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M20.L01 — Monopoly: Formal Treatment and Welfare Loss

Module: Module 20 — Monopoly and Game Theory (Advanced) Lesson: L01 of 06 Duration: ~30 minutes Level: Intermediate Provenance: Intermediate Microeconomics with Excel (Barreto) | MIT OCW 14.01 Principles of Microeconomics

Learning Objective

Derive monopoly profit maximization conditions using calculus and analyze welfare loss.

Formal Monopoly Analysis

Consider a monopolist facing inverse demand function P(y) = a - by. The firm's cost function is C(y) = cy + F. Profit maximization requires MR = MC:

Ī€(y) = P(y)y - C(y) = (a - by)y - cy - F

dĪ€/dy = a - 2by - c = 0 → y* = (a - c)/2b

Australian example: Suppose a local electricity provider faces P(y) = 100 - 0.5y with C(y) = 20y + 500. Then:

y* = (100 - 20)/1 = 80 units P* = 100 - 0.5(80) = $60

Lerner Index: L = (60 - 20)/60 = 0.67

Worked Example

[Australian context]

Sydney Water monopoly faces P(y) = 120 - y with MC = $30. Calculate: 1. Profit-maximizing output 2. Deadweight loss

Solution: 1. MR = 120 - 2y = 30 → y_m = 45 P_m = 120 - 45 = $75 2. Competitive output where P=MC: 120 - y = 30 → y_c = 90 DWL = 0.5×(75-30)×(90-45) = $1,012.50

Common Misconception

"Monopolies always produce less than competitive markets"

While typically true, this ignores durable goods monopolies (Coase conjecture) where monopoly may produce competitively to prevent future competition.

Key Takeaways

  • MR = P + y(dP/dy) reflects quantity effect on revenue
  • Lerner index L = (P-MC)/P measures market power
  • DWL exists because P > MC reduces total surplus
  • Durable goods monopolies face time inconsistency problems

Practice

  1. Derive MR for P(y) = k/y (constant elasticity demand)
  2. Calculate DWL when P(y)=50-0.2y, MC=10, F=200
  3. Explain why L = -1/Îĩ using calculus

Further Resources

Provenance: Barreto | MIT OCW