M15.L05 — Compensating and Equivalent Variation: Welfare Measurement

Module: Module 15 — Comparative Statics and Demand Theory
Lesson: L05 of 05
Duration: ~30 minutes
Level: Intermediate
Provenance: Intermediate Microeconomics with Excel (Barreto) | MIT OCW 14.01 Principles of Microeconomics

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Learning Objective

Calculate compensating and equivalent variation to measure the welfare effect of a price change.

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Two Welfare Measures

When price changes, we want to measure the welfare gain or loss in dollar terms. Two theoretically rigorous measures exist:

Compensating Variation (CV): How much income must be taken from (given to) the consumer after the price change to return them to their original utility level?

• Price falls: CV > 0 (take away income — consumer gained)
• Price rises: CV < 0 (give income — consumer lost)
• Uses original utility u⁰ as reference, new prices p¹

CV = e(p⁰, u⁰) - e(p¹, u⁰) = m - e(p¹, u⁰)

Equivalent Variation (EV): How much income would give the consumer the same welfare change as the price change, evaluated at original prices?

• Price falls: EV > 0 (consumer equivalent to gaining this much income)
• Uses new utility u¹ as reference, original prices p⁰

EV = e(p⁰, u¹) - e(p¹, u¹) = e(p⁰, u¹) - m

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CV vs EV vs Consumer Surplus Change

For a price fall:
- EV ≥ ΔCS ≥ CV (for normal goods) - EV ≤ ΔCS ≤ CV (for inferior goods) - When income effect is zero: EV = ΔCS = CV (quasilinear preferences)

The Marshallian consumer surplus change (area under demand curve) is a convenient approximation lying between CV and EV. For small price changes or when the income effect is small, all three are close.

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Worked Example

Carbon tax welfare cost for an Australian household

Cobb-Douglas utility: U = x₁^0.3 × x₂^0.7
x₁ = electricity (kWh), x₂ = other goods
Income m = $2,000/month, p₂ = $1
Original electricity price p₁⁰ = $0.25/kWh
Carbon tax raises price to p₁¹ = $0.35/kWh

Expenditure function: e(p₁, p₂, u) = u × (p₁/0.3)^0.3 × (p₂/0.7)^0.7

Original utility: u⁰ found from x₁* = 0.3m/p₁ = 0.3×2000/0.25 = 2,400 kWh
x₂* = 0.7×2000/1 = 1,400
u⁰ = (2400)^0.3 × (1400)^0.7 ≈ 1,704

CV (how much can be taken away after tax to restore u⁰):
e(p¹, u⁰) = minimum expenditure at new prices to reach u⁰
= u⁰ × (0.35/0.3)^0.3 × (1/0.7)^0.7
CV = m - e(p¹, u⁰) ≈ $2,000 - \(2,082 = -\)82/month

The household is $82/month worse off — the CV welfare cost of the carbon tax.

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Common Misconception

"CV and EV always give the same answer."

CV and EV coincide only when there is no income effect (quasilinear preferences). For normal goods they diverge: CV uses the old utility level (more conservative welfare loss) while EV uses the new utility level. For large price changes, the gap between them can be substantial.

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Key Takeaways

• CV: income adjustment needed at NEW prices to restore ORIGINAL utility — practical for policy analysis
• EV: income change at ORIGINAL prices equivalent to the price change — preferred for comparing multiple policies
• For normal goods: EV ≥ ΔCS ≥ CV for a price fall; the gap grows with the size of income effects
• Treasury and Infrastructure Australia use variants of CV/EV in cost-benefit analysis

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Practice

1. For U = x₁^0.5 × x₂^0.5, m = $1,000, p₂ = $1: Price of x₁ rises from $2 to $4. Calculate CV and EV using the expenditure function e(p₁,p₂,u) = 2u√(p₁p₂).
2. For quasilinear U = ln(x₁) + x₂, show that CV = EV = ΔCS for a price change (no income effect on x₁).
3. A petrol excise of $0.10/L is proposed. Use CV to estimate the welfare cost per Australian household (assume average consumption 1,200 L/year and income elasticity of petrol ≈ 0.3).

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Further Resources

• 📺 Compensating and Equivalent Variation (University of Utah) — intermediate series
• 📺 Consumer Surplus and Welfare (MIT OCW)
• 📚 Barreto — Intermediate Microeconomics with Excel

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Provenance: Intermediate Microeconomics with Excel (Barreto) | MIT OCW 14.01 Principles of Microeconomics