Difficulty: Intermediate | Prerequisites: Cobb-Douglas CV/EV notes, budget constraints, corner solutions
Perfect complements and perfect substitutes are the two polar cases of consumer preferences. Where Cobb-Douglas gives smooth curves and interior solutions, these utility functions produce L-shaped indifference curves (complements) or straight-line indifference curves (substitutes). The mechanics of CV, EV, and the Slutsky decomposition change in important ways: perfect complements have zero substitution effect, and perfect substitutes can produce all-or-nothing corner solutions. This material tests whether you understand the economic intuition behind the maths, not just the formula.
With perfect complements (Leontief), the consumer always buys goods in a fixed ratio, so a price change produces no substitution effect at all: the entire demand response is an income effect. With perfect substitutes, the consumer buys whichever good gives more utility per pound, and the demand response can be entirely a substitution effect. CV and EV are computed differently for each case.
Perfect complements (Leontief utility)
A utility function of the form U(X, Y) = MIN(aX, bY). The consumer always consumes goods in the fixed ratio aX = bY. Indifference curves are L-shaped. Think of it as left shoes and right shoes: having extra left shoes without matching right shoes adds no value.
Perfect substitutes
A utility function of the form U(X, Y) = aX + bY. The consumer is willing to trade X for Y at a constant rate regardless of how much of each they have. Indifference curves are straight lines. Think of it as two brands of bottled water: the consumer simply buys whichever is cheaper per unit of utility.
Corner solution
An optimal bundle where the consumer buys zero of one good and spends everything on the other. This happens with perfect substitutes when the marginal rate of substitution does not equal the price ratio.
Interior solution
An optimal bundle where the consumer buys positive quantities of both goods. Cobb-Douglas always gives interior solutions. Perfect complements always give interior solutions (at the kink). Perfect substitutes give interior solutions only when MRS equals the price ratio exactly.
Kink point
For perfect complements, the optimal bundle sits at the corner (kink) of the L-shaped indifference curve, where aX = bY. This is where the consumer gets full value from both goods with no waste.
Given: U(X, Y) = MIN(5X, Y), M = $24, Px = $1, Py = $1. Then Px rises to $3.
At the kink of the L-shaped indifference curve: 5X = Y.
Substitute into the budget constraint: Px X + Py Y = M.
1 X + 1(5X) = 24, so 6X = 24, X* = 4.
Y* = 5(4) = 20.
U(4, 20) = MIN(5 x 4, 20) = MIN(20, 20) = 20.
The kink condition 5X = Y still holds.
3X + 1(5X) = 24, so 8X = 24, X** = 3.
Y** = 5(3) = 15.
New utility: U(3, 15) = MIN(15, 15) = 15.
At the new prices (Px = $3, Py = $1), what income restores utility to 20?
At the kink: 5X = Y and U = 5X = 20, so X = 4, Y = 20.
Required income: 3(4) + 1(20) = 12 + 20 = $32.
CV = $32 - $24 = $8.
At the old prices (Px = $1, Py = $1), what income gives utility 15?
At the kink: 5X = Y and U = 5X = 15, so X = 3, Y = 15.
Required income: 1(3) + 1(15) = $18.
EV = $24 - $18 = $6.
The compensated bundle uses new prices and compensated income ($32):
3X + 1(5X) = 32, so 8X = 32, Xc = 4, Yc = 20.
SE = Xc - X = 4 - 4 = 0.* There is no substitution effect. The consumer cannot substitute between goods because they are perfect complements.
IE = X - Xc = 3 - 4 = -1.** The entire change in demand (from 4 to 3, a decrease of 1) is due to the income effect.
This makes intuitive sense: with L-shaped indifference curves, there is no scope for substitution along the curve. The consumer is stuck at the kink.
Given: U(X, Y) = 3X + Y, M = $24, Px = $1, Py = $1. Then Px rises to $4.
The MRS = MUx / MUy = 3 / 1 = 3. The price ratio = Px / Py = 1 / 1 = 1.
Since MRS (3) > price ratio (1), X gives more utility per dollar than Y. The consumer spends everything on X.
X* = 24 / 1 = 24, Y* = 0.
U* = 3(24) + 0 = 72.
MRS = 3. Price ratio = 4 / 1 = 4.
Since MRS (3) < price ratio (4), Y now gives more utility per dollar. The consumer spends everything on Y.
X** = 0, Y** = 24 / 1 = 24.
U** = 3(0) + 24 = 24.
At new prices (Px = $4, Py = $1), what income reaches utility 72?
At these prices the consumer buys only Y (since MRS < price ratio), so U = Y = M_c / 1. Need U = 72, so M_c = $72.
CV = $72 - $24 = $48.
At old prices (Px = $1, Py = $1), what income gives utility 24?
At these prices the consumer buys only X (since MRS > price ratio), so U = 3X = 3(M_e / 1). Need U = 24, so 3 M_e = 24, M_e = $8.
EV = $24 - $8 = $16.
The consumer buys only X when MRS > Px/Py, i.e. when 3 > Px (since Py = 1). The consumer buys only Y when Px > 3.
Px = $1: MRS (3) > price ratio (1). All X. Qd = 24 / 1 = 24.
Px = $2: MRS (3) > price ratio (2). All X. Qd = 24 / 2 = 12.
Px = $4: MRS (3) < price ratio (4). All Y. Qd = 0.
Px = $8: MRS (3) < price ratio (8). All Y. Qd = 0.
At Px = $3: MRS = price ratio. The consumer is indifferent between X and Y. Any combination on the budget line gives the same utility. Qd of X can be anything from 0 to 8 (= 24/3).
Perfect complements U = MIN(aX, bY):
Optimal consumption rule: aX = bY, so Y = (a/b)X.
Substitute into budget: Px X + Py(a/b)X = M, giving X = M / (Px + Py(a/b)).
Expenditure function: E = U(Px/a + Py/b).
CV = U_old(Px_new/a + Py/b) - M.
EV = M - U_new(Px_old/a + Py/b).
SE = 0 always. IE = total change.
Perfect substitutes U = aX + bY:
Decision rule: compare MRS = a/b with Px/Py.
If a/b > Px/Py: buy only X. X = M/Px.
If a/b < Px/Py: buy only Y. Y = M/Py.
If a/b = Px/Py: any combination on the budget line is optimal.
Expenditure function: E = U x min(Px/a, Py/b).
CV = U_old x min(Px_new/a, Py/b) - M.
EV = M - U_new x min(Px_old/a, Py/b).
Perfect complements show up whenever two goods must be used together in fixed proportions: a printer and ink cartridges, a phone plan and a phone, a car and petrol. Raising the price of one good cannot cause the consumer to substitute the other, because the goods are useless alone.
Perfect substitutes arise when two goods serve the same purpose interchangeably: generic vs brand-name paracetamol, two streaming services with identical content, or regular vs premium petrol for a car that runs on either. A small price difference can flip the consumer's entire budget from one good to the other.
Students sometimes try to set MRS = price ratio for perfect complements. You cannot do this. The MRS is undefined at the kink. Instead, use the kink condition (aX = bY) and the budget constraint.
With perfect substitutes, students forget to check which good the consumer buys before and after the price change. The optimal good can flip entirely.
Students often assume there must always be both a substitution effect and an income effect. With perfect complements, SE is always zero. With perfect substitutes, when the consumer switches from one corner to the other, the decomposition can be tricky: the entire response may look like a substitution effect.
For perfect complements, CV and EV are not equal in general (they differ because of the price level at which each is evaluated), but they are closer together than for Cobb-Douglas because there is no substitution.
⚠️ Perfect complements questions often ask you to state that SE = 0. This is the key insight that distinguishes them from Cobb-Douglas.
⚠️ For perfect substitutes, you must correctly identify corner vs interior solutions at each price. Getting this wrong cascades through the entire problem.
⚠️ The demand schedule for perfect substitutes has a discontinuity: demand drops from M/Px to 0 (or jumps from 0 to M/Py for Y) at the price where MRS = Px/Py.
⚠️ Examiners love to test whether you understand that CV and EV give different numbers and why. For a price increase: CV > EV with Cobb-Douglas, and CV and EV differ with complements and substitutes too.
⚠️ Always verify: SE + IE = total change. This is your error-check.
True or False: With perfect complements, the substitution effect is always zero. (True.)
Fill in the blank: For U = MIN(aX, bY), the optimal consumption rule is ____. (aX = bY)
True or False: With perfect substitutes U = 3X + Y, if Px = $2 and Py = $1, the consumer buys only X. (True, because MRS = 3 > 2 = Px/Py.)
Fill in the blank: For perfect substitutes, the consumer switches from buying all X to all Y when Px/Py exceeds ____. (MRS = a/b)
True or False: With perfect complements, the indifference curves are straight lines. (False. They are L-shaped.)
Q: U = MIN(2X, Y), M = $30, Px = $2, Py = $1. Find X and Y.**
A: Kink condition: 2X = Y. Budget: 2X + 1(2X) = 30, so 4X = 30, X* = 7.5, Y* = 15.
Q: Using the same setup, Px rises to $4. What is the CV?
A: At new prices, need utility = MIN(2(7.5), 15) = 15. At the kink with Px = $4: 4X + 1(2X) = 6X. Need 2X = 15, so X = 7.5, Y = 15. Income needed: 6(7.5) = $45. CV = $45 - $30 = $15.
Q: U = 5X + Y, M = $20, Px = $1, Py = $1. Px rises to $10. What does the consumer buy before and after?
A: Before: MRS = 5 > 1 = Px/Py. Buy only X: X* = 20, Y* = 0, U* = 100. After: MRS = 5 < 10 = Px/Py. Buy only Y: X** = 0, Y** = 20, U** = 20.
Q: For perfect complements, why is SE always zero?
A: The indifference curves have a right-angle kink. The consumer is always at the kink, so there is no scope to slide along the curve to a different combination of goods. Any movement along the L-shaped curve away from the kink wastes money on a good that does not increase utility.
These three utility types (Cobb-Douglas, perfect complements, perfect substitutes) form a spectrum. Cobb-Douglas sits in the middle with smooth substitution, perfect complements allow none, and perfect substitutes allow infinite substitution at a constant rate. Understanding all three gives you the full picture of how the shape of indifference curves determines the breakdown between substitution and income effects. This connects to consumer surplus and deadweight loss: the substitution effect is what creates deadweight loss from a tax, so a good with no substitution effect (perfect complements) generates no deadweight loss from taxation.
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