M13.L04 — Utility Functions: Ordinal Representation of Preferences

Module: Module 13 — Budget Constraints, Preferences, and Utility Lesson: L04 of 05 Duration: ~30 minutes Level: Intermediate Provenance: MIT OCW 14.01 Principles of Microeconomics | University of Utah Intermediate Microeconomics

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Learning Objective

Transform utility functions while preserving preference orderings and calculate marginal rates of substitution.

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Utility Functions

A utility function U(x₁,x₂) assigns numbers to bundles such that: U(A) > U(B) ⇔ A ≻ B

Key Properties: 1. Ordinal (only ranking matters) 2. Preserved under monotonic transformations 3. MRS = -MU₁/MU₂

Common Forms: 1. Cobb-Douglas: U = x₁ᵃx₂ᵇ 2. Perfect substitutes: U = ax₁ + bx₂ 3. Perfect complements: U = min(ax₁, bx₂)

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Worked Example

Australian Energy Preferences

Consider U = √(electricity)×√(gas):

1. Show V = ln(U) represents same preferences: V = 0.5ln(x₁) + 0.5ln(x₂) → monotonic transformation

2. Calculate MRS: MU₁ = 0.5/√x₁, MU₂ = 0.5/√x₂ MRS = -MU₁/MU₂ = -√x₂/√x₁

3. At (4,9): MRS = -3/2

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Common Misconception

"Higher utility numbers mean greater happiness"

Correction: Utility is ordinal - only rankings matter, not absolute values.

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Key Takeaways

• Utility functions numerically represent preference orderings
• Monotonic transformations preserve preference rankings
• MRS depends only on preference structure, not utility scale
• Marginal utilities measure local tradeoff rates

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Practice

1. Show U=x₁x₂ and V=lnx₁+lnx₂ yield same MRS
2. Find MRS for U=x₁²x₂ at (2,3)
3. Explain why U=100+x₁ and V=eˣ¹ represent identical preferences

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Further Resources

• 📺 Utility Functions and MRS — University of Melbourne
• 📚 Varian Ch.4: Utility

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Provenance: MIT OCW 14.01 Principles of Microeconomics | University of Utah Intermediate Microeconomics