 fewer out of 2,000 is a sliver. A 1-dollar rise on a snack is huge, and on a laptop it's nothing. To compare reactions, you need percentages.
So economists define the price elasticity of demand as a measure of how strongly quantity demanded responds to a change in a good's own price: the percentage change in quantity demanded divided by the percentage change in price.
There's a catch. A percentage needs a base. If you divide by the starting value, going up gives you one number and coming down gives you another. This course fixes that with the midpoint formula: each change is divided by the average of its starting and ending values, so the result is the same whichever direction the price moves.
Think of it like splitting a taxi fare fairly. You don't charge the person who got in first; you split it down the middle.
Here's a small case from the chapter. A bike-rental stand rents 30 bikes a day at 10 dollars and 20 at 15 dollars. Rentals fall by 10 around a midpoint of 25, which is 40 percent. The price rises by 5 around a midpoint of 12.5, which is also 40 percent. So the elasticity is minus 40 divided by 40, or minus 1.
Pause and try this. If the bike stand's price fell from 15 back to 10 dollars, would the midpoint formula give a different elasticity?
Think about what the midpoint is the middle of.
Here's the answer. No, it gives minus 1 again. Both changes flip sign, but the midpoints stay the same, so the ratio doesn't change.
2. The schedule and its middle segment
Now we need a demand curve to measure. This week uses a second illustrative schedule, separate from the running market of Weeks 3 and 4. Its rule: quantity demanded equals 200 minus 20 times the price.
Let's fill it in out loud. At 2 dollars, buyers want 160 units. At 4 dollars, 120. At 6 dollars, 80. At 8 dollars, 40. Every 2-dollar step cuts quantity by the same 40 units. Hold on to that sameness.
Let's measure the middle step, from 4 to 6 dollars. Quantity falls by 40 around a midpoint of 100. That's minus 40 percent. The price rises by 2 around a midpoint of 5. That's plus 40 percent. Minus 0.4 divided by 0.4 is minus 1.
Two facts live in that one number. The sign is negative because price and quantity demanded move in opposite directions. That's the law of demand. The size, with the sign dropped, is 1. It says the two percentages match. Demand like that has a name: unit-elastic demand, where the magnitude of the price elasticity equals exactly 1, so the percentage changes in quantity demanded and in price are the same size and total revenue does not change.
Pause and try this. On this middle segment, which part of minus 1 tells you buyers moved opposite to the price, and which part tells you how strongly?
Split the number into its two pieces.
Here's the answer. The minus sign gives the direction, and the 1 gives the strength. You read the size by dropping the sign and comparing it with 1.
3. Elastic and inelastic
Now measure the other two steps on the same line.
From 2 to 4 dollars, quantity falls by 40 around a midpoint of 140. The price rises by 2 around a midpoint of 3. The elasticity comes out to about minus 0.43. From 6 to 8 dollars, quantity falls by 40 around a midpoint of 60, and the price rises by 2 around a midpoint of 7. That gives about minus 2.33.
So we have three labels. Elastic demand is demand where the magnitude of the price elasticity is greater than 1, so quantity demanded changes by a larger percentage than the price. Inelastic demand is demand where the magnitude is less than 1, so quantity changes by a smaller percentage than the price. And at the far end of the scale, perfectly inelastic demand is demand whose quantity stays the same whatever the price, so its elasticity is 0 and its curve is a vertical line.
Here's a way to keep the words straight. Think of a rubber band. An elastic band stretches a lot when you pull it. Elastic demand stretches a lot when the price pulls on it. An inelastic one hardly gives.
Pause and try this. The segment from 6 to 8 dollars has an elasticity of about minus 2.33. Is demand there elastic or inelastic?
Drop the sign, then compare with 1.
Here's the answer. Elastic. Its size, about 2.33, is bigger than 1, so buyers change their quantity by more than twice the percentage the price rose.
4. Elasticity is not slope
Here's the puzzle. Every 2-dollar step cut quantity by the same 40 units. The line is straight. Its slope is minus 20 units per dollar, everywhere. Yet the elasticity went from about minus 0.43, to minus 1, to about minus 2.33. How can one straight line have three elasticities?
The key is that slope counts raw units, and elasticity counts percentages. Losing 40 units when about 140 are being bought is a small share. Losing the same 40 units when only about 60 are being bought is a big share. And a 2-dollar rise is a big share of a low price but a small share of a high one. Both effects push the same way.
Picture the line in words. Quantity runs along the bottom, price up the side. The line starts high on the left, at 8 dollars and 40 units, and falls steadily to 2 dollars and 160 units on the right. Its steepness never changes. The upper-left stretch, with high prices and small quantities, is elastic. The lower-right stretch is inelastic. The unit-elastic part sits in the middle.
So a straight-line demand curve is elastic at high prices, inelastic at low prices, and unit-elastic in between.
Pause and try this. A friend says, "This line has one slope, so it has one elasticity." What would you tell them?
Think about where the 40 units are being measured from.
Here's the answer. One slope, three elasticities. The same 40-unit drop is a small percentage at low prices and a large one at high prices, so elasticity changes along the line while the slope stays at minus 20.
5. The total-revenue test
Now the seller's question: if I raise my price, do I take in more money or less? The money a seller takes in is total revenue, the money sellers take in from a good, found by multiplying its price by the quantity sold at that price.
Let's run it along the schedule. At 2 dollars, 2 times 160 is 320 dollars. At 4 dollars, 4 times 120 is 480. At 6 dollars, 6 times 80 is also 480. At 8 dollars, 8 times 40 is 320.
Now line those up with the elasticities. From 2 to 4 dollars, where demand is inelastic, revenue rises from 320 to 480. From 4 to 6, where demand is unit-elastic, revenue stays at 480. From 6 to 8, where demand is elastic, revenue falls from 480 to 320.
That's the total-revenue test: on an elastic segment a price rise lowers total revenue, on an inelastic segment a price rise raises it, and on a unit-elastic segment total revenue does not change.
Why? A price rise does two things. Each unit brings in more, and fewer units are sold. When buyers barely react, the first effect wins. When they react strongly, the second one does.
Pause and try this. A seller charging 6 dollars wants more revenue. Should it raise the price to 8?
Check which segment it's on first.
Here's the answer. No. That segment is elastic, so revenue would fall from 480 to 320 dollars. Buyers would leave faster than the higher price could make up.
6. Responses to income
Elasticity is a general tool. It's a percentage change in quantity divided by a percentage change in whatever caused it. Income is one cause. The income elasticity of demand is the percentage change in quantity demanded divided by the percentage change in buyers' income, with its sign kept. It's positive for a normal good, above 1 for an income-elastic good, and negative for an inferior good.
Here's the illustrative case. A household's income rises from 40,000 to 60,000 dollars a year. By the midpoint formula, that's a 40 percent rise. Its restaurant meals rise from 6 to 10 a month, which is 50 percent. So the income elasticity is 50 divided by 40, or 1.25. Meals are a normal good for this household, and income-elastic, because they rose by a bigger percentage than income.
Its bus rides fall from 30 to 20 a month, which is minus 40 percent. Minus 40 divided by 40 is minus 1. Bus rides are an inferior good for this household. That says nothing about the bus being bad. It says that with more income, this household rides it less.
Pause and try this. Why must you keep the sign on an income elasticity instead of dropping it?
Think about what the sign told you about bus rides.
Here's the answer. Because the sign is what sorts the good. Positive means normal and negative means inferior, so dropping it would erase the very thing you measured.
7. Responses to another good's price
The same tool works across two goods. Picture a café. When coffee gets more expensive, some customers switch to tea, and some skip the muffin they'd have with their coffee. The cross-price elasticity of demand is the percentage change in one good's quantity demanded divided by the percentage change in another good's price, with its sign kept: positive when the goods are substitutes and negative when they are complements.
Coffee rises from 2 to 3 dollars a cup. Around a midpoint of 2.5, that's a 40 percent rise. Cups of tea rise from 40 to 60 a day, which is 40 percent, so tea's cross-price elasticity is 1. Positive, so tea and coffee are substitutes. Muffins fall from 50 to 30 a day, which is minus 50 percent, so their cross-price elasticity is minus 50 divided by 40, or minus 1.25. Negative, so muffins and coffee are complements.
Pause and try this. A classmate sees the muffins' minus 1.25 and says, "One went up, the other went down, so they must be substitutes." Are they right?
Ask whether people buy muffins instead of coffee, or with it.
Here's the answer. No. A negative cross-price elasticity means complements. Muffins go with coffee, so a dearer coffee sells fewer muffins.
8. What makes demand elastic, and supply's version
Why is demand for some things stretchy and for others stiff? Compare table salt with one brand of sparkling water. Salt is cheap, you need a little, and nothing really replaces it. One brand of sparkling water has a dozen near-copies on the same shelf.
Those differences have a name: the determinants of price elasticity, the features that make buyers respond more or less to a price change. Close substitutes, a luxury rather than a necessity, a large share of the budget, a narrowly defined market, and more time to adjust all make demand more elastic. The time one matters a lot. In the first week after a price rise you may have few options. A year later, you may have changed your whole routine.
Sellers respond to price too. The price elasticity of supply is a measure of how strongly sellers respond to a change in a good's own price: the percentage change in quantity supplied divided by the percentage change in price. By the law of supply, it's positive. We'll keep it in words this week. It comes back in Week 7, because when a tax lands on a market, the side that responds less to price bears more of it, whoever writes the check.
Pause and try this. Which demand is likely more elastic: rent for an apartment next month, or rent over three years?
Think about what renters can change if they have time.
Here's the answer. Rent over three years. With more time, renters can move, take a roommate, or relocate for work, so they respond more to the price.
9. The classic wrong turns
Most elasticity mistakes are small slips with big effects. A sign gets dropped. Two words get swapped. The wrong number ends up on the bottom of a fraction. Here's a three-step routine that catches all of them.
First, compute both percentages at the midpoint, dividing each change by the average of its two values. Second, divide the quantity percentage by the percentage change in whatever caused it: the price, income, or another good's price. Third, report the sign, then read the size, and name the label.
Run it on two questions. Should the seller at 6 dollars raise the price to 8? The elasticity is about minus 2.33, which is elastic, so revenue falls. Are muffins a substitute for coffee? Their cross-price elasticity is minus 1.25, negative, so no, they're complements.
One more honest note. Every number this week is illustrative, built for this course. Real elasticities are estimated from data, and estimates differ. And whether a seller should care only about revenue is a normative question. The model answers the positive one.
Pause and try this. Someone computes an elasticity on the bike stand using starting values and gets two different numbers for the same step. Which step of the routine did they skip?
Picture what goes on the bottom of each fraction.
Here's the answer. The first step. Dividing by the midpoint instead of the starting value makes the two directions agree, at minus 1.
Closing
Let me pull it together. The price elasticity of demand compares percentage changes, and the midpoint formula makes the answer the same in both directions. The sign gives the direction and the size gives the strength: above 1 is elastic, below 1 is inelastic, exactly 1 is unit-elastic. Slope and elasticity are different things, and a straight line is elastic at high prices and inelastic at low ones.
The total-revenue test tells a seller that a price rise pays only where demand is inelastic. Income elasticity sorts normal goods from inferior ones, and cross-price elasticity sorts substitutes from complements, so you keep those signs. Substitutes, luxuries, big budget shares, narrow markets, and time make demand more elastic.
Next in this module come the slide deck, the Readings & Resources page, the lecture tutorial, the practice set, and the quiz. Then, in Week 6, Consumer & Producer Surplus & Market Efficiency, we measure who gains from a market and by how much.
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