M13.L05 โ Special Preferences: Perfect Substitutes, Complements, and Cobb-Douglas
Module: Module 13 โ Budget Constraints, Preferences, and Utility Lesson: L05 of 05 Duration: ~30 minutes Level: Intermediate Provenance: Principles of Microeconomics (Curtis and Irvine) | Khan Academy Microeconomics
Learning Objective
Distinguish between major classes of utility functions and their implications for consumer choice.
Special Preference Classes
- Perfect Substitutes:
- U = axโ + bxโ
- Linear ICs, constant MRS = -a/b
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Example: Different brands of rice
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Perfect Complements:
- U = min(axโ, bxโ)
- L-shaped ICs, MRS undefined at kink
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Example: Left/right shoes
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Cobb-Douglas:
- U = xโแตxโแต
- Smooth convex ICs, MRS = -a/b ร xโ/xโ
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Example: General consumer goods
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Quasilinear:
- U = v(xโ) + xโ
- Parallel ICs, MRS depends only on xโ
- Example: Goods with no income effect
Worked Example
Australian Beverage Choices
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Perfect substitutes: Tea (xโ) and coffee (xโ) for caffeine: U = 2xโ + xโ โ MRS = -2 always
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Perfect complements: Coffee (xโ) and sugar (xโ) in fixed ratio: U = min(xโ, 5xโ) โ 1 coffee : 5 sugars
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Cobb-Douglas: U = โxโโxโ โ MRS = -xโ/xโ
Common Misconception
"Cobb-Douglas preferences imply equal spending on both goods"
Correction: Spending shares equal exponent ratios (a:b), which needn't be 1:1.
Key Takeaways
- Perfect substitutes have linear ICs and constant MRS
- Perfect complements have L-shaped ICs with optimal bundles at fixed ratios
- Cobb-Douglas yields smooth convex ICs with variable MRS
- Quasilinear preferences eliminate income effects for one good
Practice
- Sketch ICs for U=2xโ + 3xโ
- Derive MRS for U=xโ^(0.3)xโ^(0.7)
- Explain why perfect complements lead to corner solutions off the fixed ratio
Further Resources
- ๐บ Types of Utility Functions โ UNSW Economics
- ๐ Varian Ch.4: Utility
Provenance: Principles of Microeconomics (Curtis and Irvine) | Khan Academy Microeconomics