M15.L02 — Hicksian (Compensated) vs Marshallian (Ordinary) Demand

Module: Module 15 — Comparative Statics and Demand Theory
Lesson: L02 of 05
Duration: ~30 minutes
Level: Intermediate
Provenance: Intermediate Microeconomics with Excel (Barreto) | University of Utah Intermediate Microeconomics

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Learning Objective

Distinguish Hicksian compensated demand from Marshallian ordinary demand and derive the expenditure function.

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Two Concepts of Demand

Marshallian (ordinary) demand x(p, m) holds nominal income constant. When price changes, the consumer's real purchasing power also changes — the demand response reflects both substitution and income effects.

Hicksian (compensated) demand h(p, u) holds utility constant at some reference level u. As price changes, income is hypothetically adjusted to keep the consumer on the same indifference curve. Only the substitution effect is captured.

The key difference: Marshallian demand is what we observe in markets (income fixed). Hicksian demand is a theoretical construct useful for welfare analysis.

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The Expenditure Function

The expenditure function e(p, u) gives the minimum expenditure needed to achieve utility level u at prices p:

e(p, u) = min{p₁x₁ + p₂x₂ : U(x₁,x₂) ≥ u}

Shephard's Lemma: The partial derivative of the expenditure function with respect to price equals the Hicksian demand:

∂e(p,u)/∂pᵢ = hᵢ(p, u)

This is one of the most useful results in demand theory — it lets us recover Hicksian demands from the expenditure function without solving the full minimisation problem.

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The Slutsky Relationship

The Slutsky equation connects Marshallian and Hicksian demands:

∂xᵢ/∂pⱼ = ∂hᵢ/∂pⱼ - xⱼ × ∂xᵢ/∂m

Substituting the expenditure function: hᵢ(p, u) = xᵢ(p, e(p, u))

For own-price effects (i = j):

∂x₁/∂p₁ = ∂h₁/∂p₁ - x₁ × ∂x₁/∂m = (substitution effect) + (income effect)

The substitution effect ∂h₁/∂p₁ ≤ 0 always (Law of Compensated Demand). The income effect depends on whether the good is normal or inferior.

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Worked Example

Electricity tariff reform in Australia

A household has Cobb-Douglas utility U = x₁^0.5 × x₂^0.5 where x₁ = electricity (kWh), x₂ = other goods.

Expenditure function: e(p₁, p₂, u) = 2u(p₁p₂)^0.5

Hicksian demand: h₁ = ∂e/∂p₁ = u(p₂/p₁)^0.5

Marshallian demand: x₁ = m/(2p₁)

Electricity price rises from p₁ = 0.30 to p₁ = 0.40 ($/kWh). Income m = $200.

Marshallian response: x₁ falls from 200/(2×0.30) = 333 to 200/(2×0.40) = 250 kWh.

Hicksian response (holding u constant at original level): - Original utility: U = (333)^0.5 × (333)^0.5 ≈ 100 (assuming p₂=1, x₂=100) - h₁ at new price: 100 × (1/0.40)^0.5 ≈ 250 kWh

Here income and substitution effects nearly coincide because electricity is a small budget share — a common real-world pattern.

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Common Misconception

"Hicksian demand is just demand with a lower income."

Hicksian demand is not simply demand at a reduced income level. It adjusts income precisely to keep utility constant — the amount of compensation depends on the shape of the indifference curves and can vary nonlinearly with price changes. For large price changes, Marshallian and Hicksian demands can diverge substantially.

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Key Takeaways

• Marshallian demand holds income constant; Hicksian demand holds utility constant
• The expenditure function e(p,u) gives minimum cost to achieve utility u at prices p
• Shephard's Lemma: ∂e/∂pᵢ = Hicksian demand hᵢ(p,u)
• Hicksian demand satisfies the law of compensated demand (negative own-price derivative always)

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Practice

1. For U = min(x₁, x₂) (perfect complements), derive the expenditure function e(p₁, p₂, u) and verify Shephard's Lemma.
2. For Cobb-Douglas U = x₁^a × x₂^(1-a), derive the Hicksian demand h₁(p,u) using the expenditure function e = u × p₁^a × p₂^(1-a) / (a^a(1-a)^(1-a)).
3. A government carbon tax raises electricity prices. Using Marshallian vs Hicksian demand, explain why the Marshallian estimate of the welfare cost exceeds the Hicksian (compensating variation) estimate.

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Further Resources

• 📺 Expenditure Function and Hicksian Demand (University of Utah) — intermediate micro series
• 📺 Compensated vs Uncompensated Demand (Khan Academy) — conceptual overview
• 📚 Barreto — Intermediate Microeconomics with Excel

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Provenance: Intermediate Microeconomics with Excel (Barreto) | University of Utah Intermediate Microeconomics