Other Elasticities — AP Microeconomics Study Guide

For: AP Microeconomics candidates sitting AP Microeconomics.

Covers: This chapter covers income elasticity of demand (YED), cross-price elasticity of demand (XED), and price elasticity of supply (PES), including their formulas, sign interpretation, good categorization, and exam-style problem solving.

You should already know: Price elasticity of demand calculation, the midpoint (arc elasticity) method, basic supply and demand shifters.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the AP Microeconomics style for educational use. They are not reproductions of past College Board / Cambridge / IB papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official mark schemes for grading conventions.

1. What Is Other Elasticities?

After covering price elasticity of demand (PED) in introductory elasticity topics, the AP Microeconomics Course and Exam Description (CED) lists this topic as Unit 2 Topic 2.7: Other Elasticities, accounting for 1-4% of total exam score weight. It appears regularly in both multiple-choice questions (MCQ) and as an early part of free-response questions (FRQ), often testing calculation, interpretation, or application of elasticity concepts to market changes. "Other elasticities" extend the core elasticity idea: measuring the percentage responsiveness of one economic variable to a 1% change in another, beyond just quantity demanded’s response to own price. The three core elasticities covered here are income elasticity of demand, cross-price elasticity of demand, and price elasticity of supply. Standard AP notation uses the abbreviations YED, XED, and PES respectively, and the topic builds directly on the midpoint method already introduced for PED. Mastery of these elasticities is required to correctly categorize goods and predict market responses to shocks like income changes, input price changes, or price changes for related goods.

2. Income Elasticity of Demand (YED)

Income elasticity of demand measures the responsiveness of quantity demanded of a good to a change in consumer income, holding all other variables constant. It allows economists to categorize goods based on how consumer demand changes as purchasing power rises or falls, which is useful for predicting demand shifts during economic expansions or recessions. The general formula for YED is: $$YED=\frac{\%\Delta Q_d}{\%\Delta Y}$$ where $\%\Delta Q_d$ is the percentage change in quantity demanded, and $\%\Delta Y$ is the percentage change in consumer income. The sign of YED is the most important value for categorizing goods: a positive YED means demand rises with income, so the good is normal. A negative YED means demand falls as income rises, so the 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 to calculate percentage changes for discrete values, same as for PED.

Worked Example

When average consumer income in a city rises from $40,000peryearto$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 values: $Q_1=100,000$, $Q_2=90,000$, $Y_1=40,000$, $Y_2=45,000$.
2. Calculate $\%\Delta Q_d$ 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 $\%\Delta Y$ 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 justifcation connecting the sign to the good type, even if your calculation is correct.

3. Cross-Price Elasticity of Demand (XED)

Cross-price elasticity of demand measures the responsiveness of quantity demanded of Good A to a change in the price of Good B, holding all else constant. It is used to identify whether two goods are related as substitutes or complements, which helps predict how demand shifts when the price of a related good changes. The formula for XED is: $$XED=\frac{\%\Delta Q_{d,A}}{\%\Delta P_B}$$ where $\%\Delta Q_{d,A}$ is the percentage change in quantity demanded of Good A, and $\%\Delta P_B$ is the percentage change in price of Good B. Like YED, the sign of XED determines the relationship between the two goods: a positive XED means an increase in price of B leads to an increase in quantity demanded of A, so the goods are substitutes (consumers switch to A when B gets more expensive). A negative XED means an increase in price of B leads to a decrease in quantity demanded of A, so the goods are complements (they are used together, so higher price of B reduces demand for A). The magnitude of XED indicates how strong the relationship is: a larger absolute value means the goods are closer substitutes or stronger complements.

Worked Example

When the price of movie theater tickets falls from $15to$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 all values: Good A = streaming subscriptions, so $Q_{A1}=22,000$, $Q_{A2}=20,000$; Good B = movie tickets, so $P_{B1}=15$, $P_{B2}=12$.
2. Calculate $\%\Delta Q_A$ 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 $\%\Delta P_B$ 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.

4. Price Elasticity of Supply (PES)

Price elasticity of supply measures the responsiveness of quantity supplied of a good to a change in its own price, holding all else constant. It describes how easily firms can adjust output when the market price changes, which depends on factors like input availability, time horizon, and storage capacity. The formula for PES is: $$PES=\frac{\%\Delta Q_s}{\%\Delta P}$$ where $\%\Delta Q_s$ is the percentage change in quantity supplied, and $\%\Delta P$ is the percentage change in price. 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. a mass-produced good with constant per-unit cost). Key determinants of PES are: longer time horizons make supply more elastic (firms can adjust capacity), easier access to inputs makes supply more elastic, and better storage capacity makes supply more elastic.

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 $4to$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 $\%\Delta Q_s$ via midpoint: $\%\Delta Q_s=\frac{140-100}{(100+140)/2}=\frac{40}{120}\approx 0.333=33.3\%$.
3. Calculate $\%\Delta P$ 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.

5. Common Pitfalls (and how to avoid them)

• Wrong move: Calling a negative YED good a normal good, or a positive YED good an inferior good. Why: 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 move: Always write the sign of YED explicitly after calculation, and memorize: negative = inferior, positive = normal.
• Wrong move: Mixing up numerator and denominator for XED, calculating $\%\Delta P_B/\%\Delta Q_{d,A}$ instead of the reverse. Why: Students mix up which variable responds to which, since cross-price involves two separate goods. Correct move: Always write the full formula and label each good explicitly before plugging in numbers.
• Wrong move: Claiming negative XED means two goods are substitutes. Why: Students mix up XED sign rules with YED sign rules, or misremember the direction of change for complements. Correct move: Use the mnemonic Substitutes are Positive (SP), Complements are Negative (CN) to check your sign every time.
• Wrong move: Using the point method (dividing by the original value) instead of the midpoint method when the question explicitly requires midpoint. Why: Students get lazy and use the simpler calculation after repeated practice. Correct move: Circle the phrase "use the midpoint method" in the question prompt before starting your calculation to avoid missing it.
• Wrong move: Claiming PES can be negative because higher prices reduce supply. Why: Students confuse PES with PED and mix up the law of supply with the law of demand. Correct move: 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.

6. Practice Questions (AP Microeconomics Style)

Question 1 (Multiple Choice)

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

Worked Solution: First, calculate YED using the given percentage changes: $YED=\frac{\%\Delta Q_d}{\%\Delta Y}=\frac{15\%}{10\%}=1.5$. A positive YED means the good is normal, so we can eliminate options A and B, which incorrectly label it as inferior. YED = 1.5 is greater than 1, so it is income elastic and a luxury good, eliminating option D. The correct answer is C.

Question 2 (Free Response)

When the average price of automobiles falls from $30,000to$25,000, the quantity of tires demanded changes from 1,000,000 units per year to 1,200,000 units per year. (a) Calculate the cross-price elasticity of demand for tires with respect to automobile price, using the midpoint method. Show your work. (b) Based on your calculation, what is the relationship between automobiles and tires? Explain. (c) If incomes in the market rise by 20% and the quantity of automobiles demanded rises by 15%, what is the income elasticity of demand for automobiles, and are automobiles a normal good?

Worked Solution: (a) Good A = tires, Good B = automobiles: $Q_{A1}=1,000,000$, $Q_{A2}=1,200,000$, $P_{B1}=30,000$, $P_{B2}=25,000$. $\%\Delta Q_A=\frac{1,200,000-1,000,000}{(1,000,000+1,200,000)/2}=\frac{200,000}{1,100,000}\approx 0.1818$. $\%\Delta P_B=\frac{25,000-30,000}{(30,000+25,000)/2}=\frac{-5,000}{27,500}\approx -0.1818$. $XED=\frac{0.1818}{-0.1818}=-1.0$.

(b) Automobiles and tires are complementary goods. XED is negative: when the price of automobiles falls, quantity demanded of automobiles rises, and consumers also demand more tires (which are used with automobiles), matching the negative XED result.

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

Question 3 (Application / Real-World Style)

A small family farm grows strawberries, which spoil within 3 days of picking and cannot be stored long-term. When the market price of strawberries rises from $2perpoundto$3 per pound, the farm increases its daily quantity supplied from 100 pounds to 150 pounds. Calculate PES for strawberries using the midpoint method, and explain what the result means for the farm’s ability to adjust output to price changes in the short run.

Worked Solution: Step 1: Identify values: $Q_1=100$, $Q_2=150$, $P_1=2$, $P_2=3$. Step 2: Calculate percentage changes via midpoint: $\%\Delta Qs=\frac{150-100}{(100+150)/2}=0.4=40\%$, $\%\Delta P=\frac{3-2}{(2+3)/2}=0.4=40\%$. Step 3: Calculate PES: $PES=\frac{40\%}{40\%}=1$. Interpretation: Supply of strawberries is unit elastic in the short run: a 1% increase in price leads to a 1% increase in quantity supplied. The farm has limited ability to adjust output quickly, because strawberries cannot be stored to sell later at a higher price, and the farm has fixed growing capacity in the short run.

7. Quick Reference Cheatsheet

8. What's Next

This topic completes the core set of elasticity concepts introduced in 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 international trade shapes relative prices. 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 taxes and transfer programs, so they underpin most applied microeconomic analysis later in the course.

Taxes and subsidies, Consumer choice, Market efficiency, Theory of the firm