M14.L03 — Deriving the Demand Curve from Utility Maximisation

Module: Module 14 — Consumer Choice and Demand Lesson: L03 of 05 Duration: ~30 minutes Level: Intermediate Provenance: Intermediate Microeconomics with Excel (Barreto) | Khan Academy Microeconomics

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Learning Objective

Derive individual demand functions from utility maximisation problems.

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Demand Function Derivation

From the utility maximisation solution, we obtain demand functions x₁*(p₁,p₂,m). For Cobb-Douglas utility U = x₁ᵃx₂ᵇ:

x₁* = (a/(a+b))·(m/p₁)

Key properties: 1. Homogeneous of degree 0: x₁*(tp₁,tp₂,tm) = x₁*(p₁,p₂,m) 2. Downward sloping in own-price (law of demand) 3. Engel curve shows x₁* vs m

Australian example: The demand for avocado toast (x₁) follows x₁* = 0.6m/p₁ in Sydney cafes.

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Worked Example

Melbourne Coffee Demand

Suppose utility over coffee (x₁) and tea (x₂) is U = x₁⁰·⁷⁵x₂⁰·²⁵ with m = $50:

1. Demand functions: x₁* = (0.75/1)·(50/p₁) = 37.5/p₁ x₂* = (0.25/1)·(50/p₂) = 12.5/p₂
2. At p₁ = $4.50 (typical Melbourne flat white): x₁* ≈ 8.33 coffees/week
3. Demand curve: x₁* = 37.5/p₁ (rectangular hyperbola)

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Common Misconception

"Demand curves are linear in practice"

Most Australian demand curves (housing, education, healthcare) show non-linear price sensitivity, especially at income thresholds.

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Key Takeaways

• Demand functions emerge from utility maximisation solutions
• Cobb-Douglas yields constant budget shares
• Homogeneity means demand responds to real (not nominal) values
• Price-consumption curve traces optimal bundles as price changes
• Australian examples include: petrol demand, NBN internet plans

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Practice

1. Derive demand functions for U = x₁ + ln(x₂)
2. Show that x₁* = 10/p₁ is homogeneous of degree 0
3. Draw the price-consumption curve for perfect complements

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Further Resources

• 📺 Deriving Demand Curves (Uni of Sydney) — Australian lecturer demo
• 📚 Varian Ch.6: Demand

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