Difficulty: Intermediate | Prerequisites: Budget constraints, indifference curves, understanding of the min function.
Perfect complements (also called Leontief preferences) describe consumers who want goods in fixed proportions, like left and right shoes. The min function means there is no smooth trade-off between goods, so the tangency method from Cobb-Douglas does not apply. Instead, you use the kink condition. This is one of the three core preference types in intermediate micro (alongside Cobb-Douglas and perfect substitutes), and exams regularly test all three in a single paper.
Perfect complements are consumed in fixed ratios (like 2 shorts for every 1 pair of leggings). There is no tangency condition to set, because the indifference curves have a kink, not a smooth slope. You find the kink condition from the utility function, substitute it into the budget constraint, and solve. Both Engel and demand curves follow the same logic as Cobb-Douglas, with the kink condition replacing the tangency condition.
Perfect complements
Preferences where two goods are always consumed in a fixed ratio. Having more of one good without more of the other adds no utility. Think of it as needing exactly two socks per pair of shoes: extra socks without shoes are useless.
Leontief preferences
Another name for perfect complements, after economist Wassily Leontief. The terms are interchangeable in this course.
Min function
The utility function U(x, y) = min(ax, by), which returns the smaller of the two arguments. This is what creates the L-shaped indifference curves.
Kink point
The corner of an L-shaped indifference curve where the consumer optimally locates. At the kink, the two arguments of the min function are equal: ax = by. This replaces the tangency condition used for smooth preferences.
Kink condition
The equation ax = by derived from setting the two arguments of the min function equal. For U(S, L) = min(S, 2L), the kink condition is S = 2L. This is your first equation in two unknowns.
Engel curve (perfect complements)
Still a function of income with prices fixed. Under perfect complements it is linear through the origin, just as with Cobb-Douglas.
Demand curve (perfect complements)
A function of the good's own price with income and other prices fixed. Under perfect complements, demand has the form quantity = income / (weighted sum of prices), which is a hyperbola but with a different structure from the Cobb-Douglas case.
For U(S, L) = min(S, 2L), the consumer is at the kink when S = 2L. This is not a tangency condition. It comes from setting the two arguments of the min function equal to each other. The rule: whatever is inside the min, set those expressions equal.
The method parallels Cobb-Douglas, with the kink condition replacing the tangency condition.
Step 1: Write the kink condition. From U(S, L) = min(S, 2L), set S = 2L.
Step 2: Substitute into the budget constraint. With P_S = 12, P_L = 18, m = 42: 12S + 18L = 42. Replace S with 2L: 12(2L) + 18L = 42, so 42L = 42, giving L = 1.
Step 3: Back-substitute. S = 2(1) = 2. The optimal bundle is S* = 2, L* = 1.
When income rises to $63, the kink condition S = 2L is unchanged. Substituting into 12(2L) + 18L = 63 gives 42L = 63, so L = 1.5 and S = 3. Both goods increase proportionally with income, confirming they are normal goods.
To express S as a function of income, rearrange the kink condition to L = S/2 (because we want to eliminate L, not S). Substitute into the budget with prices plugged in but income left as m:
12S + 18(S/2) = m
12S + 9S = m
21S = m
S(m) = m/21
This is a straight line through the origin.
To express S as a function of P_S, leave P_S as a variable but plug in P_L = 18 and m = 42.
Kink condition: S = 2L, so L = S/2.
Budget: P_S · S + 18(S/2) = 42, which simplifies to P_S · S + 9S = 42.
Factor: S(P_S + 9) = 42.
Solve: S(P_S) = 42 / (P_S + 9).
Notice the structure: the denominator includes P_S plus a constant that comes from the other good's price weighted by the kink ratio. This is characteristic of perfect complements demand. When graphed with S on the horizontal axis and P_S on the vertical axis, you rearrange to P_S(S) = (42 - 9S) / S. The curve is downward-sloping and hits the horizontal axis at S = 42/9 = 4.67.
U(S, L) = \min(S,\; 2L)\text{Kink condition: } S = 2L\text{Budget constraint: } P_S \cdot S + P_L \cdot L = m\text{Engel curve for shorts: } S(m) = \frac{m}{21}\text{Demand curve for shorts: } S(P_S) = \frac{42}{P_S + 9}\text{Inverse demand (for graphing): } P_S(S) = \frac{42 - 9S}{S}The general pattern for U = min(ax, by): the kink condition is ax = by, and demand for x is x = m / (P_x + (a/b) · P_y).
Students often try to differentiate the min function to find the MRS. You cannot do this, because the min function has a kink (it is not differentiable at the kink point). Use the kink condition instead.
A frequent mistake is writing the kink condition backwards. For U = min(S, 2L), the condition is S = 2L, not 2S = L. Set the two arguments equal exactly as they appear.
When deriving the demand curve for shorts, students sometimes leave both P_S and P_L as variables. For a demand curve, you leave only the price of the good in question as a variable and plug in the other price.
Students sometimes confuse which variable to eliminate when deriving Engel or demand curves. The rule: keep the good you are solving for and the variable the curve is a function of (income for Engel, own price for demand). Eliminate everything else.
Exams often place a Cobb-Douglas problem and a perfect complements problem side by side to test whether you know which method to apply. If the utility function uses min(), you use the kink condition. If it has the x^a · y^b form, you use tangency.
The demand curve for perfect complements, S(P_S) = 42/(P_S + 9), has a different algebraic form from Cobb-Douglas demand. Both are hyperbolas, but the perfect complements version shifts the denominator by a constant. Know how to recognise the difference.
Graphing the demand curve requires rearranging to get price as a function of quantity (the inverse demand). This is commonly tested.
The Engel curve S(m) = m/21 is still linear through the origin, so shorts are a normal good. Exams may ask you to verify this or explain why both goods must be normal under perfect complements.
True or false: For perfect complements, you set the MRS equal to the price ratio to find the optimal bundle. (False. You use the kink condition.)
Fill in the blank: For U(S, L) = min(S, 2L), the kink condition is S = ______. (2L.)
True or false: Under perfect complements, it is possible for the consumer to optimally buy more of one good without buying more of the other when income rises. (False. Both goods always increase together in fixed proportions.)
Fill in the blank: The Engel curve for shorts in this problem is S(m) = ______. (m/21.)
True or false: The demand curve S(P_S) = 42/(P_S + 9) implies that as P_S approaches infinity, quantity demanded approaches zero. (True.)
Q: Suppose U(S, L) = min(S, 2L), P_S = 12, P_L = 18, m = 42. What is the optimal bundle?
A: Kink condition: S = 2L. Substitute into 12(2L) + 18L = 42, giving 42L = 42, so L = 1 and S = 2.
Q: If income rises to $63, what is the new optimal bundle?
A: Same kink condition. Substitute into 12(2L) + 18L = 63, giving 42L = 63, so L = 1.5 and S = 3.
Q: Derive the Engel curve for shorts.
A: From S = 2L, get L = S/2. Substitute into 12S + 18(S/2) = m, giving 21S = m, so S(m) = m/21.
Q: Derive the demand curve for shorts using m = 42.
A: From S = 2L, get L = S/2. Substitute into P_S · S + 18(S/2) = 42, giving S(P_S + 9) = 42, so S(P_S) = 42/(P_S + 9).
Q: At the original prices and income, how much does Selena spend on each good?
A: On shorts: 12 x 2 = $24. On leggings: 18 x 1 = $18. Total: $42, which exhausts the budget. Unlike Cobb-Douglas, the spending shares here (24/42 and 18/42) depend on prices, not just the exponents.
This connects directly to Cobb-Douglas optimisation because the overall framework is identical: two equations, two unknowns. The difference is only in where the first equation comes from (kink condition vs tangency condition). Studying them together makes both clearer.
Perfect substitutes are the third case. There, the consumer typically buys only one good (a corner solution), so neither tangency nor kink conditions apply. Knowing all three cases and when each method activates is the core skill of consumer theory at this level.
The fixed-proportions idea returns in producer theory as the Leontief production function, where inputs must be combined in fixed ratios. The cost-minimisation problem with Leontief technology mirrors the consumer problem here.
Perfect complements, Leontief preferences, min function utility, kink point, kink condition, fixed proportions, L-shaped indifference curves, Engel curve, demand curve, normal good, budget constraint, ECN 312, intermediate microeconomics, consumer choice, Wassily Leontief, complementary goods, inverse demand