When two goods must be consumed in fixed proportions (like left and right shoes), the utility function takes the form u(x₁, x₂) = min{x₁, x₂}. The consumer maximises utility by choosing the point where the budget line meets the 45-degree line x₁ = x₂, because any extra of one good without the other adds nothing. Solving is algebra: substitute x₁ = x₂ into the budget constraint and solve for the quantities.
Difficulty: Intermediate | Prerequisites: Budget constraints (Lecture 4-5), basic indifference curve concepts, understanding of the min function.
This lecture sits in the consumer theory block of intermediate microeconomics. Earlier lectures introduced preferences, indifference curves, and budget constraints. Here those pieces come together: given a budget, which bundle does the consumer actually choose? The specific case tested is perfect complements, where the min function forces the consumer onto the 45-degree line. If you are comfortable with Cobb-Douglas utility maximisation (tangency condition), this is the companion case where that tangency method does not apply and a different approach is needed.
Budget line (budget constraint)
The set of all bundles (x₁, x₂) that exactly exhaust the consumer's income: p₁x₁ + p₂x₂ = I. Everything on or below this line is affordable; everything above it is not.
Think of it as the boundary of what you can buy with your pay cheque.
Utility function
A mathematical rule that assigns a number to each bundle, ranking them by the consumer's preference. Higher number means the consumer prefers that bundle.
In simple terms, it is a scoring system for combinations of goods.
Perfect complements (Leontief preferences)
Preferences where goods are consumed in fixed proportions and one good cannot substitute for the other at all. The utility function is u(x₁, x₂) = min{x₁, x₂}.
Think of it as left shoes and right shoes: an extra left shoe with no matching right shoe does nothing for you.
Indifference curve
The set of bundles that give the consumer exactly the same utility level. For perfect complements, these are L-shaped with the kink on the line x₁ = x₂.
In simple terms, every bundle on one indifference curve feels equally good to the consumer.
Optimal consumption bundle (x₁, x₂)**
The affordable bundle that gives the highest utility. For perfect complements, it sits at the kink of the highest reachable indifference curve, which is where x₁ = x₂ meets the budget line.
Think of it as the best deal you can get with the money you have.
Given I = 8, p₁ = 2, p₂ = 1, the budget line is:
2x_1 + x_2 = 8Solving for x₂ gives the slope-intercept form: x₂ = 8 - 2x₁.
x₁-intercept (set x₂ = 0): x₁ = 4
x₂-intercept (set x₁ = 0): x₂ = 8
Slope = -p₁/p₂ = -2
The budget line is a straight line from (4, 0) to (0, 8). Every point on or below it is affordable.
The utility function u(x₁, x₂) = min{x₁, x₂} means:
Only the smaller quantity counts. If x₁ = 3 and x₂ = 7, utility is 3, not 7 or 10.
Extra units of one good alone do not increase utility.
The consumer always wants x₁ = x₂. Any bundle off this line wastes money on the surplus good.
Each indifference curve is L-shaped:
The kink (corner) sits on the line x₁ = x₂.
The horizontal arm extends to the right of the kink (adding more x₁ with x₂ fixed does nothing).
The vertical arm extends upward from the kink (adding more x₂ with x₁ fixed does nothing).
For utility level k, the kink is at (k, k). The horizontal arm is all points (x₁, k) where x₁ ≥ k. The vertical arm is all points (k, x₂) where x₂ ≥ k.
Three example indifference curves from the quiz:
u = 2: kink at (2, 2)
u = 8/3 ≈ 2.67: kink at (8/3, 8/3), the optimal bundle
u = 4: kink at (4, 4), not affordable at this income
Step 1: Recognise that optimality requires x₁ = x₂. Any point where x₁ ≠ x₂ wastes income on the good in surplus.
Step 2: Substitute x₁ = x₂ into the budget constraint.
2x_1 + x_1 = 83x_1 = 8x_1^* = \frac{8}{3}Step 3: Since x₁* = x₂*, both quantities equal 8/3.
The optimal bundle is (x₁*, x₂*) = (8/3, 8/3), and the maximum utility is u = min{8/3, 8/3} = 8/3.
With Cobb-Douglas preferences, you set MRS = price ratio and solve. With perfect complements, the indifference curves have a kink at the optimal point, so the MRS is undefined there. The slope jumps from 0 (on the horizontal arm) to infinity (on the vertical arm). You cannot set MRS equal to anything, so you use the x₁ = x₂ condition instead.
Budget constraint (general form)
p_1 x_1 + p_2 x_2 = IBudget constraint (this problem)
2x_1 + x_2 = 8Utility function (perfect complements)
u(x_1, x_2) = \min\{x_1, x_2\}Optimal bundle (general, when p₁x₁ + p₂x₂ = I and x₁ = x₂)
x_1^* = x_2^* = \frac{I}{p_1 + p_2}This general formula works for any perfect complements problem with 1:1 proportions. Substitute in the numbers and you get 8/(2+1) = 8/3.
Graph checklist (what the exam expects on your sketch):
Budget line from (0, 8) to (4, 0), labelled with slope = -2
The 45-degree line x₁ = x₂ drawn through the origin
Optimal bundle marked at the intersection: (8/3, 8/3)
At least three L-shaped indifference curves, kinks on x₁ = x₂, one passing through (8/3, 8/3)
Axes labelled x₁ (horizontal) and x₂ (vertical)
Perfect complements show up whenever goods must be used together in fixed proportions. Car tyres come in sets of four: a fifth spare has some value, but you cannot drive on three. Printer ink and paper work the same way: 500 sheets with no ink is as useless as a full cartridge with no paper. The model also applies to recipe ingredients (one cup of flour per egg in a specific recipe) and to complementary tech products (a phone case designed for exactly one phone model).
Students often try to set MRS = p₁/p₂ for perfect complements. This does not work because the MRS is undefined at the kink point where the consumer optimises. Use x₁ = x₂ instead.
Students sometimes think min{x₁, x₂} means you add the two quantities. It does not. The min function returns whichever quantity is smaller.
A common graphing error is drawing smooth, curved indifference curves. Perfect complements produce L-shaped indifference curves with sharp right-angle corners, not curves.
Some students confuse perfect complements with perfect substitutes. Perfect substitutes have straight-line indifference curves and u = x₁ + x₂ (or similar). Perfect complements have L-shaped curves and u = min{x₁, x₂}. They are opposite cases.
⚠️ You will almost certainly be asked to solve a utility maximisation problem with either Cobb-Douglas or perfect complements. Know both methods and when each applies.
⚠️ Graphing is heavily tested. Expect to draw the budget line, at least two or three indifference curves, and mark the optimum clearly with coordinates.
⚠️ The general formula x₁* = x₂* = I/(p₁ + p₂) saves time on exams for 1:1 perfect complements. Memorise it.
⚠️ If the problem changes the ratio (e.g., u = min{x₁, 2x₂}), the kink line changes to x₁ = 2x₂ instead of x₁ = x₂. The same substitution method applies, but with the new ratio.
True or False: For u = min{x₁, x₂}, the indifference curves are smooth and convex.
Answer: False. They are L-shaped with a sharp corner.
Fill in the blank: The slope of the budget line equals ____.
Answer: -p₁/p₂ (in this problem, -2).
True or False: At the optimal bundle for perfect complements, x₁ must equal x₂.
Answer: True (for 1:1 perfect complements).
Fill in the blank: If I = 12, p₁ = 3, p₂ = 1, then x₁* = x₂* = ____.
Answer: 12/(3+1) = 3.
True or False: You can solve perfect complements by setting MRS = p₁/p₂.
Answer: False. The MRS is undefined at the kink, so you use x₁ = x₂ instead.
Q: Given I = 10, p₁ = 3, p₂ = 2 and u(x₁, x₂) = min{x₁, x₂}, find the optimal bundle.
A: Set x₁ = x₂ and substitute into the budget constraint: 3x₁ + 2x₁ = 10, so 5x₁ = 10, giving x₁* = x₂* = 2. Utility = 2.
Q: Why does the consumer never choose a point where x₁ ≠ x₂ with perfect complements?
A: Because the surplus good contributes nothing to utility. The consumer could reallocate spending from the surplus good to the deficit good and increase utility without spending more.
Q: Suppose u(x₁, x₂) = min{x₁, 2x₂}. What condition replaces x₁ = x₂ at the optimum?
A: The consumer optimises where the arguments of the min function are equal: x₁ = 2x₂. The kink line shifts to reflect the new consumption ratio.
Q: On a graph, how do you identify the optimal bundle for perfect complements?
A: It is the intersection of the budget line and the line x₁ = x₂ (the line through the kinks of the indifference curves). Graphically, it is the kink of the highest L-shaped indifference curve that still touches the budget line.
Q: If income doubles from 8 to 16 while prices stay the same, what happens to the optimal bundle?
A: x₁* = x₂* = 16/(2+1) = 16/3 ≈ 5.33. Both quantities double. With perfect complements at 1:1, the income expansion path is the 45-degree line, so both goods increase proportionally.
This connects to Cobb-Douglas utility maximisation (the other major case in consumer theory) because the two cases illustrate when tangency works and when it does not. It also connects to the income and substitution effects unit: with perfect complements, the substitution effect is always zero because the consumer never trades one good for the other along an indifference curve. Finally, this is the consumer-side analogue of fixed-proportions production functions (Leontief production), which appears later in producer theory.
Utility maximisation, perfect complements, Leontief preferences, Leontief utility function, min utility function, L-shaped indifference curves, budget constraint, budget line, optimal consumption bundle, consumer choice, ECON 3310, intermediate microeconomics, fixed-proportions preferences, complementary goods, kink point, 45-degree line, income expansion path, MRS undefined, non-differentiable utility