Difficulty: Intermediate | Prerequisites: Budget constraints, indifference curves, understanding of Cobb-Douglas optimisation (see companion notes).
Perfect complements (Leontief preferences) are the polar opposite of smooth substitution. The consumer always wants goods in a fixed ratio, so indifference curves are L-shaped. This topic also introduces two important welfare measures, compensating variation (CV) and equivalent variation (EV), which put a dollar value on how much a price change hurts or helps a consumer. These concepts appear on virtually every intermediate micro exam.
With U = min(aX, bY), the consumer always chooses the kink of the L-shaped indifference curve, where aX = bY. There is no substitution effect for perfect complements. CV asks "how much extra income would the consumer need at new prices to be as well off as before?" EV asks "how much income could you take away at old prices to make the consumer as badly off as they will be after the price change?"
Perfect complements (Leontief preferences)
Preferences where goods are consumed in a fixed ratio. U(X, Y) = min(aX, bY). The consumer gains nothing from having more of one good without more of the other in the correct proportion. Think of left shoes and right shoes.
L-shaped indifference curves
The graphical signature of perfect complements. The kink sits where aX = bY. Utility increases only by moving to a higher L (further from the origin).
Compensating variation (CV)
The amount of additional income a consumer would need, at the new prices, to reach the same utility as before the price change. When the price goes up, CV is positive: the consumer needs compensation.
Equivalent variation (EV)
The amount of income you could take away from the consumer, at the old prices, that would reduce their utility to the level they will have after the price change. When the price goes up, EV is positive: the price change is equivalent to losing that much income.
Indirect utility function V(Px, Py, M)
The maximum utility achievable given prices and income. For min(aX, bY): V = aM / (Px + (a/b)Py). You get this by substituting the optimal bundle back into the utility function.
Expenditure function M(U, Px, Py)
The minimum income needed to achieve utility level U at given prices. It is the inverse of the indirect utility function, solved for M.
The key condition. At the optimum, aX = bY (the consumer sits at the kink). Any other point wastes money on a good that does not increase utility.
Solving for X and Y. From aX = bY, get Y = (a/b)X. Substitute into the budget constraint Px·X + Py·Y = M:
Px·X + Py·(a/b)·X = M
X·(Px + (a/b)·Py) = M
X* = M / (Px + (a/b)·Py)
Y* = (a/b)·X* = aM / (b·(Px + (a/b)·Py))
Worked example (from the homework). U = min(3X, Y), so a = 3, b = 1. M = 30, Px = 2, Py = 1.
At the kink: 3X = Y
Substitute: 2X + 1·(3X) = 30, so 5X = 30, X* = 6
Y* = 3·6 = 18
U(6, 18) = min(18, 18) = 18
Why there is no tangency condition. The indifference curve has a kink at the optimal point, so the MRS is undefined there. You cannot set MRS = price ratio. Instead, you use aX = bY plus the budget constraint.
CV answers: at the new prices, how much extra income does the consumer need to reach the original utility?
Step 1. Find the original utility: U = 18 (from the worked example above).
Step 2. Find the indirect utility or, equivalently, the expenditure function. For U = min(3X, Y) with a = 3, b = 1:
V(Px, Py, M) = M / (Px/3 + Py) = 3M / (Px + 3Py)
Expenditure function (invert for M): M(U, Px, Py) = U·(Px + 3Py) / 3
Step 3. Now Px rises to 3. Compute the income needed at new prices for U = 18:
M^c = 18·(3 + 3·1) / 3 = 18·6/3 = 36
Step 4. CV = M^c - M = 36 - 30 = 6. The consumer needs $6 of compensation.
Check. At the new prices with M^c = 36: X^c = 36/(3 + 3) = 6, Y^c = 18. U = min(18, 18) = 18. Correct.
EV answers: at the old prices, how much income could you take away to leave the consumer as badly off as after the price change?
Step 1. Find the new utility after the price change. With Px = 3, Py = 1, M = 30:
X** = 30/(3 + 3) = 5, Y** = 15
U_new = min(15, 15) = 15
Step 2. Find the income at old prices that gives U = 15:
M^E = 15·(2 + 3·1) / 3 = 15·5/3 = 25
Step 3. EV = M - M^E = 30 - 25 = 5. Losing $5 at old prices is equivalent to the price increase.
Check. At old prices with M^E = 25: X^E = 25/5 = 5, Y^E = 15. U = min(15, 15) = 15. Correct.
For a normal good whose price rises, CV > EV. Here CV = 6 > EV = 5. This is a general result. CV uses the new (higher) prices to value the loss, so it yields a larger dollar amount.
For U(X, Y) = min(aX, bY) with prices Px, Py and income M:
Optimal bundle (at the kink, aX = bY):
X^* = \frac{M}{P_X + \frac{a}{b}P_Y}, \quad Y^* = \frac{a}{b} \cdot X^*Indirect utility:
V(P_X, P_Y, M) = \frac{aM}{P_X + \frac{a}{b}P_Y}Expenditure function:
M(U, P_X, P_Y) = \frac{U}{a}\left(P_X + \frac{a}{b}P_Y\right)Welfare measures (price increase):
CV = M(U_{\text{old}}, P_X^{\text{new}}, P_Y) - MEV = M - M(U_{\text{new}}, P_X^{\text{old}}, P_Y)Perfect complements model situations where goods must be used together in fixed proportions: a printer and ink cartridges, a phone plan and a handset, or components in a recipe. CV and EV are used in cost-benefit analysis of public policy. When a government considers a tax that raises the price of a necessity, CV measures how much it would need to transfer back to households to leave them no worse off. EV converts the harm into a lump-sum-at-old-prices equivalent, which is useful for comparing policies on a common basis.
Students try to set MRS = price ratio for perfect complements. There is no MRS at the kink. Use the condition aX = bY instead.
Confusing CV and EV. CV uses new prices and asks for extra income. EV uses old prices and asks how much income to take away. A mnemonic: CV compensates you after the change; EV is equivalent to the change before it happens.
Forgetting to check the direction. When a price increases, both CV and EV are positive. Students sometimes get a negative number and do not realise they have swapped the subtraction.
Assuming CV = EV. They are equal only for quasilinear preferences (no income effect on the good in question). For normal goods, CV > EV when the price rises.
⚠️ Perfect complements questions are almost guaranteed on intermediate micro exams. The algebra is simpler than Cobb-Douglas, so professors often pair them with CV/EV calculations.
⚠️ You will be asked to graph three budget lines (original, new, and either compensated or equivalent) with L-shaped indifference curves passing through each optimal bundle.
⚠️ With perfect complements there is zero substitution effect. The entire change in demand is an income effect. If an exam asks you to decompose SE and IE for a Leontief consumer, SE = 0.
⚠️ Know both the CV and EV formulas cold, and be ready to compute both in a single problem. The homework asks for both.
For U = min(3X, Y), what condition must hold at the optimum? (3X = Y)
True or false: The MRS is well-defined at the optimal bundle for perfect complements. (False)
If CV = 6 and EV = 5 for a price increase, which is larger and why? (CV, because it is valued at the higher new prices)
True or false: For perfect complements, the substitution effect is always zero. (True)
The expenditure function for min(3X, Y) at prices Px = 3, Py = 1 and U = 18 equals ______. (36)
Q: A consumer has U = min(2X, Y), M = 60, Px = 4, Py = 1. What is the optimal bundle?
A: At the kink, 2X = Y. Budget: 4X + Y = 60. Substitute: 4X + 2X = 60, so X* = 10, Y* = 20. U = min(20, 20) = 20.
Q: Now Px rises to 6. What is the new optimal bundle?
A: 6X + 2X = 60, so X** = 7.5, Y** = 15. U_new = 15.
Q: Calculate the compensating variation.
A: M^c = U_old · (Px_new/a + Py/b) = 20 · (6/2 + 1/1) = 20 · 4 = 80. CV = 80 - 60 = 20.
Q: Calculate the equivalent variation.
A: M^E = U_new · (Px_old/a + Py/b) = 15 · (4/2 + 1/1) = 15 · 3 = 45. EV = 60 - 45 = 15.
Q: Why is CV > EV here?
A: Both goods are normal. CV is calculated at the new, higher prices, so the dollar cost of restoring the consumer's welfare is larger than the dollar equivalent of the loss at the old, lower prices.
CV and EV bridge consumer theory and welfare economics. They reappear when you study consumer surplus, deadweight loss from taxation, and the welfare effects of price controls. For quasilinear utility (no income effect), CV = EV = change in consumer surplus, which simplifies welfare analysis considerably.
Perfect complements also connect to production theory: a fixed-proportions (Leontief) production function works the same way, with inputs instead of consumption goods.
Compare with the Cobb-Douglas notes to see the contrast: smooth substitution (Cobb-Douglas) versus zero substitution (Leontief). Most exam questions test both in the same assignment.
Tags: perfect complements, Leontief preferences, min function, L-shaped indifference curves, compensating variation, CV, equivalent variation, EV, indirect utility function, expenditure function, welfare measurement, consumer surplus, normal good, fixed proportions, ECON 302, intermediate microeconomics