Difficulty: Intermediate | Prerequisites: Chapter 11, Section 12.1–12.2 notes on conditions and perfect price discrimination
When a firm cannot identify every individual's willingness to pay, it can still increase profit by sorting customers into identifiable groups (students vs. adults, domestic vs. foreign) and charging each group a different price. The group with more elastic demand pays less. The optimal rule is to set marginal revenue equal across all groups and equal to marginal cost: MR1 = MC = MR2.
Group price discrimination (third-degree price discrimination)
The firm charges different groups of customers different prices but charges a given customer the same price for every unit sold. This is the most common form of price discrimination in practice.
Elasticity of demand (e)
The percentage change in quantity demanded divided by the percentage change in price. In the context of group pricing, the group with demand closer to perfectly elastic (larger |e|) pays the lower price.
Marginal revenue equality condition
The profit-maximising rule for a firm selling to multiple groups: set MR1 = MC = MR2. Each group's marginal revenue must equal the common marginal cost.
Price-elasticity pricing rule
The ratio of prices across groups depends solely on the elasticities at the optimum: p2/p1 = (1 + 1/e1) / (1 + 1/e2). The group with less elastic demand (|e| closer to 1) is charged more.
Switchers
Customers who buy whichever brand is cheapest at the moment. In the Heinz ketchup example, switchers buy Heinz during sales and generic brands otherwise.
Self-selection
A method of identifying groups by designing pricing schemes that let consumers sort themselves. Examples include coupons (price-sensitive consumers clip them), advance-purchase airline fares (leisure travellers book early), and rebates (only price-sensitive buyers bother to apply).
The firm divides customers into two or more groups with different demand curves. It charges each group a separate price but a uniform price within each group. As with all price discrimination, the firm needs market power, the ability to identify groups, and the ability to prevent resale.
Suppose a monopoly sells to two groups. It sells Q1 to group 1 (revenue R1) and Q2 to group 2 (revenue R2). Total cost is C(Q1 + Q2). The firm maximises:
profit = R1(Q1) + R2(Q2) - C(Q1 + Q2)
The first-order conditions require that the marginal revenue from each group equals the marginal cost of total output:
MR1 = MC
MR2 = MC
Therefore MR1 = MC = MR2
With a constant marginal cost, the firm can treat each group as an independent single-price monopoly problem.
Since MR = p(1 + 1/e), the condition MR1 = MR2 implies:
p1(1 + 1/e1) = p2(1 + 1/e2)
Rearranging gives the price ratio: p2/p1 = (1 + 1/e1) / (1 + 1/e2).
The group with less elastic demand (|e| closer to 1) pays the higher price. This is intuitive: if a group is not very responsive to price, the firm can charge them more without losing many sales.
Observable characteristics. The firm divides buyers based on traits it can see: age (movie theatre senior discounts), location (Tesla charging more in Europe than the US), student status (WSJ student subscriptions).
Self-selection through actions. The firm designs mechanisms that let price-sensitive consumers reveal themselves. Coupons (only price-sensitive people clip them), advance-purchase airline fares (leisure travellers plan ahead, business travellers do not), rebates (only price-sensitive buyers bother with the paperwork), and reverse auctions (Priceline bidders accept inconvenience for a lower price) all serve this purpose.
Group discrimination vs. competition. Consumer surplus is lower and deadweight loss is positive under group discrimination. The firm charges above marginal cost in both markets, so output is below the competitive level in each.
Group discrimination vs. single-price monopoly. Theory alone cannot tell you which is better for welfare. Both charge above MC. Output may rise if groups that did not buy at the single price start buying under discrimination. But group discrimination also introduces consumption inefficiency (different consumers pay different prices for the same good, so they value the last unit differently) and search costs (consumers may waste time looking for lower prices).
The closer the firm comes to perfect price discrimination (by dividing into many small groups), the less deadweight loss and the higher the output.
\max_{Q_1, Q_2} \pi = R_1(Q_1) + R_2(Q_2) - C(Q_1 + Q_2)\frac{\partial \pi}{\partial Q_1} = \frac{dR_1}{dQ_1} - \frac{dC}{dQ} = 0 \quad \Rightarrow \quad MR^1 = MC\frac{\partial \pi}{\partial Q_2} = \frac{dR_2}{dQ_2} - \frac{dC}{dQ} = 0 \quad \Rightarrow \quad MR^2 = MCCombined: MR1 = MC = MR2 (Equation 12.7).
\frac{p_2}{p_1} = \frac{1 + 1/\varepsilon_1}{1 + 1/\varepsilon_2}The group with less elastic demand (|e| closer to 1) pays the higher price.
Given: p1 = 60 - 5Q1 (seniors), p2 = 90 - 10Q2 (adults), MC = 10.
With price discrimination:
Seniors: MR1 = 60 - 10Q1 = 10, so Q1 = 5, p1 = 35.
Adults: MR2 = 90 - 20Q2 = 10, so Q2 = 4, p2 = 50.
Without price discrimination (single price):
Total demand has a kink at p = 60 (seniors only buy below 60).
Below p = 60, the total inverse demand is p = 70 - (1/0.3)Q.
MR = 70 - (2/0.3)Q = 10, so Q = 9, p = 40.
The single price (40) lies between the two group prices (35 < 40 < 50).
Given: p1 = 100 - Q1, p2 = 100 - 2Q2, MC = 20.
Country 1: MR1 = 100 - 2Q1 = 20, so Q1 = 40, p1 = 60.
Country 2: MR2 = 100 - 4Q2 = 20, so Q2 = 20, p2 = 60.
Same price in both countries. Why? For a linear inverse demand p = a - bQ, the profit-maximising price is p = (a + m)/2. It depends on the intercept a, not the slope b. Since both markets share a = 100, the price is the same. The elasticities at the optimal price are equal, so discrimination provides no benefit.
Tesla Model S. Tesla charges about $80,000 in the US and $130,000 in Europe, with MC around $30,000. Using the elasticity formula: eA = 80/(30 - 80) = -1.6, eE = 130/(30 - 130) = -1.3. European demand is less elastic, so Europeans pay 62.5% more. Resale is prevented because Tesla honours warranties only in the region of purchase.
Greyhound bus lines. Greyhound offers senior discounts because seniors have lower willingness to pay for long-distance travel. The single price, if forced to charge one, falls between the two group prices.
Tinder age-based pricing. Tinder Plus charged $19.99/month for users over 30 and $9.99-$14.99 for users under 30. A California court ruled this was unlawful age discrimination under state law. This illustrates that while group price discrimination is generally legal for firms, discrimination based on certain protected characteristics can attract legal challenges.
Airline tickets. Airlines separate business travellers (inelastic demand, last-minute booking, no restrictions) from leisure travellers (elastic demand, advance purchase, many restrictions). The average price difference between high and low fares on the same US route is 36% of the average ticket price.
Coupons. In 2018, firms distributed 143 billion coupons; only about 0.6% were redeemed. People who clip coupons reveal themselves as price-sensitive. The firm effectively charges clippers less and non-clippers more.
Rebates. A rebate of $5 is not the same as a $5 price cut, because claiming the rebate requires effort. Only price-sensitive consumers bother. According to Consumer Reports, 25% of customers never apply for rebates.
"The group charged a lower price always benefits from price discrimination." Compared to competition, both groups pay more than marginal cost. The lower-priced group may still be worse off than under competition. Group discrimination transfers surplus from all consumers to the firm.
"Group price discrimination always reduces welfare compared to a single-price monopoly." It can go either way. If discrimination opens up a market to a group that would not buy at the single price, total output rises and welfare may increase.
"Different prices always mean the firm is price discriminating." The demand intercept matters more than the slope. Solved Problem 12.3 shows two markets with different demand curves where the firm charges the same price because the vertical intercepts are equal.
"A firm should always charge a higher price in the bigger market." Market size alone does not determine price. What matters is elasticity. The firm charges more to the group with less elastic demand, regardless of market size.
⚠️ The condition MR1 = MC = MR2 is the single most important equation in this section. You will be asked to apply it numerically.
⚠️ Be able to derive and use the price-ratio formula p2/p1 = (1 + 1/e1) / (1 + 1/e2). An exam may give you prices and ask you to infer elasticities, or vice versa.
⚠️ Know how to solve a group-pricing problem when the total demand curve has a kink (as in the Greyhound problem). The MR curve is discontinuous at the kink.
⚠️ Understand why a firm might not price discriminate even when it can (Solved Problem 12.3). This is a common conceptual exam question.
⚠️ Be prepared to compare welfare under group discrimination, single-price monopoly, and competition. The key insight: group discrimination vs. single-price monopoly is ambiguous in theory.
True or false: A group-discriminating monopoly charges the group with more elastic demand a higher price. (False. It charges the more elastic group less.)
Fill in the blank: The profit-maximising condition for a firm selling to two groups is MR1 = ____ = MR2. (MC.)
True or false: Group price discrimination always increases total welfare compared to a single-price monopoly. (False. The welfare comparison is ambiguous in theory.)
Fill in the blank: For linear demand p = a - bQ, the monopoly price depends on the intercept ____ and marginal cost, not on the slope. (a.)
True or false: Coupons are a method of self-selection that allows firms to group price discriminate. (True.)
Q: A monopoly sells in two markets with constant MC = 15. In market 1, inverse demand is p1 = 80 - 2Q1. In market 2, inverse demand is p2 = 60 - Q2. Find the profit-maximising price and quantity in each market.
A: Market 1: MR1 = 80 - 4Q1 = 15, so Q1 = 16.25, p1 = 80 - 2(16.25) = 47.50. Market 2: MR2 = 60 - 2Q2 = 15, so Q2 = 22.5, p2 = 60 - 22.5 = 37.50. The firm charges more in market 1 because demand there is less elastic at the optimum.
Q: Using the Tesla example, if pA = $80k, pE = $130k, and MC = $30k, what is the elasticity of demand in each market?
A: eA = pA/(MC - pA) = 80/(30 - 80) = -1.6. eE = pE/(MC - pE) = 130/(30 - 130) = -1.3. European demand is less elastic (closer to -1), so Tesla charges Europeans more.
Q: Why does the Greyhound MR curve have a discontinuity in the single-price case?
A: The total demand curve has a kink at p = 60 (the price at which seniors begin buying). A kink in the demand curve produces a vertical gap (discontinuity) in the MR curve at the corresponding quantity.
Q: A firm faces two markets with demand p1 = 100 - Q1 and p2 = 100 - 3Q2 and MC = 20. Does it price discriminate?
A: No. Both demand functions share the same intercept (a = 100). Since the monopoly price is (a + m)/2 = (100 + 20)/2 = 60 for both markets, the firm charges the same price. The equilibrium elasticities happen to be equal.
Q: Explain how rebates serve as a self-selection mechanism for group price discrimination.
A: Claiming a rebate requires time and effort. Only price-sensitive consumers bother to apply. The firm effectively charges two prices: the shelf price (for price-insensitive buyers) and the shelf price minus the rebate (for price-sensitive buyers who self-select by completing the rebate process).
Group price discrimination builds on the marginal revenue = marginal cost rule from Chapter 11's monopoly analysis. The Lerner Index, p(1 + 1/e) = MC, is exactly the condition used to derive the price-ratio formula here.
This section connects forward to nonlinear price discrimination (Section 12.4), which covers cases where the firm cannot identify groups but can vary price by quantity. Two-part pricing (Section 12.5) is another approach when group identification is limited.
The welfare analysis here parallels the deadweight-loss discussion from Chapter 8 (competitive markets) and Chapter 11 (monopoly). The new wrinkle is that welfare comparisons between group discrimination and single-price monopoly are ambiguous.
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