Source: Varghese practice problems, textbook chapters
Tags: price elasticity of demand, elastic, inelastic, unit elastic, demand curve shifts, demand equation, complements, substitutes, price discrimination, first degree, second degree, third degree, hurdle model, markup pricing, Lerner index, network externalities, information goods, ECON 323, Varghese
This section ties together demand curves, elasticity, and pricing strategy. You need to know how to calculate and interpret price elasticity, understand what shifts a demand curve versus what causes movement along it, and apply the different forms of price discrimination. The "rule of thumb" pricing formula (using elasticity and marginal cost) is a workhorse for exam problems.
Price elasticity of demand (Ed)
A measure of how responsive quantity demanded is to a change in price. Formally: Ed = (%ΔQ) / (%ΔP). Usually negative for a downward-sloping demand curve. Problems often work with the absolute value.
Elastic demand
|Ed| > 1. Quantity is highly responsive to price changes. A price increase causes total revenue (expenditure) to fall.
Inelastic demand
|Ed| < 1. Quantity is not very responsive to price. A price increase causes total revenue to rise.
Unit elastic demand
|Ed| = 1. The percentage change in quantity exactly offsets the percentage change in price. Total revenue is unchanged.
Perfectly inelastic demand
|Ed| = 0. A perfectly vertical demand curve. Quantity demanded does not respond to price at all.
Perfectly elastic (infinitely elastic) demand
|Ed| = ∞. A perfectly horizontal demand curve. Any price increase causes quantity demanded to drop to zero.
Point elasticity (point-slope method)
Calculated at a specific point on the demand curve. For a linear demand Q = a − bP: Ed = (dQ/dP) × (P/Q) = −b × (P/Q). For inverse demand P = a − bQ: Ed = (1/(dP/dQ)) × (P/Q) = (−1/b) × (P/Q).
Arc elasticity (direct method)
Calculated between two points. Ed = [(Q₂ − Q₁)/Q₁] / [(P₂ − P₁)/P₁], using the initial point as the base, or the midpoint formula for a more symmetric measure.
Demand curve shift
A change in any variable other than the good's own price (income, prices of related goods, preferences, etc.) shifts the entire demand curve to a new position.
Movement along the demand curve
A change in the good's own price causes movement along the existing curve, not a shift.
Substitute goods
Goods where an increase in the price of one leads to an increase in demand for the other. The cross-price coefficient in the demand equation is positive.
Complementary goods
Goods where an increase in the price of one leads to a decrease in demand for the other. The cross-price coefficient is negative.
First-degree price discrimination (perfect price discrimination)
Charging each customer the maximum they are willing to pay (their reservation price). Captures the entire consumer surplus.
Second-degree price discrimination
Offering different prices for different quantities or bundles, and letting customers self-select. Volume discounts are a classic example.
Third-degree price discrimination
Charging different prices to different identifiable market segments (e.g. students vs. adults, domestic vs. foreign).
Hurdle model
A form of price discrimination where customers must clear a "hurdle" (clip a coupon, wait for a sale, buy at an inconvenient time) to get a lower price. Self-selection separates price-sensitive customers from less sensitive ones.
Information goods
Products characterised by high fixed costs of creation but very low (near-zero) marginal costs of reproduction. Software, digital media, and databases are examples.
Network externalities (positive)
A situation where the value of a good to one user increases as more people use it. Social media platforms and communication tools are standard examples.
Lerner index / margin
(P − MC) / P. Measures the firm's markup as a fraction of price. At the profit-maximising price: margin = 1/|Ed|.
Markup
(P − MC) / MC. Measures how much above marginal cost the firm charges, as a fraction of cost.
A demand equation such as Q = 60 − 60P + 2Y tells you how quantity depends on price (P) and income (Y).
The coefficient on P (here, −60) tells you how much Q changes per $1 increase in P, holding everything else constant.
The coefficient on Y (here, +2) tells you how much Q changes per $1 increase in income.
To find the combined effect of a $2 price increase and an $80 income increase: ΔQ = −60(2) + 2(80) = −120 + 160 = +40 units.
The demand curve plots Q against the good's own price, holding all other variables fixed. Changes in these other variables shift the curve:
Consumer income
Prices of related goods (substitutes and complements)
Consumer preferences and expectations
Number of buyers
A change in the good's own price causes movement along the existing curve, not a shift.
Look at the sign of the cross-price coefficient.
Motorcycles and motorcycle helmets are complements. More motorcycle demand raises helmet demand. In the helmet demand equation, the coefficient on motorcycle quantity (or a variable representing motorcycle demand) is positive.
The coefficient on the price of a complement is negative (higher price of the complement reduces demand for both goods).
The coefficient on the price of a substitute is positive (higher price of the substitute drives consumers toward your good).
Point-slope method (most common on this exam):
Given an inverse demand P = a − bQ, at a specific (P, Q) point:
Ed = (1 / slope of inverse demand) × (P / Q) = (−1/b) × (P/Q)
Worked example (hot dog vendor, Problem 3.15):
Q = 180 − 15P, so inverse demand: P = 12 − (1/15)Q.
At Q = 30: P = 12 − 30/15 = 12 − 2 = $10.
Revenue = P × Q = 10 × 30 = $300.
Ed = (dQ/dP) × (P/Q) = −15 × (10/30) = −5.
|Ed| = 5, which is elastic.
Since demand is elastic, the vendor should decrease price to increase revenue. When demand is elastic, a price cut raises total revenue because the percentage gain in quantity outweighs the percentage loss in price.
Direct (arc) method:
Worked example (Tod's peanut butter, Problem 3.16):
P = 32 − 3Qd. At P = $3: Q = (32 − 3)/3 = 29/3 ≈ 9.67. At P = $10: Q = (32 − 10)/3 = 22/3 ≈ 7.33.
Arc elasticity = [(Q₂ − Q₁)/Q₁] / [(P₂ − P₁)/P₁].
ΔQ = 7.33 − 9.67 = −2.33, %ΔQ = −2.33/9.67 ≈ −24.1%.
ΔP = 10 − 3 = 7, %ΔP = 7/3 ≈ 233.3%.
Ed = −24.1% / 233.3% ≈ −0.103.
Point-slope at P = $3: Ed = (−1/3)(3/9.67) ≈ −0.103. At P = $10: Ed = (−1/3)(10/7.33) ≈ −0.455.
This is a high-frequency exam topic.
Elastic demand (|Ed| > 1): price increase causes revenue to decrease. Price decrease causes revenue to increase.
Inelastic demand (|Ed| < 1): price increase causes revenue to increase. Price decrease causes revenue to decrease.
Unit elastic (|Ed| = 1): revenue is at its maximum. Any price change reduces revenue.
Practical implication: if your demand is elastic and you want more revenue, lower the price. If inelastic, raise the price.
The relationship between price, marginal revenue, and elasticity:
MR = P × (1 − 1/|Ed|)
Rearranged to solve for P:
P = MR / (1 − 1/|Ed|)
Worked example (Problem 3.7): MR = $20, |Ed| = 5.
P = 20 / (1 − 1/5) = 20 / (4/5) = 20 × 5/4 = $25.
At the profit-maximising output, MR = MC. Combined with the elasticity-MR formula:
P = MC / (1 − 1/|Ed|)
This is the "rule of thumb" pricing formula. It tells you the optimal price given your marginal cost and elasticity.
Margin = (P − MC) / P = 1/|Ed|.
Markup = (P − MC) / MC.
Worked example (shoe seller, Problem 3.12): |Ed| = 3, MC = $20.
P = 20 / (1 − 1/3) = 20 / (2/3) = $30.
Margin = (30 − 20)/30 = 1/3 ≈ 33.3%.
Markup = (30 − 20)/20 = 1/2 = 50%.
Worked example (clothing store, Problem 3.13a): MC = $2, |Ed| = 4.
P = 2 / (1 − 1/4) = 2 / (3/4) = $2.67.
Margin = 1/4 = 25%.
Markup = (2.67 − 2)/2 = 0.67/2 = 33.3%.
Finding MC from price and elasticity (Problem 3.13c): P = $5, |Ed| = 3.
MC = P × (1 − 1/|Ed|) = 5 × (1 − 1/3) = 5 × 2/3 = $3.33.
If |Ed| < 1 (inelastic), the pricing formula produces a negative denominator (since 1 − 1/|Ed| < 0), which would imply a negative price. This signals that the firm is on the wrong part of its demand curve.
A profit-maximising firm should never operate in the inelastic region of demand, because it could raise price, sell fewer units, and simultaneously increase revenue while reducing cost. The correct action is to raise price until demand becomes elastic.
Conditions for effective price discrimination:
The firm must be able to identify and separate different market segments with different price elasticities.
Customers in the low-price segment must not be able to resell to customers in the high-price segment (no arbitrage).
Charge the higher price in the segment with more inelastic demand. Those customers are less responsive to price, so the revenue loss from a higher price is smaller.
Examples of price discrimination (Problem 3.8): discount coupons, airline super-saver fares, cheap movie matinees. All of these qualify.
Professional journal subscriptions (Problem 3.14):
Libraries have more inelastic demand (they need the journals for their collections regardless of price). Individual subscribers are more price-sensitive.
The two segments are identifiable and cannot easily trade with each other (a library subscription is institutional).
The hurdle model applies: individual subscribers "clear a hurdle" by being willing to accept personal-use restrictions, slower delivery, or other inconveniences in exchange for a lower price.
Parking garage example: to maximise revenue across short-term and long-term segments, set prices so that marginal revenue is equal across segments.
First degree: each customer pays their individual maximum willingness to pay. Captures all consumer surplus. Rare in practice.
Second degree: different prices for different quantities purchased. Customers self-select into the tier that suits them. Examples: bulk discounts, tiered data plans.
Third degree: different prices for identifiable groups. Examples: student discounts, senior pricing, foreign vs. domestic markets.
Worked example (Juice Shop, Problem 12): Q = 1,000 − 240P + 80Pc.
At P = $1.50, Pc = $1.20: Q = 1,000 − 240(1.50) + 80(1.20) = 1,000 − 360 + 96 = 736 drinks.
Demand curve when Pc = $1.20: Q = 1,000 − 240P + 80(1.20) = 1,096 − 240P.
If the competitor's price rises to $1.50: Q = 1,000 − 240P + 80(1.50) = 1,120 − 240P. The demand curve shifts outward (to the right), meaning more quantity demanded at every price. This is consistent with the cafe being a substitute.
Worked example (Model-It software, Problem 13): Qm = 1,200 − 8Pm + 4Ps.
At Pm = $200, Ps = $300: Qm = 1,200 − 8(200) + 4(300) = 1,200 − 1,600 + 1,200 = 800 units.
Demand curve when Ps = $300: Qm = 1,200 − 8Pm + 4(300) = 2,400 − 8Pm.
Inverse demand: 8Pm = 2,400 − Qm, so Pm = 300 − 0.125Qm.
Information goods have high fixed costs and negligible marginal costs. A firm should not price at average fixed cost or give the product away for free. Optimal pricing still uses marginal analysis, but the very low MC means high margins.
Positive network externalities mean the product becomes more valuable as more people adopt it. An online dating service restricted to registered users is a good example: more users means better matches for everyone.
Worked example (Problem 5 from text): point elasticity = −1.5, initial P = $20, Q = 10. Price drops to $17.50.
%ΔP = (17.50 − 20)/20 = −12.5%.
%ΔQ = Ed × %ΔP = −1.5 × (−12.5%) = +18.75%.
Price elasticity of demand (point method): Ed = (dQ/dP) × (P/Q)
MR-elasticity relationship: MR = P × (1 − 1/|Ed|)
Rule-of-thumb pricing: P = MC / (1 − 1/|Ed|)
Margin (Lerner index): (P − MC) / P = 1/|Ed|
Markup over cost: (P − MC) / MC
Total revenue: TR = P × Q
Revenue and elasticity: Elastic (|Ed| > 1): price ↑ causes TR ↓. Inelastic (|Ed| < 1): price ↑ causes TR ↑.
⚠️ If demand is elastic and price increases, expenditure (revenue) decreases. This is one of the most commonly tested relationships.
⚠️ A perfectly vertical demand curve is completely inelastic (|Ed| = 0). A perfectly horizontal demand curve is infinitely elastic. Do not mix these up.
⚠️ A profit-maximising firm never operates in the inelastic portion of its demand curve. If |Ed| < 1, the firm should raise price.
⚠️ Charging each customer their maximum willingness to pay is first-degree price discrimination. Volume-based pricing is second-degree. Group-based pricing is third-degree.
⚠️ For third-degree price discrimination, the higher price goes to the segment with more inelastic demand, not the larger segment or the one with higher quantity.
⚠️ To maximise revenue across market segments, set prices so that marginal revenues are equal across segments.
⚠️ A change in the good's own price is movement along the demand curve. A change in income, preferences, or the price of a related good shifts the demand curve.
⚠️ In the demand equation, a positive coefficient on another good's price means the goods are substitutes. A negative coefficient means complements.
⚠️ Know the rule-of-thumb formula cold: P = MC / (1 − 1/|Ed|). Be able to rearrange it to find MC, margin, or markup.
Q: If demand for hamburgers is elastic, what happens to expenditures when the price increases?
A: Expenditures decrease. With elastic demand, the percentage drop in quantity exceeds the percentage rise in price.
Q: A perfectly vertical demand curve is what type of elasticity?
A: Completely inelastic (|Ed| = 0).
Q: MR = $20 and |Ed| = 5. What is the profit-maximising price?
A: $25. P = 20 / (1 − 1/5) = 20 / 0.8 = 25.
Q: A firm charges each customer the maximum they are willing to pay. What degree of price discrimination is this?
A: First-degree price discrimination.
Q: Which market segment gets the higher price under third-degree price discrimination?
A: The segment with more inelastic demand.
Q: |Ed| = 3 and MC = $20. What price should the firm charge?
A: $30. P = 20 / (1 − 1/3) = 20 / (2/3) = 30.
Q: At a price of $5 and |Ed| = 3, what is the marginal cost?
A: $3.33. MC = P × (1 − 1/|Ed|) = 5 × (2/3) = 3.33.
Q: If |Ed| = 0.5, what should a profit-maximising firm do?
A: Raise the price. The firm is operating in the inelastic portion of demand. Raising price will increase revenue and reduce costs simultaneously, increasing profit.
Q: Point elasticity is −1.5, P = $20, Q = 10. If price drops to $17.50, by what percentage does quantity increase?
A: 18.75%. The price change is −12.5%, so %ΔQ = −1.5 × (−12.5%) = +18.75%.
Q: What are the two conditions required for effective price discrimination?
A: (1) The firm must be able to identify segments with different elasticities. (2) Resale between segments must be prevented (no arbitrage).
Q: In Q = 1,000 − 240P + 80Pc, what happens to the demand curve if the competitor's price (Pc) rises?
A: The demand curve shifts outward (rightward). Higher competitor price drives more customers to this firm. The positive coefficient on Pc confirms the two products are substitutes.
Q: Which correctly defines second-degree price discrimination?
A: The seller offers different prices for varying amounts purchased, and customers choose the price and quantity combination that best suits them.
Q: For a parking garage with short-term and long-term segments, how should the owner set prices to maximise revenue?
A: Set prices so that marginal revenues from the two segments are equal.
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