Difficulty: Intermediate | Prerequisites: Sections 12.1–12.3 notes on price discrimination conditions, perfect and group price discrimination
When a firm cannot sort customers into groups, it can still price discriminate by varying the price with quantity purchased (nonlinear pricing). Block pricing charges different per-unit prices for successive quantity ranges. Two-part pricing charges a lump-sum access fee plus a per-unit price, so that heavy buyers pay less per unit on average. Both methods let the firm extract more surplus than a single price, and the more price tiers the firm sets, the closer it gets to perfect price discrimination.
Nonlinear price discrimination (second-degree price discrimination)
The price varies with the quantity purchased, but all customers who buy the same quantity pay the same price. The consumer's total expenditure does not rise linearly with quantity. Think of it as "buy more, pay less per unit" pricing.
Block pricing
Charging one price for the first "block" of units and a different (usually lower) price for subsequent blocks. Utilities commonly use this. A declining-block schedule charges less per unit as consumption increases; an increasing-block schedule charges more.
Quantity discount
A lower per-unit price for larger purchases. The 48-pack of batteries costs 40 cents each while the 24-pack costs 51 cents each. This is nonlinear pricing unless the price difference reflects a cost difference.
Two-part pricing (two-part tariff)
The firm charges a lump-sum access fee for the right to buy, plus a per-unit price for each unit purchased. The average price per unit falls as the customer buys more, because the fixed access fee is spread over more units.
Access fee
The lump-sum payment a consumer must make before purchasing any units. Examples: gym membership fee, Costco annual membership, Dallas Cowboys personal seat licence.
Personal seat licence (PSL)
A one-time fee giving a fan the right to buy season tickets for a set period. The Dallas Cowboys PSLs sell for $10,000 to $125,000.
The firm cannot identify which customers have high or low reservation prices, but it knows that the typical customer's demand curve slopes downward (they value the first unit more than the second). The firm lets price vary with quantity: everyone faces the same pricing schedule, but how much each person pays per unit depends on how much they buy.
Requirements: market power, and the ability to prevent customers who buy at a low per-unit price from reselling to those who would otherwise pay more.
Setup: each identical customer has demand p = 90 - Q, and the firm has constant MC = m = 30.
The firm sets two blocks:
Block 1: a higher price p1 on the first Q1 units.
Block 2: a lower price p2 on units from Q1 to Q2.
Both prices lie on the demand curve: p1 = 90 - Q1 and p2 = 90 - Q2.
The firm chooses Q1 and Q2 to maximise profit. The solution gives Q1 = 20, Q2 = 40, p1 = 70, p2 = 50.
Block Pricing | Single-Price Monopoly | |
|---|---|---|
Consumer Surplus | $400 (A + C) | $450 (E) |
Producer Surplus / Profit | $1,200 (B) | $900 (F) |
Total Welfare | $1,600 | $1,350 |
Deadweight Loss | $200 (D) | $450 (G) |
Key points:
The firm is better off with block pricing ($1,200 vs. $900 profit).
Society is better off (DWL falls from $450 to $200).
But consumers as a group are worse off ($400 vs. $450 CS).
The lower per-unit price on the second block lets the firm sell more units (40 vs. 30), but the higher first-block price captures surplus that would otherwise go to consumers.
The more blocks the firm uses, the closer it approaches perfect price discrimination, eventually capturing all consumer surplus and eliminating all deadweight loss.
The firm charges a lump-sum access fee (f) plus a per-unit price (p). The average cost per unit to the consumer is p + f/Q, which falls as Q rises.
With identical consumers: the optimal strategy is to set the per-unit price equal to marginal cost (p = MC) and then set the access fee equal to the entire consumer surplus at that price. This extracts all surplus, replicating perfect price discrimination. Output is at the competitive level, so there is no deadweight loss.
With different consumers: the firm faces a trade-off. Setting a very high access fee drives away low-demand consumers. Setting a low access fee leaves money on the table from high-demand consumers. The firm must balance these effects. The per-unit price may be set above MC, and the access fee below the maximum possible.
Requirements for two-part pricing: market power, prevention of resale, and knowledge of how individual demand curves vary across customers.
With inverse demand p = 90 - Q, MC = 30, two blocks:
\pi = p(Q_1) \cdot Q_1 + p(Q_2) \cdot (Q_2 - Q_1) - m \cdot Q_2Expanding: profit = (90 - Q1)Q1 + (90 - Q2)(Q2 - Q1) - 30Q2.
Take partial derivatives and set to zero:
With respect to Q1: Q2 - 2Q1 = 0
With respect to Q2: Q1 - 2Q2 + 60 = 0
Solving simultaneously: Q1 = 20, Q2 = 40.
Block prices: p1 = 90 - 20 = 70, p2 = 90 - 40 = 50.
First block revenue: 70 x 20 = $1,400. Second block revenue: 50 x (40 - 20) = $1,000. Total revenue: $2,400. Total cost: 30 x 40 = $1,200. Profit: $1,200.
Compare to single-price monopoly: MR = 90 - 2Q = 30, so Q = 30, p = 60. Profit = (60 - 30) x 30 = $900.
Set per-unit price p = MC. The access fee f equals the consumer surplus at that price:
f = CS = \frac{1}{2}(a - MC) \cdot Q_cwhere Qc is the competitive quantity (where demand meets MC) and a is the demand intercept.
With p = 90 - Q, MC = 30: Qc = 60, f = (1/2)(90 - 30)(60) = $1,800.
Total payment per consumer: f + MC x Q = 1,800 + 30 x 60 = $3,600.
Profit per consumer: f = $1,800 (all consumer surplus captured).
Battery pricing. In 2018, a 48-pack of Duracell AA batteries cost $18.98 (40 cents each), while a 24-pack cost $12.34 (51 cents each). The per-unit price falls with quantity, which is nonlinear pricing unless the cost difference explains the discount.
Utility block pricing. Gas, electric, and water utilities commonly use block-pricing schedules: one rate for the first block of usage, a different rate for subsequent blocks. Some use declining-block pricing (lower rates for higher usage), while others use increasing-block pricing (higher rates to discourage heavy usage).
Fitness clubs. Many gyms charge a yearly membership fee (access fee) plus a per-session charge. The membership fee is the two-part pricing access component; the per-session charge is the per-unit price.
Warehouse stores (Costco, Sam's Club). Customers pay an annual membership fee to access lower-priced goods. The membership fee functions as the access fee in a two-part pricing scheme.
Dallas Cowboys personal seat licences. Fans must purchase a PSL ($10,000 to $125,000) for the right to buy season tickets for the next 30 years. The PSL is the access fee; the season ticket price is the per-unit charge.
Car rental firms. Some charge a daily rental fee (access) plus a per-mile charge (per-unit price). This is textbook two-part pricing.
"Quantity discounts always benefit consumers." The textbook flags this as a common confusion. Block pricing lets the firm charge a higher first-block price than it would under single pricing. Consumers as a group can end up with less surplus (in the textbook example, CS falls from $450 to $400).
"Two-part pricing is the same as bundling." They are distinct. Two-part pricing has an access fee plus a per-unit price. Bundling sells multiple products together for a single price. Both are forms of nonuniform pricing, but the mechanics differ.
"The access fee should always equal the consumer surplus." This is only optimal when all consumers are identical. With heterogeneous consumers, the firm must balance attracting low-demand buyers (low access fee) against extracting surplus from high-demand buyers (high access fee).
"More blocks always make consumers better off." More blocks push the firm closer to perfect price discrimination, which means less consumer surplus. Welfare (CS + PS) rises, but consumers lose.
⚠️ Be able to solve the two-block pricing problem: set up the profit function, take partial derivatives with respect to Q1 and Q2, solve simultaneously for the optimal block quantities and prices.
⚠️ Know the welfare comparison table for block pricing vs. single-price monopoly. The firm gains, society gains, but consumers lose. This is a common short-answer topic.
⚠️ For two-part pricing with identical consumers, remember: set p = MC, then set the access fee equal to the entire consumer surplus. This replicates perfect price discrimination.
⚠️ Understand why two-part pricing with heterogeneous consumers is more complex, and be able to explain the trade-off in words even if you are not asked to solve it formally.
⚠️ Do not confuse nonlinear pricing (second-degree, same schedule for everyone, price varies by quantity) with group pricing (third-degree, different prices for different groups, same price per unit within a group).
True or false: Under nonlinear price discrimination, different customers who buy the same quantity pay different prices. (False. Everyone who buys the same quantity pays the same price.)
Fill in the blank: In the textbook's block-pricing example, the firm earns profit of ____ with two blocks, compared to ____ with a single price. (1,200; 900.)
True or false: With identical consumers, the optimal two-part pricing strategy sets the per-unit price equal to marginal cost. (True.)
Fill in the blank: As the number of pricing blocks increases, the scheme approaches ____ price discrimination. (Perfect/first-degree.)
True or false: Quantity discounts always increase consumer surplus compared to a single price. (False. In the textbook example, consumer surplus falls from 450 to 400.)
Q: A monopoly faces identical consumers with demand p = 100 - 2Q and MC = 20. If it uses two-part pricing with identical consumers, what per-unit price and access fee should it set?
A: Set p = MC = 20. At p = 20, each consumer buys Q = (100 - 20)/2 = 40 units. Consumer surplus = (1/2)(100 - 20)(40) = 1,600. Set the access fee f = 1,600. The firm captures all surplus.
Q: In the textbook's block-pricing example (p = 90 - Q, MC = 30), explain why consumer surplus is lower with block pricing than with a single price even though total welfare is higher.
A: The first-block price (70) is higher than the single monopoly price (60), extracting more surplus from early units. The second-block price (50) lets the firm sell more units (40 vs. 30), but that extra surplus goes mostly to the firm, not consumers. Total welfare rises because 10 more units are sold (DWL shrinks), but the gains accrue to the producer.
Q: Why is nonlinear price discrimination also called second-degree price discrimination?
A: Because the consumer's total expenditure is a nonlinear (not proportional) function of quantity. With a constant price, spending = p x Q (linear). With quantity-dependent pricing, spending curves away from a straight line.
Q: A gym charges a 500 per year membership plus 10 per session. If a member attends 50 sessions, what is their average price per session?
A: Total payment = 500 + (10 x 50) = 1,000. Average per session = 1,000/50 = 20. The access fee raises the average price above the per-session charge, and this gap narrows as the member attends more sessions.
Q: A utility uses two-block pricing. The first 100 kWh cost 0.15 per kWh, and additional kWh cost 0.08. What does a customer who uses 250 kWh pay in total?
A: First block: 100 x 0.15 = 15. Second block: 150 x 0.08 = 12. Total = 27. Average price per kWh = 27/250 = 0.108.
Nonlinear pricing and two-part pricing are extensions of the price discrimination framework from Sections 12.1-12.3. The key difference is that here the firm does not need to identify groups directly; instead, the pricing schedule itself sorts consumers by quantity.
Two-part pricing with identical consumers is equivalent to perfect price discrimination (Section 12.2): set p = MC, capture all surplus via the access fee.
The next sections in the chapter cover tie-in sales (Section 12.6), which is yet another form of nonuniform pricing, and advertising (Section 12.7), which shifts the demand curve rather than extracting more from the existing one.
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