Difficulty: Intermediate-Advanced | Prerequisites: Cobb-Douglas and Leontief utility maximisation, budget constraints, indirect utility functions
When a price changes, we know the consumer is better or worse off, but by how much? Compensating variation (CV) and equivalent variation (EV) answer that question in dollar terms. The Slutsky decomposition then splits the total demand change into a substitution effect (the change due to relative prices shifting) and an income effect (the change due to purchasing power shifting). These are the core welfare-measurement tools in intermediate micro and appear on virtually every ECON 302 exam.
CV asks: how much extra money does the consumer need at the new prices to reach their original utility? EV asks: how much money could you take away at the original prices to make the consumer as badly off as they will be at the new prices? The substitution effect isolates the price-ratio change by giving the consumer enough money to stay on the original indifference curve at the new prices. The income effect is whatever demand change remains after that.
Compensating variation (CV)
The amount of additional income a consumer would need, at the new prices, to achieve the same utility they had before the price change. Think of it as the money required to "compensate" the consumer for a price increase so they are no worse off than before.
Equivalent variation (EV)
The amount of income you could take away from the consumer, at the original prices, that would leave them exactly as badly off as the price change does. Think of it as the maximum the consumer would pay to prevent the price increase from happening.
Substitution effect (SE)
The change in quantity demanded that results purely from the change in relative prices, holding utility constant. Graphically, it is the movement along the original indifference curve from the old tangency to the tangency on the compensated budget line. For a normal good with a price increase, SE is always negative (less of the good whose price rose).
Income effect (IE)
The change in quantity demanded that results from the change in real purchasing power, holding relative prices constant at their new level. It is the shift from the compensated bundle to the final bundle. For a normal good, the income effect reinforces the substitution effect (both reduce demand when price rises).
Compensated budget constraint
The hypothetical budget line at the new prices, but with enough income added (the CV) so that the consumer can just reach their original utility level. It is tangent to the original indifference curve at the new price ratio.
Slutsky decomposition
The method of breaking the total change in demand into the substitution effect and the income effect. Total change = SE + IE. Named after economist Eugen Slutsky.
Indirect utility function
Utility expressed as a function of prices and income: V(Px, Py, M). Obtained by substituting the optimal demands back into the utility function. Used to compute CV and EV by solving for the income needed to hit a target utility level.
Setup: U = X^(1/2) · Y^(1/2), M = $24, Px = $1, Py = $1. Px rises to $4.
Original bundle: X* = 12, Y* = 12. Original utility: U* = 12.
New bundle: X** = 3, Y** = 12. New utility: U** = 6.
To find CV, ask: what income M' at the new prices (Px = 4, Py = 1) gives utility = 12?
At new prices, optimal demands are X = M'/(2·4) = M'/8 and Y = M'/(2·1) = M'/2.
Utility: (M'/8)^(1/2) · (M'/2)^(1/2) = (M'^2/16)^(1/2) = M'/4 = 12, so M' = 48.
CV = M' - M = 48 - 24 = $24. The consumer needs $24 extra to reach their old utility at the new prices.
Setup: U = min(3X, Y), M = $36, Px = $1, Py = $1. Px rises to $3.
Original bundle: X* = 9, Y* = 27. Original utility: V* = 27.
New utility at new prices: V** = (3·1·36)/(1·3 + 3·1) = 108/6 = 18.
To find EV, ask: what income M'' at the original prices (Px = 1, Py = 1) gives utility = 18?
At original prices: V = (3·1·M'')/(1·1 + 3·1) = 3M''/4 = 18, so M'' = 24.
EV = M - M'' = 36 - 24 = $12. You could take $12 away at the old prices and the consumer would be exactly as badly off as after the price increase.
Same setup: U = min(3X, Y), M = $36, Px rises from $1 to $3, Py = $1.
Original utility: V* = 27.
At new prices, V = (3·1·M')/(1·3 + 3·1) = 3M'/6 = M'/2 = 27, so M' = 54.
CV = 54 - 36 = $18.
Setup: same as the CV example above. Px rises from $1 to $4.
Original bundle: (X*, Y*) = (12, 12).
New bundle: (X**, Y**) = (3, 12).
Total change in X: ΔX = 3 - 12 = -9.
To decompose:
Compensated bundle (X^c, Y^c): the bundle at new prices (Px = 4) with enough income to reach original utility (U = 12). We found M' = 48. So X^c = 48/8 = 6, Y^c = 48/2 = 24.
Substitution effect: X^c - X* = 6 - 12 = -6.
Income effect: X** - X^c = 3 - 6 = -3.
Check: SE + IE = -6 + (-3) = -9 = total change. Correct.
Both effects are negative because X is a normal good and its price increased. The substitution effect (-6) accounts for two-thirds of the total demand drop, and the income effect (-3) accounts for the remaining third.
The exam commonly asks you to draw three budget lines on one graph:
Original budget constraint: M = 24, Px = 1, Py = 1. Intercepts at (24, 0) and (0, 24). The optimal bundle (12, 12) sits on this line.
New budget constraint: M = 24, Px = 4, Py = 1. Intercepts at (6, 0) and (0, 24). Steeper because X is now more expensive. The new bundle (3, 12) sits here.
Compensated budget constraint: M' = 48, Px = 4, Py = 1. Intercepts at (12, 0) and (0, 48). Same slope as the new budget line, but shifted outward. Tangent to the original indifference curve at the compensated bundle (6, 24).
CV formula (general):
Find M' such that V(Px_new, Py, M') = V(Px_old, Py, M). Then CV = M' - M.
EV formula (general):
Find M'' such that V(Px_old, Py, M'') = V(Px_new, Py, M). Then EV = M - M''.
CV for Cobb-Douglas U = X^(1/2) · Y^(1/2):
Indirect utility: V = M / (2 · Px^(1/2) · Py^(1/2))
Set M' / (2 · Px_new^(1/2) · Py^(1/2)) = V_old and solve for M'.
CV and EV for Leontief U = min(aX, bY):
Indirect utility: V = abM / (bPx + aPy)
Set abM' / (bPx_new + aPy) = V_old for CV; set abM'' / (bPx_old + aPy) = V_new for EV.
Slutsky decomposition:
Total change: ΔX = X** - X*
Substitution effect: SE = X^c - X*
Income effect: IE = X** - X^c
Check: SE + IE = ΔX
Students frequently confuse CV and EV. A useful mnemonic: CV uses new prices (how much money compensates you after the change), EV uses old prices (what equivalent income loss would hurt you the same amount before the change).
A common sign error: for a price increase, CV is positive (the consumer needs more money) and EV is also positive (you could take money away). The problem set may tell you the sign convention. Read carefully.
Students sometimes compute the substitution effect by moving along the budget line rather than along the indifference curve. The substitution effect holds utility constant, not income.
For perfect complements, the substitution effect is zero. Students who try to compute a nonzero SE for Leontief preferences have made an error. The entire demand change is the income effect.
When graphing, students often draw the compensated budget constraint tangent to the wrong indifference curve. It should be tangent to the original indifference curve (the one through the original bundle), not the new one.
⚠️ CV and EV questions are high-value on exams (6+ points each in this problem set). The computation follows a fixed recipe: find original utility, set up the indirect utility equation at the relevant prices, solve for the required income, subtract.
⚠️ The graphing question (original, new, and compensated budget constraints with labelled bundles) is worth 14 points in this problem set. Label every line with M, Px, and Py values. Mark the three bundles (X*, Y*), (X**, Y**), and (X^c, Y^c) with their coordinates.
⚠️ For Cobb-Douglas, CV ≠ EV in general (they are equal only when income effects are zero, which they are not here). For Leontief, CV and EV also differ. Know why: they use different price vectors.
⚠️ Always verify SE + IE = total change. If they do not add up, you have an error somewhere.
True or False: CV is computed at the new prices.
True or False: The substitution effect for perfect complements is zero.
Fill in the blank: For Cobb-Douglas, the compensated bundle is found by optimising at the new prices with income equal to ____.
True or False: CV and EV are always equal for Cobb-Douglas utility.
Fill in the blank: If SE = -6 and total change = -9, then IE = ____.
Answers: 1. True. 2. True. 3. M' (the income required to reach the original utility at new prices, i.e. M + CV). 4. False (they differ because Cobb-Douglas has nonzero income effects). 5. IE = -3.
Q: A consumer has U = X^(1/3) · Y^(2/3), M = $90, Px = $1, Py = $2. Px rises to $3. What is the CV?
A: Original demands: X* = (1/3)(90/1) = 30, Y* = (2/3)(90/2) = 30. U* = 30^(1/3) · 30^(2/3) = 30. At new prices: X** = (1/3)(90/3) = 10, Y** = (2/3)(90/2) = 30. For CV, find M' such that (M'/9)^(1/3) · (M'/6)^(2/3) = 30. This simplifies to solving for M'; the indirect utility is V = (1/3)^(1/3) · (1/3)^(2/3) · M / (Px^(1/3) · Py^(2/3)). At new prices: M'/((3)^(1/3)·(3)^(1/3)·(2)^(2/3)·3) = 30... The cleanest approach is: V_old = 30, and V at new prices = M' · (1/3)^(1/3) · (1/3)^(2/3) / (3^(1/3) · 2^(2/3)) = 30. Solve for M'.
Q: For a Leontief consumer U = min(2X, Y) with M = $100, Px = $5, Py = $2, what is the substitution effect when Px rises to $10?
A: Zero. For perfect complements, the substitution effect is always zero because the consumer cannot substitute between goods. The entire demand change is the income effect.
Q: In the Slutsky decomposition, what does the compensated bundle represent economically?
A: It represents what the consumer would buy if they faced the new price ratio but were given just enough income to maintain their original utility level. It isolates the pure price-ratio effect by removing the purchasing-power loss.
Q: A consumer's SE for Good X is -6 and their IE for Good X is -3. Is Good X a normal good, an inferior good, or a Giffen good?
A: Normal good. Both the SE and IE are negative (reducing demand when price rises), which is the defining pattern for a normal good. An inferior good would have a positive IE partly offsetting the negative SE. A Giffen good would have a positive IE large enough to outweigh the SE entirely.
CV and EV connect to consumer surplus (the area under the demand curve), which you will study soon. Consumer surplus is a rough approximation; CV and EV are the exact welfare measures. The Slutsky decomposition connects to the distinction between normal, inferior, and Giffen goods, which is determined entirely by the sign and magnitude of the income effect. These tools also reappear in general equilibrium and public finance later in the course.
compensating variation, CV, equivalent variation, EV, substitution effect, income effect, Slutsky decomposition, compensated budget constraint, Hicksian decomposition, indirect utility, expenditure function, normal good, inferior good, Giffen good, welfare measurement, price change, ECON 302, intermediate microeconomics