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M17.L03 — Competitive Equilibrium in an Exchange Economy

Module: Module 17 — Exchange Economy and General Equilibrium Lesson: L03 of 05 Duration: ~30 minutes Level: Intermediate Provenance: Intermediate Microeconomics with Excel (Barreto) | MIT OCW 14.01 Principles of Microeconomics

Learning Objective

Determine competitive equilibrium prices and allocations in a pure exchange economy.

Walrasian Equilibrium

In a competitive exchange economy:

  1. Consumers take prices (p_x, p_y) as given
  2. Markets clear when supply = demand for both goods
  3. Walras' Law: If n-1 markets clear, the nth must also clear
  4. Equilibrium allocation is Pareto efficient (First Welfare Theorem)

Australian context: The Emissions Reduction Fund creates a market where p_x = price per ton CO₂, with equilibrium emerging from bid/ask spreads.

Mathematically, solve:

  1. Consumer A's problem: max U_A(x_A,y_A) s.t. p_x x_A + p_y y_A = p_x ω_xA + p_y ω_yA
  2. Consumer B's problem: similarly
  3. Market clearing: x_A + x_B = X̄, y_A + y_B = Ȳ

Worked Example

[Australian context]

Two regions trade renewable energy (X) and coal (Y): - U_A = x_A^0.7 y_A^0.3, ω_A = (100,50) - U_B = x_B^0.3 y_B^0.7, ω_B = (50,100) - Total endowments: X̄ = 150, Ȳ = 150

  1. Set p_y = 1 (numeraire), find p_x
  2. Solve demand functions: x_A = 0.7*(100p_x + 50)/p_x y_A = 0.3*(100p_x + 50)
  3. Market clearing: x_A + x_B = 150 → p_x = 1
  4. Equilibrium allocation: x_A = 105, y_A = 45, x_B = 45, y_B = 105

Common Misconception

"Equilibrium prices are always positive"

Correction: Free goods can have zero price (e.g., Australia's solar surplus periods when p_x ≈ 0).

Key Takeaways

  • Competitive equilibrium requires market clearing and utility maximization
  • Walras' Law simplifies finding equilibrium prices
  • Numeraire normalization reduces degrees of freedom
  • Australian energy markets show real-world price discovery

Practice

  1. Find equilibrium prices for U_A = min(x_A, y_A), U_B = x_B + y_B with ω_A = (2,0), ω_B = (0,2)
  2. Derive excess demand functions for the worked example
  3. Explain how Australia's Renewable Energy Target affects equilibrium prices

Further Resources

Provenance: University of Utah Intermediate Microeconomics | Principles of Microeconomics (Curtis and Irvine)