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M20.L02 — Price Discrimination: Formal Analysis

Module: Module 20 — Monopoly and Game Theory (Advanced) Lesson: L02 of 06 Duration: ~30 minutes Level: Intermediate Provenance: Principles of Microeconomics (Curtis and Irvine) | University of Utah Intermediate Microeconomics

Learning Objective

Formulate optimal price discrimination conditions using calculus.

Third-Degree Price Discrimination

A monopolist serving two markets (1,2) maximizes:

π = P₁(y₁)y₁ + P₂(y₂)y₂ - C(y₁+y₂)

FOCs: ∂π/∂y₁ = MR₁ - MC = 0 ∂π/∂y₂ = MR₂ - MC = 0

Thus MR₁ = MR₂ = MC

Using MR = P(1 - 1/|ε|), we get: P₁/P₂ = (1 - 1/|ε₂|)/(1 - 1/|ε₁|)

Australian example: Qantas charges business travelers (ε = 1.2) 3× tourist class fares (ε = 3).

Worked Example

[Australian context]

An Australian textbook publisher sells to: 1. Students: P₁ = 50 - 0.5q₁, ε₁ = 1.5 2. Libraries: P₂ = 200 - 2q₂, ε₂ = 1.2 MC = $10 per book. Find optimal prices.

Solution: P₁(1 - 1/1.5) = P₂(1 - 1/1.2) = 10 → P₁ = 10/(1/3) = $30 → P₂ = 10/(1/6) = $60

Common Misconception

"Price discrimination always reduces welfare"

Third-degree PD may increase total output, potentially improving welfare when it brings output closer to competitive levels.

Key Takeaways

  • First-degree PD extracts all CS (P=MWTP)
  • Third-degree PD sets MR equal across markets
  • Two-part tariffs combine access fee with per-unit price
  • Welfare effects depend on output changes

Practice

  1. Derive the two-part tariff solution when P=100-Q, MC=20
  2. Calculate optimal prices for ε₁=2, ε₂=4 when MC=15
  3. Explain why first-degree PD achieves competitive output

Further Resources

Provenance: Curtis and Irvine | Utah Economics