ECOS2001 · Intermediate Microeconomics
Slutsky, Income Effects & Applications
The Slutsky decomposition splits the total effect of a price change into a substitution effect — the move when relative prices change but the consumer is just able to afford the old bundle (purchasing power held constant) — and an income effect, the change in real purchasing power. The substitution effect always opposes the price change; the income effect's sign depends on whether the good is normal or inferior, and a Giffen good needs an inferior income effect large enough to dominate. The same machinery powers the applications: lump-sum versus per-unit taxes, the endowment economy, intertemporal choice with interest, and the labour–leisure trade-off behind a backward-bending supply curve.
What this chapter covers
- 01Slutsky decomposition: total effect = substitution effect + income effect
- 02Sign rules: substitution effect opposes the price change; income effect signs by normal/inferior
- 03Giffen goods need a strong inferior income effect
- 04Taxes and subsidies on the budget: lump-sum vs per-unit
- 05Endowment economy: budget p₁x₁ + p₂x₂ = p₁ω₁ + p₂ω₂ and net demands
- 06Intertemporal choice: slope −(1+r), savers vs borrowers
- 07Labour–leisure and the backward-bending labour supply
Slutsky: substitution vs income effect
- 3 marksCobb-Douglas with equal exponents spends half of income on each good, so x = (1/2)·m/pₓ. Original demand: x₀ = (1/2)·144/4 = 18. New demand: x₂ = (1/2)·144/1 = 72. Total effect = 72 − 18 = +54.
- 1 markFind the original y to fix the old bundle: y₀ = (1/2)·m/p_y = (1/2)·144/1 = 72. So the old bundle is (18, 72).
- 2 marksSlutsky pivot: give the consumer just enough income m′ to still afford the old bundle at the new prices: m′ = pₓ′·x₀ + p_y·y₀ = 1·18 + 1·72 = 90.
- 1 markSubstitution-step demand at the new price with income m′: x₁ = (1/2)·90/1 = 45.
- 1 markSplit the total: substitution effect = x₁ − x₀ = 45 − 18 = +27; income effect = x₂ − x₁ = 72 − 45 = +27.
Key terms
- Substitution effect
- The change in quantity from a price change holding purchasing power constant (the old bundle just affordable); it always moves opposite to the price change.
- Income effect
- The change in quantity from the change in real purchasing power caused by a price change; positive for a normal good, negative for an inferior good.
- Endowment economy
- A consumer who owns bundles ω₁, ω₂ rather than money income, with budget p₁x₁ + p₂x₂ = p₁ω₁ + p₂ω₂ and net demand xᵢ − ωᵢ.
- Backward-bending labour supply
- When a higher wage's income effect (wanting more leisure) outweighs its substitution effect (working more), labour supply falls as the wage rises at high wages.
Slutsky, Income Effects & Applications FAQ
What is held constant in the Slutsky substitution effect?
Purchasing power, in the sense that the consumer is given just enough income to afford the original bundle at the new prices. The Slutsky version pivots the budget through the old bundle; the related Hicks version holds utility (the original indifference curve) constant instead.
Why does a Giffen good need an inferior income effect?
The substitution effect always lowers quantity when price rises, so demand can only rise with price if the income effect is both opposite (inferior good) and large enough to overturn the substitution effect. That makes Giffen goods a rare, strongly inferior special case.
Exam move
Practise the full pivot: original bundle → m′ that just affords it at new prices → substitution demand → income effect by subtraction. Memorise the sign rules so you can sanity-check each step rather than only trusting the algebra.
