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Intermediate Microeconomics Unit 4: Income and Substitution Effects (Slutsky)

Every price change has two parts: a substitution effect from changed relative prices and an income effect from changed purchasing power; separating them explains the shape of demand curves and how to measure welfare.

Covers: Slutsky decomposition, Hicks decomposition and compensated demand, Normal, inferior and Giffen goods, Slutsky equation, Expenditure function, CV and EV, Consumer surplus, Labor supply and endowment effects

Study guide

When the price of x changes, the consumer reacts for two reasons. First, x is now relatively cheaper or dearer than other goods, which is the substitution effect. Second, her income buys more or less than before, which is the income effect. The Slutsky decomposition separates these by giving the consumer just enough income to afford her original bundle at the new prices, Δm = x₀Δp₁. The move from the original choice to the choice with this compensated budget is the substitution effect; the move from there to the final choice is the income effect. The Hicks version instead compensates to keep utility constant, which uses the compensated demand curve h(p, u).

The substitution effect always moves demand opposite to the price change: compensated demand curves slope down. The income effect depends on the type of good. For a normal good, a price rise lowers real income and further reduces demand, so the two effects reinforce each other. For an inferior good, the income effect works against the substitution effect. If the good is inferior and the income effect is large enough to dominate, demand rises with price: a Giffen good. Giffen goods must therefore be inferior, and in practice usually take up a large share of a poor household's budget.

The Slutsky equation states the decomposition exactly: ∂x/∂p₁ = ∂h/∂p₁ − x(∂x/∂m). The first term is the substitution effect and the second the income effect, which is scaled by x because a $1 price rise costs the consumer x dollars. In elasticity form, ε = εʰ − s·η, where s is the budget share and η the income elasticity. Special cases help: perfect complements have no substitution effect, and quasilinear utility has no income effect. When the consumer owns an endowment of the good, an extra term appears: a net seller gains income when the price rises.

Welfare measures turn utility changes into dollars. The expenditure function e(p, u) gives the least money needed to reach utility u. Compensating variation, CV = e(p¹, u⁰) − e(p⁰, u⁰) for a price rise, is the money that would restore the old utility at new prices; equivalent variation uses the new utility at old prices. Consumer surplus, the area under ordinary demand above price, is a convenient approximation. With no income effects, CV, EV and ΔCS are equal; for a normal good facing a price fall, CV < ΔCS < EV. Labor supply applies the same logic: a higher wage raises the price of leisure but also raises income, so hours can rise or fall.

Key ideas

• Total effect = substitution effect + income effect.
• Slutsky compensation keeps the old bundle affordable; Hicks compensation keeps old utility attainable.
• The own-price substitution effect is never positive.
• Normal goods have reinforcing effects; inferior goods have opposing effects; Giffen goods have a dominant income effect.
• Perfect complements have zero substitution effect; quasilinear goods have zero income effect.
• CV uses new prices and old utility, EV uses old prices and new utility; with no income effects they equal ΔCS.
• A wage rise has a substitution effect toward work and, if leisure is normal, an income effect toward leisure.

Formulas and rules

Common mistakes

• Adding instead of subtracting the income term in the Slutsky equation.
• Calling every inferior good a Giffen good; most inferior goods still have downward-sloping demand.
• Swapping the reference points for CV and EV.
• Computing the substitution effect with the original income instead of the compensated income.
• Assuming a higher wage always raises hours worked, ignoring the income effect on leisure.

Before the test, make sure you can

• Decompose a price change into Slutsky substitution and income effects with numbers.
• Perform a Hicks decomposition using compensated demand.
• Classify goods as normal, inferior or Giffen from the signs of the effects.
• Use the Slutsky equation in derivative and elasticity form.
• Compute CV, EV and ΔCS and rank them for normal goods.
• Explain why a lump-sum tax is better for the consumer than an equal-revenue per-unit tax.
• Analyze labor supply responses to wage changes using income and substitution effects.

Words to know

Substitution effect

Change in demand from a change in relative prices with purchasing power (or utility) held constant.

Income effect

Change in demand from the change in purchasing power caused by a price change.

Slutsky compensation

Income adjustment Δm = x₀Δp that makes the original bundle exactly affordable at new prices.

Hicks compensation

Income adjustment that keeps the consumer on her original indifference curve at new prices.

Compensated (Hicksian) demand

Demand as a function of prices and utility, h(p, u), from minimizing expenditure.

Expenditure function

The minimum spending needed to reach utility u at prices p, e(p, u).

Slutsky equation

∂x/∂p = ∂h/∂p − x ∂x/∂m: total effect equals substitution plus income effect.

Giffen good

An inferior good whose income effect outweighs its substitution effect, so demand rises with price.

Compensating variation

Money that would restore the consumer's original utility after a price change, at the new prices.

Equivalent variation

Money change at the original prices that would give the same utility change as the price change.

Consumer surplus

The area under the demand curve and above the price; the gain from being able to buy at that price.

Deadweight loss of a tax

The amount by which the consumer's loss from a tax exceeds the revenue it raises.

Full income

In labor supply, the value of all available time at the wage plus non-labor income.

Backward-bending labor supply

Labor supply that falls at high wages because the income effect on leisure dominates.

Endowment income effect

Extra income effect when the consumer owns the good; a price rise makes net sellers richer.

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.