Elasticity Notes

The Price Elasticity of Demand

• The law of demand states that when prices rise, quantity demanded falls.

• Price elasticity of demand (PED) measures how responsive quantity demanded is to price changes.

• Key intuition:

  • Some goods are elastic: quantity demanded changes a lot when price changes.

  • Some goods are inelastic: quantity demanded changes little when price changes.

• Definition:

  • Ed = %ΔP / %ΔQd​

• Interpretation:

  • If $|E_d|\gg 1$, the good is elastic.

  • If $|E_d|\ll 1$, the good is inelastic.

• Note on sign:

  • Price and quantity move in opposite directions along the demand curve, so the raw elasticity is negative.

  • In practice we report the absolute value: $|E_d|$, so higher numbers indicate greater elasticity.

• Examples and context:

  • Goods with high elasticity: when prices rise, quantity demanded drops substantially (e.g., many goods with close substitutes).

  • Goods with low elasticity: price changes have little effect on quantity demanded (e.g., necessities with few substitutes).

• Quick example from lecture:

  • If the price of rice increases by 5% and demand falls by 2.5%,

  • $|E_d| = \left|\frac{-2.5\%}{5\%}\right| = 0.5.$

  • Indicates inelastic demand in that example.

Factors Influencing the Price Elasticity of Demand

• Several factors increase elasticity. The lecture highlights the following attributes (each paired with a contrasting example):

  • Greater availability of close substitutes

  • Margarine vs. eggs (margarine has closer substitutes than eggs in many contexts).

  • A good being a luxury (vs. a necessity)

  • Sailboats vs. a doctor’s visit (sailboats are more elastic; visits are more inelastic).

  • Narrowly defined market

  • Ice cream vs. food (ice cream in a narrowly defined market is more elastic).

  • Longer time horizon

  • Gasoline over three years vs. gasoline in the first few months (long-run elasticity is higher).

Computing the Price Elasticity of Demand

• Base formula:

  • Ed = %ΔP / %ΔQd​

• Sign convention:

  • Price and quantity move in opposite directions on the demand curve, so data produce a negative elasticity.

  • Practically we take the absolute value:

  • $E<em>d = \left| \frac{\%\Delta Q</em>d}{\%\Delta P} \right|.$

• Simple example (rice):

  • Price up by 5% (\$P\uparrow\,+5\%)), demand down by 2.5% (\$Q_d\downarrow\,-2.5\%)).

  • $|E_d| = \left|\frac{-2.5\%}{5\%}\right| = 0.5.$

Ambiguity of Elasticity: Endpoints vs. Midpoint

• Endpoint method (uses starting base or ending base) can yield different results:

  • Example with A: price from \$4 to \$6; quantity from 120 to 80.

  • Using end points:

  • $\%\Delta P = \frac{6-4}{4} = 0.50$

  • $\%\Delta Q = \frac{80-120}{120} = -0.333…$

  • Elasticity (endpoint, base = starting point) = $\left| \frac{-0.333}{0.50} \right| \approx 0.667.$

  • If you reverse bases (endpoints with the other starting point):

  • $\%\Delta P = \frac{6-4}{6} = 0.333…$

  • $\%\Delta Q = \frac{80-120}{80} = -0.500…$

  • Elasticity (endpoint, base = ending point) = $\left| \frac{-0.500}{0.333…} \right| \approx 1.5.$

  • This shows endpoint elasticity depends on the base chosen.

• Midpoint (arc) formula (remedy to the base-dependence issue):

  • For any variable X, the midpoint percent change is:

  • $\%\Delta X = \frac{X<em>2 - X</em>1}{\dfrac{X<em>1 + X</em>2}{2}} = \frac{X<em>2 - X</em>1}{(X<em>1 + X</em>2)/2}.$

  • The price elasticity of demand using the midpoint formula:

  • $\text{PED}<em>{\text{mid}} = \left| \frac{\dfrac{Q</em>2 - Q<em>1}{(Q</em>1 + Q<em>2)/2}}{\dfrac{P</em>2 - P<em>1}{(P</em>1 + P<em>2)/2}} \right| = \left| \frac{Q</em>2 - Q<em>1}{Q</em>1 + Q<em>2} \cdot \frac{P</em>1 + P<em>2}{P</em>2 - P_1} \right|.$

• Example with A and B using midpoint:

  • P1 = 4, Q1 = 120; P2 = 6, Q2 = 80.

  • $\%\Delta P = \frac{6 - 4}{(4 + 6)/2} = \frac{2}{5} = 0.4,$

  • $\%\Delta Q = \frac{80 - 120}{(120 + 80)/2} = \frac{-40}{100} = -0.4,$

  • $\text{PED}_{\text{mid}} = \left|\frac{-0.4}{0.4}\right| = 1.$

• Summary:

  • Endpoint method can be inconsistent depending on base choice.

  • Midpoint formula provides a symmetric measure and avoids the base-dependency issue.

Types of Price Elasticity of Demand

• Perfectly Inelastic: PED = 0

  • Price changes do not change quantity demanded.

• Inelastic Demand: 0 < PED < 1

• Unit Elastic Demand: PED = 1

• Elastic Demand: PED > 1

• Perfectly Elastic Demand: PED = ∞

Revenue and Elasticity

• Revenue function: $R = P \times Q_d.$

• Relationship between elasticity and revenue:

  • In the elastic portion of the demand curve, a fall in price leads to a larger percentage increase in quantity demanded, so revenue tends to rise.

  • In the inelastic portion, a fall in price increases quantity only slightly, so revenue tends to fall.

• Key implication: Elasticity helps predict whether changing price will raise or reduce total revenue.

Elasticity and Revenue: Numerical Illustration

• Example given in the lecture:

  • Price = \$14, Quantity = 30, Revenue = $R = P \times Q = 14 \times 30 = 420.$

  • This illustrates how revenue is computed from price and quantity, and how elasticity contexts influence revenue responses to price changes.

Elasticity and Changes in Revenue (Graphical Intuition)

• Relative magnitudes of elasticity determine revenue response to price changes:

  • Relatively inelastic demand: lowering price reduces revenue because quantity response is small.

  • Relatively elastic demand: lowering price increases revenue because quantity response is large.

• The lecture uses labeled regions (Relatively Inelastic vs Relatively Elastic) to show how price changes affect revenue depending on elasticity.

Elasticity and Linear Demand

• On a linear demand curve (not perfectly elastic or inelastic overall), elasticity varies along the curve:

  • Upper-left portion is elastic.

  • Lower-right portion is inelastic.

• It is incorrect to label an entire linear demand curve as elastic or inelastic.

• Check elasticity between sections A–B, B–E, E–F to see how elasticity changes along the curve.

Elasticity and Industry Examples (Linear Demand Illustration)

• A representative linear demand example: price per ride graph shows regions of elasticity and unit elasticity.

• The takeaway: elasticity is not constant on a linear demand curve; it varies with quantity and price along the curve.

Income Elasticity of Demand

• Definition:

  • $E<em>y = \frac{\%\Delta Q</em>d}{\%\Delta \text{Income}}.$

• Interpretation:

  • Normal goods: E_y > 0.

  • Inferior goods: E_y < 0.

• Use cases:

  • Staples (rice, bread, eggs, rent) tend to have positive but smaller income elasticities.

Cross-Price Elasticity of Demand

• Definition:

  • $E<em>{XY} = \frac{\%\Delta Q</em>X}{\%\Delta P_Y}.$

• Interpretation:

  • Positive $E_{XY}$ indicates substitutes (as price of good Y rises, quantity demanded of good X rises).

  • Negative $E_{XY}$ indicates complements (as price of good Y rises, quantity demanded of good X falls).

Elasticity of Supply

• Price elasticity of supply (PES) measures how responsive quantity supplied is to price changes:

  • $E<em>s = \frac{\%\Delta Q</em>s}{\%\Delta P}.$

• Intuition:

  • When PES is elastic, producers increase quantity supplied a lot in response to price increases.

  • When PES is inelastic, producers increase quantity only slightly in response to price increases.

Determinants of Elasticity of Supply

• Supply is more elastic when producers can easily adjust production in response to price changes:

  • Beach-front property: inelastic supply (hard to increase stock quickly).

  • Manufactured goods (books, cars, televisions): more elastic supply (can adjust production more readily).

• Time horizon:\nLonger time horizons increase the ability of firms to adjust quantity supplied, making supply more elastic.

Types of Supply Elasticity (Diagrammatic Intuition)

• Perfectly Inelastic Supply: price changes have no effect on quantity supplied; supply curve is vertical.

• Inelastic Supply: quantity supplied changes little with price changes.

• Elastic Supply: quantity supplied changes significantly with price changes.

• Perfectly Elastic Supply: any change in price leads to an infinite change in quantity supplied; supply curve is horizontal; elastic supply is effectively unbounded at a given price.

Applications of Supply, Demand, and Elasticity

Farming Technology

• Scenario: technological improvements reduce wheat price from \$3 to \$2 and increase quantity from 100 to 110.

• Questions: what is the new revenue? what was the elasticity of demand?

• Takeaway points:

  • Revenue calculation: initial revenue $3 \times 100 = 300; new revenue $2 \times 110 = 220$ (revenue falls).

  • Elasticity of demand context determines whether farmers are better off by adopting the technology.

  • In a competitive market, adoption spreads: if one farmer adopts, others must follow to remain competitive.

  • Historical note: in 1950, about 17% of the labor force worked on farms; today about 2%—implying shifts in production efficiency and scale.

  • Public policy: some programs pay farmers not to farm certain crops to manage inelastic demand and surplus.

OPEC and Oil Prices

• Historical question: Why did oil prices rise sharply around 1980 and then fall back by 1990?

• Explanation:

  • OPEC doubled the price from 1979 to 1981; due to the inelastic demand for oil, revenues increased for OPEC.

  • Over longer time horizons, demand and supply for oil become more elastic, so the price increase was not sustained; the plan backfired in the long run.

Drug Interdiction vs Drug Education

• Policy question: Is it better to increase penalties for supplying drugs (interdiction) or to increase drug education?

• Key insight:

  • Interdiction reduces supply; education reduces demand.

  • If demand is more inelastic than supply, education reduces the quantity of drugs more than interdiction.

Summary of Core Concepts

• Elasticity measures responsiveness; prices and quantities move inversely along the demand curve.

• PED is calculated as the percent change in quantity demanded over the percent change in price, typically reported as a nonnegative number using absolute value.

• Several factors drive elasticity: substitutes, luxuries vs necessities, market definition, and time horizon.

• There are multiple ways to compute percent changes (endpoints vs midpoint); the midpoint formula is preferred to avoid base bias.

• Elasticity types (elastic, inelastic, unit, perfectly elastic/inelastic) describe how responsive demand is to price changes.

• Elasticity interacts with revenue: in elastic regions, lower prices can raise revenue; in inelastic regions, lower prices can lower revenue.

• Elasticity concepts extend to income (income elasticity) and cross-price effects (cross-price elasticity).

• Elasticity of supply mirrors the demand side in spirit, with determinants including production flexibility and time horizon.

• Real-world applications (farming tech, OPEC, drug policy) illustrate how elasticity shapes decisions and policy.

Key Formulas to Remember

• Price elasticity of demand: $E<em>d = \frac{\%\Delta Q</em>d}{\%\Delta P}$

• Absolute value convention for reporting: $|E_d|$ (positive number for interpretation)

• Midpoint formula for percent changes:

  • $\%\Delta X = \frac{X<em>2 - X</em>1}{(X<em>1 + X</em>2)/2}$

  • Hence, $\text{PED}<em>{\text{mid}} = \left| \frac{\dfrac{Q</em>2 - Q<em>1}{(Q</em>1 + Q<em>2)/2}}{\dfrac{P</em>2 - P<em>1}{(P</em>1 + P_2)/2}} \right|.$

• Revenue: $R = P \times Q_d$

• Income elasticity of demand: $E<em>y = \frac{\%\Delta Q</em>d}{\%\Delta \text{Income}}$

• Cross-price elasticity of demand: $E<em>{XY} = \frac{\%\Delta Q</em>X}{\%\Delta P_Y}$

• Price elasticity of supply: $E<em>s = \frac{\%\Delta Q</em>s}{\%\Delta P}$