Questions: The Slutsky Equation

Question 1

The price of bread rises sharply. A very poor family spends 60% of its budget on bread. Bread is an inferior good for this family. Using the Slutsky decomposition, what happens to their bread consumption?

• A. Both the substitution and income effects push consumption down, so demand clearly falls
• B. The substitution effect pushes consumption down, but the income effect pushes it up — if the income effect dominates, this is a Giffen good and consumption rises
• C. The substitution effect pushes consumption up while the income effect pushes it down
• D. The income effect is zero for inferior goods, so only the substitution effect matters

Answer: (B)

For any own-price change, the substitution effect (∂hᵢ/∂pᵢ) is always non-positive — holding utility constant, a higher price always reduces compensated demand. The income effect is −xⱼ·(∂xᵢ/∂m). For an inferior good, ∂xᵢ/∂m < 0, and since a price increase effectively reduces real income, the income effect pushes consumption *up*. If the family spends a large fraction of income on bread, xⱼ is large and the income effect can dominate — producing a Giffen good. Option A describes a normal good. Option C reverses the directions.

Question 2

The Slutsky equation tells us that the substitution effect for an own-price change (∂hᵢ/∂pᵢ) is always non-positive. What mathematical property guarantees this?

• A. Diminishing marginal utility, which implies consumers always prefer variety
• B. The concavity of the expenditure function, which makes the Slutsky matrix negative semidefinite
• C. The strict convexity of indifference curves, which guarantees interior solutions
• D. The symmetry of the Slutsky matrix, combined with the assumption that goods are substitutes

Answer: (B)

The expenditure function e(p, u) is concave in prices — its Hessian (the Slutsky matrix S with entries ∂²e/∂pᵢ∂pⱼ = ∂hᵢ/∂pⱼ) is negative semidefinite. A negative semidefinite matrix has non-positive diagonal entries, so ∂hᵢ/∂pᵢ ≤ 0 always. This is a mathematical consequence of constrained expenditure minimization, not an assumption about preferences per se. Diminishing marginal utility (option A) is neither necessary nor sufficient for this result.

Question 3

For a normal good, the substitution and income effects work in opposite directions when its own price rises — the substitution effect reduces demand while the income effect increases it.

• T. True
• F. False

Answer: False

For a *normal* good, both effects push in the same direction. The substitution effect (always non-positive) reduces compensated demand when own-price rises. The income effect: a price rise reduces real income, and for a normal good (∂xᵢ/∂m > 0), lower income reduces demand further. So both channels reduce quantity demanded — this is why normal goods reliably obey the law of demand. It is *inferior* goods where the effects oppose each other.

Question 4

The substitution effect in the Slutsky equation is always non-positive for own-price changes, regardless of whether the good is normal, inferior, or Giffen.

• T. True
• F. False

Answer: True

This is exactly right. The sign of the substitution effect (∂hᵢ/∂pᵢ) is determined by the negative semidefiniteness of the Slutsky matrix — a property of the expenditure function's mathematical structure. It holds for all goods regardless of income effects. What distinguishes normal, inferior, and Giffen goods is the *income* effect. A Giffen good's demand slopes upward not because the substitution effect flips, but because a large, positive income effect (inferior good with large budget share) overwhelms the always-negative substitution effect.

Question 5

Explain why a Giffen good's demand curve slopes upward, using the substitution and income effects from the Slutsky decomposition.

Model answer: For a Giffen good, the income effect dominates and works in the opposite direction from the substitution effect. When price rises: the substitution effect (always negative) reduces compensated demand. But because the good is inferior (∂xᵢ/∂m < 0) and consumes a large share of the budget, the effective income loss is large — and for an inferior good, lower real income *increases* consumption. If this income effect exceeds the substitution effect in magnitude, total demand rises with price. A Giffen good requires two conditions: it must be inferior, and the consumer must spend enough on it that the real-income effect is large.

The Slutsky equation makes this precise: ∂xᵢ/∂pᵢ = ∂hᵢ/∂pᵢ − xᵢ·(∂xᵢ/∂m). For a Giffen good, the second term is positive (inferior good: ∂xᵢ/∂m < 0, so the negative of a negative times xᵢ > 0 is positive), and it exceeds the magnitude of the first term (which is always ≤ 0). Classic examples include staple foods like bread or potatoes for very poor consumers.