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M17.L02 — Pareto Efficiency and the Contract Curve

Module: Module 17 — Exchange Economy and General Equilibrium Lesson: L02 of 05 Duration: ~30 minutes Level: Intermediate Provenance: Intermediate Microeconomics with Excel (Barreto) | University of Utah Intermediate Microeconomics

Learning Objective

Identify Pareto efficient allocations using the contract curve in an Edgeworth box.

Pareto Efficiency Defined

An allocation is Pareto efficient if no reallocation can make one consumer better off without making the other worse off. In the Edgeworth box:

  1. Pareto improvements move northeast/southwest within the lens-shaped area
  2. Efficiency occurs where indifference curves are tangent (MRS_A = MRS_B)
  3. The contract curve connects all such efficient points

Australian example: The National Electricity Market achieves Pareto efficiency when state energy trades balance marginal benefits.

Mathematically, for utility functions U_A(x_A,y_A) and U_B(x_B,y_B):

∂U_A/∂x_A / ∂U_A/∂y_A = ∂U_B/∂x_B / ∂U_B/∂y_B

Worked Example

[Australian context]

Two Australian states trade electricity (X) and gas (Y) with: - U_A = ln(x_A) + 2ln(y_A) - U_B = 2ln(x_B) + ln(y_B) - Total endowments: X̄ = 3000 GWh, Ȳ = 2000 TJ

  1. Calculate MRS for both states: MRS_A = y_A/(2x_A) MRS_B = (2y_B)/x_B
  2. Set MRS_A = MRS_B and use x_B = 3000 - x_A, y_B = 2000 - y_A
  3. Solve to find contract curve: y_A = (4000x_A)/(3000 + x_A)

Common Misconception

"Any point on the contract curve is equally fair"

Correction: While all points are efficient, they represent different distributions of welfare. Australia's energy market rules explicitly consider equity alongside efficiency.

Key Takeaways

  • Pareto efficiency requires MRS_A = MRS_B
  • The contract curve contains all efficient allocations
  • Efficiency ≠ fairness - Australia's carbon policy debates highlight this distinction
  • Core allocations are those where no coalition can do better on their own

Practice

  1. Find the contract curve for U_A = x_A + y_A and U_B = min(x_B, y_B)
  2. Draw the contract curve for two Australian farmers with Cobb-Douglas utilities
  3. Explain why the core shrinks as the number of traders increases

Further Resources

Provenance: Principles of Microeconomics (Curtis and Irvine) | MIT OCW 14.01