Difficulty: Intermediate to Advanced | Prerequisites: Hicksian and Marshallian demand, envelope theorems, duality
When a price changes, two things happen at once. The good becomes relatively cheaper or more expensive compared to other goods (the substitution effect), and the consumer's real purchasing power changes (the income effect). The Slutsky equation separates these two forces analytically. It is the central decomposition in demand theory and explains why demand usually slopes down, but can, under specific conditions, slope up. This material ties together everything from the Hicksian/Marshallian demand framework and duality.
The Slutsky equation decomposes the total effect of a price change on Marshallian demand into a substitution effect (always negative, always pushes demand in the opposite direction to price) and an income effect (whose sign depends on whether the good is normal or inferior). Demand slopes down whenever the income effect reinforces the substitution effect or is too small to overcome it. Demand can slope up only for inferior goods where the income effect dominates, producing a Giffen good.
Slutsky equation
The identity that decomposes the total price effect on Marshallian demand into a substitution effect and an income effect: ∂xᵢ*/∂pⱼ = ∂hᵢ/∂pⱼ − xⱼ* · (∂xᵢ*/∂m).
In simple terms, it says: the total change in demand when a price moves equals the change holding utility fixed, minus an adjustment for the change in purchasing power.
Substitution effect
The change in demand due to the change in relative prices, holding utility constant (moving along the indifference curve). This is ∂hᵢ/∂pᵢ, the slope of the Hicksian demand. For own-price changes, the substitution effect is always non-positive: when a good gets relatively more expensive, the consumer substitutes away from it.
Think of it as: "Holding happiness fixed, I swap toward the good that just got cheaper."
Income effect
The change in demand due to the change in real purchasing power caused by the price change. When pᵢ rises, the consumer is effectively poorer (their income buys less). The income effect is −xᵢ* · (∂xᵢ*/∂m): the quantity currently consumed times the response of demand to income.
Think of it as: "The price rise made me poorer, so I adjust my consumption the way I would if I had lost income."
Giffen good
A good whose Marshallian demand increases when its own price rises. This requires the good to be inferior (∂xᵢ*/∂m < 0) and the income effect to be large enough in magnitude to overwhelm the substitution effect.
In simple terms, it is the rare case where the "I'm poorer" effect is so strong that it overrides the "it's relatively more expensive" effect.
Law of demand
The empirical regularity that demand slopes down (quantity demanded falls when price rises). The Slutsky equation shows this holds whenever the substitution and income effects work in the same direction (normal goods) or the income effect is too small to reverse the substitution effect.
The Slutsky equation (own-price)
∂x₁*/∂p₁ = ∂h₁/∂p₁ − x₁* · (∂x₁*/∂m)
Total effect = Substitution effect + Income effect
The left side is the slope of the Marshallian demand curve. The first term on the right is the slope of the Hicksian (compensated) demand curve. The second term adjusts for the fact that a price change alters real income.
Graphical decomposition (price of good 1 rises)
Start at the original optimum, bundle A, on the original budget line tangent to indifference curve U₀.
The price rise rotates the budget line inward (pivoting at the x₂-intercept). The new optimum is bundle C on a lower indifference curve U₁. The move from A to C is the total effect.
To isolate the substitution effect, imagine giving the consumer just enough income to stay on U₀ at the new prices. This creates a hypothetical budget line parallel to the new one but tangent to U₀. The tangency point is bundle B. The move from A to B (along U₀) is the substitution effect.
The move from B to C (shifting from the hypothetical budget line to the actual new one) is the income effect. This is a parallel shift of the budget line with no change in relative prices, so it is a pure income change.
Why the substitution effect is always negative (for own-price)
Moving along a strictly convex indifference curve, when good 1 becomes relatively more expensive, the tangency point shifts toward less x₁ and more x₂. This follows from the convexity of the indifference curve: the MRS is diminishing, so a steeper price line (higher p₁/p₂) is tangent further to the left.
Starting point: the duality identity
At the optimum, Marshallian and Hicksian demands coincide: hᵢ(p, U) = xᵢ*(p, e(p, U)).
Differentiate both sides with respect to pⱼ
∂hᵢ/∂pⱼ = ∂xᵢ*/∂pⱼ + (∂xᵢ*/∂m) · (∂e/∂pⱼ)
The left side is the Hicksian (compensated) price response. The right side uses the chain rule: pⱼ affects Marshallian demand directly (first term) and through its effect on expenditure (second term).
Apply Shephard's lemma
By Shephard's lemma, ∂e/∂pⱼ = hⱼ(p, U) = xⱼ* (at the optimum). Substitute:
∂hᵢ/∂pⱼ = ∂xᵢ*/∂pⱼ + xⱼ* · (∂xᵢ*/∂m)
Rearrange
∂xᵢ*/∂pⱼ = ∂hᵢ/∂pⱼ − xⱼ* · (∂xᵢ*/∂m)
This is the Slutsky equation. For the own-price case (i = j):
∂xᵢ*/∂pᵢ = ∂hᵢ/∂pᵢ − xᵢ* · (∂xᵢ*/∂m)
The derivation rests entirely on the duality identity and Shephard's lemma. No additional assumptions beyond the smoothness and strict concavity of u are needed.
Conditions for demand to slope down (∂xᵢ/∂pᵢ < 0)*
From the Slutsky equation: ∂xᵢ*/∂pᵢ = ∂hᵢ/∂pᵢ − xᵢ*(∂xᵢ*/∂m).
The substitution effect ∂hᵢ/∂pᵢ is always ≤ 0 (strictly negative with strictly convex preferences).
Demand slopes down when:
The good is normal (∂xᵢ*/∂m > 0). Then the income effect term −xᵢ*(∂xᵢ*/∂m) is negative, reinforcing the substitution effect. Both terms push in the same direction.
The good is inferior (∂xᵢ*/∂m < 0) but the income effect is smaller in magnitude than the substitution effect. The income effect pushes demand up, but not enough to overcome the substitution effect pushing it down.
Conditions for demand to slope up (∂xᵢ/∂pᵢ > 0, a Giffen good)*
This requires:
The good must be inferior: ∂xᵢ*/∂m < 0
The income effect must dominate: |xᵢ*(∂xᵢ*/∂m)| > |∂hᵢ/∂pᵢ|
Both conditions must hold simultaneously. Being inferior is necessary but not sufficient for Giffen behaviour.
Practically, Giffen goods are rare. They tend to arise with goods that are both a large share of the budget (making xᵢ* large, which amplifies the income effect) and strongly inferior. The classic textbook example is a staple food (like potatoes or rice) that a very poor household spends most of its income on.
Why income effects are critical in demand theory
Without income effects, demand theory would be simple: the substitution effect always pushes demand in the direction you would expect (buy less of what got expensive), and every demand curve would slope down. Income effects are what make demand theory interesting and empirically relevant, because they allow for the possibility that demand curves can slope up.
The income effect also explains why Marshallian and Hicksian demands differ. The Hicksian demand isolates the substitution effect by holding utility fixed. The Marshallian demand includes both effects. The gap between them is precisely the income effect from the Slutsky equation.
Can a good be normal over some income range and Giffen over others?
Yes, and this is an important conceptual point. A good can be normal at moderate incomes (demand rises with income) but inferior at high incomes (the consumer switches to a superior substitute). Since Giffen behaviour requires the good to be inferior, a good can be normal over one income range and Giffen over another.
The key insight: "normal," "inferior," and "Giffen" are local properties of the demand function, not global ones. They can change as income or prices change. A good that is Giffen at very low incomes (where it dominates the budget) can become a perfectly ordinary normal good at higher incomes (where it is a small share of spending and the income effect is negligible).
The Slutsky decomposition is used in tax policy analysis. When a government raises a sales tax on a good, the substitution effect captures the distortion (consumers shift to less-preferred alternatives), while the income effect captures the burden (consumers are poorer). Distinguishing the two matters because the deadweight loss of a tax comes from the substitution effect, not the income effect.
The Jensen and Miller (2008) study of rice consumption among poor Chinese households is the most cited empirical evidence for Giffen behaviour in a real-world setting.
Students often think all inferior goods are Giffen goods. They are not. Giffen requires inferiority plus a dominant income effect. Most inferior goods have small income effects and perfectly normal downward-sloping demand.
A frequent error is writing the Slutsky equation with the wrong sign on the income effect. The correct form has a minus sign: ∂xᵢ*/∂pⱼ = ∂hᵢ/∂pⱼ minus xⱼ*(∂xᵢ*/∂m).
Students sometimes confuse the Slutsky decomposition with the Hicks decomposition. Both decompose a price change into substitution and income effects, but they differ in how they define the hypothetical "compensated" budget. The Slutsky version holds purchasing power of the original bundle fixed; the Hicks version holds utility fixed. In this course, you are using the Hicks version (based on the Hicksian demand).
Some students think the substitution effect can be positive for own-price changes. It cannot (with convex preferences). The compensated demand curve always slopes down.
⚠️ Be able to state the Slutsky equation, identify each term, and explain its sign.
⚠️ Know the graphical decomposition cold: original bundle → compensated bundle (substitution effect along the indifference curve) → new bundle (income effect, parallel shift).
⚠️ Be prepared to explain the necessary and sufficient conditions for a Giffen good.
⚠️ Expect a question asking whether a good can be normal at some incomes and Giffen at others. The answer is yes, with reasoning about local vs. global properties.
⚠️ The calculus derivation (using the duality identity and Shephard's lemma) is a standard proof question.
True or False: The substitution effect for an own-price change can be positive. (False. It is always ≤ 0 with convex preferences.)
Fill in the blank: A Giffen good must be ________, but not all such goods are Giffen. (Inferior)
True or False: For a normal good, the income effect reinforces the substitution effect. (True.)
Fill in the blank: The Slutsky equation is derived by differentiating the identity hᵢ(p,U) = xᵢ*(p, ________). (e(p,U))
True or False: "Normal" and "inferior" are global properties that hold at all income levels. (False. They are local.)
Q: State the Slutsky equation and identify each term.
A: ∂xᵢ*/∂pᵢ = ∂hᵢ/∂pᵢ − xᵢ*(∂xᵢ*/∂m). The left side is the total effect of a price change on Marshallian demand. The first term on the right is the substitution effect (compensated price response). The second term is the income effect (quantity consumed times the income response of demand).
Q: Why is the substitution effect always non-positive for own-price changes?
A: The Hicksian demand moves along a fixed indifference curve. With convex preferences, as the price of good i rises (making it relatively more expensive), the tangency point on the indifference curve shifts toward less of good i. The expenditure function is concave in prices, which implies ∂hᵢ/∂pᵢ ≤ 0.
Q: What conditions are necessary and sufficient for a good to be Giffen?
A: The good must be inferior (∂xᵢ*/∂m < 0), and the magnitude of the income effect xᵢ*|∂xᵢ*/∂m| must exceed the magnitude of the substitution effect |∂hᵢ/∂pᵢ|.
Q: Draw the Slutsky decomposition for a normal good when p₁ rises.
A: Start at bundle A (original tangency). Rotate the budget line inward (new tangency at C on a lower indifference curve). Draw a compensating budget line parallel to the new one but tangent to the original indifference curve (tangency at B). A to B is the substitution effect (less x₁, more x₂). B to C is the income effect (for a normal good, also less x₁). Both move x₁ in the same direction.
Q: Give an example of how a good could be normal at one income level and Giffen at another.
A: Rice for a very poor household. At very low incomes, rice dominates the budget. A rice price increase makes the household so much poorer that they can no longer afford meat and must eat even more rice (Giffen). At higher incomes, rice is a small budget share, the income effect is negligible, and demand for rice slopes down normally.
The Slutsky equation is the bridge between the Hicksian and Marshallian frameworks. It connects to welfare economics (compensating and equivalent variation use the Hicksian demand to measure welfare changes without the confounding income effect). In econometrics, the Slutsky symmetry condition (∂hᵢ/∂pⱼ = ∂hⱼ/∂pᵢ) is used as a testable restriction on demand systems. The concept of income and substitution effects carries into labour economics (the backward-bending labour supply curve) and public finance (the excess burden of taxation).
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