Difficulty: Intermediate-Advanced | Prerequisites: Utility maximisation, tangency condition, Cobb-Douglas demand, budget constraints, indifference curves.
Big picture: When a price changes, the consumer's behaviour shifts for two distinct reasons: the good became relatively cheaper or more expensive (the substitution effect), and the consumer's purchasing power changed (the income effect). Disentangling these two is the Slutsky decomposition, one of the most important analytical tools in consumer theory. It appears on nearly every intermediate micro exam, and it is the foundation for understanding Giffen goods, compensating variation, and welfare analysis.
TL;DR: A price drop makes the consumer buy more of the good for two reasons: it is now a better deal relative to the other good (substitution effect), and the consumer's real income has risen (income effect). The decomposition finds a hypothetical "compensated" income that holds utility constant at the old level, isolating the pure substitution effect from the income effect.
Substitution effect (SE)
The change in quantity demanded due solely to the change in relative prices, holding utility constant at the original level. Think of it as: "How would the consumer adjust if prices changed but we compensated them to keep them just as happy as before?"
Income effect (IE)
The change in quantity demanded due to the change in real purchasing power, moving from the compensated bundle to the final bundle. In simple terms, a price drop is like getting a raise, and this measures how much of the extra purchasing power goes toward the cheaper good.
Hicksian (compensating variation) decomposition
The version used in this course: find the income needed to reach the original utility level at the new prices, then compute the bundle the consumer would choose with that compensated income. The SE is the move from the original bundle to the compensated bundle; the IE is the move from the compensated bundle to the final bundle.
Compensated income (M')
The hypothetical income that, at the new prices, would allow the consumer to reach exactly the same utility as before the price change. This is the key intermediate calculation in the decomposition.
Normal good (in this context)
A good for which the income effect reinforces the substitution effect. When P_X falls, both effects push X upward.
Inferior good
A good for which the income effect works against the substitution effect. When P_X falls, the substitution effect raises X, but the income effect reduces it. If the income effect dominates, you have a Giffen good (rare in practice, but examinable in theory).
Total effect
The overall change in quantity demanded. Total effect = substitution effect + income effect.
Given: U(X, Y) = XY. Income M = 180. Original prices: P_X = 4.50, P_Y = 2. Price change: P_X falls to 2.
This is a Cobb-Douglas utility function with a = b = 1, so the consumer spends half of income on each good.
Spending on X = M/2 = 90. So X = 90 / 4.50 = 20.
Spending on Y = M/2 = 90. So Y = 90 / 2 = 45.
Original bundle: (20, 45).
U = X · Y = 20 · 45 = 900.
New prices: P_X = 2, P_Y = 2.
Spending on X = 180/2 = 90. X = 90/2 = 45.
Spending on Y = 180/2 = 90. Y = 90/2 = 45.
New bundle: (45, 45). Total effect on X = 45 - 20 = +25 units.
We need the income M' such that, at the new prices (P_X = 2, P_Y = 2), the consumer reaches exactly U = 900.
At prices P_X = P_Y = 2, the optimal bundle is X = M'/(2 · 2) = M'/4 and Y = M'/(2 · 2) = M'/4. (Since the spending shares are equal and both prices are 2.)
So utility = (M'/4)(M'/4) = (M')²/16 = 900.
Solving: (M')² = 14,400, so M' = 120.
The compensated bundle at M' = 120: X = 120/4 = 30, Y = 120/4 = 30.
Check: U = 30 · 30 = 900. Correct.
Substitution effect: Change in X from original to compensated bundle = 30 - 20 = +10.
Income effect: Change in X from compensated to final bundle = 45 - 30 = +15.
Total effect: 10 + 15 = +25. This checks out.
Both effects are positive, confirming X is a normal good. The substitution effect (the consumer switches toward the now-cheaper good) accounts for 10 of the 25-unit increase. The income effect (the price drop effectively raised real income) accounts for the remaining 15.
The five-step recipe for Hicksian decomposition:
Find the original optimal bundle (X₁, Y₁) using tangency + budget constraint.
Compute original utility: U₁ = U(X₁, Y₁).
Find the new optimal bundle (X₂, Y₂) at the new prices with the same income.
Find the compensated income M': the income that, at new prices, yields U₁. For Cobb-Douglas U = X^a · Y^b:
M' = \left( U_1 \cdot \left(\frac{P_X'}{a}\right)^a \cdot \left(\frac{P_Y}{b}\right)^b \cdot (a+b)^{a+b} \right)^{\frac{1}{a+b}}For U = XY (a = b = 1): the compensated bundle has X = M'/(2P_X') and Y = M'/(2P_Y), with XY = U₁. Solve for M'.
Find the compensated bundle (X_c, Y_c) at new prices with income M'. Then:
SE = X_c - X_1, \qquad IE = X_2 - X_c, \qquad TE = X_2 - X_1 = SE + IEQuick check: If the good is normal, SE and IE have the same sign. If inferior, they have opposite signs. The total effect should equal SE + IE.
The Slutsky decomposition is how economists separate the effect of a petrol tax into "people drive less because petrol is relatively more expensive" (substitution) and "people drive less because they are poorer" (income). Governments designing tax rebates or lump-sum transfers care about this distinction: a rebate can offset the income effect but not the substitution effect, which is the whole point of the tax.
Students often compute the compensated income by plugging the old bundle into the new budget constraint. That is the Slutsky (not Hicksian) approach and gives a different number. This course uses the Hicksian method: find the income that reaches the old utility at the new prices.
Students sometimes compute the SE and IE for the wrong good. The question asks about X, so all your differences should be in X-values. The Y-values change too, but the question is about X.
Students forget to verify that SE + IE = total effect. This is a free error check, and failing to do it is throwing away marks.
Students mix up the direction of effects. For a price decrease in a normal good: SE is positive (buy more of the cheaper good), IE is positive (higher real income increases demand). For an inferior good, the IE would be negative.
⚠️ The decomposition is worth 10-12 points on a typical exam. The compensated income is the hardest step and where most errors occur.
⚠️ Always state the SE and IE separately and label them. Partial credit depends on it.
⚠️ Know that for perfect complements, the SE is zero (no substitution possible), and for perfect substitutes the IE may be zero (corner solutions). The decomposition is most interesting for Cobb-Douglas, where both effects are non-zero.
True or False: The substitution effect of a price decrease always increases quantity demanded of that good.
Fill in the blank: The compensated income is the income needed to reach the ______ utility level at the ______ prices.
True or False: For a normal good, the income effect of a price decrease reduces quantity demanded.
Fill in the blank: Total effect = ______ + ______.
True or False: For U = XY with M = 180, original P_X = 4.50, P_Y = 2, the original optimal X is 20.
Answers: 1. True (by the law of demand applied to compensated demand). 2. Original, new. 3. False (it increases quantity demanded for a normal good). 4. Substitution effect, income effect. 5. True (spending on X = 90, X = 90/4.50 = 20).
Q: A consumer has U(X, Y) = XY, M = 200, P_X = 5, P_Y = 2. P_X falls to 2. Find the SE and IE for X.
A: Original: X = 200/(2·5) = 20, Y = 200/(2·2) = 50. U = 1000. New: X = 200/(2·2) = 50, Y = 50. Total effect = +30. Compensated: at new prices, X = M'/4 and Y = M'/4, so XY = (M')²/16 = 1000, giving M' = 126.49 (approx). X_c = 126.49/4 = 31.62. SE = 31.62 - 20 = 11.62. IE = 50 - 31.62 = 18.38. SE + IE = 30. Check.
Q: Why is the compensated income always less than the original income when a price falls?
A: When a price falls, the consumer can reach the same utility with less money, because the goods are cheaper. So M' < M.
Q: A consumer with perfect complements utility U = MIN[X, Y] faces a price drop in X. Decompose the total effect into SE and IE.
A: The SE is zero, because the consumer cannot substitute between goods (fixed ratio). The entire total effect is the income effect. The price drop raises real income, and the consumer buys more of both goods in the fixed ratio.
Q: In the worked example, the compensated income was 120 and the original income was 180. What is the compensating variation?
A: The compensating variation is M - M' = 180 - 120 = 60. This is the amount of income the consumer would be willing to give up and still be as well off as before the price drop.
The Slutsky decomposition connects directly to the theory of demand. The substitution effect alone generates the compensated (Hicksian) demand curve, which is always downward-sloping. The ordinary (Marshallian) demand curve includes both effects, which is why it can theoretically slope upward for a Giffen good.
This also ties into welfare economics: the compensating variation (how much income the consumer would give up to keep the same utility after a price drop) is a money measure of welfare change used in cost-benefit analysis and tax policy.
The decomposition for perfect complements (SE = 0, everything is IE) and for quasilinear preferences (IE = 0 for the good entering linearly) are important special cases that the course often tests as contrasts to the Cobb-Douglas case worked here.
Slutsky decomposition, Hicksian decomposition, substitution effect, income effect, compensating variation, equivalent variation, compensated demand, Hicksian demand, Marshallian demand, normal good, inferior good, Giffen good, Cobb-Douglas, utility maximisation, price change, real income, purchasing power, ECON 302, Penn State, intermediate microeconomics, consumer theory, indifference curve analysis