CHAPTER 11: PRICING WITH MARKET POWER

Complete Solution Manual — IPMX Managerial Economics

QUESTIONS FOR REVIEW

Q1. Perfect First-Degree Price Discrimination: Lowest Price and Total Output

Concept: Perfect (first-degree) price discrimination means charging every consumer exactly their reservation price — the maximum they are willing to pay.

Why this applies: The question asks for the floor price and total output under perfect discrimination.

Key Logic:

• The firm keeps producing as long as any consumer is willing to pay at least the marginal cost (MC).
• There is no single "price" — every unit sold has a different price.
• The firm captures all consumer surplus as profit.

Answer:

• Lowest price charged = Marginal Cost (MC)
• The firm sells to every consumer whose reservation price ≥ MC.
• Total output = the same as a perfectly competitive market (where P = MC).

Economic Intuition: Under perfect discrimination, the firm has no incentive to restrict output. It sells to everyone who values the good above cost. The demand curve itself becomes the firm's marginal revenue curve.

Managerial Implication: Perfect discrimination maximizes profit but eliminates all consumer surplus. It requires perfect information about every buyer's willingness to pay — practically impossible but conceptually useful.

Q2. How Does a Car Salesperson Practice Price Discrimination?

Concept: Third-degree price discrimination — charging different prices based on observable or inferred characteristics.

How it works:

• The salesperson observes signals about buyer's willingness to pay: urgency, emotional attachment to the car, wealth signals (clothes, watch, car they arrived in), financing needs.
• Negotiation allows the salesperson to test reservation prices through back-and-forth offers.
• Each customer pays a different price for the essentially same product.

Sorting Mechanism: Observing buyer characteristics and negotiating skill.

Arbitrage Prevention: Cars are durable and traceable — a buyer cannot easily resell to another person without transaction costs eliminating the margin.

Effect on earnings:

• A salesperson who discriminates well extracts more consumer surplus, meaning higher transaction prices, which translates into higher commission.
• Poor discriminators either lose sales (price too high) or leave money on the table (price too low).

Q3. Why Might Second-Degree Price Discrimination by Electric Utilities Improve Consumer Welfare?

Concept: Second-degree price discrimination — charging different prices for different quantities (block pricing).

Structure: Electric utilities typically charge:

• High price per unit for the first block (low usage)
• Lower price per unit for additional blocks (higher usage)

Why it can improve consumer welfare:

1. Low-volume consumers (e.g., poor households) pay a higher per-unit rate, but their total bill may still be lower than if the utility charged a uniform average cost price — because fixed costs are spread over more units.
2. High-volume consumers get a discount for scale, making them better off than under a single high price.
3. The utility captures more revenue from those with high willingness to pay, allowing it to remain viable without government subsidies.
4. More consumers are served — output is higher than under pure monopoly pricing.

Caveat: Whether consumer welfare actually rises depends on the specific structure. Some low-income consumers may be worse off if the first-block price is very high.

Q4. Examples of Third-Degree Price Discrimination; Same Elasticities Across Groups

Concept: Third-degree price discrimination — charging different prices to groups with different demand curves.

Examples:

Can it work when groups have the same elasticity?

No — it cannot be effective. The optimal pricing rule is:

MR₁ = MR₂ = MC

And since: MR = P(1 + 1/Ed)

If both groups have the same Ed, then to set MR₁ = MR₂, the firm must set P₁ = P₂ — no discrimination occurs.

Conclusion: Price discrimination requires different price elasticities across groups. Same elasticity → same optimal price → no discrimination.

Q5. Why MR₁ = MR₂ = MC in Optimal Third-Degree Discrimination; Response to Demand Shift

Concept: Profit maximization under market segmentation.

Formula: For a firm selling in markets 1 and 2 with total cost C(Q₁+Q₂):

Maximize π = P₁Q₁ + P₂Q₂ - C(Q₁+Q₂)
∂π/∂Q₁ = MR₁ - MC = 0  →  MR₁ = MC
∂π/∂Q₂ = MR₂ - MC = 0  →  MR₂ = MC

Therefore: MR₁ = MR₂ = MC

Intuition: If MR₁ > MR₂, the firm should reallocate output from market 2 to market 1 — sell more where the marginal unit earns more. Equality means no profitable reallocation remains.

If demand in Group 1 shifts outward:

• MR₁ increases at every quantity.
• Now MR₁ > MR₂ = MC.
• The firm should increase output (and price) in market 1 and can reduce output (lower price) in market 2 if needed to maintain equality.
• Total output rises; the firm serves the stronger-demand market more intensively.

Q6. Why Do American Car Companies Charge High Markups on Luxury Options?

Concept: Third-degree price discrimination via product feature bundling; elasticity-based pricing.

The Lerner Markup Rule:

(P - MC)/P = -1/Ed

Higher markup → lower price elasticity of demand.

Why luxury options have high markups:

1. Lower price elasticity: Buyers who purchase a $60,000 car and are considering leather seats are less sensitive to a $2,000 vs. $3,000 price difference — it is a small fraction of total cost.
2. Sorting mechanism: Luxury option buyers reveal themselves as high-income, lower-elasticity customers.
3. Arbitrage is impossible: You cannot buy leather seats separately and install them — the option must come from the manufacturer.
4. Basic options (power steering, automatic) are now expected by most buyers, and competitive pressure keeps their prices low; demand for these is more elastic.

Managerial Lesson: Segment your product line and charge the highest markup where customers are least price-sensitive.

Q7. Peak-Load Pricing as Price Discrimination; Can It Make Consumers Better Off?

Concept: Peak-load pricing charges higher prices during high-demand periods and lower prices during low-demand periods.

Why it is a form of price discrimination:

• Different customers face different prices for essentially the same good.
• Peak users are charged more (their demand is less elastic at peak times).
• Off-peak users pay less.

Can it make consumers better off? Yes:

• Off-peak consumers benefit directly — they pay less than if a uniform price were charged.
• Even peak consumers may benefit — the higher peak price signals real resource cost; without it, capacity would be overbuilt and all consumers would pay more through taxes or regulation.
• Overall efficiency improves — resources are allocated more efficiently when prices reflect true costs.

Example: Electricity pricing — peak daytime rates are higher, nighttime rates lower. This discourages wasteful peak consumption and encourages energy-efficient behavior.

Example: Hotel rooms priced higher on weekends in tourist cities; Uber surge pricing.

Q8. Determining Optimal Two-Part Tariff with Two Customers and Different Demands

Concept: Two-part tariff = entry fee (T) + per-unit usage fee ℗. Maximize profit by choosing T and P together.

Setup: Two customers with different demand curves D₁℗ and D₂℗, where D₁ > D₂ (Customer 1 has higher demand).

Step-by-step procedure:

Step 1: Set usage fee P. Step 2: Calculate consumer surplus for each type at that P:

• CS₁℗ = area under D₁ above P
• CS₂℗ = area under D₂ above P

Step 3: Set entry fee T ≤ min(CS₁, CS₂) — limited by the lower-demand customer to keep both in market.

Step 4: Profit = 2T + (P - MC)(Q₁ + Q₂)

Step 5: Optimize over P. This is a trade-off:

• Higher P → higher profit per unit but lower CS₂ → lower maximum T.
• Lower P (toward MC) → more consumer surplus captured through T.

Result: Optimal P > MC (to extract more from the high-demand customer), and T = CS₂℗ — set at the low-demand customer's CS.

General Rule: If customers are identical, set P = MC and T = entire CS. If they differ, there is a trade-off and the optimal solution involves P > MC.

Q9. Gillette Razor as Two-Part Tariff; Does Gillette Need a Blade Monopoly?

Concept: Razor-and-blade pricing = two-part tariff where the razor is the "entry fee" and blades are the "usage fee."

Structure:

• Razor (Entry fee): Priced low — sometimes at or below cost to get customers into the system.
• Blades (Usage fee): Priced high — this is where Gillette makes its profit.

Does Gillette need to monopolize blades?

• Yes, effectively. If rival firms sell compatible blades cheaply, the usage fee revenue disappears.
• Gillette uses patents and proprietary blade-cartridge design to prevent compatible blades.
• Without a blade monopoly, the strategy fails: consumers would buy cheap blades and Gillette cannot recoup its subsidy on razors.

Pricing Procedure for Gillette:

1. Estimate the demand curve for blades per customer (as a function of blade price).
2. Compute CS at various blade prices.
3. Set blade price P_B above MC.
4. Set razor price P_R = -(CS lost from P_B > MC) or subsidize razor price.
5. Total profit = margin on blades × expected lifetime volume - any razor subsidy.

Q10. Senior Discounts: Dental vs. Eye Exams in Woodland, CA

Concept: Price discrimination requires market power. Without market power, price discrimination cannot persist.

Structure of the question:

• Many dentists → competitive market → no market power.
• One eye doctor → monopoly → market power exists.

Answer: Senior citizens are more likely to receive discount prices for eye exams, not dental exams.

Reason:

• The monopoly eye doctor can price discriminate — seniors with lower income are likely more price-elastic and the eye doctor can profitably offer them a lower price.
• Dentists cannot price discriminate sustainably — if any dentist charges seniors more, they switch to another dentist. Competition drives prices to cost for everyone.

Lesson: Price discrimination requires market power. A competitive market eliminates a firm's ability to charge different prices.

Q11. Why Did MGM Bundle Gone with the Wind and Getting Gertie's Garter?

Concept: Pure bundling is profitable when consumer reservation prices for two goods are negatively correlated.

The Key Characteristic:

• Consumer A values GWTW highly but not GGG.
• Consumer B values GGG highly but not GWTW.
• If sold separately, both consumers pay the minimum of what each will accept, leaving lots of surplus on the table.
• Bundled together, their total valuations are more similar — bundling reduces the variance in reservation prices, allowing the seller to capture more total revenue.

MGM's Logic: The two films had audiences with opposite preferences. By bundling, MGM could charge a single bundle price that extracted more from both types than selling each film separately.

General Condition for Bundling to Work: Reservation prices across products must be negatively correlated across consumers.

Q12. Mixed Bundling vs. Pure Bundling

Concept: Pure bundling = offer only the bundle. Mixed bundling = offer bundle AND individual goods.

Key Differences:

When is mixed bundling preferable?

1. When marginal costs are significant — selling a bundle to someone who doesn't value one product means selling it below MC.
2. When reservation prices are not perfectly negatively correlated — some consumers value only one product and it's worth selling to them separately.

Why restaurants use mixed bundling:

• Some customers want only the main course or only dessert.
• A set menu (pure bundle) would exclude them.
• By offering both set menus and à la carte, the restaurant captures both types of customers and maximizes revenue.

Q13. Tying vs. Bundling

Concept: Tying = requiring purchase of one product with another (not necessarily in fixed ratios). Bundling = selling two or more products only as a package (fixed combination).

Difference:

• Bundling is a subset of tying.
• Tying often involves metering — the tied good is purchased in proportion to usage (e.g., IBM requiring proprietary punch cards with its computers).
• Bundling typically involves a fixed package (movies 1 and 2 sold together).

Why a firm might practice tying:

1. Price discrimination via metering: Heavy users buy more of the tied good, paying more in total — extracting more consumer surplus.
2. Brand protection: Ensures customers use quality complementary products (e.g., Kodak cameras with Kodak film) — preserves brand reputation.
3. Entry barriers: Prevents rival firms from gaining a foothold in the tied good market.
4. Competitive advantage: Locks customers into the ecosystem.

Q14. Why Is It Incorrect to Advertise Until the Last Dollar of Advertising Generates $1 of Sales?

Concept: Advertising rule — the Dorfman-Steiner condition.

The Wrong Rule: "Last dollar of ads = $1 of sales"

This ignores profit margin. If the profit margin is 40%, then $1 of additional sales generates only $0.40 of additional profit. Spending $1 on advertising to get $1 of sales (not profit) is wasteful.

The Correct Rule:

Marginal Revenue from advertising = Marginal Cost of advertising

(ΔQ/ΔA) × (P - MC) = 1

Or equivalently (the Dorfman-Steiner condition):

A/PQ = -(E_A / E_P)

Where:

• A = advertising expenditure
• PQ = total revenue (sales)
• E_A = advertising elasticity of demand
• E_P = price elasticity of demand

Correct Rule: Advertise until the marginal profit generated by the last advertising dollar equals $1 — not until the last dollar of sales generated equals $1.

Q15. How Can a Firm Check Its Advertising-to-Sales Ratio?

Concept: The Dorfman-Steiner condition provides a benchmark.

Formula:

Optimal A/PQ = |E_A| / |E_P|

Information needed:

1. Price elasticity of demand (E_P): How much does quantity demanded change with price? Estimated via regression analysis, market research, or historical data.
2. Advertising elasticity of demand (E_A): How much does quantity demanded change with advertising? Also estimated via regression on historical advertising-sales data.
3. Current advertising-to-sales ratio (A/PQ): Available from accounting records.

Check:

• If actual A/PQ > |E_A|/|E_P|: Advertising too much — cut back.
• If actual A/PQ < |E_A|/|E_P|: Under-advertising — increase spend.
• If equal: Optimal level.

EXERCISES

Exercise 1: Price Discrimination Schemes — Sorting and Arbitrage

Concept: Price discrimination requires (1) ability to sort customers by willingness to pay, and (2) ability to prevent arbitrage between customer groups.

(a) Saturday Night Stay Requirement for Low Airfare

Sorting: Business travelers (less elastic demand) need to return by Friday. Leisure travelers (more elastic) are happy to stay over Saturday.

• The rule perfectly sorts travelers without asking them directly.

Arbitrage Prevention: Tickets are non-transferable and issued with personal ID. Resale is impractical and often illegal per ticket terms.

Discrimination: Business travelers pay high fares; leisure travelers pay low fares.

(b) Delivering Cement Based on Buyer's Location

Sorting: Buyers near the plant are competing with local suppliers and have more elastic demand. Distant buyers have fewer alternatives and less elastic demand.

• Location-based pricing extracts more from buyers with fewer options.

Arbitrage Prevention: Cement is heavy, perishable (it sets), and expensive to transport. A distant buyer cannot resell to a nearby buyer profitably.

© Food Processor Coupons — $10 Rebate

Sorting: Price-sensitive consumers are willing to invest time to clip, mail, and track coupons. Price-insensitive consumers (higher WTP) won't bother.

• The coupon self-selects the elastic-demand group.

Arbitrage Prevention: Rebates are typically one-per-household and require proof of purchase — limits bulk buying and resale.

(d) Temporary Price Cuts on Bathroom Tissue

Sorting: Price-conscious consumers buy more during sales and stockpile. Time-constrained, price-insensitive consumers buy at regular price when needed.

Arbitrage Prevention: Bathroom tissue is bulky and low-value relative to storage space. Resale is not practical. Geographic constraints.

(e) Charging High-Income Patients More for Plastic Surgery

Sorting: Income is a signal for willingness to pay. The doctor explicitly charges based on ability to pay (verified via tax returns or financial disclosure).

Arbitrage Prevention: Medical services cannot be resold. The service is consumed at the moment of delivery by the specific patient.

Exercise 2: Drive-In Movie — Couples vs. Singles

Question: If couples have more elastic demand than singles, is it optimal to charge per driver plus extra per passenger? True or false?

Answer: True.

Reasoning:

• Singles: Less elastic demand → higher willingness to pay per person → charge higher effective per-person price.
• Couples: More elastic demand → more price-sensitive → need lower per-person price to get them in.

The pricing scheme:

• Flat fee per car (driver): captures price from the inelastic single driver as well.
• Extra fee per passenger: couples pay more total, but the per-person effective price is moderated by spreading fixed costs.

Wait — let me re-examine. The question says couples are more elastic. So the firm should charge couples less per person than singles. The way to do this:

• Charge one admission for the driver (used by both singles and the driver of a couple — same fee).
• Charge a smaller incremental fee per passenger than a full second admission would imply.

This way:

• Singles pay the driver fee only (higher per-person).
• Couples pay driver fee + smaller passenger fee (lower per-person).

This is effective price discrimination: lower per-person cost for the more elastic group (couples), higher per-person cost for less elastic group (singles).

Conclusion: TRUE — charging per driver plus a lower extra fee per passenger effectively gives couples a discount per person, consistent with optimal third-degree discrimination.

Exercise 3: German Coupon Ban — Effects on Consumers and Producers

Concept: Coupons enable third-degree price discrimination: elastic consumers use coupons (lower price), inelastic consumers don't (pay full price).

(a) Are German Consumers Better Off or Worse Off?

Worse off overall, though results vary by type:

• Price-elastic consumers who would have used coupons: clearly worse off — they lose access to lower prices. They either don't buy or pay the higher regular price.
• Price-inelastic consumers (non-coupon users): ambiguous — they were already paying full price. If prohibition leads producers to lower the regular price somewhat, they might benefit. But producers are more likely to keep regular prices high.

Net effect: Most consumers are worse off. The elastic segment loses its discount; the inelastic segment is unaffected.

(b) Are German Producers Better Off or Worse Off?

Worse off:

• Coupons allow producers to serve both market segments profitably (high WTP consumers at full price; low WTP consumers at discounted price).
• Banning coupons forces a single price. Producers either:
  • Set price high → lose elastic consumers → lower revenue.
  • Set price low → serve everyone but at lower profit margin → less profit.
• Discrimination always weakly increases producer profit compared to uniform pricing. Removing it reduces profit.

Conclusion: Both consumers (especially elastic ones) and producers are worse off when coupon-based discrimination is banned.

Exercise 4: BMW — Third-Degree Price Discrimination (Europe vs. USA)

Given:

• MC = $20,000 (constant)
• Fixed cost = $10 billion
• Q_E = 4,000,000 − 100P_E (Europe)
• Q_U = 1,000,000 − 20P_U (USA)

(a) Optimal Prices and Quantities in Separate Markets

Step 1: Invert demand curves.

Europe: Q_E = 4,000,000 − 100P_E → P_E = 40,000 − Q_E/100

USA: Q_U = 1,000,000 − 20P_U → P_U = 50,000 − Q_U/20

Step 2: Marginal Revenue.

MR_E = 40,000 − Q_E/50 (derivative of P_E × Q_E)

MR_U = 50,000 − Q_U/10

Step 3: Set MR = MC = 20,000.

Europe: 40,000 − Q_E/50 = 20,000 Q_E/50 = 20,000 Q_E* = 1,000,000 P_E* = 40,000 − 1,000,000/100 = 40,000 − 10,000 = $30,000

USA: 50,000 − Q_U/10 = 20,000 Q_U/10 = 30,000 Q_U* = 300,000 P_U* = 50,000 − 300,000/20 = 50,000 − 15,000 = $35,000

Step 4: Profit.

Total Revenue = (30,000)(1,000,000) + (35,000)(300,000) = 30,000,000,000 + 10,500,000,000 = $40,500,000,000

Total Cost = 20,000 × (1,000,000 + 300,000) + 10,000,000,000 = 20,000 × 1,300,000 + 10,000,000,000 = 26,000,000,000 + 10,000,000,000 = $36,000,000,000

π = $40,500,000,000 − $36,000,000,000 = $4,500,000,000

Summary:

(b) Single Price Across Both Markets

Step 1: Combined demand (add horizontally).

For P ≤ 40,000 (both markets active): Q = Q_E + Q_U = (4,000,000 − 100P) + (1,000,000 − 20P) = 5,000,000 − 120P

Inverse: P = (5,000,000 − Q)/120 = 41,667 − Q/120

Step 2: MR from combined demand.

MR = 41,667 − Q/60

Step 3: Set MR = MC = 20,000.

41,667 − Q/60 = 20,000 Q/60 = 21,667 Q* = 1,300,000

P* = 41,667 − 1,300,000/120 = 41,667 − 10,833 = $30,833

Step 4: Allocate between markets at P* = $30,833.

Q_E = 4,000,000 − 100(30,833) = 4,000,000 − 3,083,300 = 916,700 Q_U = 1,000,000 − 20(30,833) = 1,000,000 − 616,660 = 383,340

Check: 916,700 + 383,340 ≈ 1,300,000 ✓

Step 5: Profit.

TR = 30,833 × 1,300,000 = $40,082,900,000 TC = 20,000 × 1,300,000 + 10,000,000,000 = 36,000,000,000

π = $40,082,900,000 − $36,000,000,000 = $4,082,900,000

Price discrimination profit ($4.5B) > Single price profit ($4.08B) — discrimination is more profitable by ~$417 million.

Exercise 5: Two Geographic Markets — Optimal Discrimination vs. Single Price

Given:

• P₁ = 15 − Q₁, MR₁ = 15 − 2Q₁ (East Coast)
• P₂ = 25 − 2Q₂, MR₂ = 25 − 4Q₂ (Midwest)
• C = 5 + 3(Q₁ + Q₂), so MC = 3

(i) With Price Discrimination

Set MR₁ = MC: 15 − 2Q₁ = 3 → Q₁* = 6, P₁* = 15 − 6 = $9

Set MR₂ = MC: 25 − 4Q₂ = 3 → Q₂* = 5.5, P₂* = 25 − 2(5.5) = $14

Profit: π = P₁Q₁ + P₂Q₂ − C = (9)(6) + (14)(5.5) − [5 + 3(6 + 5.5)] = 54 + 77 − [5 + 34.5] = 131 − 39.5 = $91.50

MR₁ = 15 − 2(6) = 3 = MC ✓ MR₂ = 25 − 4(5.5) = 3 = MC ✓

(ii) Without Price Discrimination (Single Price)

Step 1: Combine demands.

Q₁: Q₁ = 15 − P (for P ≤ 15) Q₂: Q₂ = (25 − P)/2 (for P ≤ 25)

For P ≤ 15: Q = Q₁ + Q₂ = (15 − P) + (25 − P)/2 = 15 − P + 12.5 − 0.5P = 27.5 − 1.5P

Inverse: P = (27.5 − Q)/1.5 = 18.33 − Q/1.5

MR = 18.33 − Q/0.75 = 18.33 − 4Q/3

Set MR = MC = 3: 18.33 − 4Q/3 = 3 4Q/3 = 15.33 Q* = 11.5

P* = 18.33 − 11.5/1.5 = 18.33 − 7.67 = $10.67

Q₁ = 15 − 10.67 = 4.33 Q₂ = (25 − 10.67)/2 = 7.17

Check: 4.33 + 7.17 = 11.5 ✓

Profit: π = 10.67 × 11.5 − [5 + 3(11.5)] = 122.7 − [5 + 34.5] = 122.7 − 39.5 = $83.2

Deadweight Loss: Competitive output: where P = MC = 3 Q₁_comp = 15 − 3 = 12; Q₂_comp = (25−3)/2 = 11; Q_comp = 23

DWL from discrimination vs. competition is measured by welfare loss relative to P = MC outcome.

Results Table:

Interesting: Under discrimination, Market 1 (lower willingness to pay) gets a lower price, while Market 2 gets a higher price — discrimination benefits East Coast consumers but hurts Midwest consumers.

Exercise 6: Elizabeth Airlines — Pricing, Discrimination, Viability

Given:

• Demand: Q = 500 − P → P = 500 − Q
• Cost: 30,000 + 100Q (FC = $30,000; MC = $100)

(a) Profit-Maximizing Price, Quantity, Profit (FC = $30,000)

MR = MC:

MR = 500 − 2Q 500 − 2Q = 100 2Q = 400 Q* = 200 P* = 500 − 200 = $300

Profit: π = TR − TC = 300(200) − [30,000 + 100(200)] = 60,000 − 50,000 = $10,000 per flight

(b) Fixed Cost Rises to $41,000 — Will EA Stay in Business?

New profit: π = 60,000 − [41,000 + 20,000] = 60,000 − 61,000 = −$1,000 per flight

Average Cost at Q = 200: AC = (41,000 + 100×200)/200 = 61,000/200 = $305

Since P* = $300 < AC = $305, the airline makes a loss on every flight.

Short-run decision: Should continue if P > AVC. AVC = $100 (just variable per-passenger cost).

• P = $300 >> AVC = $100 → continue in short run.

Long-run: If losses persist with no adjustment, EA will exit unless it can increase revenue.

Graphical note: The AC curve with FC = $41,000 lies above the demand curve at the profit-maximizing quantity — the minimum AC is not reached below P = $300.

© Price Discrimination: Type A (Business) and Type B (Students)

Type A: Q_A = 260 − 0.4P → P_A = 650 − Q_A/0.4 = 650 − 2.5Q_A Type B: Q_B = 240 − 0.6P → P_B = 400 − Q_B/0.6 = 400 − 5Q_B/3

Note: Total demand at P must match combined: Q_A + Q_B = (260 − 0.4P) + (240 − 0.6P) = 500 − P ✓ (consistent with part a)

Optimal Type A price (MR_A = MC = 100): MR_A = 650 − 5Q_A 650 − 5Q_A = 100 → Q_A* = 110 P_A* = 650 − 2.5(110) = 650 − 275 = $375

Optimal Type B price (MR_B = MC = 100): MR_B = 400 − 10Q_B/3 400 − 10Q_B/3 = 100 → 10Q_B/3 = 300 → Q_B* = 90 P_B* = 400 − (5/3)(90) = 400 − 150 = $250

Totals: Q = 110 + 90 = 200 (same total passengers as before)

(d) EA's Profit with Price Discrimination (FC = $41,000)

Revenue: 375(110) + 250(90) = 41,250 + 22,500 = $63,750 Cost: 41,000 + 100(200) = 41,000 + 20,000 = $61,000

π = $63,750 − $61,000 = $2,750 per flight

Yes, EA stays in business. Discrimination makes the airline viable even with higher fixed costs.

Consumer Surplus:

• Type A CS = 0.5 × (650 − 375) × 110 = 0.5 × 275 × 110 = $15,125
• Type B CS = 0.5 × (400 − 250) × 90 = 0.5 × 150 × 90 = $6,750
• Total CS = $21,875

(e) Consumer Surplus Before Discrimination

Before discrimination (P = $300, Q = 200):

• CS = 0.5 × (500 − 300) × 200 = 0.5 × 200 × 200 = $20,000

Type A portion: At P = $300, Q_A = 260 − 0.4(300) = 140 CS_A = 0.5 × (650 − 300) × 140 = 0.5 × 350 × 140 = $24,500

Type B portion: At P = $300, Q_B = 240 − 0.6(300) = 60 CS_B = 0.5 × (400 − 300) × 60 = 0.5 × 100 × 60 = $3,000

Total CS before discrimination = 24,500 + 3,000 = $27,500 ≠ $20,000...

This is because the demand curves are not simply summed — the reservation prices differ. The correct total CS before discrimination = $20,000 (from the combined demand curve).

Why total CS declined after discrimination (from ~$27,500 under combined to ~$21,875 under discrimination):

• Type A was paying $300 and getting Q_A = 140 with CS = $24,500 → now pays $375 for only 110 units: lost significant CS.
• Type B was paying $300 for only 60 units → now pays $250 for 90 units: gained CS ($6,750 > $3,000).
• Net: Type A's large CS loss outweighs Type B's gain → total CS falls.

Discrimination transfers surplus from business travelers to the airline while only partially compensating students with lower prices.

Exercise 7: Video Store Two-Part Tariff — Logic and Choice of Plans

Concept: Two-part tariff captures consumer surplus through the membership fee and serves high-usage customers efficiently through low per-unit fees.

Logic of Two-Part Tariff for Video Stores:

Plan 1: $40 annual membership + $2/film/day Plan 2: No membership + $4/film/day

• Heavy users (10+ films/year): Plan 1 is cheaper. They pay $40 up front but save $2 per film.
• Light users (fewer than 20 films/year): Plan 2 may be better — they avoid the $40 fixed fee.

Why offer both plans instead of just one two-part tariff?

Customers have heterogeneous demands. If only Plan 1 existed:

• Light users would not join — the $40 fee exceeds their total consumer surplus.
• The store loses revenue from this group.

By offering both plans:

• Heavy users self-select into Plan 1 (worth it for them).
• Light users self-select into Plan 2 (they pay more per film but avoid the membership fee).

The store captures surplus from both types. This is a form of second-degree price discrimination (quantity-based — high quantity users get lower per-unit price via membership).

Exercise 8: Sal's Satellite — Third-Degree Price Discrimination

Given:

• Q_NY = 60 − 0.25P_NY → P_NY = 240 − 4Q_NY
• Q_LA = 100 − 0.50P_LA → P_LA = 200 − 2Q_LA
• C = 1000 + 40Q, so MC = 40

(a) Separate Market Pricing (Price Discrimination)

New York: MR_NY = 240 − 8Q_NY = 40 8Q_NY = 200 Q_NY* = 25 P_NY* = 240 − 4(25) = $140

Los Angeles: MR_LA = 200 − 4Q_LA = 40 4Q_LA = 160 Q_LA* = 40 P_LA* = 200 − 2(40) = $120

Profit: Q_total = 25 + 40 = 65 π = P_NY × Q_NY + P_LA × Q_LA − C = 140(25) + 120(40) − [1000 + 40(65)] = 3500 + 4800 − [1000 + 2600] = 8300 − 3600 = $4,700

(b) Single Price Required (No Discrimination)

Combined demand (add quantities for same P): Q = Q_NY + Q_LA = (60 − 0.25P) + (100 − 0.5P) = 160 − 0.75P

Inverse: P = (160 − Q)/0.75 = 213.3 − Q/0.75 = 213.3 − 4Q/3

MR = 213.3 − 8Q/3

Set MR = MC = 40: 213.3 − 8Q/3 = 40 8Q/3 = 173.3 Q* = 65

P* = 213.3 − 4(65)/3 = 213.3 − 86.7 = $126.67

Q_NY at P = 126.67: Q_NY = 60 − 0.25(126.67) = 60 − 31.67 = 28.33 Q_LA at P = 126.67: Q_LA = 100 − 0.5(126.67) = 100 − 63.33 = 36.67

Profit: π = 126.67(65) − [1000 + 40(65)] = 8,233.55 − 3,600 = $4,633.55

© Comparison and Consumer Surplus

Sal is better off with price discrimination: $4,700 vs. $4,633.55

New York consumers:

• With discrimination: P_NY = $140, Q_NY = 25 CS_NY = 0.5 × (240 − 140) × 25 = $1,250
• With single price: P = $126.67, Q_NY = 28.33 CS_NY = 0.5 × (240 − 126.67) × 28.33 = $1,605

New York prefers single price (lower price, more quantity, higher CS).

Los Angeles consumers:

• With discrimination: P_LA = $120, Q_LA = 40 CS_LA = 0.5 × (200 − 120) × 40 = $1,600
• With single price: P = $126.67, Q_LA = 36.67 CS_LA = 0.5 × (200 − 126.67) × 36.67 = $1,344

Los Angeles prefers price discrimination (lower price, more quantity, higher CS).

Intuition: NY has more inelastic demand (higher Ed in absolute terms) → pays higher price under discrimination. LA has more elastic demand → gets lower price under discrimination.

Exercise 9: Super Computer Inc. — Two-Part Tariff with Two Customer Types

Given:

• 10 Business firms: Q_B = 10 − P each
• 10 Academic institutions: Q_A = 8 − P each
• MC = 2 cents/second

(a) Separate Two-Part Tariffs for Each Group

Business firms: At usage fee P_B, CS_B = 0.5 × (10 − P_B)²

Set P_B = MC = 2 (capture all CS as rental fee): CS_B = 0.5 × (10 − 2)² = 0.5 × 64 = $32 per firm Q_B per firm = 10 − 2 = 8

Rental fee T_B = $32 per business firm. Profit from businesses = 10 × [T_B + (P_B − MC) × Q_B] = 10 × [32 + 0] = $320

Academic institutions: At P_A = MC = 2: CS_A = 0.5 × (8 − 2)² = 0.5 × 36 = $18 per academic Q_A per academic = 8 − 2 = 6

Rental fee T_A = $18. Profit from academics = 10 × 18 = $180

Total profit = $320 + $180 = $500

(b) Unable to Separate Groups — Zero Rental Fee, Maximize on Usage Fee

If rental fee = 0, profit comes only from markup on usage: π = (P − MC) × Total Q = (P − 2) × [10(10 − P) + 10(8 − P)] = (P − 2) × [100 − 10P + 80 − 10P] = (P − 2)(180 − 20P)

Expand: π = 180P − 20P² − 360 + 40P = 220P − 20P² − 360

dπ/dP = 220 − 40P = 0 → P* = 5.5

Q_total = 10(10 − 5.5) + 10(8 − 5.5) = 10(4.5) + 10(2.5) = 45 + 25 = 70

π = (5.5 − 2)(70) = 3.5 × 70 = $245

This is much less than the $500 achieved with separate groups.

© Single Two-Part Tariff (One Rental Fee + One Usage Fee for Everyone)

Strategy: Must set T ≤ CS of the lower-demand customer (academic), otherwise academics won't join.

At usage fee P:

• CS_A℗ = 0.5(8 − P)² (binding constraint — lower demand)
• CS_B℗ = 0.5(10 − P)²
• Set T = CS_A℗ = 0.5(8 − P)²

Profit: π = 20 × T + (P − MC) × [10(10 − P) + 10(8 − P)] = 20 × 0.5(8−P)² + (P−2) × [180 − 20P] = 10(8−P)² + (P−2)(180−20P)

Let f℗ = 10(64 − 16P + P²) + (P−2)(180−20P) = 640 − 160P + 10P² + 180P − 20P² − 360 + 40P = 280 + 60P − 10P²

dπ/dP = 60 − 20P = 0 → P* = 3

T* = 0.5(8 − 3)² = 0.5 × 25 = $12.50

Q_B per firm = 10 − 3 = 7; Q_A per institution = 8 − 3 = 5 Total Q = 10(7) + 10(5) = 120

π = 20(12.50) + (3 − 2)(120) = 250 + 120 = $370

Why P ≠ MC: If P = MC = 2, T = 0.5(8−2)² = $18. π = 20(18) + 0 = $360 < $370. There is a benefit to charging P above MC because the increased profit per unit more than compensates for the reduced T (due to lower academic CS). The optimal P = 3 > MC = 2.

Summary:

Separating groups is best; failing that, a two-part tariff beats zero rental fee.

Exercise 10: Tennis Club — Two-Part Tariff with Two Player Types

Given:

• Serious players: Q₁ = 10 − P (1000 players)
• Occasional players: Q₂ = 4 − 0.25P (1000 players)
• MC = 0, FC = $10,000/week
• Players look alike → must charge same prices.
• P is court fee per hour; T is membership dues (annual → convert to weekly: T_weekly = T_annual/52).

(a) Limit Membership to Serious Players Only

Set T = CS of serious player (they must be just indifferent to join):

At court fee P: CS_serious = 0.5(10 − P)²

To exclude occasional players: occasional player's CS must be ≤ T. CS_occasional = 0.5 × (4 − 0.25P) × (4/0.25 − P) ... let me use: at P, Q₂ = 4 − 0.25P; max WTP = 16 (when Q₂ = 0). CS_occasional = 0.5 × (16 − P) × (4 − 0.25P) = 0.5 × (4 − 0.25P)²/0.25 ...

Actually: Demand Q₂ = 4 − 0.25P means inverse demand P₂ = 16 − 4Q₂. CS_occasional = 0.5 × (16 − P) × Q₂ = 0.5(16 − P)(4 − 0.25P)

Constraint to exclude occasional players: T > CS_occasional

If we set T = CS_serious = 0.5(10 − P)², then we need: 0.5(10 − P)² > 0.5(16 − P)(4 − 0.25P)

This constraint binds at some P. For simplicity, find optimal P for serious-only membership:

Profit (1000 serious players): π = 1000 × [T + P × Q₁] − FC = 1000 × [0.5(10−P)² + P(10−P)] − 10,000 = 1000 × (10−P)[0.5(10−P) + P] − 10,000 = 1000 × (10−P)(0.5×10 + 0.5P) − 10,000 = 1000 × (10−P)(5 + 0.5P) − 10,000

Let f℗ = (10−P)(5+0.5P) = 50 + 5P − 5P − 0.5P² = 50 − 0.5P²

dπ/dP = 1000 × (−P) = 0 → P = 0

So optimal court fee = P* = 0 (since MC = 0).

T = 0.5(10 − 0)² = $50 per week per member (or $2,600/year annual dues).

Weekly Profit: π = 1000 × 50 + 0 − 10,000 = 50,000 − 10,000 = $40,000/week

(b) Include Both Types — Maximize Profit

Constraint: T ≤ min(CS_serious, CS_occasional). Must set T ≤ CS_occasional = 0.5(16 − P)(4 − 0.25P) at optimal P.

At P = 0: CS_occasional = 0.5 × 16 × 4 = $32; CS_serious = $50. Set T = $32 (bind on occasional player).

Profit = 2000 × T + 0 − 10,000 = 2000 × 32 − 10,000 = 64,000 − 10,000 = $54,000/week

This exceeds $40,000 from serious-only strategy!

Your friend is right — including both types is more profitable at P = 0, T = $32.

But let's verify this is truly optimal. At general P with both types: T = CS_occasional = 0.5(16−P)(4−0.25P)

π = 2000 × T + P × [1000(10−P) + 1000(4−0.25P)] − 10,000 = 2000 × 0.5(16−P)(4−0.25P) + P × 1000(14 − 1.25P) − 10,000

At P = 0: π = 2000 × 32 + 0 − 10,000 = $54,000

Checking P slightly above 0 — any deviation reduces profit here since MC = 0. P* = 0, T* = $32/week ($1,664/year).

Weekly Profit = $54,000

© 3000 Serious + 1000 Occasional Players

Option 1: Serve both types at P = 0, T = $32: π = 4000 × 32 − 10,000 = 128,000 − 10,000 = $118,000/week

Option 2: Serious only at P = 0, T = $50: π = 3000 × 50 − 10,000 = 150,000 − 10,000 = $140,000/week

Serious-only strategy wins when there are many serious players.

Answer: P* = 0, T = $50/week ($2,600/year), serious-only → Profit = $140,000/week.

Catering to occasional players is no longer optimal — losing $18/week per serious player (dropping T from $50 to $32) to gain 1000 occasional players at $32 each: gain = $32,000, loss = 3000 × $18 = $54,000. Net loss of $22,000. Not worth it.

Exercise 11: Bundling with Three Consumers (Figure 11.12)

Given (from standard Figure 11.12): Three consumers with reservation prices:

MC = 0 for both goods.

Selling Separately:

• Good 1: Charge $10 → sell 1 unit © → Revenue = $10. Or charge $8.25 → sell 2 units (B,C) → $16.50. Or charge $3.25 → sell all 3 → $9.75. Best: P₁ = $8.25, Revenue = $16.50.
• Good 2: Charge $6 → sell 1 (A) → $6. Or charge $3.25 → sell 2 (A,B) → $6.50. Or charge $2 → sell all 3 → $6. Best: P₂ = $3.25, Revenue = $6.50.

Total revenue (separate) = $16.50 + $6.50 = $23.00

Pure Bundling: Bundle values:

• A: 3.25 + 6 = $9.25
• B: 8.25 + 3.25 = $11.50
• C: 10 + 2 = $12.00

Charge $9.25 → sell to all 3 → Revenue = $27.75 Charge $11.50 → sell to B and C → Revenue = $23.00 Charge $12 → sell to C only → Revenue = $12.00

Best pure bundle: P_B = $9.25, Revenue = $27.75

Mixed Bundling: Set P₁ = $10 (sell to C alone), P₂ = $6 (sell to A alone), P_B = $9.25 (sell bundle to A, B, C):

• A: Bundle value $9.25 ≥ P_B = $9.25 (buys bundle); P₂ = $6 < $9.25 (might prefer bundle) A buys bundle: net surplus = 0. Pays $9.25.
• B: Bundle value = $11.50 > $9.25 (buys bundle). Pays $9.25.
• C: Bundle value = $12 > $9.25; also P₁ = $10 → surplus from buying just Good 1: 10−10=0; bundle: 12−9.25 = $2.75 > 0. Buys bundle.

Revenue from mixed bundling here = $9.25 × 3 = $27.75 (same as pure bundling in this case).

Best Strategy: Pure Bundling at $9.25 — Revenue = $27.75.

Exercise 12: Figure 11.17 — Pure Bundling with Zero Marginal Costs

Concept: When c₁ = c₂ = 0, pure bundling is optimal when reservation prices are negatively correlated.

Setup: Figure 11.17 shows consumers distributed with negative correlation between r₁ and r₂.

With zero MC:

• No consumer will be sold a good below their reservation price = no loss from including them in a bundle.
• The bundle price P_B captures value from consumers who value the goods differently but whose total reservation prices cluster around a common sum.
• The option to sell individually (mixed bundling) adds complexity but not profit when MC = 0 — anyone who would prefer to buy just one good at the individual price would have been captured by the bundle anyway.

Pure Bundling Pricing:

• Set P_B = r₁ + r₂ for the lowest-value consumer you wish to include (the binding constraint).
• All consumers with total reservation price ≥ P_B buy the bundle.

Profit: P_B × (number of consumers buying bundle)

Why pure dominates mixed at MC = 0:

• With mixed bundling, some consumers buy only one good at P₁ or P₂ — but since MC = 0, the firm would prefer to sell them the bundle at their total reservation price.
• Pure bundling forces all consumers to reveal their total valuation, allowing the firm to capture it.

Exercise 13: IBM — Leasing vs. Selling

Concept: Intertemporal pricing, market power maintenance, and commitment problems.

(a) Argument FOR Selling (Getting Customers to Purchase More)

1. Revenue recognition: Sales generate immediate, large revenue — better for cash flow and accounting.
2. Eliminates used-market competition with oneself: Under leasing, IBM must compete with its own previously leased machines in the secondary market.
3. Avoids obsolescence risk: Once sold, the obsolescence risk transfers to the buyer.
4. Larger competitor discouragement: Locking in customers via outright ownership makes switching harder — competitors must overcome installed base.

(b) Argument AGAINST Selling (In Favor of Leasing)

1. Maintains market power: By retaining ownership, IBM can upgrade customers and prevent a durable goods monopoly problem.
2. The Coase Conjecture: A durable goods monopolist facing rational buyers cannot sustain monopoly prices — buyers anticipate future price cuts and wait. Leasing avoids this problem by never transferring ownership.
3. Better discrimination: Leasing prices can be adjusted to different customers' usage patterns (heavy vs. light users pay differently over time).
4. Ongoing revenue stream: Leasing generates predictable recurring income.

© Factors Determining Which is Preferable

Exercise 14: Bundling Three Consumers — Unit Cost $30

Given:

MC = $30 per unit.

(i) Selling Separately

Good 1 options:

• P₁ = $100 → sell to C → Revenue = $100, Profit = $70
• P₁ = $60 → sell to B, C → Revenue = $120, Profit = $60 (= 2×30)
• P₁ = $20 → sell to all → Revenue = $60, Profit = $0

Best: P₁ = $60, Profit₁ = $60

Good 2 options:

• P₂ = $100 → sell to A → Revenue = $100, Profit = $70
• P₂ = $60 → sell to A, B → Revenue = $120, Profit = $60
• P₂ = $20 → sell to all → Revenue = $60, Profit = $0

Best: P₂ = $60, Profit₂ = $60

Total separate profit = $60 + $60 = $120

(ii) Pure Bundling

Bundle reservation prices:

• A: 20 + 100 = $120
• B: 60 + 60 = $120
• C: 100 + 20 = $120

Conveniently identical! Set P_B = $120.

• All three buy.
• Revenue = 3 × $120 = $360
• Cost = 3 × (30 + 30) = $180

Pure bundle profit = $180

(iii) Mixed Bundling

With P_B = $120 (all buy bundle), individual prices don't matter if everyone prefers the bundle. But let's check if mixed bundling can do better:

At P₁ = $100: C buys Good 1 alone (surplus = 100 − 100 = 0; bundle surplus = 120 − 120 = 0 → indifferent). Revenue from C either way = $100 + $20 = $120 (bundle) or just $100 (individual).

Actually with unit cost $30, mixed bundling here cannot beat pure bundling of $120 since all consumers have the same total valuation of $120.

Mixed bundling profit = $180 (same result here).

(b) Most Profitable Strategy?

Pure bundling (or mixed — same here) at P_B = $120 → Profit = $180.

This beats separate selling ($120) because bundling reduces the variance in reservation prices — all three consumers have the same total willingness to pay of $120, making pure bundling perfect.

Exercise 15: Four Consumers, Zero MC — Optimal Bundling Strategy

Given:

MC = 0.

(a) Zero Marginal Cost

Selling Separately:

Good 1:

• P₁ = $100 → D only → $100
• P₁ = $80 → C, D → $160
• P₁ = $40 → B, C, D → $120
• P₁ = $25 → all → $100

Best: P₁ = $80, Revenue = $160

Good 2:

• P₂ = $100 → A only → $100
• P₂ = $80 → A, B → $160
• P₂ = $40 → A, B, C → $120
• P₂ = $25 → all → $100

Best: P₂ = $80, Revenue = $160

Total separate revenue = $320

Pure Bundling: Bundle values: A = $125, B = $120, C = $120, D = $125

• P_B = $125 → A and D buy → Revenue = $250
• P_B = $120 → all four buy → Revenue = $480

Best: P_B = $120, Revenue = $480

Mixed Bundling (try P₁ = $100, P₂ = $100, P_B = $120):

• A: r₁ = $25 < $100, r₂ = $100 = $100 → Buy Good 2 at $100 (surplus = 0). Or buy bundle: surplus = 125 − 120 = $5. Buys bundle.
• B: r₁ = $40 < $100, r₂ = $80 < $100. Buy bundle: surplus = 120 − 120 = $0. Indifferent; let's say buys bundle.
• C: same as B by symmetry. Buys bundle.
• D: r₁ = $100 = $100, r₂ = $25 < $100. Individual Good 1 surplus = $0. Bundle surplus = $5. Buys bundle.

Revenue = 4 × $120 = $480 (same as pure bundling).

Try P₁ = $100, P₂ = $100, P_B = $120: Check if D prefers P₁ = $100: surplus = 100 − 100 = 0 vs. bundle surplus = 5. Still buys bundle.

Best strategy: Pure bundling at P_B = $120, Revenue = $480.

(b) With MC = $30 per Good

Selling Separately (each unit costs $30 to produce):

Good 1:

• P₁ = $80 → C, D → Revenue = $160, Cost = $60, Profit = $100
• P₁ = $100 → D only → Profit = $70

Best: P₁ = $80, Profit₁ = $100

Good 2:

• P₂ = $80 → A, B → Profit = $100

Total separate profit = $200

Pure Bundling at P_B = $120 (all four buy): Profit = 4 × ($120 − $60) = 4 × $60 = $240

Mixed Bundling: Set P₁ = $100, P₂ = $100, P_B = $120:

• A: r₂ = $100, buys Good 2 alone (surplus = 0); bundle surplus = $5. Bundle preferred.
• D: r₁ = $100, buys Good 1 alone (surplus = 0); bundle surplus = $5. Bundle preferred.
• B: neither individual. Bundle (surplus = $0). Buys bundle.
• C: same. Buys bundle.

All buy bundle → Revenue = $480, Cost = $240, Profit = $240 (same as pure).

Try P₁ = $100, P₂ = $100 as options and see if anyone buys individually: No one buys individually (all prefer bundle or are indifferent). So mixed bundling = pure bundling here.

Actually, try P₁ = $90, P₂ = $90, P_B = $120:

• D: r₁ = $100, individual surplus = $10 vs. bundle surplus = $5. D buys Good 1 alone.
• A: r₂ = $100, individual surplus = $10 > $5. A buys Good 2 alone.
• B: bundle only (r₁ = $40, r₂ = $80 < $90 each). Bundle surplus = $0. Buys bundle.
• C: same as B. Bundle surplus = $0.

Revenue = $90 (A) + $90 (D) + $120 (B) + $120 © = $420 Cost = $30 (A's Good 2) + $30 (D's Good 1) + $60 (B bundle) + $60 (C bundle) = $180 Profit = $420 − $180 = $240 (still same).

With MC = $30: Pure bundling, mixed bundling, and even some separate selling yield $240. Optimal is still pure bundling at $120.

Why is MC = $30 a different situation conceptually? With MC > 0, selling goods to consumers who barely value them (below marginal cost) would be a loss. With pure bundling, if a consumer values the bundle at exactly the bundle price, the firm is still profitable since the bundle price covers both marginal costs. But mixed bundling becomes useful when some consumers' individual valuations greatly exceed the individual MC.

Exercise 16: Cable TV Mixed Bundling (Figure 11.21 Analysis)

Setup: Sports Channel (Product 1) at P₁; Movie Channel (Product 2) at P₂; Bundle at P_B. The figure divides consumer space into regions based on reservation prices r₁ and r₂.

(a) Purchases in Each Region

Region I (r₁ < P_B − P₂ and r₂ < P_B − P₁): Neither individual good is worth buying; bundle isn't worth it either. → No purchase.

Region II (r₁ ≥ P₁ but r₂ < P_B − P₁): Consumer values Sports Channel highly enough to buy individually, but not the bundle. → Buy Sports Channel (Product 1) only.

Region III (r₂ ≥ P₂ but r₁ < P_B − P₂): Values Movie Channel highly enough individually. → Buy Movie Channel (Product 2) only.

Region IV (r₁ + r₂ ≥ P_B and in the bundle-preferred zone): Total value exceeds bundle price. → Buy the bundle.

(b) Are Reservation Prices Negatively Correlated for Cable TV?

Yes, plausibly. People who love sports tend to have lower interest in movies, and vice versa. Sports enthusiasts dedicate time and emotional energy to sports content; movie lovers prefer drama and film. This negative correlation is what makes bundling profitable — by bundling, the firm captures revenue from both types.

However: Not necessarily — some consumers love both. The correlation is imperfect, justifying mixed bundling over pure bundling.

© Does Zero MC Make Mixed Bundling = Pure Bundling?

Disagree with the VP.

When MC = 0, pure bundling is only optimal if reservation prices are perfectly negatively correlated. If some consumers only value one channel (Figure 11.21 suggests consumers in regions II and III exist), selling them only that channel at P₁ or P₂ captures revenue that pure bundling would miss:

• A consumer in Region II with r₁ = $15, r₂ = $2 would not buy the bundle at P_B = $12 (total value = $17 > $12 → actually would buy bundle).

Wait — if r₁ + r₂ ≥ P_B, they're in Region IV and buy the bundle. If r₁ < P_B − P₂, they're in Region I or II.

The VP is wrong if there are consumers in Regions II and III who would buy an individual product but NOT the bundle. In that case, mixed bundling captures this revenue; pure bundling does not.

Conclusion: Mixed bundling is better when the distribution of reservation prices is not perfectly negatively correlated — even with MC = 0. The VP's claim is incorrect.

(d) Should the Cable Company Alter Prices?

Looking at Figure 11.21: If there are many consumers near the boundaries of each region, the company should:

• Consider lowering P_B to capture more Region IV consumers.
• Consider raising P₁ or P₂ if many individual buyers remain after moving to the bundle.
• The distribution of X's (consumer reservation prices) guides the optimal adjustment.

Key principle: If the cluster of X's is dense near the bundle price cutoff, lowering P_B slightly brings in many more buyers and can increase total profit.

Exercise 17: Monopolist with Advertising — Profit Maximization and Lerner Index

Given:

• P = 100 − 3Q + 4A^(1/2)
• C = 4Q² + 10Q + A
• Where A = advertising expenditure.

(a) Find Optimal A, Q, and P

Step 1: Profit function.

π = PQ − C = (100 − 3Q + 4A^(1/2))Q − (4Q² + 10Q + A) = 100Q − 3Q² + 4A^(1/2)Q − 4Q² − 10Q − A = 90Q − 7Q² + 4A^(1/2)Q − A

Step 2: First-order conditions.

∂π/∂Q = 90 − 14Q + 4A^(1/2) = 0 ... (1)

∂π/∂A = 2A^(-1/2)Q − 1 = 0 → 2Q/A^(1/2) = 1 → A^(1/2) = 2Q ... (2)

Step 3: Substitute (2) into (1).

90 − 14Q + 4(2Q) = 0 90 − 14Q + 8Q = 0 90 − 6Q = 0 Q* = 15

Step 4: Find A.

From (2): A^(1/2) = 2(15) = 30 → A* = 900

Step 5: Find P.

P* = 100 − 3(15) + 4(30) = 100 − 45 + 120 = $175

Step 6: Verify profit.

π = 175(15) − [4(225) + 10(15) + 900] = 2625 − [900 + 150 + 900] = 2625 − 1950 = $675

(b) Lerner Index

MC = dC/dQ = 8Q + 10 = 8(15) + 10 = 130

Lerner Index:

L = (P − MC)/P = (175 − 130)/175 = 45/175 = 9/35 ≈ 0.257

Interpretation: About 25.7% of the price represents the markup over marginal cost. This reflects the degree of market power. The advertising investment allows the firm to raise demand sufficiently to sustain a significant markup.

Final Answer Box:

CHAPTER 11 LEARNING ENHANCEMENTS

Key Formula Sheet

| Concept | Formula | |---|---| | MR = MC condition | Basic profit max for all pricing strategies | | Lerner Index | L = (P − MC)/P = −1/E_d | | 3rd-degree optimal | MR₁ = MR₂ = MC | | Two-part tariff profit | π = n·T + (P−MC)·Q | | Entry fee (homogeneous) | T = CS at usage price P | | Dorfman-Steiner condition | A/PQ = |E_A| / |E_P| | | Bundle profit | P_B × (# buying) − cost | | Advertising FOC | 2Q/√A = 1 (at optimum with √A in demand) |

Important Equilibrium Conditions

1. Optimal discrimination: MR in each market = MC
2. Two-part tariff (homogeneous customers): P = MC, T = entire CS
3. Two-part tariff (heterogeneous): T = CS of low-demand customer; P > MC
4. Bundling: Charge bundle price = minimum total reservation price of target group
5. Advertising: MR from advertising = MC of advertising (= $1)

Common IPMX Exam Traps

1. Forgetting arbitrage: If a student says discrimination exists but doesn't address how resale is prevented, the answer is incomplete.
2. Confusing two-part tariff optimum: When customers are heterogeneous, P > MC at optimum — NOT equal to MC. Many students wrongly set P = MC.
3. Bundling condition: Bundling is NOT always better. It requires negatively correlated reservation prices. If positively correlated, separate selling may be better.
4. Lerner Index = elasticity: Remember L = −1/E_d — directly linked to price elasticity, not markup percentage in casual terms.
5. Advertising rule: The Dorfman-Steiner says A/PQ = |E_A|/|E_P|, NOT "last dollar of ads = last dollar of sales."
6. Consumer surplus after discrimination: Many students assume all consumers benefit from discrimination. Often, the inelastic group loses (pays higher price) while the elastic group gains.

Concept Comparison Table: Pricing Strategies

Quick Revision Summary

Core idea of Chapter 11: A firm with market power can do better than a single uniform price by using pricing strategies that extract consumer surplus.

• 1st-degree: Perfect — charge each person their exact WTP. Theoretical maximum profit.
• 2nd-degree: Block pricing — quantity discounts. Self-selection mechanism.
• 3rd-degree: Group pricing — different prices for identifiable groups. Requires no arbitrage.
• Two-part tariff: Fixed fee + per-unit fee. Best when customers are homogeneous.
• Bundling: Package multiple goods. Works when valuations are negatively correlated.
• Tying: Link purchase of product A to product B. Used for metering and brand protection.
• Advertising: Shift demand outward. Optimal when E_A is high relative to E_P.