Firm Strategy & Market Dynamics: Problem Set Insights

This step-by-step analysis covers Problem Sets 6-9, emphasizing key concepts from Problem Sets 7 and 8, essential for your final exam.

Problem Set 6: Product Differentiation & Merger Impacts

1. Why Bertrand Does Not Equal Marginal Cost in Reality

• Firms may experience:

  • Capacity constraints

  • Brand loyalty (differentiated products)

  • Reputational concerns or switching costs

2. Bertrand Competition with Differentiated Products

• Demand:

  • Q_M = 1000 - 200P_M + 100P_B

  • Q_B = 1000 - 200P_B + 100P_M

• Steps:

  1. Plug in rival's price to derive inverse demand.

  2. Derive Marginal Revenue (MR); set MR = Marginal Cost (MC) = 4.

  3. Solve for the best response price.

  4. Set both best responses equal to solve for the Nash Equilibrium (NE).

  5. Calculate quantity, profit, and price-cost margin.

3. Analyzing Merger Effects

• Post-merger, the firm internalizes cross-price effects, leading to higher prices.

• Compare profit at different price points:

  • $5: Profit = (5-4) * 500 = 1000

  • $7: Profit = (7-4) * 300 = 1800

• A merger likely increases prices, resulting in a higher profit margin (43%).

Problem Set 7: Mergers, HHI, and Welfare Impacts

1. HHI and Market Concentration

• HHI = Sum of (market shares in %)^2

• DOJ and FTC Thresholds:

  • Unconcentrated: HHI < 1500

  • Moderate: 1500-2500

  • High: > 2500

  • Red flag: Change in HHI > 200 in a concentrated market.

Calculation Steps:

1. Square each firm's market share.

2. Sum the squared shares for total HHI.

3. Post-merger: Recompute the new HHI.

4. Change in HHI = Post-merger HHI - Pre-merger HHI.

5. Interpret the results against merger guidelines.

2. Cournot Merger and Welfare Analysis

• Demand: Q = 200 - 10P; Inverse Demand: P = 20 - Q/10

• Marginal Cost (MC): $5 (pre-merger) → $3 (post-merger).

Cournot Nash Equilibrium (Pre-Merger):

• Use formula:

  • Q_i = (a - MC) / (3b)

  • Total Q = 100, P = $10

• Consumer Surplus (CS) = 0.5 * (20 - 10) * 100 = 500

• Producer Surplus (PS) = (10 - 5) * 100 = 500

Monopoly Outcome (Post-Merger):

• Marginal Revenue (MR) = 20 - 2Q/10; set MR = $3 → Q = 85, P = $11.5.

• CS decreases to $361.25, PS increases to $722.50, Aggregate Surplus (AS) increases to $1083.75.

Calculation Steps:

1. Determine Quantity (Q) and Price (P) for both market structures.

2. Apply triangle/rectangle formulas to calculate CS and PS.

3. Compute changes:

   • Change in CS = -138.75

   • Change in PS = +222.5

   • Change in AS = +83.75

Policy Implications:

• If the regulator prioritizes Consumer Surplus (CS), the merger should be blocked.

• If Aggregate Surplus (AS) is the primary concern, the merger should be approved.

3. Bertrand Merger Comparison

• Duopoly Price (P_duo) = $5 (Marginal Cost), Quantity (Q) = 150 → CS = 1125, PS = 0.

• After merger: Price = $11.5, Quantity = 85 → CS = $361.25, PS = $722.50.

• Aggregate Surplus (AS) drops, indicating the merger is not justified under a CS or AS focus.

Problem Set 8: Collusion, Entry Games, Predatory Pricing

1. Cartel Sustainability

• Cournot Nash Equilibrium: Ben & Jerry produce 200 units each, Price = $10, Profit = $1000.

• Collusion (150 units each): Price = $12.5, Profit = $1125.

• If Ben cheats: Price = $11.25, Ben's Profit = $1250, Jerry's Profit = $937.50.

Discounted Payoff Check:

• If 2 periods:

  • Collude = 1125 + 0.8 * 1125 = 2025

  • Cheat = 1250 + 0.8 * 1000 = 2050 → Not sustainable.

• If 3 periods:

  • Collude = 2745

  • Cheat = 2690 → Sustainable!

Rule of Thumb for Sustainability:

Discount Factor (δ) ≥ (Profit_Defect - Profit_Collude) / (Profit_Defect - Profit_Nash)

2. Entry Deterrence Game Theory

• The entrant chooses to enter (paying fixed cost F) or not, after the incumbent's move.

• Analysis using Backward Induction:

  • If Fixed Cost (F) = 500: The entrant will not enter unless the incumbent chooses a 'small' strategy.

  • If Fixed Cost (F) = 300: The entrant enters even if the incumbent chooses 'small,' leading the incumbent to likely choose 'small' to prevent a larger loss.

• Optimal strategy depends on the fixed cost and the threat of the incumbent's response.

Step-by-Step Backward Induction:

1. Start at the end of the game tree (bottom) for backward induction.

2. Determine the entrant's best move at each decision node.

3. Use this information to determine the incumbent's optimal choice.

3. Predatory Pricing and Exclusive Contracts

• Common Entry Deterrence Strategies:

  • Build excess capacity

  • Exclusive contracts with retailers

  • Predatory pricing (price below cost to drive out rival)

• Predatory pricing is illegal only if both conditions are met:

  • Price is below Marginal Cost (MC), and

  • The firm expects to recoup its losses after the rival exits the market.

Problem Set 9: Innovation & Willingness to Pay for Tech

1. Willingness to Pay for Cost-Reducing Technology

• Monopoly Scenario:

  • Old Profit = $500 (Marginal Cost = $10)

  • New Profit = $1125 (Marginal Cost = $5)

  • Willingness to Pay (WTP) = $625

2. Willingness to Pay When Facing Entry

• If the entrant acquires the technology, Bertrand competition ensues, and Smoothie-Queen undercuts prices.

• Smoothie-Queen (SQ) Profit ≈ $1000 (captures the market).

• Smoothie-King (SK) Profit = $0.

• Therefore: SK's WTP = $1125 (to avoid zero profit); SQ's WTP = $1000.

Key Insight:

A monopolist is willing to pay more to protect its market position than a duopolist is willing to pay to improve its profit.