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M13.L03 — Properties of Preferences: Convexity, Monotonicity, and More

Module: Module 13 — Budget Constraints, Preferences, and Utility Lesson: L03 of 05 Duration: ~30 minutes Level: Intermediate Provenance: Intermediate Microeconomics with Excel (Barreto) | Khan Academy Microeconomics

Learning Objective

Analyze how preference properties shape indifference curves and consumer behavior.

Fundamental Properties

  1. Completeness: Can compare any two bundles (A≿B or B≿A)
  2. Transitivity: If A≿B and B≿C, then A≿C
  3. Monotonicity: More is better (if A has ≥ quantities than B, then A≿B)
  4. Convexity: Averages preferred to extremes (if A∼B, then (A+B)/2 ≿ A)

Special Cases: - Strict convexity: Averages strictly preferred - Satiation: Bliss point exists

Worked Example

Australian Food Preferences

Consider a consumer choosing between meat pies (x₁) and vegemite toast (x₂):

  1. Show convexity holds for U=min(x₁, 2x₂)
  2. Take A=(2,1) and B=(4,2) → U=2 for both
  3. Average bundle C=(3,1.5) → U=min(3,3)=3

  4. 3 > 2 → convexity satisfied

  5. Show MRS is undefined at kinks (perfect complements)

Common Misconception

"Convex preferences mean the utility function is convex"

Correction: The utility function is quasiconcave - convex preferences correspond to convex lower contour sets.

Key Takeaways

  • Well-behaved preferences satisfy completeness, transitivity, monotonicity, and convexity
  • Convexity implies diminishing MRS
  • Strict convexity guarantees unique optimal bundles
  • Non-monotonic preferences have bliss points

Practice

  1. Prove that U=x₁x₂ represents convex preferences
  2. Sketch indifference curves for a satiated preference
  3. Explain why transitivity prevents preference cycles

Further Resources

Provenance: Intermediate Microeconomics with Excel (Barreto) | Khan Academy Microeconomics