CHAPTER 12: MONOPOLISTIC COMPETITION AND OLIGOPOLY

Complete Solution Manual — IPMX Managerial Economics

QUESTIONS FOR REVIEW

Q1. Characteristics of Monopolistically Competitive Markets; Effect of New Product Entry

Characteristics:

1. Many sellers — no single firm dominates.
2. Differentiated products — each firm's product is a close but imperfect substitute for others.
3. Free entry and exit — no barriers; profits attract entry, losses cause exit.
4. Some market power — each firm faces a downward-sloping demand curve (unlike perfect competition).

Effect of a new, improved product by one firm:

• Short run: The innovating firm captures market share. Its demand curve shifts right (more consumers). Price and profit rise.
• Other firms: Their demand curves shift left — they lose customers. Their profits fall.
• Long run: If profits persist, more firms enter with their own differentiated products. Each firm's demand curve shifts back left until zero economic profit is restored (price = average cost, P = AC, not P = MC).

Q2. Why Is the Firm's Demand Curve Flatter Than Total Market Demand?

Because products are differentiated but close substitutes.

• The total market demand is relatively inelastic — consumers' overall demand for the product category doesn't change much with price.
• Each individual firm's demand is highly elastic — if one firm raises its price even slightly, many consumers will switch to close substitutes from rival firms.

Mathematical intuition: The cross-price elasticity between individual brands is high, making each firm's residual demand very flat.

Long-run adjustment from short-run profits:

If a monopolistically competitive firm earns positive profit:

• New firms enter with differentiated products.
• Each existing firm's demand curve shifts LEFT (loses customers to entrants).
• Demand also becomes more elastic (more substitutes available).
• Entry continues until the demand curve is tangent to the AC curve — profits = 0.

Q3. Too Many Brands of Breakfast Cereal — For and Against

Argument FOR (Too Many Brands):

1. Excess capacity: Each firm produces below its minimum efficient scale — AC > minimum AC. This is the "excess capacity theorem" of monopolistic competition.
2. Wasteful resource use: Advertising, packaging, and product differentiation costs are social wastes from a welfare perspective.
3. Consumer confusion: Too many choices may increase search costs and reduce decision quality.
4. Duplication of fixed costs: Multiple firms each bearing R&D, setup, and marketing costs for very similar products is inefficient.

Argument AGAINST (Not Too Many Brands):

1. Consumer preference for variety: Consumers value having choices — one brand cannot satisfy all tastes.
2. Innovation: Competition through differentiation drives quality improvement.
3. Lower prices than monopoly: Even with many brands, competition keeps prices lower than a single-brand monopoly.
4. The market reflects revealed preference: If consumers didn't value variety, they wouldn't buy niche brands — the market responds to real demand.

Q4. Why Is Cournot Equilibrium Stable? Why Don't Firms Collude?

Why Cournot is stable:

At Cournot equilibrium, each firm is producing its best response to the other's output. By definition:

• If Firm 1 takes Q₂ as given, Q₁* maximizes its profit.
• If Firm 2 takes Q₁ as given, Q₂* maximizes its profit.
• Neither firm can increase profit by unilaterally changing output — this is the Nash equilibrium condition.

If a firm were to deviate:

• Increasing output → drives down market price → reduces profit (the MR = MC condition is violated).
• Decreasing output → gives up profitable sales.

Why don't firms collude to maximize joint profit?

1. Incentive to cheat: At the collusive (joint monopoly) output, each firm's marginal revenue exceeds its marginal cost — it would be individually profitable to produce more.
2. Legal barriers: Explicit collusion is illegal in most jurisdictions (antitrust law).
3. Detection and retaliation: Even secret collusion risks detection and punishment.
4. Prisoner's dilemma: Both firms are better off colluding, but each is individually better off defecting. The Nash equilibrium of the one-shot game is to produce the Cournot quantity.

Q5. Why Does the Stackelberg Leader Have an Advantage?

Concept: In Stackelberg, Firm 1 (leader) commits to an output level before Firm 2 (follower) decides.

The leader's advantage:

1. Commitment power: By choosing first and credibly committing, the leader shapes the follower's best response.
2. The leader exploits the reaction curve: The leader knows Firm 2's reaction function R₂(Q₁) and chooses Q₁ to maximize its own profit subject to Firm 2 acting optimally.
3. Leader produces more than Cournot: Since the leader knows the follower will reduce output in response (reaction curves are downward-sloping), the leader can produce a large quantity and the follower accommodates with less.
4. Higher total output: The Stackelberg outcome has more total output than Cournot, which means a lower price — but the leader captures more market share.

Result: The leader earns more than it would in Cournot; the follower earns less.

Q6. Cournot vs. Bertrand — Similarities and Differences

Common elements:

• Both are simultaneous-move duopoly models.
• Both involve Nash equilibrium — each firm optimizes given the other's choice.
• Both result in outcomes between monopoly (less output, higher price) and perfect competition.

Key difference — the strategic variable:

Cournot is more realistic for industries where firms set capacity in advance (e.g., airlines, oil refineries).

Bertrand is more realistic for industries where goods are homogeneous and production is flexible (e.g., commodity spot markets, some retail).

Q7. Nash-Bertrand Equilibrium — Meaning, Stability, and Why No Collusion

Meaning:

• In Bertrand competition with homogeneous goods, the Nash equilibrium is P₁ = P₂ = MC.
• Each firm prices at marginal cost. No firm can profitably undercut further, and if either firm raises price, it loses all customers.

Why it's stable:

• At P = MC, there is no incentive to deviate. Raising price → lose all customers. Cutting price → sell more but at a loss.
• This is the Nash equilibrium: best response to the other firm pricing at MC is also to price at MC.

Why don't firms raise prices to joint-profit-maximizing level?

• Individual incentive to defect: If Firm 1 raises price, Firm 2 can slightly undercut and steal all customers. Firm 1 knows this.
• This is the Bertrand paradox — even with only two firms, competition drives price to MC, eliminating all profits.
• Collusion is unstable: each firm has an incentive to slightly undercut the cartel price and capture the entire market.

Q8. Kinked Demand Curve — How It Works, Limitations, and Price Rigidity

How the model works:

The kinked demand curve explains price rigidity in oligopoly:

• If a firm raises price, rivals do NOT follow — customers switch to rivals → demand is elastic above current price.
• If a firm lowers price, rivals match it to avoid losing market share → demand is inelastic below current price.
• Result: Demand curve has a kink at the current price.
• The MR curve has a gap (discontinuity) at the kink.
• As long as MC intersects the MR gap, the firm won't change price even if costs shift moderately.

Limitations:

1. Doesn't explain how the current price was determined — only why it is sticky once established.
2. Empirically weak — price rigidity is not always observed in oligopolies.
3. Assumes rivals follow price cuts but not increases — this assumption is not always correct.
4. No strategic basis — it's an ad hoc observation, not derived from Nash equilibrium.

Why price rigidity occurs: Oligopolists fear that price wars will harm all firms, and fear losing market share from unilateral price hikes. The result is a reluctance to change prices — better to let MC fluctuate within the MR gap.

Q9. Why Does Price Leadership Evolve? How Does the Price Leader Maximize Profit?

Why price leadership evolves:

In oligopolistic markets with one dominant firm and competitive fringe:

• Coordination on price is needed to avoid destructive price wars.
• The dominant firm (lowest cost, largest market share) has the best information and most to gain from setting the industry price.
• Smaller firms (fringe) accept the dominant firm's price and act as price-takers within that price.

How the price leader determines profit-maximizing price:

1. Identify the residual demand facing the dominant firm:
   • D_dominant = Market Demand − Fringe Supply
   • The fringe supplies whatever it wants at the leader's price.
2. The dominant firm finds its MR from the residual demand curve.
3. Sets MR_dominant = MC_dominant to find optimal Q.
4. Reads off the price from the residual demand curve.
5. Fringe firms then supply the remainder at that price.

Q10. OPEC Success vs. CIPEC Failure — Conditions for Successful Cartelization

OPEC (Oil): Relatively successful in raising prices (especially 1973, 1979). CIPEC (Copper): Failed to significantly raise prices.

Why the difference?

General Conditions for Successful Cartelization:

1. Inelastic demand — buyers can't easily substitute.
2. Large market share among cartel members.
3. Few non-cartel producers — limits competitive supply.
4. Ability to monitor and punish cheaters.
5. Cost similarity across members — reduces incentive to undercut.
6. Political cohesion — helps sustain agreement.

Organizational problems a cartel must overcome:

• Allocating output quotas fairly among members.
• Detecting and punishing cheating.
• Maintaining cohesion when members have different costs and objectives.

EXERCISES

Exercise 1: Merger of All Monopolistically Competitive Firms

Question: If all firms merged into one, would it produce as many brands? Would it produce only one brand?

Answer: It would produce FEWER brands than the sum of individual firms, but NOT just one brand.

Reasoning:

• Before merger: Each firm maximizes its own profit. Multiple firms produce multiple brands, including some with overlapping appeal — leading to "excess brands" from a social standpoint.
• After merger: The merged firm internalizes the cannibalization effect — each additional brand it produces takes sales from its own existing brands. A single firm wouldn't launch a new brand if it just steals sales from another of its own brands without reducing rival sales.
• Result: The merged firm eliminates brands that are too close substitutes for each other. It retains brands that target distinct market segments.
• Why not one brand? Different consumers have genuinely different preferences. Serving them all with one product would lose customers at the tails of the preference distribution. The merged firm maximizes profit by maintaining some variety — just less than the competitive fragmentation.

Managerial Implication: Post-merger brand rationalization is profit-maximizing, not just cost-cutting.

Exercise 2: Two-Firm Cournot — Joint Max, Nash, and Takeover Value

Given:

• P = 50 − 5Q, where Q = Q₁ + Q₂
• C₁(Q₁) = 20 + 10Q₁; C₂(Q₂) = 10 + 12Q₂
• MC₁ = 10, MC₂ = 12

(a) Joint Profit-Maximizing Output

Step 1: Joint profit = (50 − 5Q)Q − 20 − 10Q₁ − 10 − 12Q₂

To maximize, treat it as one firm. Joint MR = 50 − 10Q.

But the marginal cost depends on which firm produces: use the lower-MC firm first.

Joint MR = MC₁ = 10 (use Firm 1 as long as MC₁ < MC₂): 50 − 10Q = 10 → Q* = 4

Check: Marginal cost of Q₁ is 10; marginal cost of Q₂ is 12. Since MR at Q = 4 is 10 = MC₁, it's optimal to have Firm 1 produce all Q = 4 units.

At Q = 4: P = 50 − 5(4) = $30

Revenue = 30 × 4 = $120 Cost = (20 + 10×4) + (10 + 12×0) = 60 + 10 = $70 π_joint = $50

If firms have entered: Firm 1 produces Q₁* = 4, Firm 2 produces Q₂* = 0. If not entered: Firm 2 would choose not to enter (it would earn negative profit at any positive Q at P = $30, since AC₂ > $30 for small Q).

If firms have not yet entered: Only Firm 1 would enter (lower MC). Firm 2 would not enter since its minimum AC > monopoly price.

(b) Cournot Equilibrium

Step 1: Profit functions.

π₁ = (50 − 5Q₁ − 5Q₂)Q₁ − 20 − 10Q₁ = 50Q₁ − 5Q₁² − 5Q₁Q₂ − 20 − 10Q₁

π₂ = (50 − 5Q₁ − 5Q₂)Q₂ − 10 − 12Q₂

Step 2: Reaction curves.

∂π₁/∂Q₁ = 50 − 10Q₁ − 5Q₂ − 10 = 0 → R₁: Q₁ = (40 − 5Q₂)/10 = 4 − Q₂/2

∂π₂/∂Q₂ = 50 − 5Q₁ − 10Q₂ − 12 = 0 → R₂: Q₂ = (38 − 5Q₁)/10 = 3.8 − Q₁/2

Step 3: Solve simultaneously.

Substitute R₁ into R₂: Q₂ = 3.8 − (4 − Q₂/2)/2 Q₂ = 3.8 − 2 + Q₂/4 Q₂ − Q₂/4 = 1.8 3Q₂/4 = 1.8 Q₂* = 2.4

Q₁* = 4 − 2.4/2 = 4 − 1.2 = 2.8

Q* = 2.8 + 2.4 = 5.2 P* = 50 − 5(5.2) = $24

Profits: π₁ = (24 − 10)(2.8) − 20 = 14(2.8) − 20 = 39.2 − 20 = $19.20 π₂ = (24 − 12)(2.4) − 10 = 12(2.4) − 10 = 28.8 − 10 = $18.80

© Takeover Value of Firm 2 for Firm 1

If Firm 1 acquires Firm 2 (and they collude): Joint monopoly profit (from part a) = $50 (all produced by Firm 1 at Q₁ = 4). Firm 2's fixed cost would be eliminated: savings = $10. Total value to Firm 1 of merged entity ≈ $50 (net of fixed costs) vs. $19.20 standalone.

Maximum Firm 1 would pay for Firm 2: = (Joint monopoly π) − (Firm 1's standalone Cournot π) = $50 − $19.20 = $30.80

But Firm 2's current Cournot profit = $18.80. The takeover value must exceed $18.80 (otherwise Firm 2 won't sell).

Firm 1 would pay between $18.80 and $30.80 for Firm 2.

Exercise 3: Monopolist then Cournot Duopoly; N-Firm Cournot

Given:

• MC = AC = $5
• Q = 53 − P → P = 53 − Q

(a) Monopoly

MR = 53 − 2Q = 5 → Q* = 24, P* = 53 − 24 = $29 π_monopoly = (29 − 5)(24) = 24 × 24 = $576

(b) Profit Functions with Two Firms

P = 53 − Q₁ − Q₂

π₁ = (53 − Q₁ − Q₂ − 5)Q₁ = (48 − Q₁ − Q₂)Q₁

π₂ = (48 − Q₁ − Q₂)Q₂

© Reaction Curves

∂π₁/∂Q₁ = 48 − 2Q₁ − Q₂ = 0 → R₁: Q₁ = (48 − Q₂)/2 = 24 − Q₂/2

∂π₂/∂Q₂ = 48 − Q₁ − 2Q₂ = 0 → R₂: Q₂ = (48 − Q₁)/2 = 24 − Q₁/2

(d) Cournot Equilibrium

By symmetry (identical firms): Q₁ = Q₂ = Q*

Q* = 24 − Q/2 3Q/2 = 24 Q* = 16 each

Q_total = 32 P* = 53 − 32 = $21 π_each = (21 − 5)(16) = 16 × 16 = $256

(e) N-Firm Cournot

Each firm's reaction: Q_i = (48 − ΣQ_j)/2 where ΣQ_j = total output of others.

By symmetry: Q_i = Q* for all firms. Q* = [48 − (N−1)Q]/2 2Q = 48 − (N−1)Q* Q(2 + N − 1) = 48 Q(N + 1) = 48 Q_i* = 48/(N + 1)

Total output: Q_total = 48N/(N+1)

Market price: P* = 53 − 48N/(N+1) = 53 − 48N/(N+1) = [53(N+1) − 48N]/(N+1) = [53N + 53 − 48N]/(N+1) = (5N + 53)/(N + 1)

Profit each: π = (P* − 5) × Q_i = [48/(N+1)] × [48/(N+1)] = (48/(N+1))²

As N → ∞: P* = (5N + 53)/(N + 1) → 5 (as N dominates) = P = MC = 5 (perfectly competitive price) ✓

This confirms that as the number of Cournot competitors grows, the market outcome approaches perfect competition.

Exercise 4: Stackelberg — Continuation of Exercise 3

Setup: Firm 1 is the leader. Same costs: MC = $5.

(a) Reaction Curves

From Exercise 3: R₂: Q₂ = 24 − Q₁/2 (follower's reaction — same as Cournot reaction)

Firm 1's reaction: same formula, but as leader it optimizes knowing R₂.

Firm 1 maximizes: π₁ = (48 − Q₁ − Q₂)Q₁ Substitute Q₂ = 24 − Q₁/2: π₁ = (48 − Q₁ − 24 + Q₁/2)Q₁ = (24 − Q₁/2)Q₁ = 24Q₁ − Q₁²/2

dπ₁/dQ₁ = 24 − Q₁ = 0 → Q₁* = 24

Q₂* = 24 − 24/2 = 24 − 12 = 12

Q_total = 36 P* = 53 − 36 = $17

Profits: π₁ = (17 − 5)(24) = 12 × 24 = $288 π₂ = (17 − 5)(12) = 12 × 12 = $144

Comparison:

Leader (Firm 1) benefits from going first: $288 > $256 (Cournot). Follower is hurt: $144 < $256.

Exercise 5: Environmental Regulation — Firm 2's MC Rises to $15; Does Price Rise to Monopoly Level?

Given:

• P = 30 − Q; Q = Q₁ + Q₂
• MC₁ = 0, MC₂ = $15

Claim: Market price will rise to monopoly level. True or False?

Step 1: Find new Cournot equilibrium.

π₁ = (30 − Q₁ − Q₂)Q₁ → FOC: 30 − 2Q₁ − Q₂ = 0 → R₁: Q₁ = (30 − Q₂)/2

π₂ = (30 − Q₁ − Q₂ − 15)Q₂ = (15 − Q₁ − Q₂)Q₂ → FOC: 15 − Q₁ − 2Q₂ = 0 → R₂: Q₂ = (15 − Q₁)/2

Solve: Q₁ = (30 − Q₂)/2 = (30 − (15 − Q₁)/2)/2 Q₁ = 15 − (15 − Q₁)/4 4Q₁ = 60 − 15 + Q₁ 3Q₁ = 45 Q₁* = 15

Q₂* = (15 − 15)/2 = 0

Q* = 15, P* = 30 − 15 = $15

Step 2: What is the monopoly price if Firm 1 were alone (MC = 0)?

MR = 30 − 2Q = 0 → Q_monopoly = 15, P_monopoly = $15

The statement is TRUE (in a specific sense): The market price does equal Firm 1's monopoly price of $15. But this happens because Firm 2 is effectively driven out (Q₂ = 0), not because the firms collude. Firm 1 simply operates as a monopolist since Firm 2 cannot profitably produce at P = $15 with MC₂ = $15.

Nuance: Price rose not through collusion but through Firm 2's competitive exit. This is a Cournot equilibrium, not a cartel outcome. The claim is technically true but for the wrong reason if "monopoly level" implies collusion.

Exercise 6: Cournot, Cartel, Monopoly, and Cheating

Given:

• C₁ = 60Q₁, C₂ = 60Q₂ → MC = 60 for both
• P = 300 − Q, Q = Q₁ + Q₂

(a) Cournot-Nash Equilibrium

π₁ = (300 − Q₁ − Q₂ − 60)Q₁ = (240 − Q₁ − Q₂)Q₁ FOC: 240 − 2Q₁ − Q₂ = 0 → R₁: Q₁ = (240 − Q₂)/2 = 120 − Q₂/2

By symmetry: **Q₁ = Q₂ = Q***

Q* = 120 − Q/2 → 3Q/2 = 120 → Q* = 80 each

Q_total = 160 P* = 300 − 160 = $140 π_each = (140 − 60)(80) = 80 × 80 = $6,400

(b) Cartel (Joint Profit Maximization)

Joint MR = 300 − 2Q = 60 → Q_total = 120

Each firm: Q₁ = Q₂ = 60 P* = 300 − 120 = $180 π_each = (180 − 60)(60) = 120 × 60 = $7,200

© Monopoly (Only Firm 1)

Same calculation: Q = 120, P = $180, π = (180−60)(120) = $14,400

Compared to cartel: Monopolist earns $14,400 vs. total cartel profit of $14,400. Same total output, but now all profit goes to Firm 1.

Difference from cartel: Output and price are the same. The only difference is profit allocation — monopolist gets 100% vs. 50% each under cartel.

(d) Firm 1 Abides, Firm 2 Cheats (Cartel Quantity = 60, but Firm 2 Maximizes Given Q₁ = 60)

Firm 2's best response to Q₁ = 60: Q₂ = 120 − Q₁/2 = 120 − 30 = 90

Q_total = 60 + 90 = 150 P = 300 − 150 = $150

π₁ = (150 − 60)(60) = 90 × 60 = $5,400 (punished for honoring the cartel) π₂ = (150 − 60)(90) = 90 × 90 = $8,100 (Firm 2 cheats and profits)

The prisoner's dilemma in action: If Firm 1 stays honest and Firm 2 cheats: Firm 2 earns $8,100 > cartel $7,200. Firm 1 earns $5,400 < cartel $7,200. This incentive to cheat undermines cartel stability.

Exercise 7: Firms A and B, MC = $50 — Three Equilibria Under Cost Changes

Setup: Two identical firms, MC = $50. Assume demand P = a − bQ.

For each scenario, analyze Cournot, Collusive, and Bertrand equilibria.

(a) Firm A's MC Increases to $80

Cournot:

• Firm A's reaction curve shifts left (higher costs → lower optimal output).
• Q_A falls, Q_B rises (B optimally produces more as A contracts).
• Total output changes ambiguously — A reduces output, B partially compensates.
• Price rises (net effect: total output falls; P rises).

Collusive:

• The cartel would now shift more production to lower-cost Firm B.
• Q_A falls significantly; Q_B rises; total output = monopoly level.
• Price remains at cartel level (unchanged), but Firm A's share of production drops.

Bertrand:

• Firm B (lower cost at $50) undercuts Firm A.
• P = $50 (the lower firm's MC).
• Firm A exits or prices at $50 and earns zero.
• Firm B captures the entire market at P = MC_B = $50.

(b) MC of Both Firms Increases

Cournot:

• Both reaction curves shift inward.
• Both firms produce less; total output falls; price rises.

Collusive:

• Monopoly output falls; price rises by more than in Cournot.
• Both firms share production at the new cartel quantity.

Bertrand:

• Price rises to the new MC level. Both firms price at the new (higher) MC.
• With homogeneous goods and P = MC, both still earn zero profit.

© Demand Shifts Right

Cournot:

• Both reaction curves shift outward.
• Both firms produce more; total output rises; price may rise or fall depending on the demand shift.
• Typically, price rises with a rightward demand shift (higher willingness to pay).

Collusive:

• Monopoly output rises; price rises (higher demand at every quantity).

Bertrand:

• Output rises to meet demand at P = MC.
• Price remains at MC — no market power, just higher quantity sold.

Exercise 8: Airlines — Cournot, Stackelberg, Cost Reduction

Given:

• C(q) = 40q for both, MC = $40
• P = 100 − Q, Q = Q_A + Q_T (American and Texas Air)

(a) Cournot-Nash Equilibrium

π_A = (100 − Q_A − Q_T − 40)Q_A = (60 − Q_A − Q_T)Q_A

FOC: 60 − 2Q_A − Q_T = 0 → R_A: Q_A = (60 − Q_T)/2

By symmetry: **Q_A = Q_T = Q***

Q* = 30 − Q/2 → **Q = 20 each**

Q_total = 40 P* = 100 − 40 = $60 π_each = (60 − 40)(20) = 20 × 20 = $400

(b) Texas Air MC = $25, American MC = $40 (Stackelberg-like Asymmetric Cournot)

Reaction curves:

Texas Air (T): π_T = (100 − Q_T − Q_A − 25)Q_T = (75 − Q_T − Q_A)Q_T FOC: 75 − 2Q_T − Q_A = 0 → R_T: Q_T = (75 − Q_A)/2

American (A): π_A = (60 − Q_T − Q_A)Q_A FOC: R_A: Q_A = (60 − Q_T)/2

Solve simultaneously: Q_T = (75 − Q_A)/2 and Q_A = (60 − Q_T)/2

Substitute R_A into R_T: Q_T = (75 − (60 − Q_T)/2)/2 2Q_T = 75 − (60 − Q_T)/2 4Q_T = 150 − 60 + Q_T 3Q_T = 90 Q_T* = 30

Q_A* = (60 − 30)/2 = 15

Q_total = 45 P* = 100 − 45 = $55 π_T = (55 − 25)(30) = $900 π_A = (55 − 40)(15) = $225

© Investment in Cost Reduction — Texas Air's Willingness to Pay

Texas Air: Reduce MC from $40 to $25 (American stays at $40).

From part (b): Texas Air earns $900. Without investment (symmetric Cournot at $40): Texas Air earns $400.

Texas Air's gain = $900 − $400 = $500. Maximum willingness to invest = $500.

American: Reduce MC from $40 to $25 (Texas Air already has $25).

If American reduces to $25: symmetric Cournot with MC = $25 for both. Q* = (75 − Q*)/2 − ...

Actually: if both have MC = $25, symmetric Cournot: Q_i = (75 − Q_i)/2 × ...

Both firms' reaction: Q = (75 − Q)/2 → Q = 25 each → Q_total = 50, P = $50 π_each = (50 − 25)(25) = $625

American's gain from reducing to $25 (vs. earning $225 in the asymmetric case): = $625 − $225 = $400. Maximum willingness to invest = $400.

Texas Air invests up to $500; American up to $400. Texas Air has stronger incentive to invest since it gains a larger competitive advantage.

Exercise 9: Light Bulbs — Perfect Competition, Cournot, Stackelberg, Collusion

Given:

• Q = 100 − P → P = 100 − Q
• C_i = 10Q_i + ½Q_i², so MC_i = 10 + Q_i
• Q = Q_E + Q_D

(a) Perfect Competition (P = MC for each firm)

Each firm: P = MC_i = 10 + Q_i → Q_i = P − 10

Total supply (2 firms): Q_s = 2(P − 10) = 2P − 20

Set Q_s = Q_d: 2P − 20 = 100 − P 3P = 120 P* = $40 Q_i = 40 − 10 = 30 each; Q_total = 60

π_each = TR − TC = 40(30) − [10(30) + 0.5(900)] = 1200 − [300 + 450] = $450 each

Wait — under perfect competition, π should be near zero with free entry. But with fixed costs = 0 here, firms earn producer surplus. This is short-run competitive equilibrium without free entry.

(b) Cournot Equilibrium

π_E = (100 − Q_E − Q_D − 10)Q_E − ½Q_E² = (90 − Q_E − Q_D)Q_E − ½Q_E²

FOC w.r.t. Q_E: 90 − 2Q_E − Q_D − Q_E = 0 → 90 − 3Q_E − Q_D = 0 → R_E: Q_E = (90 − Q_D)/3

By symmetry: **Q_E = Q_D = Q***

Q* = (90 − Q)/3 3Q = 90 − Q* 4Q* = 90 Q* = 22.5 each

Q_total = 45 P* = 100 − 45 = $55 π_each = (55)(22.5) − [10(22.5) + 0.5(22.5)²] = 1237.5 − [225 + 253.125] = 1237.5 − 478.125 = $759.375 each

© Stackelberg (Everglow Leads)

Everglow knows D_imlit's reaction: Q_D = (90 − Q_E)/3

Everglow maximizes: π_E = (90 − Q_E − (90−Q_E)/3)Q_E − ½Q_E² = (90 − Q_E − 30 + Q_E/3)Q_E − ½Q_E² = (60 − 2Q_E/3)Q_E − ½Q_E² = 60Q_E − 2Q_E²/3 − Q_E²/2

FOC: 60 − 4Q_E/3 − Q_E = 0 60 = Q_E(1 + 4/3) = 7Q_E/3 Q_E* = 60 × 3/7 = 180/7 ≈ 25.71

Q_D* = (90 − 180/7)/3 = (630/7 − 180/7)/3 = (450/7)/3 = 150/7 ≈ 21.43

Q_total = 330/7 ≈ 47.14 P* = 100 − 330/7 = 370/7 ≈ $52.86

π_E = (P − MC_E) × Q_E − ½Q_E² MC_E at Q_E = 25.71: MC = 10 + 25.71 = 35.71 π_E = (52.86 − 35.71)(25.71) − ½(25.71)² ...

Actually using TR − TC: TR_E = 52.86 × 25.71 ≈ $1,359.2 TC_E = 10(25.71) + 0.5(25.71)² = 257.1 + 330.5 = $587.6 π_E ≈ $771.6

π_D: TR_D = 52.86 × 21.43 ≈ $1,132.9 TC_D = 10(21.43) + 0.5(21.43)² = 214.3 + 229.6 = $443.9 π_D ≈ $689.0

Everglow earns more as Stackelberg leader than in Cournot ($771.6 > $759.4).

(d) Collusion

Maximize joint profit: π = (100 − Q − 10)Q − Q² [both firms split Q equally, same cost structure]

Actually with symmetric costs: Joint MR = MC. Joint profit = PQ − C_total

With Q = Q_E + Q_D and minimizing cost (equal split since same MC): Q_E = Q_D = Q/2; C_total = 2[10(Q/2) + 0.5(Q/2)²] = 10Q + Q²/4

π_joint = (100 − Q)Q − 10Q − Q²/4 = 90Q − Q² − Q²/4 = 90Q − 5Q²/4

FOC: 90 − 5Q/2 = 0 → Q_total* = 36

Each firm: Q_E = Q_D = 18 P* = 100 − 36 = $64

π_each = (64)(18) − [10(18) + 0.5(324)] = 1152 − [180 + 162] = 1152 − 342 = $810 each

Summary Table:

Exercise 10: WW and BBBS — Cournot, Collusion, Payoff Matrix, Stackelberg

Given:

• C(q) = 30q + 1.5q²; MC = 30 + 3q
• P = 300 − 3Q, Q = q₁ + q₂

(a) Cournot Equilibrium

π₁ = (300 − 3q₁ − 3q₂ − 30 − 3q₁)q₁ = (270 − 6q₁ − 3q₂)q₁

Wait — revenue minus cost: π₁ = P × q₁ − C(q₁) = (300 − 3q₁ − 3q₂)q₁ − 30q₁ − 1.5q₁²

FOC: ∂π₁/∂q₁ = 300 − 6q₁ − 3q₂ − 30 − 3q₁ = 0 270 − 9q₁ − 3q₂ = 0 → R₁: q₁ = (270 − 3q₂)/9 = 30 − q₂/3

By symmetry: **q₁ = q₂ = q***

q* = 30 − q/3 4q/3 = 30 q* = 22.5 each

Q* = 45 P* = 300 − 3(45) = 300 − 135 = $165 π_each = 165(22.5) − [30(22.5) + 1.5(22.5)²] = 3712.5 − [675 + 759.375] = 3712.5 − 1434.375 = $2,278.13 each

(b) Collusion — Joint Profit Maximization

Maximize joint profit: π_joint = (300 − 3Q)Q − 2(30q + 1.5q²) where Q = 2q (by symmetry)

π_joint = (300 − 6q)(2q) − 60q − 3q² = 600q − 12q² − 60q − 3q² = 540q − 15q²

FOC: 540 − 30q = 0 → q* = 18 each

Q_total = 36 P* = 300 − 3(36) = $192 π_each = 192(18) − [30(18) + 1.5(324)] = 3456 − [540 + 486] = 3456 − 1026 = $2,430 each

© Payoff Matrix

Need payoffs for all four combinations:

Case 1: Both play Cournot (q = 22.5 each): π_WW = π_BBBS = $2,278

Case 2: WW plays Cournot (22.5), BBBS plays Cartel (18):

WW's best response to q₂ = 18: q₁ = 30 − 18/3 = 30 − 6 = 24

P = 300 − 3(24 + 18) = 300 − 126 = $174 π_WW = 174(24) − [30(24) + 1.5(576)] = 4176 − [720 + 864] = 4176 − 1584 = $2,592 π_BBBS = 174(18) − [30(18) + 1.5(324)] = 3132 − 1026 = $2,106

Case 3: WW plays Cartel (18), BBBS plays Cournot (22.5): By symmetry: π_WW = $2,106, π_BBBS = $2,592

Case 4: Both play Cartel (18 each): π_WW = π_BBBS = $2,430

Payoff Matrix:

Analysis: This is a prisoner's dilemma:

• Each firm is better off cheating (playing Cournot) if the other plays cartel: $2,592 > $2,430.
• If both cheat: $2,278 < $2,430.
• Dominant strategy: Play Cournot. Nash equilibrium is (Cournot, Cournot) at ($2,278, $2,278).

(d) Stackelberg — WW Chooses Output First

WW maximizes knowing BBBS follows reaction: q₂ = 30 − q₁/3

π_WW = (300 − 3q₁ − 3q₂)q₁ − 30q₁ − 1.5q₁² Substitute q₂ = 30 − q₁/3: = (300 − 3q₁ − 3(30 − q₁/3))q₁ − 30q₁ − 1.5q₁² = (300 − 3q₁ − 90 + q₁)q₁ − 30q₁ − 1.5q₁² = (210 − 2q₁)q₁ − 30q₁ − 1.5q₁² = 210q₁ − 2q₁² − 30q₁ − 1.5q₁² = 180q₁ − 3.5q₁²

FOC: 180 − 7q₁ = 0 → q₁* = 180/7 ≈ 25.71

q₂* = 30 − 25.71/3 = 30 − 8.57 = 21.43

Q_total = 47.14 P* = 300 − 3(47.14) = 300 − 141.43 = $158.57

π_WW = 158.57(25.71) − [30(25.71) + 1.5(25.71)²] = 4077.1 − [771.4 + 991] = 4077.1 − 1762.4 = $2,314.7

π_BBBS = 158.57(21.43) − [30(21.43) + 1.5(21.43)²] = 3398.6 − [642.9 + 689.0] = 3398.6 − 1331.9 = $2,066.7

Is WW better off going first? WW earns $2,314.7 as Stackelberg leader vs. $2,278.1 in Cournot → Yes, slightly better off. BBBS earns $2,066.7 as follower vs. $2,278.1 in Cournot → Worse off.

First-mover advantage exists but is modest in this case.

Exercise 11: Bertrand Competition with Differentiated Products

Given:

• Q₁ = 20 − P₁ + P₂; Q₂ = 20 + P₁ − P₂
• MC = 0

Note: If both firms charge the same price P, Q₁ = Q₂ = 20 (demand independent of P when prices equal). Firms can extract infinite profits if both raise prices together — hence the unusual note in the problem.

(a) Simultaneous Nash-Bertrand Equilibrium

Maximize π₁ with respect to P₁: π₁ = P₁ × Q₁ = P₁(20 − P₁ + P₂)

∂π₁/∂P₁ = 20 − 2P₁ + P₂ = 0 → R₁: P₁ = (20 + P₂)/2 = 10 + P₂/2

By symmetry: R₂: P₂ = 10 + P₁/2

Solve: P₁ = 10 + (10 + P₁/2)/2 = 10 + 5 + P₁/4 P₁ − P₁/4 = 15 3P₁/4 = 15 P₁* = P₂* = $20

Q₁ = Q₂ = 20 − 20 + 20 = 20 each

π₁ = π₂ = 20 × 20 = $400 each

(b) Sequential Pricing — Firm 1 Sets Price First

Firm 2 sets price knowing P₁: R₂: P₂ = 10 + P₁/2

Firm 1 maximizes π₁ given Firm 2's reaction: P₂ = 10 + P₁/2

π₁ = P₁(20 − P₁ + P₂) = P₁(20 − P₁ + 10 + P₁/2) = P₁(30 − P₁/2) = 30P₁ − P₁²/2

dπ₁/dP₁ = 30 − P₁ = 0 → P₁* = $30

P₂* = 10 + 30/2 = $25

Q₁ = 20 − 30 + 25 = 15 Q₂ = 20 + 30 − 25 = 25

π₁ = 30 × 15 = $450 π₂ = 25 × 25 = $625

© Which Game Would You Prefer?

You would prefer the rival to set price first → You earn $625.

Why? With differentiated products and upward-sloping reaction curves (prices are strategic complements), being the follower is advantageous. The leader sets a high price ($30), and you optimally respond with a slightly lower but still high price ($25) — capturing more customers while still pricing high. This is the second-mover advantage in Bertrand competition with differentiation.

This is opposite to Cournot (quantities) where being the leader is advantageous (quantities are strategic substitutes).

Exercise 12: OPEC Oil Cartel — Dominant Firm Model

Given:

• World demand: W = 160P^(−1/2)
• Non-OPEC supply: S = (13⅓)P^(1/2) = (40/3)P^(1/2)
• OPEC's net demand: D = W − S
• OPEC MC ≈ 0

(a) Diagram Description

• W is a downward-sloping demand curve.
• S is an upward-sloping non-OPEC supply curve.
• D = W − S is OPEC's residual demand (more elastic than W; shifts with S).
• OPEC's MR is below D.
• Optimal OPEC production: where MR_OPEC = MC = 0 (i.e., where MR = 0).
• Optimal OPEC price: read from D at that quantity.
• Non-OPEC production: read from S at that price.

If non-OPEC oil becomes more expensive (reserves run out):

• S shifts left (less supply at each price).
• D shifts right (OPEC's residual demand increases).
• OPEC's MR shifts right.
• OPEC's optimal price rises.

(b) OPEC's Optimal Price (MC = 0 → Maximize Revenue)

OPEC revenue = P × D = P × (W − S) = P × [160P^(−1/2) − (40/3)P^(1/2)] = 160P^(1/2) − (40/3)P^(3/2)

Maximize with respect to P: dR/dP = 80P^(−1/2) − (40/3)(3/2)P^(1/2) = 0 80P^(−1/2) = 20P^(1/2) 80 = 20P P* = $4 per barrel

Verify: D at P = 4: W = 160(4)^(−1/2) = 160/2 = 80; S = (40/3)(4)^(1/2) = (40/3)(2) = 80/3 ≈ 26.67 D = 80 − 26.67 = 53.33 mb/d (OPEC production)

Revenue = 4 × 53.33 = $213.3 million/day

© Buyers' Cartel (Monopsony Power)

What we can say:

• A buyers' cartel acts as a monopsonist — it reduces the quantity it buys to drive the price down below competitive levels.
• It would face OPEC's residual supply as an upward-sloping supply curve.
• The buyers' cartel sets MFC (marginal factor cost) = marginal value of oil.
• Price would fall — the buyers' cartel successfully exercises countervailing power.

What we cannot say:

• Exactly how much the price would fall — this depends on the specific demand, supply, and cartel cohesion.
• Whether the buyers' cartel would hold together — member countries have incentives to cheat (buy more oil individually if the cartel price is below their willingness to pay).
• The ultimate outcome also depends on whether OPEC adjusts its behavior in response.

Conclusion: A buyers' cartel could reduce the oil price, but its effectiveness depends on cohesion, member incentives, and OPEC's strategic response.

Exercise 13: Dominant Firm and Fringe — Tennis Shoes

Given:

• Market demand: Q = 400 − 2P → P = 200 − Q/2
• Dominant firm MC = $20 (constant)
• 5 fringe firms each with MC = 20 + 5q

(a) Fringe Supply Curve

Each fringe firm: P = MC → P = 20 + 5q → q = (P − 20)/5

Total fringe supply (5 firms): Q_f = 5q = 5 × (P − 20)/5 = P − 20 ✓

Active when P ≥ 20.

(b) Dominant Firm's Demand Curve

D_dominant = Market Demand − Fringe Supply = (400 − 2P) − (P − 20) = 400 − 2P − P + 20 = 420 − 3P

Inverse: P = 140 − Q_D/3

© Dominant Firm's Optimal Q and P; Fringe Quantities

MR_dominant: MR = 140 − 2Q_D/3

Set MR = MC = 20: 140 − 2Q_D/3 = 20 2Q_D/3 = 120 Q_D* = 180

P*** = 140 − 180/3 = 140 − 60 = **$80

Fringe supply at P = $80: Q_f = 80 − 20 = 60 (total for all 5 fringe firms) Each fringe firm: q = 60/5 = 12

Check market: Q_D + Q_f = 180 + 60 = 240; Market Q = 400 − 2(80) = 400 − 160 = 240 ✓

Profits:

• Dominant firm: π = (80 − 20)(180) = 60 × 180 = $10,800
• Each fringe firm: π = P × q − TC = 80(12) − [20(12) + 5(12)²/2... ]

Wait, fringe firm TC: if MC = 20 + 5q, then TC = 20q + 5q²/2 (assuming no fixed cost). At q = 12: TC = 20(12) + 2.5(144) = 240 + 360 = $600 Revenue = 80(12) = $960 Fringe firm π = $960 − $600 = $360 each

(d) With 10 Fringe Firms

Total fringe supply: Q_f = 10 × (P − 20)/5 = 2(P − 20) = 2P − 40

D_dominant = (400 − 2P) − (2P − 40) = 440 − 4P

Inverse: P = 110 − Q_D/4

MR = 110 − Q_D/2

Set MR = MC = 20: 110 − Q_D/2 = 20 → Q_D* = 180 (same!)

P* = 110 − 180/4 = 110 − 45 = $65

Q_f = 2(65) − 40 = 90 (more fringe firms producing more)

More fringe firms → lower market price ($65 vs. $80) → dominant firm has less market power.

(e) With 5 Fringe Firms but MC = 20 + 2q (Lower Fringe Costs)

Each fringe: q = (P − 20)/2 Total fringe (5 firms): Q_f = 5(P − 20)/2 = 2.5P − 50

D_dominant = (400 − 2P) − (2.5P − 50) = 450 − 4.5P

Inverse: P = 100 − Q_D/4.5 = 100 − 2Q_D/9

MR = 100 − 4Q_D/9

Set MR = 20: 100 − 4Q_D/9 = 20 4Q_D/9 = 80 Q_D* = 180 (interesting — dominant firm output unchanged again)

P* = 100 − 2(180)/9 = 100 − 40 = $60

Q_f = 2.5(60) − 50 = 150 − 50 = 100

Lower fringe MC → More fringe supply → Lower market price → Dominant firm loses market share.

Exercise 14: Lemon Cartel — Cost Tables, Optimal Allocation, Incentive to Cheat

Given:

• TC₁ = 20 + 5Q₁², MC₁ = 10Q₁
• TC₂ = 25 + 3Q₂², MC₂ = 6Q₂
• TC₃ = 15 + 4Q₃², MC₃ = 8Q₃
• TC₄ = 20 + 6Q₄², MC₄ = 12Q₄

(a) Total, Average, Marginal Cost Tables

Firm 1: TC = 20 + 5Q₁²

Firm 2: TC = 25 + 3Q₂²

Firm 3: TC = 15 + 4Q₃²

Firm 4: TC = 20 + 6Q₄²

(b) Allocating 10 Cartons at P = $25

Optimal cartel allocation = equalize MC across firms (where MC = cartel price = $25).

Set each firm's MC ≤ $25 and find how many units each should produce:

• Firm 1: MC = 10Q₁ = 25 → Q₁* = 2.5
• Firm 2: MC = 6Q₂ = 25 → Q₂* = 4.17
• Firm 3: MC = 8Q₃ = 25 → Q₃* = 3.125
• Firm 4: MC = 12Q₄ = 25 → Q₄* = 2.08

Total = 2.5 + 4.17 + 3.125 + 2.08 = 11.87 > 10. Need to scale back.

For exactly 10 cartons, find the common MC (shadow price λ) such that:

Q₁ + Q₂ + Q₃ + Q₄ = 10 λ/10 + λ/6 + λ/8 + λ/12 = 10

Find common denominator (120): 12λ/120 + 20λ/120 + 15λ/120 + 10λ/120 = 10 57λ/120 = 10 λ = 1200/57 ≈ $21.05

Optimal allocations:

• Q₁ = 21.05/10 ≈ 2.1
• Q₂ = 21.05/6 ≈ 3.5
• Q₃ = 21.05/8 ≈ 2.6
• Q₄ = 21.05/12 ≈ 1.75

Total ≈ 9.95 ≈ 10 ✓ (rounding to integers: Q₁=2, Q₂=4, Q₃=3, Q₄=1 → total = 10)

Using integer approximation: Q₁ = 2, Q₂ = 4, Q₃ = 3, Q₄ = 1: Total = 10 ✓ MC₁=20, MC₂=24, MC₃=24, MC₄=12 — not all equal, but best feasible integer solution.

© Which Firm Has the Most Incentive to Cheat?

At cartel price P = $25 per carton, each firm has an incentive to cheat if its MC at the current quota < P.

At the optimal allocation:

• Firm 1 (Q₁ ≈ 2): MC₁ = 20 < 25 → Incentive to produce more. Potential gain = $25 − $20 = $5 per additional carton.
• Firm 2 (Q₂ ≈ 4): MC₂ = 24 < 25 → Small incentive. Gain = $1/carton.
• Firm 3 (Q₃ ≈ 3): MC₃ = 24 < 25 → Small incentive.
• Firm 4 (Q₄ ≈ 1): MC₄ = 12 << 25 → Largest gap. Gain = $25 − $12 = $13 per additional carton. Firm 4 has the strongest incentive to cheat.

Firm 4 has the most incentive to cheat — it has the lowest MC relative to the cartel price.

Does any firm lack incentive to cheat?

In principle, all firms whose MC at quota < cartel price have incentive to cheat. However:

• The firm closest to the point where MC = P has the weakest incentive.
• If any firm's MC at quota = P exactly, it has no incentive — but this is rare in practice.

At P = $25:

• No firm here has MC exactly equal to $25 at their quota — all have incentive to some degree.
• Firm 2 and Firm 3 have very small incentives ($1 margin); Firm 4 has the largest incentive.

CHAPTER 12 LEARNING ENHANCEMENTS

Key Formula Sheet

N-Firm Cournot (symmetric):

• Each firm: Q_i = (a − c)/[b(N+1)]
• Market Q = N(a−c)/[b(N+1)]
• P = c + (a−c)/(N+1)
• π_i = [(a−c)/(N+1)]² / b
• As N → ∞: P → c (competitive)

Important Equilibrium Conditions

1. Cournot: Each firm maximizes profit treating rival output as fixed. Nash equilibrium at intersection of reaction functions.
2. Stackelberg: Leader uses follower's reaction function in optimization. Leader produces more; follower less than Cournot.
3. Bertrand (homogeneous): P = MC. Undercutting drives price to cost.
4. Bertrand (differentiated): Price > MC possible. Prices are strategic complements — reaction curves slope upward.
5. Monopolistic competition LR: P = AC; MR = MC; excess capacity exists.
6. Cartel: MC equalized across all members. Total output at monopoly level.

Concept Comparison Tables

Cournot vs. Bertrand vs. Stackelberg vs. Collusion

Monopolistic Competition vs. Oligopoly

Common IPMX Exam Traps

1. Stackelberg output vs. Cournot: Leader produces MORE than Cournot; follower produces LESS. Total output is HIGHER under Stackelberg than Cournot. Many students get this backwards.

2. Bertrand paradox: Even with 2 firms, P = MC under Bertrand with homogeneous goods. Students often forget this and write P = monopoly price.

3. Differentiated Bertrand: Second mover has advantage (opposite of Cournot). Be clear about the type of competition.

4. Cartel instability: The cartel maximizes joint profit, but individual incentive to cheat exists for every member whose MC < cartel price. This is the prisoner's dilemma.

5. N-firm Cournot as N → ∞: P → MC (competitive). Students must know this limiting result.

6. Monopolistic competition LR: The demand curve is tangent to AC — NOT to MC. So P > MC (unlike perfect competition) and P = AC (unlike monopoly). This creates excess capacity.

7. Dominant firm model: Dominant firm faces the RESIDUAL demand (market demand minus fringe supply), not the full market demand.

8. Price leadership: The dominant firm sets the price; fringe firms take it as given. Don't confuse with Stackelberg (which is a quantity leadership model).

Quick Revision Summary

Monopolistic competition:

• Short run: can earn profits (like monopoly) or losses.
• Long run: entry eliminates profits; P = AC; excess capacity.
• Too many brands from social efficiency standpoint; valued by consumers.

Oligopoly models:

• Cournot: Simultaneous quantity choice → intermediate outcome between monopoly and competition.
• Bertrand (homogeneous): Simultaneous price choice → competitive outcome (P = MC) with only 2 firms. The Bertrand paradox.
• Stackelberg: Sequential quantity choice → leader benefits from first-mover advantage; produces more than Cournot.
• Bertrand (differentiated): Simultaneous price with differentiation → P > MC; second mover advantage (prices are complements).

Cartels:

• Set output at joint monopoly level; equalize MC across members.
• Inherently unstable: each member has incentive to cheat.
• Success requires: inelastic demand, large cartel market share, ability to monitor and punish.

Dominant firm:

• Sets price based on residual demand (market − fringe).
• Fringe takes price as given and supplies competitively.
• More fringe firms → lower dominant firm's market power.