Week 5 · Chapter

Chapter 5 — Elasticity

Principles of Microeconomics Free sample edition

The module's primary reading · ~40–50 minutes · Objective 3 (compute and interpret price, income, and cross-price elasticities of demand by the midpoint formula, and relate price elasticity to total revenue), building on Objective 2 (explain how demand and supply determine a market's equilibrium price and quantity, and predict how a shift in demand or supply changes them) · Read it first this week; the lecture, the slide deck, the tutorial, the quiz, and the Model Workshop all build on it.

In plain language

This week is about how strongly buyers react to a change. Week 3 told you which way buyers move when a price rises; this week measures how far. Economists call that strength the price elasticity of demand (a number that says how much buyers cut back, in percentage terms, for each percent the price goes up). When buyers react a lot, demand is elastic (stretchy); when they barely react, it is inelastic (stiff). You will also see why that number tells a seller whether a price rise brings in more money or less, and how the same idea measures responses to income and to the prices of other goods.

Why you already care about this

Think about two prices that went up on you. Maybe your music streaming plan rose by 2 dollars a month, and you grumbled but kept it. Maybe the sandwich shop near your job raised a sandwich from 9 dollars to 12, and you started packing lunch four days a week. Both times the law of demand held: you bought less, or no more, at the higher price. But the strength of your reaction was very different.

Now picture the people setting those prices. The streaming service wants to know whether 2 more dollars will lose it a few subscribers or thousands. A city transit agency wants to know whether a higher bus fare will raise its fare money or empty its buses. The direction is not the hard part. The size is, and the size has a name.

By the end of this chapter you will be able to:

compute a price elasticity of demand with the midpoint formula, showing every step;
state its sign and read its magnitude as elastic, inelastic, or unit-elastic;
explain why elasticity is not the slope and changes along a straight-line demand curve;
use the total-revenue test to predict whether a price change raises or lowers revenue;
compute and interpret an income elasticity and a cross-price elasticity, keeping their signs.

Carry this sentence with you: the sign says which way; the size says how much.

Where we are

Already covered: Week 1 taught opportunity cost, models, and ceteris paribus ("all else equal"). Week 2 showed the gains from trade. Week 3 built demand: a change in a good's own price is a movement along the demand curve, and income and related goods' prices shift it. Week 4 added supply and found the market's equilibrium price and quantity.
This week: how strongly quantity demanded responds to the good's own price, to income, and to another good's price, and what that means for a seller's revenue.
How they connect: every price elasticity is a measurement of a movement along one demand curve, the Week 3 idea, now with a number attached.
Where we're going next: Week 6 measures who gains from a market with consumer and producer surplus, and Week 7 uses this week's responsiveness to explain who really pays a tax.

Week 3 is the chapter this one leans on most. Recall that a normal good is one buyers want more of when their incomes rise, and an inferior good is one they want less of; the label says nothing about quality. Recall too that substitutes are used in place of each other and complements are used together. Those four labels come back this week, and each now gets a sign you can compute.

1. Measure the response: percentages at the midpoint

Start with a question about size. If a price rises by 1 dollar and buyers want 20 fewer units, is that a big reaction? It depends. A 1-dollar rise on a 2-dollar snack is huge. A 1-dollar rise on a 900-dollar laptop is barely a rounding error. Twenty fewer units out of 40 is half the market; twenty fewer out of 2,000 is a sliver. Raw changes cannot be compared across goods. Percentages can.

The price elasticity of demand is 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 is one more choice to make. A percentage change needs a base. If you divide by the starting value, you get a different number going up than coming down. Economists in this course avoid that by dividing each change by the average of its two values, its midpoint.

The midpoint formula is the way this course computes an elasticity between two points: each change is divided by the average of its starting and ending values, so the result is the same whichever direction the price moves.

Written on one line, with P1 and P2 for the starting and ending prices and Q1 and Q2 for the quantities demanded at those prices, the formula reads:

E_d = [(Q2 − Q1) ⁄ ((Q1 + Q2) ⁄ 2)] ÷ [(P2 − P1) ⁄ ((P1 + P2) ⁄ 2)]

Read it as two fractions. The first is the percentage change in quantity. The second is the percentage change in price. Here E_d stands for the price elasticity of demand, a pure number with no units.

Worked example — a bike-rental stand (an illustrative case). A stand rents 30 bikes a day at 10 dollars and 20 bikes a day at 15 dollars.

Change in quantity. 20 − 30 = −10 bikes, around a midpoint of (30 + 20) ⁄ 2 = 25, so the quantity change is −10 ⁄ 25 = −40%.
Change in price. 15 − 10 = 5 dollars, around a midpoint of (10 + 15) ⁄ 2 = 12.5, so the price change is 5 ⁄ 12.5 = +40%.
Elasticity. −40 ÷ 40 = −1.0.

In words: for this stand, a price change of a given percentage moves rentals by the same percentage in the other direction. If the price fell from 15 back to 10 dollars, both changes would flip sign and the result would still be −1.0. That is the payoff of the midpoint.

Misconception check. Students often compute the percentages on the starting value, because that is how a store's "percent off" sign works. Watch what happens at the bike stand. Going up from 10 to 15 dollars, rentals fall 10 out of 30 while the price rises 5 out of 10, which gives about −0.67. Going down from 15 to 10, rentals rise 10 out of 20 while the price falls 5 out of 15, which gives −1.5. One curve, two answers. The cure: divide by the midpoint, and the two directions agree at −1.0.

2. Read the schedule and compute the middle segment

To see elasticity at work, you need a demand curve to measure. This chapter uses a second illustrative schedule, separate from the running market of Weeks 3 and 4. Its numbers are constructed for this course, not taken from a real market. Its rule is Qd = 200 − 20P, where P is the good's own price in dollars and Qd is quantity demanded, in units.

Worked example — building the elasticity schedule. Put each price into the rule.

At 2 dollars. Qd = 200 − 20 × 2 = 200 − 40 = 160 units.
At 4 dollars. Qd = 200 − 20 × 4 = 200 − 80 = 120 units.
At 6 dollars. Qd = 200 − 20 × 6 = 200 − 120 = 80 units.
At 8 dollars. Qd = 200 − 20 × 8 = 200 − 160 = 40 units.

The elasticity schedule: price in dollars and quantity demanded in units (illustrative)
Price (dollars)	Quantity demanded (units)
2	160
4	120
6	80
8	40

Every 2-dollar step in price cuts quantity demanded by the same 40 units. Hold on to that sameness, because the elasticity is about to change while it stays put.

Worked example — the middle segment, from 4 to 6 dollars. Read both rows from the table and work the formula one part at a time.

Change in quantity. 80 − 120 = −40, around a midpoint of (120 + 80) ⁄ 2 = 100.
Change in price. 6 − 4 = 2, around a midpoint of (4 + 6) ⁄ 2 = 5.
Elasticity. E_d = (−40 ⁄ 100) ÷ (2 ⁄ 5) = −0.4 ÷ 0.4 = −1.0.

In words: on this segment, a 40% rise in price brings a 40% fall in quantity demanded. Two facts sit in that one number. The sign is negative because price and quantity demanded move in opposite directions, which is the law of demand. The magnitude, written |E_d| and read as the value with the sign dropped, is 1. It says the two percentages are the same size.

Unit-elastic demand is demand on a segment 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.

In practice (your tools). In Google Sheets, type the prices 2, 4, 6, and 8 in cells A2 to A5. In B2, type =200-20*A2 and copy it down to B5. In C3, type =((B3-B2)/((B2+B3)/2))/((A3-A2)/((A2+A3)/2)) and copy it down to C5. Cell C4, the segment from 4 to 6 dollars, shows −1.

Misconception check. Students often drop the minus sign and then forget what it meant, or keep it and call −1.0 "less than 1, so inelastic." The cure is to read the two parts separately. The sign tells you the direction, and for a price elasticity of demand it is negative. The magnitude, 1 here, is what you compare with 1.

3. Sort the segments: elastic, inelastic, unit-elastic

Now measure the other two steps on the same line. One sits at low prices and one at high prices.

Worked example — the lower segment, from 2 to 4 dollars. Use the 2-dollar and 4-dollar rows.

Change in quantity. 120 − 160 = −40, around a midpoint of (160 + 120) ⁄ 2 = 140.
Change in price. 4 − 2 = 2, around a midpoint of (2 + 4) ⁄ 2 = 3.
Elasticity. E_d = (−40 ⁄ 140) ÷ (2 ⁄ 3) = −0.4286.

In words: here buyers change their quantity by a smaller percentage than the price changed. Demand on this segment is inelastic, because its magnitude is below 1.

Worked example — the upper segment, from 6 to 8 dollars. Use the 6-dollar and 8-dollar rows.

Change in quantity. 40 − 80 = −40, around a midpoint of (80 + 40) ⁄ 2 = 60.
Change in price. 8 − 6 = 2, around a midpoint of (6 + 8) ⁄ 2 = 7.
Elasticity. E_d = (−40 ⁄ 60) ÷ (2 ⁄ 7) = −2.3333.

In words: here buyers react strongly, cutting their quantity by more than twice the percentage the price rose. Demand on this segment is elastic, because its magnitude is above 1.

The three labels, set off for your notes:

Elastic demand is demand on a segment 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 on a segment where the magnitude of the price elasticity is less than 1, so quantity demanded changes by a smaller percentage than the price.
Perfectly inelastic demand is demand whose quantity stays the same whatever the price, so its price elasticity is 0 and its curve is a vertical line.

In symbols, demand is elastic when |E_d| > 1, inelastic when |E_d| < 1, and unit-elastic when |E_d| = 1. A perfectly inelastic curve is the far end of the scale: whatever happens to the price, buyers take the same amount.

Misconception check. Students often swap the two words, because "elastic" sounds like "barely moves" to some ears. 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. The cure: a big magnitude, above 1, is elastic. A small one, below 1, is inelastic.

Pause and reflect — choose one. Pick either prompt; they count the same.

(a) Name one thing you buy whose price could rise a lot before you bought less. In two sentences, explain why your demand for it is inelastic.
(b) Name one thing you would drop quickly if its price rose. In two sentences, explain why your demand for it is elastic.

4. Separate elasticity from slope

Here is the puzzle the schedule set up. Every 2-dollar step cut quantity by the same 40 units, yet the elasticity ran from −0.4286 to −1.0 to −2.3333. How can one straight line have three elasticities?

The key is in the word "percentage." Slope counts raw units. Elasticity counts percentages, and a percentage depends on where you start.

Worked example — the slope stays put. On every segment, quantity changes by −40 units for a 2-dollar rise. So the slope is −40 ÷ 2 = −20 units per dollar on all three segments, while E_d runs −0.4286, −1.0, −2.3333.

In words: the same 40-unit drop is a small share of a big quantity at low prices, where about 140 units are being bought, and a large share of a small quantity at high prices, where about 60 are. At the same time, a 2-dollar rise is a large share of a low price and a small share of a high one. Both effects push the same way. A straight-line demand curve is elastic at high prices, inelastic at low prices, and unit-elastic in between.

Described as a graph, the line runs from (40, 8) at the upper left to (160, 2) at the lower right, each point written as (quantity, price). Its steepness never changes. The upper-left part, where prices are high and quantities small, is the elastic part. The lower-right part is the inelastic part. The unit-elastic stretch sits in the middle, between (120, 4) and (80, 6).

A caveat about this model: real demand curves are rarely perfect straight lines. The straight line is a classroom model, and it is useful precisely because its slope cannot change. Whatever changes along it has to be the elasticity.

Misconception check. Students often read elasticity off the slope: "a steep curve is inelastic, a flat one is elastic." That shortcut can mislead you badly, because one line with one slope has elastic, unit-elastic, and inelastic stretches. The cure: compute the percentages. Slope is in units per dollar; elasticity has no units at all.

5. Run the total-revenue test

Now the question every seller asks. If I raise my price, do I take in more money or less? The money a seller takes in has a name.

Total revenue is the money sellers take in from a good, found by multiplying its price by the quantity sold at that price.

Worked example — total revenue along the schedule. Multiply each price by its quantity.

At 2 dollars. 2 × 160 = 320 dollars.
At 4 dollars. 4 × 120 = 480 dollars.
At 6 dollars. 6 × 80 = 480 dollars.
At 8 dollars. 8 × 40 = 320 dollars.

Total revenue and the price elasticity of demand on each segment of the elasticity schedule (illustrative)
Segment (dollars)	Revenue before and after (dollars)	Change in revenue (dollars)	E_d	Label
2 to 4	320 to 480	+160	−0.4286	inelastic
4 to 6	480 to 480	0	−1.0	unit-elastic
6 to 8	480 to 320	−160	−2.3333	elastic

Read the rows. From 2 to 4 dollars, demand is inelastic and revenue rises by 480 − 320 = 160. From 4 to 6 dollars, demand is unit-elastic and revenue changes by 480 − 480 = 0. From 6 to 8 dollars, demand is elastic and revenue falls by 480 − 320 = 160. Raising the price from 2 to 4 dollars brings in 480 dollars versus 320, one and a half times as much; raising it from 6 to 8 brings in 320 versus 480.

The total-revenue test is the rule linking price elasticity to revenue: 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 does it work? A price rise has two effects on revenue. Each unit sold brings in more, which pushes revenue up. Fewer units are sold, which pushes it down. When demand is inelastic, the quantity effect is the smaller one, so revenue rises. When demand is elastic, the quantity effect wins, so revenue falls.

In practice (your tools). In the same sheet, type =A2*B2 in D2 and copy it down to D5. Column D shows 320, 480, 480, and 320. Change the rule in column B and watch the revenue column and the elasticity column move together.

Misconception check. Students often think a higher price always means more money. A seller on an elastic segment who raises the price from 6 to 8 dollars loses buyers so fast that revenue falls from 480 to 320 dollars. The cure: check the elasticity first. Raise the price for more revenue only where demand is inelastic.

Pause and reflect — choose one. Pick either prompt; they count the same.

(a) Imagine you run a small food stall and are thinking about raising a price. In two or three sentences, say what you would want to know about your buyers first, and why.
(b) Think of a business that cut its price and seemed busier afterward. In two or three sentences, say what the total-revenue test predicts about its revenue if its demand was elastic, and if it was inelastic.

6. Measure responses to income: the income elasticity

Elasticity is a general tool: a percentage change in quantity divided by a percentage change in something that causes it. Income is one such cause. When a household earns more, it buys more of some things and less of others.

The income elasticity of demand is the percentage change in quantity demanded divided by the percentage change in buyers' income, with its sign kept: positive for a normal good, above 1 for an income-elastic good, and negative for an inferior good.

For income elasticity the sign is the whole story of the good's type. Do not drop it.

Worked example — a household's income rises (illustrative). Income rises from 40,000 to 60,000 dollars a year.

Change in income. 60,000 − 40,000 = 20,000 dollars, around a midpoint of (40,000 + 60,000) ⁄ 2 = 50,000, so 20,000 ÷ 50,000 = +40%.
Restaurant meals. They rise from 6 to 10 a month: 10 − 6 = 4 more, around a midpoint of (6 + 10) ⁄ 2 = 8, so 4 ÷ 8 = +50%. Income elasticity = 50 ÷ 40 = 1.25.
Bus rides. They fall from 30 to 20 a month: 20 − 30 = −10, around a midpoint of (30 + 20) ⁄ 2 = 25, so −10 ÷ 25 = −40%. Income elasticity = −40 ÷ 40 = −1.0.

In words: restaurant meals are a normal good for this household, and an income-elastic one, since meals rose by a bigger percentage than income. Bus rides are an inferior good for it: with more income, it rides the bus less, perhaps because it can now afford a car. That is a statement about the income response, not about the quality of the bus.

Misconception check. Students often call any good with a negative income elasticity "cheap" or "bad." The bus might be clean, fast, and well run. The cure: the sign reports what buyers do when their income changes. Negative means inferior in that one sense, and nothing more.

7. Measure responses to another good's price: the cross-price elasticity

The same tool measures how one good's buyers respond to a different good's price. Picture a café. When coffee gets more expensive, some customers switch to tea, and some skip the muffin they used to 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.

Worked example — the price of coffee rises (illustrative). Coffee rises from 2 to 3 dollars a cup.

Change in the price of coffee. 3 − 2 = 1 dollar, around a midpoint of (2 + 3) ⁄ 2 = 2.5, so 1 ÷ 2.5 = +40%.
Tea. Cups of tea rise from 40 to 60 a day: 60 − 40 = 20 more, around a midpoint of (40 + 60) ⁄ 2 = 50, so 20 ÷ 50 = +40%. Cross-price elasticity = 40 ÷ 40 = 1.0.
Muffins. They fall from 50 to 30 a day: 30 − 50 = −20, around a midpoint of (50 + 30) ⁄ 2 = 40, so −20 ÷ 40 = −50%. Cross-price elasticity = −50 ÷ 40 = −1.25.

In words: tea's positive cross-price elasticity says customers use it in place of coffee, so tea and coffee are substitutes. The muffins' negative one says customers buy them with coffee, so muffins and coffee are complements. The café manager now knows that a coffee price rise sells more tea and fewer muffins.

Misconception check. Students often read a negative cross-price elasticity as "substitutes," since one good went up and the other went down. Look at the café: the coffee price rose and muffin sales fell, because muffins go with coffee. The cure: positive means substitutes, negative means complements. Keep the sign.

Pause and reflect — choose one. Pick either prompt; they count the same.

(a) Pick two goods you buy together. In two sentences, predict the sign of their cross-price elasticity and explain your reasoning.
(b) Pick two goods you treat as substitutes. In two sentences, predict the sign of their cross-price elasticity and say what would happen to one when the other's price rose.

8. Name what makes demand elastic, and meet supply's version

Why is demand for some goods stretchy and for others stiff? Compare table salt with one brand of sparkling water. Salt is cheap, you need a little of it, and there is no real stand-in, so a price rise barely changes what you buy. A single brand of sparkling water has a dozen near-copies on the same shelf, so a price rise sends buyers to the next bottle.

The determinants of price elasticity are 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.

Each one has an everyday version. Close substitutes: one brand of sparkling water versus water in general. Luxury or necessity: a weekend trip versus a needed prescription. Budget share: a new car versus a pack of gum. Market definition: "this café's latte" versus "coffee." Time: in the first week after a price rise, a commuter has few options; a year later, the same commuter may have moved, joined a carpool, or changed jobs.

Worked example — ranking two goods. Decide which demand is likely more elastic, and name the determinant.

A movie theater's popcorn versus all snack food. The theater's popcorn is a narrow market with substitutes outside the theater, so its demand is more elastic.
Rent for an apartment next month versus over three years. With more time, renters can move, take a roommate, or relocate for work, so demand over three years is more elastic.

Sellers respond to prices too, and their response has its own elasticity.

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 from Week 4, it is positive: a higher price raises quantity supplied. Like demand, supply is called elastic when its quantity changes by a larger percentage than its price and inelastic when it changes by a smaller one. This chapter keeps supply's version in words. It matters again in Week 7: when a tax is placed on a good, the side of the market that responds less to price bears more of it, whoever writes the check.

Misconception check. Students often think a "necessity" is perfectly inelastic, so its sellers can charge anything. Even for needed goods, buyers cut back at the margin, look for substitutes, and adjust over time. The cure: treat the determinants as pushing elasticity up or down, never as switching responsiveness off.

9. Check the classic wrong turns

Most elasticity mistakes are not deep confusions. They are a sign dropped, a word swapped, or the wrong number on the bottom of a fraction. Run this short routine every time.

Step one: compute both percentages at the midpoint, dividing each change by the average of its two values.
Step two: divide quantity's percentage by the cause's percentage, price, income, or another good's price.
Step three: report the sign, then read the magnitude, and name the label: elastic, inelastic, or unit-elastic for price; normal or inferior for income; substitutes or complements for cross-price.

Worked example — one routine, two questions. Use the elasticity schedule and the café.

Should a seller at 6 dollars raise the price to 8? E_d = (−40 ⁄ 60) ÷ (2 ⁄ 7) = −2.3333, which is elastic, so revenue falls from 480 to 320 dollars. The total-revenue test says no, if more revenue is the goal.
Are muffins a substitute for coffee? Cross-price elasticity = −50 ÷ 40 = −1.25, which is negative, so muffins and coffee are complements.

In words: the seller would lose 160 dollars of revenue by raising the price on that elastic segment, and the café's muffins move with coffee, not against it. Whether the seller should care only about revenue is a different, normative question; the model answers the positive one.

There is one honesty note to keep. Every number in this chapter is illustrative. Real elasticities for real products are estimated from data, they differ across studies, and this course never grades you on one.

Misconception check. Students often treat a quick slope reading, a dropped sign, and a starting-value percentage as small slips. Each one can flip a conclusion. A slope cannot sort elastic from inelastic; a dropped sign can turn complements into substitutes; and a starting-value percentage gives two answers for one segment. The cure is the three-step routine above, every time.

Key terms

Price elasticity of demand — 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. (first used: 1. Measure the response: percentages at the midpoint)
Midpoint formula — The way this course computes an elasticity between two points: each change is divided by the average of its starting and ending values, so the result is the same whichever direction the price moves. (first used: 1. Measure the response: percentages at the midpoint)
Unit-elastic demand — Demand on a segment 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. (first used: 2. Read the schedule and compute the middle segment)
Elastic demand — Demand on a segment where the magnitude of the price elasticity is greater than 1, so quantity demanded changes by a larger percentage than the price. (first used: 3. Sort the segments: elastic, inelastic, unit-elastic)
Inelastic demand — Demand on a segment where the magnitude of the price elasticity is less than 1, so quantity demanded changes by a smaller percentage than the price. (first used: 3. Sort the segments: elastic, inelastic, unit-elastic)
Perfectly inelastic demand — Demand whose quantity stays the same whatever the price, so its price elasticity is 0 and its curve is a vertical line. (first used: 3. Sort the segments: elastic, inelastic, unit-elastic)
Total revenue — The money sellers take in from a good, found by multiplying its price by the quantity sold at that price. (first used: 5. Run the total-revenue test)
Total-revenue test — The rule linking price elasticity to revenue: 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. (first used: 5. Run the total-revenue test)
Income elasticity of demand — The percentage change in quantity demanded divided by the percentage change in buyers' income, with its sign kept: positive for a normal good, above 1 for an income-elastic good, and negative for an inferior good. (first used: 6. Measure responses to income: the income elasticity)
Cross-price elasticity of demand — 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. (first used: 7. Measure responses to another good's price: the cross-price elasticity)
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. (first used: 8. Name what makes demand elastic, and meet supply's version)
Price elasticity of supply — 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. (first used: 8. Name what makes demand elastic, and meet supply's version)

Summary

The price elasticity of demand is the percentage change in quantity demanded over the percentage change in price.
The midpoint formula divides each change by the average of its two values, so direction does not matter.
Report the sign, then read the magnitude: above 1 is elastic, below 1 is inelastic, exactly 1 is unit-elastic.
Elasticity is not slope; a straight-line demand curve is elastic at high prices and inelastic at low ones.
Total-revenue test: a price rise raises revenue when demand is inelastic and lowers it when demand is elastic.
Income elasticity is positive for a normal good and negative for an inferior good.
Cross-price elasticity is positive for substitutes and negative for complements.
Substitutes, luxuries, big budget shares, narrow markets, and time all make demand more elastic.

Check your understanding

Work these before looking at the answers — they are ungraded, and the answers (with reasoning) are at the bottom of the page.

An illustrative car wash sells 50 washes a day at 8 dollars and 30 washes a day at 12 dollars. Compute the price elasticity of demand by the midpoint formula, label it, and use total revenue to check your label.
A classmate says, "This demand curve is a straight line, so its elasticity is the same everywhere." Explain in two or three sentences why that is wrong.
An illustrative household's income rises from 30,000 to 50,000 dollars a year, and its purchases of store-brand cereal fall from 9 to 7 boxes a month. Compute the income elasticity and say whether the cereal is a normal or an inferior good for this household.
At an illustrative movie theater, the ticket price rises from 9 to 11 dollars, and popcorn sales fall from 130 to 70 tubs a night. Compute the cross-price elasticity of popcorn with respect to the ticket price, and say whether the two are substitutes or complements.

Answers (worked reasoning — read after attempting)

Quantity: 30 − 50 = −20 around a midpoint of (50 + 30) ⁄ 2 = 40, so −20 ÷ 40 = −50%. Price: 12 − 8 = 4 around a midpoint of (8 + 12) ⁄ 2 = 10, so 4 ÷ 10 = +40%. E_d = −50 ÷ 40 = −1.25, whose magnitude is above 1, so demand is elastic on this segment. Revenue check: 8 × 50 = 400 dollars and 12 × 30 = 360 dollars, so the price rise lowered revenue by 40 dollars, which is what elastic demand predicts.
A straight line has one slope, so each equal price step changes quantity by the same number of units. Elasticity uses percentages, and the same change in units is a small percentage of a large quantity and a large percentage of a small one. So a straight-line demand curve is inelastic at low prices and elastic at high prices, with a unit-elastic stretch in between.
Income: 50,000 − 30,000 = 20,000 around a midpoint of (30,000 + 50,000) ⁄ 2 = 40,000, so 20,000 ÷ 40,000 = +50%. Cereal: 7 − 9 = −2 around a midpoint of (9 + 7) ⁄ 2 = 8, so −2 ÷ 8 = −25%. Income elasticity = −25 ÷ 50 = −0.5. The sign is negative, so store-brand cereal is an inferior good for this household, which says nothing about how good the cereal tastes.
Ticket price: 11 − 9 = 2 around a midpoint of (9 + 11) ⁄ 2 = 10, so 2 ÷ 10 = +20%. Popcorn: 70 − 130 = −60 around a midpoint of (130 + 70) ⁄ 2 = 100, so −60 ÷ 100 = −60%. Cross-price elasticity = −60 ÷ 20 = −3.0. The sign is negative, so popcorn and movie tickets are complements: fewer moviegoers means fewer tubs sold.

Next up in this module: the Readings & Resources page, the lecture tutorial, the practice set, and the quiz — and next week the consumer-surplus and producer-surplus model measures who gains from a market at its equilibrium price.

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