Difficulty: Intermediate | Prerequisites: Budget constraints, utility maximisation, indifference curves
This material sits at the heart of intermediate microeconomics: once you can solve a consumer's optimisation problem, the next question is how a price change affects their welfare and their choices. Compensating and equivalent variation are the two monetary measures of welfare change, and the Slutsky decomposition splits the total demand response into substitution and income effects. You need to be comfortable with constrained optimisation (Lagrangian method) and indifference curves before tackling this.
When a price rises, a Cobb-Douglas consumer buys less of that good. The compensating variation (CV) asks how much extra income restores the old utility; the equivalent variation (EV) asks how much income you could take away at the old prices to make them equally worse off. The Slutsky decomposition then breaks the total change in quantity demanded into a substitution effect (movement along the indifference curve) and an income effect (shift to a new curve).
Cobb-Douglas utility function
A utility function of the form U(X, Y) = X^a Y^b (often with a + b = 1). It produces smooth, convex indifference curves and always yields interior solutions. In simple terms, the consumer always wants some of both goods.
Compensating variation (CV)
The amount of additional income a consumer would need, after a price change, to reach the same utility level they had before. Think of it as: "how much would we have to pay you to make the price hike tolerable?"
Equivalent variation (EV)
The amount of income you could take away from the consumer, at the original prices, to leave them as badly off as the price change would. Think of it as: "how much poorer would you need to be at the old prices to feel equally hurt?"
Substitution effect (SE)
The change in quantity demanded of a good caused purely by the change in relative prices, holding utility constant. The consumer slides along the same indifference curve to a new tangency point.
Income effect (IE)
The change in quantity demanded caused by the change in real purchasing power, after the substitution effect has been accounted for. The consumer moves from the compensated bundle to the final bundle.
Compensated (Hicksian) demand
The demand for a good as a function of prices, holding utility fixed rather than income fixed. This is what the compensated budget constraint captures.
Marshallian (ordinary) demand
The demand for a good as a function of prices and income. This is the standard demand function derived from utility maximisation.
Indirect utility function
The maximum utility achievable given prices and income. For Cobb-Douglas U = X^(1/2) Y^(1/2), the indirect utility is V = M / (2 sqrt(Px Py)).
Expenditure function
The minimum expenditure needed to reach a given utility level at given prices. It is the inverse of the indirect utility function and is used to compute CV and EV.
Given: U(X, Y) = X^(1/2) Y^(1/2), M = $18, Py = $1, Px starts at $1 and rises to $9.
For Cobb-Douglas with equal exponents, the consumer spends half of income on each good. The demand functions are:
X* = M / (2Px)
Y* = M / (2Py)
X* = 18 / (2 x 1) = 9
Y* = 18 / (2 x 1) = 9
U* = 9^(1/2) x 9^(1/2) = 3 x 3 = 9
X** = 18 / (2 x 9) = 1
Y** = 18 / (2 x 1) = 9
U** = 1^(1/2) x 9^(1/2) = 1 x 3 = 3
The total change in X: from 9 to 1, so delta X = -8.
CV answers: at the new prices (Px = $9, Py = $1), how much income is needed to restore the original utility of 9?
Using the expenditure function: E = 2 x U x sqrt(Px x Py) = 2 x 9 x sqrt(9 x 1) = 2 x 9 x 3 = $54.
The consumer currently has $18, so CV = $54 - $18 = $36.
EV answers: at the old prices (Px = $1, Py = $1), how much income would need to be taken away to reduce utility to the new level of 3?
E = 2 x U x sqrt(Px x Py) = 2 x 3 x sqrt(1 x 1) = 2 x 3 x 1 = $6.
The consumer currently has $18, so EV = $18 - $6 = $12.
The compensated bundle uses the new prices (Px = $9, Py = $1) and the compensated income ($54):
Xc = 54 / (2 x 9) = 3
Yc = 54 / (2 x 1) = 27
Check: U(3, 27) = 3^(1/2) x 27^(1/2) = sqrt(81) = 9. Correct.
Substitution effect (SE): Xc - X* = 3 - 9 = -6
Income effect (IE): X** - Xc = 1 - 3 = -2
Total change: SE + IE = -6 + (-2) = -8. This matches X** - X* = 1 - 9 = -8.
You need three lines on the same graph (X on the horizontal axis, Y on the vertical):
Original budget line (black): 1X + 1Y = 18. Intercepts at (18, 0) and (0, 18). Slope = -1. Optimal bundle: (9, 9).
New budget line (green): 9X + 1Y = 18. Intercepts at (2, 0) and (0, 18). Slope = -9. Optimal bundle: (1, 9).
Compensated budget line (red): 9X + 1Y = 54. Intercepts at (6, 0) and (0, 54). Slope = -9. Optimal bundle: (3, 27).
The compensated line is parallel to the new line (same slope of -9) but tangent to the original indifference curve (U = 9). It sits further out because the consumer has been given the extra $36 (the CV) to compensate for the price rise.
Cobb-Douglas demand (equal exponents):
X* = M / (2Px), Y* = M / (2Py)
Indirect utility function:
V(Px, Py, M) = M / (2 sqrt(Px Py))
Expenditure function:
E(Px, Py, U) = 2U sqrt(Px Py)
CV and EV formulas:
CV = E(Px_new, Py, U_old) - M
EV = M - E(Px_old, Py, U_new)
Slutsky decomposition:
SE = Xc - X* (substitution effect, movement along the original indifference curve at new prices)
IE = X** - Xc (income effect, shift from compensated bundle to final bundle)
Total change = SE + IE = X** - X*
CV and EV are how economists measure the welfare cost of a tax or a price increase. When a government considers a petrol tax, the CV tells policymakers how much a lump-sum transfer would need to be to keep consumers at their pre-tax wellbeing. The substitution and income effects matter in labour economics too: a higher wage makes leisure more expensive (substitution effect: work more) but also makes you richer (income effect: consume more leisure). Whether someone works more or fewer hours when wages rise depends on which effect dominates.
Students often confuse CV and EV. CV uses the new prices and the old utility. EV uses the old prices and the new utility. A mnemonic: CV compensates you after the change (new prices), EV is the equivalent hit at old prices.
Students sometimes forget that for a price increase, both CV and EV are positive numbers (the consumer is worse off). For a price decrease, both would be negative (or stated as positive gains).
The substitution effect for a normal good always goes in the opposite direction of the price change (price up, quantity down). The income effect reinforces this for normal goods. Students sometimes confuse themselves by expecting one of these to go the other way.
When computing the compensated bundle, students occasionally use the original income instead of the compensated income. The compensated bundle uses the new prices and the compensated income (M + CV).
⚠️ You will almost certainly be asked to compute CV, EV, SE, and IE for a Cobb-Douglas utility function. Know the expenditure function cold.
⚠️ Graphing questions require three budget lines and three labelled bundles. The compensated budget line must be parallel to the new budget line and tangent to the original indifference curve.
⚠️ For Cobb-Douglas with equal exponents (a = b = 1/2), CV > EV when the price increases. This is a standard property that examiners test.
⚠️ Always verify your decomposition: SE + IE must equal the total change in quantity demanded. If it does not add up, you have made an error somewhere.
True or False: The compensating variation uses the old prices to calculate the required income change. (False: CV uses the new prices.)
Fill in the blank: For Cobb-Douglas U = X^(1/2) Y^(1/2), the demand for X is X* = ____. (M / 2Px)
True or False: The substitution effect of a price increase on the good whose price rose is always negative. (True.)
Fill in the blank: The expenditure function for U = X^(1/2) Y^(1/2) is E = ____. (2U sqrt(Px Py))
True or False: For a normal good, the income effect reinforces the substitution effect when the price rises. (True.)
Q: A consumer has U = X^(1/2) Y^(1/2), M = $50, Px = $1, Py = $1. The price of X rises to $4. What is the CV?
A: Original utility: X* = 25, Y* = 25, U* = 25. Expenditure at new prices to reach U = 25: E = 2(25)sqrt(4 x 1) = 100. CV = 100 - 50 = $50.
Q: Using the same setup, what is the EV?
A: New utility: X** = 50/8 = 6.25, Y** = 25, U** = sqrt(6.25 x 25) = sqrt(156.25) = 12.5. Expenditure at old prices to reach U = 12.5: E = 2(12.5)sqrt(1) = 25. EV = 50 - 25 = $25.
Q: In the worked example (M = $18, Px goes from $1 to $9), why is CV ($36) larger than EV ($12)?
A: For a price increase with Cobb-Douglas preferences, the CV is always larger than the EV. The CV is evaluated at the new, higher prices, so restoring the old utility level is expensive. The EV is evaluated at the old, lower prices, so the equivalent income loss is smaller.
Q: If the substitution effect is -6 and the income effect is -2, what type of good is X?
A: X is a normal good. Both effects reduce demand when the price rises (SE and IE are both negative), which is the hallmark of a normal good.
This material connects directly to perfect complements (Leontief) and perfect substitutes utility, which handle CV, EV, and the Slutsky decomposition differently because of their distinct indifference curve shapes. With perfect complements, the substitution effect is zero. With perfect substitutes, the consumer may hit corner solutions, and the entire demand response can be a substitution effect with no income effect. These are covered in the companion study notes.
Cobb-Douglas utility, compensating variation, equivalent variation, Hicksian demand, Marshallian demand, Slutsky decomposition, substitution effect, income effect, expenditure function, indirect utility function, welfare economics, consumer surplus, budget constraint, indifference curve, price change welfare analysis, ECON 302, intermediate microeconomics, Penn State