How Menus Let Buyers Sort Themselves
A seller may know that buyers differ in how much they value a product without being able to identify each buyer’s willingness to pay. Instead of charging an individually tailored price, the seller can offer a set of quantity-price options. Buyers choose for themselves, and their choices can sort them into groups. This is second-degree price discrimination: the seller uses the quantity purchased or option selected as an indirect signal of willingness to pay.
The method depends on the options being designed so that buyers with different valuations prefer different offers. A lower total price for a larger purchase can encourage higher-value buyers to select a larger bundle, while lower-value buyers choose a smaller one. The seller’s problem is not simply to offer a discount; it is to choose prices that make the intended choices worthwhile to buyers and profitable to the seller.
For a buyer of type \(\theta\), let \(V_\theta(q)\) be the buyer’s total willingness to pay for \(q\) units, with \(V_\theta(0)=0\). If the total price for \(q\) units is \(T(q)\), the buyer’s net surplus is \(V_\theta(q)-T(q)\). The buyer selects the option with the greatest nonnegative surplus, including the option of buying nothing. If producing \(q\) units costs the seller \(C(q)\), the seller’s profit from that option is \(T(q)-C(q)\).
A quantity discount concerns the relationship between quantities and prices, not necessarily a lower total bill. For example, a two-unit block can cost more than a one-unit block while still having a lower average price per unit. It is also useful to distinguish an average price from the price of an extra unit: a buyer comparing a one-unit option with a two-unit option compares the difference in total prices with the value of the additional unit.
Worked Examples of Quantity-Based Pricing
Worked Example: Choosing Among Quantity-Discount Options
A hypothetical seller offers a product in quantities of one, two, or three units. The total prices are $9, $16, and $21, respectively. A buyer values the first, second, and third units at $12, $8, and $4. The seller’s constant marginal cost is $3 per unit. Which quantity does the buyer choose, and what is the seller’s profit from that sale?
Method. Calculate the buyer’s total value and net surplus for each available quantity, then calculate the seller’s profit at the chosen quantity.
For zero units, total value and price are both $0, so net surplus is $0.
For one unit, total value is $12. Net surplus is \(12-9=3\) dollars.
For two units, total value is \(12+8=20\). Net surplus is \(20-16=4\) dollars.
For three units, total value is \(12+8+4=24\). Net surplus is \(24-21=3\) dollars.
The buyer chooses two units because $4 is the greatest net surplus. The marginal prices implicit in the menu are $9 for the first unit, \(16-9=7\) dollars for the second, and \(21-16=5\) dollars for the third. Thus, the marginal prices decline, while the total price rises with quantity.
At two units, the seller receives $16 and incurs cost \(2(3)=6\) dollars. Profit is \(16-6=10\) dollars. Check: the buyer’s chosen bundle gives net surplus $4, and the seller’s profit on that bundle is $10. The mistake to avoid is treating the $16 total price as the price of the second unit; the second unit adds $7 to the bill.
Incentive Compatibility and Self-Selection
A menu sorts buyers only if each type prefers the option intended for it. Consider two types, low and high, and two quantities, \(q_L\) and \(q_H\), with \(q_H>q_L\). Let their total prices be \(T_L\) and \(T_H\). For type \(\theta\), the utility from an option is its total value minus its total price. The menu is incentive-compatible when the low type prefers the low-type option and the high type prefers the high-type option.
Rearranging the first inequality gives a lower bound on the price increase; rearranging the second gives an upper bound:
This is the two-type incentive-compatibility condition. It says the extra price for the larger option must be at least the low type’s extra willingness to pay for the additional quantity, or the low type would upgrade. It must also be no more than the high type’s extra willingness to pay, or the high type would choose the smaller option instead. A feasible separating menu therefore requires the high type’s incremental value to be at least as large as the low type’s.
Incentive compatibility alone does not guarantee that buyers participate. If buying nothing gives utility zero, each type must also receive nonnegative utility from its assigned option:
These are the participation conditions. A menu that satisfies both the self-selection comparisons and participation conditions can induce the desired choices, subject to the stated model assumptions.
Worked Example: Testing a Two-Option Menu
A hypothetical equipment supplier offers one unit for $8 or two units for $14. A low-valuation buyer values the first and second units at $10 and $4. A high-valuation buyer values them at $14 and $10. Determine whether each type selects the intended option if the low type is intended to buy one unit and the high type two units.
Method. Calculate each type’s net surplus from both offers and from buying nothing, then check the incremental-price condition.
The low type’s value for one unit is $10, so surplus from the one-unit option is \(10-8=2\) dollars. Its value for two units is \(10+4=14\), so surplus from the two-unit option is \(14-14=0\) dollars. Buying nothing gives $0. The low type therefore strictly prefers one unit.
The high type’s value for one unit is $14, giving surplus \(14-8=6\) dollars. Its value for two units is \(14+10=24\), giving surplus \(24-14=10\) dollars. It prefers two units and participates.
The price increase from the one-unit to the two-unit offer is \(14-8=6\) dollars. The low type values the extra unit at $4, while the high type values it at $10. The condition \(4\leq 6\leq 10\) holds, consistent with the choices just calculated.
The common mistake is to compare the $14 total price of the larger option with the low type’s $4 value for the second unit. That compares a total price with an incremental value. The correct comparison is the price increase, $6, against the value of the additional unit. Check: the low type gets $2 surplus from one unit, and the high type gets $10 from two.
Why the Menu Can Increase Profit
A seller who must charge one price per unit may have to accept a compromise: a price attractive to lower-value buyers can leave money uncollected from higher-value buyers, while a price aimed at higher-value buyers can exclude lower-value buyers. A menu can address this by charging different total amounts for different quantities. The options need not reveal a buyer’s type perfectly; they need only make the intended choices preferable.
The following comparison shows how a menu can outperform a single per-unit price. Buyers’ values and seller costs are hypothetical. The seller has one buyer of each type, and the production cost is $2 per unit.
Worked Example: Comparing a Menu with One Per-Unit Price
A low-type buyer values the first and second units at $10 and $2. A high-type buyer values them at $16 and $12. The seller offers one unit for $9.99 and two units for $21.98. Determine the choices and total profit. Then find the greatest profit available from any single per-unit price.
Method. First compare each type’s net surplus across the menu. Then divide possible uniform prices into ranges according to which marginal values they exceed.
For the low type, buying one unit gives surplus \(10-9.99=0.01\) dollars. The value of two units is \(10+2=12\), so buying two gives surplus \(12-21.98=-9.98\) dollars. The low type buys one unit.
For the high type, buying one unit gives surplus \(16-9.99=6.01\) dollars. The value of two units is \(16+12=28\), giving surplus \(28-21.98=6.02\) dollars. The high type buys two units. Both types participate.
The seller earns \(9.99-2=7.99\) dollars from the low type. From the high type, the cost is \(2(2)=4\) dollars, so profit is \(21.98-4=17.98\) dollars. Total profit is \(7.99+17.98=25.97\) dollars.
Now let \(p\) be a single price per unit. If \(p\leq 2\), both buyers purchase two units; total profit is \(4p-8\), which is at most $0 in this range. If \(2<p\leq 10\), the low type buys one unit and the high type buys two; profit is \(3p-6\), maximized at \(p=10\), giving $24. If \(10<p\leq 12\), only the high type buys two; profit is \(2p-4\), at most $20. If \(12<p\leq 16\), only the high type buys one; profit is \(p-2\), at most $14. Above $16, neither type buys. Thus the largest uniform-price profit is $24, below the menu profit of $25.97.
At the boundary prices, a buyer may be indifferent about an additional unit; the calculations use the quantity choices implied by the stated marginal values and the usual convention that a buyer purchases a unit when its value is at least its price. Check: the menu sells three units in total, costing $6, and receives $31.97, so total profit is \(31.97-6=25.97\) dollars.
Assumptions, Uses, and Common Traps
The basic menu model assumes buyers understand the offers, can choose among them, and have valuations that make the stated choices meaningful. It also assumes the seller can enforce the selected quantity-price option and that buyers cannot readily resell units to one another. If buyers can resell, a low-price option intended for one group may be purchased on behalf of another group, weakening the seller’s ability to sort buyers.
The seller must also account for costs. A larger bundle can raise revenue but also increase production cost. The relevant profit for an option is its total price less the cost of supplying its quantity, not its price less the cost of one unit. A menu that sorts buyers successfully can still be unprofitable if its prices do not cover those costs.
- Using total price instead of the price difference. To test whether a buyer upgrades from one quantity to another, compare the difference in total prices with the value of the additional units.
- Confusing average and marginal prices. A lower average price for a larger block does not by itself determine whether a buyer selects it; compare net surplus for every option.
- Forgetting the outside option. A buyer will not accept an offer that gives negative surplus when buying nothing yields zero.
- Checking only one type’s choice. A menu intended to separate two types must satisfy both types’ incentive constraints.
- Ignoring cost on additional units. Seller profit depends on the full cost of the quantity supplied.
Menus are useful when buyers differ in willingness to pay and the seller cannot directly condition prices on each buyer’s type. Their central design test is whether the difference in total prices falls between the types’ incremental valuations, while the assigned options also satisfy participation and profit requirements. These principles are aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
Check Your Understanding
For each problem, compare the available choices before checking the answer.
- A buyer values three successive units at $11, $5, and $2. The seller charges $9 for one unit and $13 for two, with buying nothing available. Which option does the buyer choose, and what is the buyer’s net surplus? (Answer: Two units; net surplus is $3.)
- A low type values the first and second units at $10 and $3; a high type values them at $15 and $8. The menu charges $7 for one unit and $13 for two. Do the types self-select as intended, with the low type choosing one unit and the high type two? (Answer: Yes. The price increment is $6, between the low type’s $3 and the high type’s $8 incremental values; the low type gets $3 surplus from one unit, and the high type gets $10 from two.)
- A low type values one and two units at $12 and $16 total; a high type values them at $17 and $28 total. A menu charges $9.50 for one unit and $18.50 for two. Which option does each type choose? (Answer: The low type chooses one unit, with $2.50 surplus; the high type chooses two, with $9.50 surplus.)
- A seller has three low-type buyers and two high-type buyers. Each unit costs $4 to produce. A menu charges $9.50 for one unit and $18.50 for two. Using the choices in Question 3, calculate total profit. (Answer: Low-type profit is \(3(9.50-4)=\$16.50\); high-type profit is \(2(18.50-8)=\$21.00\); total profit is $37.50.)
- One low-type buyer values successive units at $10 and $2; one high-type buyer values them at $16 and $12. Production costs $2 per unit. A uniform price \(p\) is charged per unit. What is the maximum profit over uniform prices, and at what price is it attained? (Answer: Maximum profit is $24 at \(p=\$10\); the low type buys one unit and the high type buys two.)