Lecture 8: Practice
Steak, ramen, and recessions: Other elasticities
Work through each problem on paper before you open its Solution.
1. Devon’s month
Devon works at a warehouse. His hours go up, and his monthly income rises from $2,000 to $2,500. His purchases for the month change as below.
| At $2,000 | At $2,500 | |
|---|---|---|
| Rides on the city bus | 40 | 30 |
| Gallons of milk | 8 | 9 |
| Rideshare trips | 2 | 3 |
(a) Find the percentage change in demand and the income elasticity of demand for each good.
Solution
Devon’s income rises by \[\frac{2{,}500 - 2{,}000}{2{,}000} \times 100 = 25\%\]
| At $2,000 | At $2,500 | % change in demand | Income elasticity | |
|---|---|---|---|---|
| Rides on the city bus | 40 | 30 | (30 − 40) / 40 × 100 = −25% | −25 / 25 = −1 |
| Gallons of milk | 8 | 9 | (9 − 8) / 8 × 100 = 12.5% | 12.5 / 25 = 0.5 |
| Rideshare trips | 2 | 3 | (3 − 2) / 2 × 100 = 50% | 50 / 25 = 2 |
There is no minus sign in the formula this time, so the sign of each answer is part of the answer.
(b) Which of the three goods are normal, and which are inferior? Which of the normal goods is a necessity, and which is a luxury?
Solution
- Bus rides are inferior for Devon. The income elasticity is negative: he buys fewer of them as he gets richer.
- Milk and rideshare trips are normal. Both elasticities are positive.
- Milk is a necessity, at 0.5, which is between 0 and 1: Devon buys more milk as his income rises, but demand grows more slowly than income.
- Rideshare trips are a luxury, at 2, which is greater than 1: demand grows faster than income.
(c) The next year Devon’s hours are cut and his income falls by 10%. Using the elasticities from part (a), what happens to each of the three goods?
Solution
The percentage change in demand is the elasticity times the percentage change in income.
- Bus rides: \(-1 \times (-10) = +10\%\), so he takes about 10% more bus rides.
- Milk: \(0.5 \times (-10) = -5\%\), so he buys about 5% less milk.
- Rideshare trips: \(2 \times (-10) = -20\%\), so he takes about 20% fewer rideshare trips.
The luxury loses the most demand and the necessity the least. Demand for the inferior good moves the other way and rises, which is why some goods sell better in a recession.
2. Carla’s grocery bill
Ground beef at Carla’s grocery store goes up from $4.00 to $5.00 a pound. Her purchases for the month change as below.
| At $4.00 | At $5.00 | |
|---|---|---|
| Pounds of chicken | 10 | 15 |
| Packs of hamburger buns | 8 | 6 |
| Bottles of laundry detergent | 2 | 2 |
(a) Find the percentage change in demand and the cross-price elasticity of demand for each good, with respect to the price of ground beef.
Solution
The price of ground beef rises by \[\frac{5.00 - 4.00}{4.00} \times 100 = 25\%\]
| At $4.00 | At $5.00 | % change in demand | Cross-price elasticity | |
|---|---|---|---|---|
| Pounds of chicken | 10 | 15 | (15 − 10) / 10 × 100 = 50% | 50 / 25 = 2 |
| Packs of hamburger buns | 8 | 6 | (6 − 8) / 8 × 100 = −25% | −25 / 25 = −1 |
| Bottles of laundry detergent | 2 | 2 | 0% | 0 / 25 = 0 |
(b) Which of these goods are substitutes for ground beef, which are complements, and which is neither?
Solution
- Chicken and ground beef are substitutes. The cross-price elasticity is positive: when beef gets more expensive, Carla buys chicken instead.
- Hamburger buns and ground beef are complements. The cross-price elasticity is negative: she buys the two together, so buying less beef means buying fewer buns.
- Laundry detergent is neither. Its cross-price elasticity is 0, so the price of beef tells us nothing about how much detergent she buys.
3. Reading the elasticities
Suppose a survey of US households gives the five estimates below.
- The income elasticity of demand for visits to a laundromat is \(-0.4\).
- The income elasticity of demand for bottled water is 0.3.
- The income elasticity of demand for cruise vacations is 2.2.
- The cross-price elasticity of demand for tea with respect to the price of coffee is 0.7.
- The cross-price elasticity of demand for bagels with respect to the price of cream cheese is \(-0.8\).
(a) What does each number tell you about the good, or about the pair of goods?
Solution
- Laundromat visits are an inferior good. The income elasticity is negative, so households use the laundromat less as their income rises, presumably because they buy a washing machine.
- Bottled water is a normal good and a necessity. The elasticity is positive but between 0 and 1.
- Cruise vacations are a normal good and a luxury. The elasticity is greater than 1, so demand grows faster than income.
- Tea and coffee are substitutes. The cross-price elasticity is positive, so a rise in the price of coffee sends some buyers to tea.
- Bagels and cream cheese are complements. The cross-price elasticity is negative, so a rise in the price of cream cheese lowers demand for bagels.
(b) Incomes fall in a recession. Which of the three goods in the list sells better, and which loses the most demand?
Solution
Laundromat visits sell better. Demand for an inferior good rises when income falls, so a 10% fall in income raises laundromat visits by about 4%.
Cruise vacations lose the most. They have the highest income elasticity, so the same 10% fall in income cuts demand by about 22%. Demand for bottled water falls by only about 3%.
4. Omar’s taco truck
Omar runs a taco truck downtown and sells 500 tacos a week.
(a) The burrito place across the street raises its prices by 10%, and Omar’s sales rise from 500 to 520 tacos a week. Find the cross-price elasticity of demand for his tacos with respect to the price of a burrito, and say what it tells you.
Solution
\[\%\text{ change in demand} = \frac{520 - 500}{500} \times 100 = 4\%\] \[\varepsilon_{\text{cross}} = \frac{4}{10} = 0.4\]
Tacos and burritos are substitutes, because the elasticity is positive. They are not especially close substitutes: a 10% price rise across the street moves only 4% more tacos, so most of the burrito place’s customers stay put.
(b) Later that year a warehouse nearby closes, and average income downtown falls by 8%. Omar’s sales rise from 500 to 520 tacos a week again. Find the income elasticity of demand for his tacos, and say what it tells you.
Solution
\[\varepsilon_{\text{income}} = \frac{4}{-8} = -0.5\]
Omar’s tacos are an inferior good for these customers. The elasticity is negative: they buy more of his tacos when their incomes fall, probably because they are eating at cheaper places than before.
5. Sorting goods
(a) In each pair below, which good has the higher income elasticity of demand? Explain in a sentence.
- A monthly bus pass, or a plane ticket for a spring break trip.
- Store-brand pasta, or dinner delivered from a restaurant.
- Rent on a shared apartment, or a weekend at a hotel.
Solution
- The plane ticket. A vacation is a luxury people take when they can afford it, while the bus pass is what gets you to work at any income.
- Dinner delivered from a restaurant. People who come into money eat out more and cook cheap pasta less, so the pasta may even be inferior.
- The weekend at a hotel. Everyone needs somewhere to live, so rent grows more slowly than income, while hotel weekends grow faster.
(b) For each pair below, say whether the cross-price elasticity of demand is positive, negative, or about zero, and whether the goods are substitutes, complements, or neither.
- Butter and margarine.
- Printers and ink cartridges.
- Gasoline and rides on the city bus.
- Notebooks and lawn mowers.
Solution
- Butter and margarine: positive, substitutes. A rise in the price of butter sends buyers to margarine.
- Printers and ink cartridges: negative, complements. They are used together, so a rise in the price of printers lowers demand for cartridges.
- Gasoline and bus rides: positive, substitutes. When gas gets expensive, some drivers take the bus instead.
- Notebooks and lawn mowers: about zero, neither. The price of one has no bearing on demand for the other.
6. Multiple choice
1. A household’s income rises by 10%, and the number of cans of soup it buys falls by 4%. The income elasticity of demand for soup is:
- 0.4, so soup is a normal good.
- \(-0.4\), so soup is an inferior good.
- \(-2.5\), so soup is an inferior good.
- \(-0.4\), so soup is a luxury.
Solution
The income elasticity is \(\dfrac{-4}{10} = -0.4\). It is negative, so soup is an inferior good. Option (c) divides the wrong way round, and a luxury has an income elasticity greater than 1. (b) is correct.
2. A family’s monthly income rises from $4,000 to $5,000, and the number of restaurant meals it buys rises from 4 to 6 a month. The income elasticity of demand for restaurant meals is:
- 0.5, so restaurant meals are a necessity.
- 2, so restaurant meals are a luxury.
- 2, so restaurant meals are a necessity.
- 0.5, so restaurant meals are a luxury.
Solution
Income rises by \(\frac{5{,}000 - 4{,}000}{4{,}000} \times 100 = 25\%\) and meals rise by \(\frac{6 - 4}{4} \times 100 = 50\%\), so the income elasticity is \(\dfrac{50}{25} = 2\). It is greater than 1, so restaurant meals are a luxury. (b) is correct.
3. Select all the statements that are correct.
- An inferior good has a negative income elasticity of demand.
- A good with an income elasticity of demand of 0.4 is a normal good and a necessity.
- A luxury has an income elasticity of demand between 0 and 1.
- Demand for an inferior good rises when income falls.
Solution
A luxury has an income elasticity greater than 1, not between 0 and 1, so (c) is wrong. The rest follow from the definitions: a negative income elasticity means demand falls as income rises, and so rises as income falls, and 0.4 is positive and below 1. (a), (b), and (d) are correct.
4. The price of good B rises by 8%, and demand for good A falls by 16%. The cross-price elasticity of demand for A with respect to the price of B is:
- 2, so A and B are substitutes.
- \(-2\), so A and B are complements.
- \(-0.5\), so A and B are complements.
- \(-2\), so A and B are substitutes.
Solution
The cross-price elasticity is \(\dfrac{-16}{8} = -2\). It is negative, so the two goods are complements: the more expensive B gets, the less of A people buy. Option (c) divides the wrong way round. (b) is correct.
5. For which pair of goods would you expect the cross-price elasticity of demand to be negative?
- Coffee and tea.
- Butter and margarine.
- Hot dogs and hot dog buns.
- Bus rides and rideshare trips.
Solution
A negative cross-price elasticity means the goods are complements, bought together, so a rise in the price of one lowers demand for the other. Hot dogs and hot dog buns are the pair that go together. The other three pairs are substitutes, with positive cross-price elasticities. (c) is correct.
6. From the lecture: the income elasticity of demand in the United States is about 1.25 for recreation and about 0.35 for food, beverages, and tobacco. A recession cuts incomes by 10%. Select all the statements that are correct.
- Demand for recreation falls by about 12.5%.
- Demand for food, beverages, and tobacco falls by about 3.5%.
- Recreation is a necessity and food is a luxury.
- Demand for an inferior good rises during the recession.
Solution
The percentage change in demand is the elasticity times the percentage change in income, so recreation falls by about \(1.25 \times 10 = 12.5\%\) and food by about \(0.35 \times 10 = 3.5\%\). Recreation is above 1, which makes it a luxury, and food is below 1, which makes it a necessity, so (c) has them backwards. An inferior good has a negative income elasticity, so its demand rises when income falls. (a), (b), and (d) are correct.