Source: Chapter 11 Q&A, Intro to Microeconomics, University of Texas at Austin
This document walks through the numerical side of price discrimination: how to build a marginal revenue table from reservation prices, find the profit-maximising quantity and price under single pricing, calculate economic profit under perfect price discrimination, and work out profits when the firm splits buyers into submarkets using coupons, rebates, or group IDs. These are the calculation types that appear on the exam.
Difficulty: Intermediate | Prerequisites: Part 1 study notes (concepts and types of price discrimination), basic algebra, comfort with marginal revenue and marginal cost.
The conceptual understanding of price discrimination (Part 1) is only half the exam. The other half is calculation: given a table of reservation prices and a cost figure, you need to determine quantity, price, and economic profit under different pricing scenarios. This document covers every calculation type from the Chapter 11 Q&A, with full worked examples. If you can reproduce these from scratch, you are well prepared for the quantitative questions.
Economic profit
Total revenue minus total economic cost (including opportunity costs). Economic profit = (Price x Quantity) - (Opportunity cost x Quantity) for a single-price seller. In simple terms, it is what is left after you account for the value of everything you gave up.
Opportunity cost
The value of the next-best alternative you forgo. In the tutoring example, Charles's opportunity cost is $20 per hour, the value of what he could be doing instead.
Marginal revenue (in a reservation-price table)
The additional revenue gained by serving one more buyer. Because a single-price seller must lower the price for all buyers to attract the next one, MR is not simply the next buyer's reservation price. MR = (new total revenue) - (old total revenue). Think of it as the net gain from adding one more customer after accounting for the price cut on everyone else.
Total revenue (TR)
Price times quantity when the same price is charged to all. When each buyer is charged a different price, TR is the sum of all individual prices charged.
Submarket
A segment of buyers the firm treats as a separate market with its own demand curve and MR schedule. Under third-degree discrimination, the firm sets MR = MC in each submarket independently.
Setup: Charles tutors students at an opportunity cost of $20/hour. Nine students have reservation prices: A=$50, B=$46, C=$42, D=$38, E=$34, F=$30, G=$26, H=$22, I=$18. He must charge the same price to everyone.
To sell to N students, you set price equal to student N's reservation price (the lowest in the group). Total revenue = price x N. Marginal revenue = change in total revenue from adding the Nth student.
Student | Reservation Price | Total Revenue | Marginal Revenue |
|---|---|---|---|
A | $50 | $50 | $50 |
B | $46 | $92 | $42 |
C | $42 | $126 | $34 |
D | $38 | $152 | $26 |
E | $34 | $170 | $18 |
F | $30 | $180 | $10 |
G | $26 | $182 | $2 |
H | $22 | $176 | -$6 |
I | $18 | $162 | -$14 |
Notice that MR falls faster than price. By student G, MR is only $2. By student H, MR is negative: the price cut needed to include H costs more on the existing students than H's reservation price adds.
Charles's MC (opportunity cost) is $20. He should keep adding students as long as MR ≥ MC.
Student D: MR = $26 > $20. Keep tutoring.
Student E: MR = $18 < $20. Stop.
So Charles tutors 4 students (A through D) and charges $38 (student D's reservation price).
Economic profit = Total revenue - Total cost = (4 x $38) - (4 x $20) = $152 - $80 = $72.
The socially efficient quantity is where every student whose reservation price exceeds the opportunity cost ($20) gets tutored. That means students A through H (8 students), since H's reservation price ($22) > $20, but I's ($18) < $20.
If Charles must charge a single price to serve all 8, he must set the price at $22 (student H's reservation price, the lowest in the group).
Economic profit at the socially efficient quantity = (8 x $22) - (8 x $20) = $176 - $160 = $16.
The key insight: the profit-maximising single price ($38, profit = $72) serves fewer students than the socially efficient level (8 students, profit = $16). The monopolist restricts output to keep the price high. This is the standard monopoly inefficiency, and it is exactly the gap that price discrimination can close.
If Charles knows each student's reservation price and can charge each one individually, he tutors every student whose reservation price exceeds his opportunity cost: students A through H (8 students).
He charges each student their own reservation price. Total revenue = $50 + $46 + $42 + $38 + $34 + $30 + $26 + $22 = $288.
Total cost = 8 x $20 = $160.
Economic profit = $288 - $160 = $128.
Compare across scenarios:
Pricing Strategy | Students Tutored | Price(s) | Economic Profit |
|---|---|---|---|
Single price (profit max) | 4 | $38 | $72 |
Single price (socially efficient) | 8 | $22 | $16 |
Perfect price discrimination | 8 | $22 to $50 | $128 |
Perfect price discrimination serves the same number of students as the socially efficient outcome but captures far more surplus, because the firm is not forced to cut the price for everyone to serve additional buyers.
Charles offers a rebate coupon. Only students with reservation prices of $34 or less bother to fill out the survey to claim it. This splits the market into two submarkets:
No-coupon submarket: Students A through D (reservation prices $50, $46, $42, $38). These students have higher willingness to pay and will not bother with the survey.
Coupon submarket: Students E through I (reservation prices $34, $30, $26, $22, $18). These are the price-sensitive students.
Treat each submarket as its own mini-monopoly. Build a separate MR table for each and apply MR ≥ MC ($20).
No-coupon submarket (students A to D):
Student | Reservation Price | TR | MR |
|---|---|---|---|
A | $50 | $50 | $50 |
B | $46 | $92 | $42 |
C | $42 | $126 | $34 |
D | $38 | $152 | $26 |
MR at D = $26 > $20. Serve all four. Price = $38.
Coupon submarket (students E to I):
Student | Reservation Price | TR | MR |
|---|---|---|---|
E | $34 | $34 | $34 |
F | $30 | $60 | $26 |
G | $26 | $78 | $18 |
MR at F = $26 > $20. MR at G = $18 < $20. Serve E and F. Price = $30.
Total profit = (4 x $38) + (2 x $30) - (6 x $20) = $152 + $60 - $120 = $92.
This is $20 more than the single-price profit of $72. The coupon let Charles capture two extra students (E and F) at a lower price without reducing the price charged to A through D.
Now the coupon submarket contains only F through I. The no-coupon market is A through E.
No-coupon submarket: same analysis as before yields 4 students at $38 (student E has MR = $18 in that group, below $20, so E is excluded even though E is now in this submarket). Coupon submarket: F and G are served at $26 (MR at G = $22 > $20; MR at H = $14 < $20). Profit = (4 x $38) + (2 x $26) - (6 x $20) = $152 + $52 - $120 = $84.
Seven customers want bungee jumping. MC = $0. Some have military IDs and some do not.
Customer | Max Willingness to Pay | Military ID |
|---|---|---|
Daisy | $200 | No |
Tulip | $120 | No |
Rose | $82 | Yes |
Dahlia | $65 | Yes |
Begonia | $55 | Yes |
Camelia | $40 | No |
Poppy | $35 | Yes |
With MC = $0, the firm wants to maximise revenue. Build the MR table for all seven customers ranked by willingness to pay and find the price that maximises total revenue. The optimal single price is $55 (serving Daisy, Tulip, Rose, Dahlia, Begonia = 5 customers). TR = 5 x $55 = $275.
Non-military submarket (Daisy, Tulip, Camelia): Best price = $120 (serve Daisy and Tulip). TR = 2 x $120 = $240.
Military submarket (Rose, Dahlia, Begonia, Poppy): Best price = $55 (serve Rose, Dahlia, Begonia). TR = 3 x $55 = $165.
Total revenue with two prices = $240 + $165 = $405.
$405 - $275 = $130 additional profit.
The answer is (e) $130.
"Marginal revenue equals the next student's reservation price." This is only true under perfect price discrimination. Under single pricing, MR accounts for the fact that the price must fall for all existing customers too. MR = new TR - old TR, not just the new buyer's reservation price.
"Serve every student whose reservation price exceeds MC." That is the rule for the socially efficient quantity, not the profit-maximising quantity under single pricing. The profit-maximising rule is MR ≥ MC.
"In the submarket analysis, use the whole original MR table." Each submarket gets its own MR table. The coupon submarket starts fresh with its own demand curve; you do not carry over the total revenue column from the combined market.
"Additional profit from discrimination = two-price profit - zero." The question asks for additional profit compared to the best single-price profit, not compared to zero. Always compute both scenarios and subtract.
⚠️ You will almost certainly be asked to build a marginal revenue table from reservation prices. Practise this until it is automatic.
⚠️ Know the difference between the profit-maximising quantity (MR = MC) and the socially efficient quantity (P = MC on the demand curve). The exam tests whether you can calculate both and explain why they differ.
⚠️ Perfect price discrimination profit calculations are straightforward (sum all reservation prices above MC, subtract total cost) but easy to rush. Double-check your addition.
⚠️ Submarket problems (coupon/rebate scenarios) require you to build a separate MR table for each group. The most common error is forgetting to restart the TR column for the second submarket.
⚠️ The bungee jumping style question ("how much additional profit") requires you to compute both the single-price optimum and the two-price optimum, then subtract. Read the question carefully: it asks for the difference, not the total.
Q: A monopolist faces five buyers with reservation prices of $100, $80, $60, $40, $20. MC = $15. If the monopolist charges a single price, what price maximises profit and what is that profit?
A: Build the MR table. At P=$60, Q=3, TR=$180, MR of 3rd unit = $180-$160=$20 > $15. At P=$40, Q=4, TR=$160, MR = $160-$180 = -$20 < $15. So charge $60, serve 3 buyers, profit = 3x$60 - 3x$15 = $180 - $45 = $135.
Q: Using the same five buyers, what is economic profit under perfect price discrimination?
A: Serve all buyers whose reservation price exceeds $15: all five. Profit = ($100+$80+$60+$40+$20) - (5x$15) = $300 - $75 = $225.
Q: Charles tutors students A to I (reservation prices $50 down to $18, in steps of $4). Opportunity cost = $20. He splits the market using a coupon: students with reservation prices $34 or below redeem it. What is his total profit?
A: No-coupon submarket (A to D): price = $38, serve 4. Coupon submarket (E to I): build a fresh MR table. At price $30, serve E and F, MR of F = $26 > $20, MR of G = $18 < $20. Total profit = (4x$38) + (2x$30) - (6x$20) = $152 + $60 - $120 = $92.
Q: In the bungee jumping problem, if the firm cannot discriminate and MC = $0, what is the profit-maximising single price?
A: Test each price point. At $55, serve 5 customers (Daisy, Tulip, Rose, Dahlia, Begonia), TR = $275. At $40, serve 6, TR = $240. At $65, serve 4, TR = $260. $275 is the maximum, so charge $55.
Q: A single-price monopolist produces where MR = MC at Q = 4 and charges $10. If it switches to perfect price discrimination, what happens to output, consumer surplus, and deadweight loss?
A: Output rises to the socially efficient level (where demand = MC). Consumer surplus falls to zero. Deadweight loss falls to zero. The entire area between the demand curve and MC becomes producer surplus.
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