# Other Elasticities

> AP Microeconomics · Unit 2: Supply and Demand
> Source: https://www.owlsprep.com/study/ap-microeconomics-u2-other-elasticities/

This guide covers the three core 'other elasticities' for AP Microeconomics: income elasticity of demand (YED), cross-price elasticity of demand (XED), and price elasticity of supply (PES), including calculation, interpretation, and good categorization.

**Prerequisites:** [Price elasticity of demand (PED) calculation and midpoint method](https://www.owlsprep.com/study/ap-microeconomics-u2-price-elasticity-of-demand/); Basic supply and demand shifters

## Learning objectives

- Calculate YED, XED, and PES using the midpoint method
- Interpret the sign and magnitude of each elasticity to categorize goods and relationships
- Predict market responses to income changes, related good price changes, and own price changes
- Avoid common exam pitfalls in elasticity calculations

## Overview of Other Elasticities

After covering price elasticity of demand (PED), AP Microeconomics CED Unit 2 Topic 2.7 introduces three additional elasticity measures that extend the core idea of responsiveness to other economic variables. This topic accounts for 1-4% of total AP exam score, appearing regularly in both multiple-choice and early free-response questions.

Elasticity generally measures the percentage change in one variable in response to a 1% change in another variable. The three elasticities covered here are YED, XED, and PES. Mastery of these concepts lets you categorize goods and predict market responses to shocks like income changes, input price changes, or related good price changes.

## Income Elasticity of Demand (YED)

**Income Elasticity of Demand (YED)** — Measures the responsiveness of quantity demanded of a good to a change in consumer income, holding all other variables constant. Used to categorize goods based on how demand changes with purchasing power.

$$YED = \frac{\% \Delta Q_d}{\% \Delta Y}$$

The sign of YED is the key to categorizing goods:
- Positive YED = demand rises with income → good is *normal*
- Negative YED = demand falls with income → good is *inferior*

For normal goods, magnitude adds further context: YED > 1 means the good is a luxury (demand responds more than proportionally to income), while 0 < YED < 1 means the good is a necessity (demand responds less than proportionally). The midpoint method is used for discrete percentage changes, same as for PED.

**Worked example:** When average consumer income in a city rises from \$40,000 per year to \$45,000 per year, annual quantity demanded of generic white bread falls from 100,000 loaves to 90,000 loaves. (a) Calculate YED using the midpoint method. (b) Categorize the good based on your result.

1. Identify all given values:

   $$Q_1 = 100{,}000, Q_2 = 90{,}000, Y_1 = 40{,}000, Y_2 = 45{,}000$$
2. Calculate percentage change in quantity demanded via midpoint:

   $$\% \Delta Q_d = \frac{Q_2 - Q_1}{(Q_1 + Q_2)/2} = \frac{-10{,}000}{95{,}000} \approx -0.105 = -10.5\%$$
3. Calculate percentage change in income via midpoint:

   $$\% \Delta Y = \frac{Y_2 - Y_1}{(Y_1 + Y_2)/2} = \frac{5{,}000}{42{,}500} \approx 0.118 = 11.8\%$$
4. Calculate YED:

   $$YED = \frac{-10.5\%}{11.8\%} \approx -0.89$$
5. Interpretation: YED is negative, so generic white bread is an inferior good.

> **Exam tip:** AP FRQ graders require you to explicitly link the sign of YED to your categorization to earn full points. Always write a 1-sentence justification connecting the sign to the good type, even if your calculation is correct.

## Cross-Price Elasticity of Demand (XED)

**Cross-Price Elasticity of Demand (XED)** — Measures the responsiveness of quantity demanded of Good A to a change in the price of Good B, holding all other variables constant. Used to identify whether two goods are substitutes or complements.

$$XED = \frac{\% \Delta Q_{d,A}}{\% \Delta P_B}$$

Like YED, the sign of XED determines the relationship between two goods:
- Positive XED: an increase in the price of B leads to an increase in quantity demanded of A → goods are substitutes
- Negative XED: an increase in the price of B leads to a decrease in quantity demanded of A → goods are complements

Larger absolute XED means a stronger relationship between the two goods.

> **XED Sign Mnemonic**
>
> Substitutes are Positive (SP), Complements are Negative (CN)

**Worked example:** When the price of movie theater tickets falls from \$15 to \$12, the quantity demanded for at-home streaming subscriptions falls from 22,000 to 20,000 per month in a small town. Calculate XED for streaming subscriptions with respect to movie ticket price, and identify the relationship between the two goods.

1. Identify values: Good A = streaming subscriptions, Good B = movie tickets:

   $$Q_{A1} = 22{,}000, Q_{A2} = 20{,}000, P_{B1} = 15, P_{B2} = 12$$
2. Calculate percentage change in quantity of A via midpoint:

   $$\% \Delta Q_A = \frac{20{,}000 - 22{,}000}{(22{,}000 + 20{,}000)/2} = \frac{-2{,}000}{21{,}000} \approx -0.095 = -9.5\%$$
3. Calculate percentage change in price of B via midpoint:

   $$\% \Delta P_B = \frac{12 - 15}{(15 + 12)/2} = \frac{-3}{13.5} \approx -0.222 = -22.2\%$$
4. Calculate XED:

   $$XED = \frac{-9.5\%}{-22.2\%} \approx 0.43$$
5. Interpretation: XED is positive, so movie tickets and streaming subscriptions are substitute goods.

> **Exam tip:** On MCQ, you can eliminate incorrect options just by checking the expected sign of XED before doing any calculation. This saves time for more complex questions later in the section.

## Price Elasticity of Supply (PES)

**Price Elasticity of Supply (PES)** — Measures the responsiveness of quantity supplied of a good to a change in its own price, holding all other variables constant. Describes how easily firms can adjust output when market price changes.

$$PES = \frac{\% \Delta Q_s}{\% \Delta P}$$

Per the law of supply, price and quantity supplied move in the same direction, so PES is always non-negative (never negative). PES is categorized by magnitude:
- PES > 1 = elastic supply: quantity supplied responds more than proportionally to price
- 0 < PES < 1 = inelastic supply
- PES = 1 = unit elastic
- PES = 0 = perfectly inelastic: vertical supply curve (e.g. original fine art)
- PES = ∞ = perfectly elastic: horizontal supply curve (e.g. mass-produced goods with constant per-unit cost)

Key determinants that make supply more elastic: longer time horizons, easier input access, and better storage capacity.

**Worked example:** A coffee shop can increase its output of lattes from 100 per day to 140 per day when the market price of lattes rises from \$4 to \$5. Calculate PES using the midpoint method, and state whether supply is elastic, inelastic, or unit elastic.

1. Identify all values:

   $$Q_{s1} = 100, Q_{s2} = 140, P_1 = 4, P_2 = 5$$
2. Calculate percentage change in quantity supplied via midpoint:

   $$\% \Delta Q_s = \frac{140 - 100}{(100 + 140)/2} = \frac{40}{120} \approx 0.333 = 33.3\%$$
3. Calculate percentage change in price via midpoint:

   $$\% \Delta P = \frac{5 - 4}{(4 + 5)/2} = \frac{1}{4.5} \approx 0.222 = 22.2\%$$
4. Calculate PES:

   $$PES = \frac{33.3\%}{22.2\%} \approx 1.5$$
5. Categorization: 1.5 > 1, so supply of lattes is elastic in this range.

> **Exam tip:** When asked to explain what determines PES for a good, always link your answer to firms’ ability to adjust output to price changes. AP graders do not award full points for just listing factors without this connection.

## Worked AP-Style Practice Problems

**Worked example:** When average household income rises by 10%, the quantity demanded of organic vegetables rises by 15%. Which of the following correctly categorizes organic vegetables?
A) Inferior good, income inelastic
B) Inferior good, income elastic
C) Normal good, luxury, income elastic
D) Normal good, necessity, income inelastic

1. Calculate YED from the given percentage changes:

   $$YED = \frac{\% \Delta Q_d}{\% \Delta Y} = \frac{15\%}{10\%} = 1.5$$
2. Eliminate incorrect options: Positive YED means the good is normal, so options A and B are eliminated. YED = 1.5 > 1, so it is income elastic and a luxury, eliminating option D.
3. The correct answer is C.

**Worked example:** When the average price of automobiles falls from \$30,000 to \$25,000, the quantity of tires demanded changes from 1,000,000 units per year to 1,200,000 units per year.
(a) Calculate XED for tires with respect to automobile price, using the midpoint method. Show your work.
(b) What is the relationship between automobiles and tires? Explain.
(c) If incomes rise by 20% and quantity of automobiles demanded rises by 15%, what is YED for automobiles, and are automobiles a normal good?

1. Part (a) identify values: Good A = tires, Good B = automobiles:

   $$Q_{A1} = 1{,}000{,}000, Q_{A2} = 1{,}200{,}000, P_{B1} = 30{,}000, P_{B2} = 25{,}000$$
2. Calculate percentage changes via midpoint:

   $$\% \Delta Q_A = \frac{200{,}000}{1{,}100{,}000} \approx 0.1818 \\ \% \Delta P_B = \frac{-5{,}000}{27{,}500} \approx -0.1818$$
3. Calculate XED:

   $$XED = \frac{0.1818}{-0.1818} = -1.0$$
4. Part (b): Automobiles and tires are complementary goods. XED is negative: when the price of automobiles falls, demand for automobiles rises, so demand for the complementary good tires also rises, matching the negative result.
5. Part (c) calculate YED:

   $$YED = \frac{15\%}{20\%} = 0.75$$
6. YED is positive, so automobiles are a normal good (specifically a necessity, since 0 < 0.75 < 1).

## Common pitfalls

- **Wrong:** Calling a negative YED good a normal good, or a positive YED good an inferior good.
  - Why it fails: Students confuse the direction of change: inferior goods have higher demand when income is lower, so rising income reduces demand, leading to negative YED.
  - Correct: Always write the sign of YED explicitly after calculation, and memorize: negative = inferior, positive = normal.
- **Wrong:** Mixing up numerator and denominator for XED, calculating %ΔP_B / %ΔQ_{d,A} instead of the reverse.
  - Why it fails: Students mix up which variable responds to which, since cross-price involves two separate goods.
  - Correct: Always write the full formula and label each good explicitly before plugging in numbers.
- **Wrong:** Claiming negative XED means two goods are substitutes.
  - Why it fails: Students mix up XED sign rules with YED sign rules, or misremember the direction of change for complements.
  - Correct: Use the mnemonic *Substitutes are Positive (SP), Complements are Negative (CN)* to check your sign every time.
- **Wrong:** Using the point method instead of the midpoint method when the question explicitly requires midpoint.
  - Why it fails: Students get lazy and use the simpler calculation after repeated practice.
  - Correct: Circle the phrase 'use the midpoint method' in the question prompt before starting your calculation to avoid missing it.
- **Wrong:** Claiming PES can be negative because higher prices reduce supply.
  - Why it fails: Students confuse PES with PED and mix up the law of supply with the law of demand.
  - Correct: Remember PES measures quantity supplied, which always moves in the same direction as price per the law of supply, so PES is always non-negative.

## Cheatsheet

| Elasticity Type | Formula | Key Notes |
| --- | --- | --- |
| Income Elasticity of Demand (YED) | $YED = \frac{\% \Delta Q_d}{\% \Delta Y}$ | Negative = inferior good; 0 < YED < 1 = normal necessity; YED > 1 = normal luxury; use midpoint for discrete changes |
| Cross-Price Elasticity of Demand (XED) | $XED = \frac{\% \Delta Q_{d,A}}{\% \Delta P_B}$ | Positive = substitutes; Negative = complements; larger absolute value = stronger relationship |
| Price Elasticity of Supply (PES) | $PES = \frac{\% \Delta Q_s}{\% \Delta P}$ | Always non-negative; PES > 1 = elastic; 0 < PES < 1 = inelastic; PES = 1 = unit elastic |
| Midpoint (Arc Elasticity) %Δ | $\% \Delta X = \frac{X_2 - X_1}{(X_1 + X_2)/2}$ | Required by AP for all discrete change elasticity calculations, unless told otherwise |
| Perfectly Inelastic Supply | PES = 0 | Vertical supply curve; quantity supplied fixed regardless of price (e.g. rare original art) |
| Perfectly Elastic Supply | PES = ∞ | Horizontal supply curve; firms can supply any quantity at market price |
| Unrelated Goods (XED) | XED = 0 | Price change of one good has no effect on quantity demanded of the other |

## What's next

This topic completes the core set of elasticity concepts introduced in AP Microeconomics Unit 2: Supply and Demand, and is a prerequisite for all upcoming topics that rely on predicting market responses to economic shocks. Next, you will apply these elasticity concepts to analyze how taxes and subsidies shift market outcomes, how consumers adjust spending to income changes, and how market efficiency is measured. Without correctly interpreting the sign and magnitude of these other elasticities, you will not be able to accurately predict how demand or supply shifts in response to common market shocks. In the bigger picture, these elasticities are used by firms to set prices and by policymakers to design policy, so they underpin most applied microeconomic analysis later in the course.

- [Market Equilibrium, Disequilibrium, and Changes in Equilibrium](https://www.owlsprep.com/study/ap-microeconomics-u2-market-equilibrium-disequilibrium-and-changes/)
- [Consumer and Producer Surplus](https://www.owlsprep.com/study/ap-microeconomics-u2-consumer-and-producer-surplus/)
- [Government Intervention: Price Controls and Taxes](https://www.owlsprep.com/study/ap-microeconomics-u2-government-intervention-price-controls-and/)

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