Source: Demand Analysis and Optimal Pricing lecture, Texas A&M
Tags: price elasticity, point elasticity, arc elasticity, elastic demand, inelastic demand, unitary elastic, income elasticity, cross-price elasticity, elasticity and revenue, elasticity determinants
Price elasticity of demand measures how strongly quantity demanded responds to a price change, expressed as a percentage-for-percentage ratio. Where |Ep| > 1 the good is elastic (quantity is highly responsive); where |Ep| < 1 it is inelastic. Elasticity changes along a straight-line demand curve and directly determines what happens to total revenue when price changes. Income and cross-price elasticities use the same logic to classify goods as normal/inferior and substitute/complement.
Price elasticity of demand (Ep)
The percentage change in quantity demanded divided by the percentage change in price. Ep = %ΔQd / %ΔP. Typically negative (law of demand), but we often refer to the absolute value.
Elastic demand
|Ep| > 1. Quantity demanded is highly responsive to price. A 1% price increase causes more than a 1% fall in quantity.
Inelastic demand
|Ep| < 1. Quantity demanded is not very responsive to price. A 1% price increase causes less than a 1% fall in quantity.
Unitary elastic demand
|Ep| = 1. The percentage change in quantity exactly matches the percentage change in price.
Point elasticity
Elasticity calculated at a specific point on the demand curve using the formula |(ΔQd/ΔP) × (P/Qd)|. Uses the slope of the demand curve (not the inverse demand).
Arc elasticity
Elasticity calculated between two points using the average of the starting and ending values in the denominator, to avoid the problem of the answer changing depending on which endpoint you treat as the base.
Perfectly inelastic demand
Ep = 0. A vertical demand curve. Quantity does not respond to price at all. Example: a good people absolutely need regardless of price.
Income elasticity (EI)
%ΔQd / %ΔY. Positive for normal goods, negative for inferior goods. If EI > 1, the good is a superior (luxury) good.
Cross-price elasticity (Exy)
%ΔQd / %ΔP0. Positive for substitutes, negative for complements.
Use observed data at two points in time where the demand curve is stable.
Formula: Ep = %ΔQd / %ΔP
%ΔQd = (Q_end – Q_start) / Q_start
%ΔP = (P_end – P_start) / P_start
Beef example (2014):
January: P = $3.467/lb, Q = 10 lbs (family of four)
October: P = $4.154/lb, Q = 8.78 lbs
%ΔQd = (8.78 – 10)/10 = –0.122 (–12.2%)
%ΔP = (4.154 – 3.467)/3.467 = 0.198 (19.8%)
Ep = –0.122 / 0.198 = –0.62, so |Ep| = 0.62
Interpretation: for every 1% increase in price, quantity demanded falls by 0.62%. Demand is inelastic.
Allows you to calculate elasticity at any point on a known demand curve, not just between two observed data points.
Derived by rearranging the percentage formula:
|Ep| = |(ΔQd/ΔP) × (P/Qd)|
The first term (ΔQd/ΔP) is the slope of the demand curve (not inverse demand).
The second term (P/Qd) is the coordinates of the specific point.
Worked example: Qd = 200 – 2P
Slope of demand curve: ΔQd/ΔP = –2
At P = 90: Qd = 200 – 2(90) = 20. Ep = |–2 × (90/20)| = |–9| = 9. Elastic.
At P = 10: Qd = 200 – 2(10) = 180. Ep = |–2 × (10/180)| = 0.11. Inelastic.
On a straight-line demand curve, the slope (ΔQd/ΔP) is constant, but the P/Qd ratio changes at every point.
Near the top of the curve (high P, low Qd): P/Qd is large, so demand is elastic.
At the midpoint: unitary elastic.
Near the bottom (low P, high Qd): P/Qd is small, so demand is inelastic.
This is sometimes called "line elasticity."
Two exceptions where elasticity does not change along the curve:
Perfectly inelastic (vertical line): Ep = 0 everywhere.
Perfectly elastic (horizontal line): Ep = ∞ everywhere.
Revenue = P × Q.
The relationship between a price change and revenue depends on elasticity:
Inelastic demand (|Ep| < 1):
Price rises → Qd falls, but by a smaller proportion → Revenue increases.
Price falls → Revenue decreases.
Elastic demand (|Ep| > 1):
Price rises → Qd falls by a larger proportion → Revenue decreases.
Price falls → Revenue increases.
Beef example check: Inelastic (|Ep| = 0.62).
January revenue: 10 × 3.467 = $34.67
October revenue: 8.78 × 4.154 = $36.47
Revenue rose when price rose. Consistent with inelastic demand.
Availability of substitutes: More substitutes → more elastic. Water has few substitutes, so |Ep| ≈ 0.5.
Portion of budget: Goods that take a larger share of your budget are more elastic. A 10% rise in the price of housing affects your spending far more than a 10% rise in salt.
Time horizon: Demand tends to be more elastic in the long run because consumers have time to find substitutes. Ferry ridership example: inelastic in the short run (no alternatives), elastic in the long run (people move, buy cars, find other routes).
EI = %ΔQd / %ΔY (do not take absolute values)
EI > 0: normal good
EI > 1: superior (luxury) good
EI < 0: inferior good
Business application: cyclical goods (EI > 0) see demand rise in expansions and fall in recessions. Counter-cyclical goods (EI < 0) do the opposite.
Exy = %ΔQd / %ΔP0 (do not take absolute values)
Exy > 0: the goods are substitutes
Exy < 0: the goods are complements
Es = %ΔQs / %ΔP
Measures how much a firm can adapt its output in response to price changes.
Direct elasticity: Ep = [(Q₂ – Q₁)/Q₁] / [(P₂ – P₁)/P₁]
Point elasticity: |Ep| = |(ΔQd/ΔP) × (P/Qd)|
Arc elasticity (midpoint method): Uses the average of start and end values in the denominator for both %ΔQd and %ΔP. Resolves the asymmetry of the direct method.
Revenue rule:
|Ep| < 1 → price and revenue move in the same direction
|Ep| > 1 → price and revenue move in opposite directions
|Ep| = 1 → revenue is at its maximum
Income elasticity: EI = %ΔQd / %ΔY
Cross-price elasticity: Exy = %ΔQd / %ΔP₀
⚠️ The slope of the demand curve (ΔQd/ΔP) is not the same as the slope of the inverse demand (ΔP/ΔQd). Point elasticity uses the demand curve slope. A common mistake is plugging in the inverse demand slope.
⚠️ Elasticity changes along a linear demand curve. Calling a whole linear demand curve "elastic" or "inelastic" is imprecise. You can say one curve is generally more elastic than another at the same P and Q, but for a single line, it depends on where you are.
⚠️ A firm should never price on the inelastic portion of the demand curve. If demand is inelastic, raising price increases revenue and (since quantity falls) reduces costs, so profit rises. The firm should keep raising price until it reaches the elastic portion.
⚠️ For income and cross-price elasticity, do not take absolute values. The sign carries the economic meaning (normal vs. inferior, substitute vs. complement).
⚠️ The direct elasticity method requires a stable demand curve. If the curve shifted between the two observations, the calculation is invalid.
Q: Given Qd = 200 – 2P, calculate point elasticity at P = 50. Is demand elastic, inelastic, or unitary at that price?
A: Qd = 200 – 2(50) = 100. Slope = –2. |Ep| = |–2 × (50/100)| = 1. Demand is unitary elastic at P = 50. This is the midpoint of the demand curve.
Q: A firm faces inelastic demand. It raises its price by 10% and quantity demanded falls by 4%. What happens to total revenue?
A: Revenue rises. The 10% price increase more than offsets the 4% quantity decrease. This is consistent with |Ep| = 0.4 < 1 (inelastic).
Q: The income elasticity of a good is –0.3. Classify the good and explain what happens to its demand during a recession.
A: The good is inferior (EI < 0). During a recession, incomes fall, so demand for this good increases. It is counter-cyclical.
Q: Cross-price elasticity between goods A and B is +2.1. What is the relationship, and what happens to demand for A if the price of B rises by 5%?
A: A and B are substitutes (Exy > 0). If B's price rises by 5%, demand for A rises by approximately 2.1 × 5% = 10.5%.
Q: Why should a firm never set its price on the inelastic portion of its demand curve?
A: On the inelastic portion, raising price increases revenue (quantity falls proportionally less). At the same time, producing less reduces costs. Both effects increase profit, so the firm should keep raising price until it reaches the elastic region.
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