Elasticities: Price, Income, and Cross-Price (Lecture Notes)

Price Elasticity of Demand

• Definition: price elasticity of demand measures how responsive quantity demanded is to a change in price.
• Movement along the demand curve:
  • A price change is an endogenous change that moves you along the demand curve to a new point.
  • It tells us how far quantity demanded changes when price changes, which relates to the slope of the demand curve.
• Interpretation of elasticity magnitude:
  • If the demand is flatter, quantity responds a lot to price changes → elastic demand.
  • If the demand is steeper, quantity responds little to price changes → inelastic demand.
  • Midpoint between elastic and inelastic is unit elastic demand.
• Sign convention:
  • Price elasticity of demand is typically negative (due to the Law of Demand).
  • Many contexts report the absolute value |E_p| for classification, unless the sign is required.
• Types of goods by price elasticity:
  • Elastic demand: |E_p| > 1
  • Inelastic demand: |E_p| < 1
  • Unit elastic demand: |E_p| = 1
• Important nuance about measurement:
  • Price elasticity of demand (E_p) is only about price changes and movement along the same demand curve.
  • It cannot tell you about how demand shifts when non-price factors change; that requires other elasticities.
• General rule to interpret results:
  • If you are told a good is elastic, that means the price elasticity of demand for that good is greater than 1 in magnitude (|E_p| > 1).
  • If asked to calculate, compute E_p from data and compare to 1 (in magnitude).

Midpoint/Percentage-change formula for price elasticity of demand

• The standard formula using midpoint (to avoid base effects):
  Ep = rac{rac{Q2 - Q1}{rac{Q1 + Q2}{2}}}{rac{P2 - P1}{rac{P1 + P_2}{2}}}
• Equivalent interpretation:
  • ext{Percentage change in quantity} = rac{Q2 - Q1}{(Q1 + Q2)/2}
  • ext{Percentage change in price} = rac{P2 - P1}{(P1 + P2)/2}
• Sign and magnitude come from the data; typical downward-sloping demand yields a negative E_p.

Income Elasticity of Demand

• Definition: income elasticity of demand measures the responsiveness of quantity demanded to changes in income, holding prices constant.
• Distinction from price elasticity: income elasticity captures a shift (endogenous change in demand due to income) rather than a movement along the demand curve.
• Formula (midpoint):
  EI = rac{rac{Q2 - Q1}{rac{Q1 + Q2}{2}}}{rac{I2 - I1}{rac{I1 + I_2}{2}}}
• Sign and interpretation:
  • EI can be positive or negative.
  • Positive EI: normal goods (demand increases as income increases).
  • Negative EI: inferior goods (demand decreases as income increases).
  • EI = 0: no income effect on demand (sticky goods).
• Special case: luxury goods vs necessities (both are normal goods)
  • Luxury goods: EI > 1 (percentage change in quantity demanded is larger than the percentage change in income).
  • Necessities (normal but not luxury): 0 < EI < 1.
• Examples and intuition:
  • Positive EI example: as income rises, you buy more vacations abroad (a luxury normal good).
  • Negative EI example: ramen noodles (an inferior good) – when income rises, you substitute up to more expensive foods.
  • Zero EI example: medications with fixed usage regardless of income (sticky goods such as electricity, gas, water are often treated as having EI ≈ 0 in some contexts).
• The “lawyer’s daughter” bullet point (conceptual shorthand):
  • Generally, as income rises, demand for normal goods increases.
  • For inferior goods, an income rise reduces demand (shift left in the demand curve for that good).
• Normal vs inferior vs luxury recap:
  • EI > 0: normal goods
  • EI < 0: inferior goods
  • EI > 1: luxury (a subset of normal goods)
  • EI = 0: no income effect (sticky goods)

Cross-Price Elasticity of Demand

• Definition: cross-price elasticity of demand measures how quantity demanded of one good (good x) responds to a change in the price of another good (good y).
• Formula:
  E{xy} = rac{rac{Q{x2} - Q{x1}}{rac{Q{x1} + Q{x2}}{2}}}{rac{P{y2} - P{y1}}{rac{P{y1} + P_{y2}}{2}}}
• Interpretation:
  • If $E_{xy} > 0$: the goods are substitutes (an increase in price of y leads to higher quantity demanded of x).
  • If $E_{xy} < 0$: the goods are complements (an increase in price of y leads to lower quantity demanded of x).
  • If $E_{xy} = 0$: the goods are independent.
• Examples:
  • Substitutes: coffee and tea. If the price of coffee rises, demand for tea tends to rise.
  • Complements: peanut butter and jelly. If the price of peanut butter rises, demand for jelly tends to fall.

How elasticities relate to shifts and movements in the classroom store exercise (conceptual)

• Demand shifts vs movements along the curve:
  • Price elasticity of demand (E_p) captures movement along a given demand curve as price changes.
  • Income elasticity of demand (E_I) captures shifts of the entire demand curve when income changes.
• Practical lesson from the activity:
  • When income changes, you can end up with two separate demand curves for the same goods at different income levels (e.g., one curve for income = $5 and another for income = $8).
  • For each income level, you analyze how a price change (e.g., Snickers from $1 to $2) affects quantity demanded along that income-specific curve.
  • You may compute cross-price effects by examining how the quantity demanded of one good responds to price changes in another (e.g., Snickers vs Coke, etc.).

Worked mini-examples (illustrative, using midpoint changes)

• Price elasticity example (downward-sloping demand):

  - Suppose P1 = 1, Q1 = 10; P2 = 2, Q2 = 6.

  ext{%ΔQ} = rac{Q2 - Q1}{(Q1 + Q2)/2} = rac{6 - 10}{(10 + 6)/2} = rac{-4}{8} = -0.50
  ext{ %ΔP} = rac{P2 - P1}{(P1 + P2)/2} = rac{2 - 1}{(1 + 2)/2} = rac{1}{1.5}

  \approx 0.667

  • E_p = rac{-0.50}{0.667} \approx -0.75
  • Interpretation: in this range, demand is inelastic (|E_p| < 1).

• Income elasticity example:

  • Suppose I1 = 5, I2 = 8; Q1 = 2, Q2 = 4.
  • 
    ext{ %ΔQ} = rac{4 - 2}{(2 + 4)/2} = rac{2}{3} \approx 0.667
    ext{ %ΔI} = rac{8 - 5}{(5 + 8)/2} = rac{3}{6.5} \approx 0.462
  • E_I = rac{0.667}{0.462} \approx 1.44
  • Interpretation: this good is a luxury (EI > 1).

• Cross-price example:

  • Suppose quantity of x changes from Qx1 to Qx2 when Py changes from Py1 to Py2.
  • 
    E{xy} = rac{rac{Q{x2} - Q{x1}}{(Q{x1} + Q{x2})/2}}{rac{P{y2} - P{y1}}{(P{y1} + P_{y2})/2}}
  • If Exy > 0, substitutes; if Exy < 0, complements.

Practice guidance for the classroom store exercise (summary of the procedure)

• Setup recap:
  • Four situations with a fixed income in some cases and a changed price for Snickers (the only price-changing good).
  • Initially, income = $5; prices: Coke $1, Snickers $1, Twinkie $1, Milk $1.
  • Situation 2: Snickers price rises to $2; income still $5.
  • Situation 3 and 4: income rises to $8; prices revert to original after the price change in 2 (i.e., Snickers back to $1 in 4 while income $8).
• Task outline:
  • For each situation, determine how many of each item you would buy given you must spend all income.
  • Sum individual quantities to obtain market quantities for each item in each situation.
  • For the Snickers analysis, draw two separate demand curves:
  • One for income = $5 (situations 1 and 2).
  • One for income = $8 (situations 3 and 4).
  • Use market quantities to answer the packet questions, including elasticity calculations.
• Practical tips from the discussion:
  • Treat the two income levels as separate demand curves when interpreting elasticity across income changes.
  • When asked for % changes, use the midpoint formula to compute
    ext{%ΔQ} = rac{Q2 - Q1}{(Q1 + Q2)/2}, \, \%ΔP = rac{P2 - P1}{(P1 + P2)/2}
  • For cross-price elasticity, identify which good’s price changed and which good’s quantity is being examined.
  • Note: price elasticity of demand is typically negative; other elasticities preserve the sign unless the context requires the absolute value.

Quick interpretations to remember for exam

• Elasticity sign conventions:
  • Price elasticity of demand: usually negative; interpret by magnitude |E_p|.
  • Income elasticity of demand: sign indicates normal vs inferior; positive = normal, negative = inferior.
  • Cross-price elasticity: positive = substitutes, negative = complements, zero = independent.
• Magnitude cutoffs:
  • Price: |Ep| > 1 elastic, |Ep| < 1 inelastic, |E_p| = 1 unit elastic.
  • Income: EI > 1 luxury, 0 < EI < 1 normal (necessity), EI > 0 are normal goods, EI < 0 inferior goods, EI = 0 zero income effect.

Final note on test readiness

• Elasticities are a core topic and are likely to appear with both definitions and calculations.
• Practice problems like the classroom-store exercise are representative of what you might see on quizzes or exams.
• Expect to compute and interpret Ep, EI, and E{xy} from data, using the midpoint (arc) method for percentage changes, and to distinguish shifts in demand (EI) from movements along a demand curve (Ep).