Price Elasticity of Demand: Midpoint Formula + Total Revenue Test

A movie theater raises its popcorn price by 20%, and it takes in more money. Then it raises ticket prices by 20%, and it takes in less. Same price jump, opposite result. How? The answer is one number, and it's all over your exam. This is Module 2.3, price elasticity of demand.

Elasticity measures how strongly buyers respond to a price change, in percent terms. Elastic: quantity changes by a bigger percent than price, like a stretchy rubber band. Inelastic: a smaller percent, like a stiff rope. The same idea works for income and other goods' prices. That's Module 2.5.

The formula: price elasticity of demand equals the percent change in quantity demanded divided by the percent change in price. Exam trap: quantity goes on top, never price. And use percent changes: raw change over raw change only measures steepness. By the law of demand, price and quantity move in opposite directions, so this number is negative. Exam trap: for price elasticity of demand, drop the minus sign and judge the size. Negative 2 is elastic, because its absolute value, 2, is bigger than 1.

Let's work the tickets. Price rises from $10 to $12, and tickets sold per night fall from 400 to 280.

Quick refresher: percent change is the change divided by the starting value, times 100.

Percent change in price: the $2 change divided by the starting price, $10. That's +20%. Percent change in quantity: 120 fewer tickets divided by the starting 400. That's -30%. Divide: -30% over +20% is -1.5. Drop the sign: 1.5. 1.5 is bigger than 1, so ticket demand is elastic. Each 1% price rise cuts sales by about 1.5%.

So what just happened with the tickets? One: price rose 20%. Two: tickets sold fell 30%. Three: -30% over 20% is -1.5, and without the sign, 1.5 is bigger than 1, so demand is elastic.

Some questions say to use the midpoint method: divide each change by the average of the two values. For tickets, 120 over 340 is about 35%, and 2 over 11 is about 18%. 35 over 18 is about 1.9, still elastic. The midpoint gives one answer whether price rises or falls.

So what just happened with the midpoint? One: we divided by averages, not starting values: 340 is halfway between 400 and 280, and $11 between $10 and $12. Two: quantity about 35%, price about 18%. Three: 35 over 18 is about 1.9, still elastic.

Released AP questions give you percents: just divide. Told midpoint, use averages. Told nothing, use starting values; if that answer isn't a choice, try the midpoint.

Now the ranges. The dividing line is 1: quantity changes by the same percent as price. Bigger than 1 is elastic. Smaller than 1 is inelastic. Exactly 1 is unit elastic. Justify it with the number or the percents: quantity fell 6%, less than the 20% price rise, so inelastic. Two extremes. Perfectly inelastic means elasticity is 0: quantity doesn't budge, so the demand curve is vertical.

Perfectly elastic means elasticity is infinite: raise the price a penny and quantity demanded drops to zero. That curve is horizontal. Exam trap: students swap these. A vertical line looks like a capital I, for Inelastic.

So what just happened with the extremes? One: vertical demand means quantity never moves: elasticity is 0, perfectly inelastic. Two: horizontal demand means a penny more and buyers vanish: elasticity is infinite, perfectly elastic. Three: vertical looks like a capital I, for Inelastic.

A straight demand curve has the same slope everywhere, but not the same elasticity. Near the top, quantity is small and price is high. One step is a big percent of quantity but a small percent of price. Big over small: elastic.

Quick refresher on those percents: change over the starting value. Quantity 2 to 3 is 1 over 2, 50%. Price $8 to $7 is 1 over 8, just 12.5%.

Near the bottom it flips: small over big, so demand there is inelastic. Exactly halfway down, it's unit elastic. So slope is not elasticity. One straight line holds all three ranges.

So what just happened on the straight line? One: the slope never changed. Two: at the top, big quantity percent over small price percent: elastic. Three: halfway down, unit elastic. Four: at the bottom, small over big: inelastic. So slope is not elasticity.

What makes demand elastic? The big one is substitutes. Tickets compete with streaming and other theaters, so buyers can walk away. Exam trap: when new substitutes appear, like a rival theater opening next door, demand becomes more elastic, because buyers can switch more easily. Say the word more. Once you're inside, the popcorn stand is the only seller. Few substitutes, so demand is inelastic. Necessities, like a medicine you need, have inelastic demand. Luxuries you can skip, like a cruise, have elastic demand.

Goods that take a big share of your budget, like a car, have more elastic demand than tiny purchases, like salt. And time: the longer buyers have to adjust, the more elastic demand gets. Stuck with a phone plan today? Next year, you can switch.

Total revenue is price times quantity: what sellers take in, and what buyers spend. A price rise pulls two ways. Each ticket earns more, but fewer sell. Elasticity decides which wins. Tickets: $10 times 400 is $4,000. After the rise, $12 times 280 is $3,360. Elastic, so revenue fell. Popcorn goes from $5 to $6, and buckets sold fall from 300 to 270. That's a 10% drop over a 20% rise: 0.5, inelastic.

Quick refresher: percent change is the change over the starting value, and elasticity is the quantity percent over the price percent. Price: $1 over $5 is 20%. Buckets: 30 over 300 is a 10% drop. Drop the sign: 10 over 20 is 0.5, less than 1, so inelastic.

Revenue rises from $1,500 to $1,620. That's the hook: ticket demand is elastic, popcorn demand is inelastic.

So what just happened at the theater? One: both prices rose 20%. Two: ticket demand is elastic, so revenue fell from $4,000 to $3,360. Three: popcorn demand is inelastic, so revenue rose: $5 times 300 is $1,500, and $6 times 270 is $1,620.

Exam trap, the total revenue test. Inelastic: price and total revenue move the same way. Elastic: they move opposite ways, so a price cut raises revenue. Unit elastic: revenue doesn't change. On a straight demand curve, that middle point is where total revenue peaks. Save this for monopoly in Unit 4: revenue peaks where demand is unit elastic, which is where marginal revenue is zero.

So what just happened on the revenue hill? One: as price falls through the elastic top, revenue climbs. Two: it peaks at the unit elastic middle, where marginal revenue is zero. Three: as price keeps falling through the inelastic bottom, revenue falls.

Quick check. Price rises 10% and quantity demanded falls 25%. A) 0.4, inelastic. B) 2.5, elastic. C) 0.4, elastic. D) -2.5, inelastic.

B. 25 over 10 is 2.5, bigger than 1, so elastic. C flipped the fraction, and so did A. D has the right number but calls it inelastic because it's negative.

Demand for city bus rides is inelastic. The city raises the fare. Total fare revenue will: A) fall, since fewer people ride. B) rise. C) stay the same. D) drop to zero.

B. A few riders quit, but the drop in rides is smaller, in percent, than the fare increase. The trap is A: it forgets every remaining rider pays more.

Recap: elasticity is percent change over percent change, judged by absolute value against 1. Vertical means perfectly inelastic. Inelastic means price and revenue move together. Next: Module 2.4, Price Elasticity of Supply.