Difficulty: Intermediate | Prerequisites: Budget Constraints study notes (Ch. 1).
Budget constraints tell you what a consumer can buy. Preferences tell you what they want to buy. This section introduces the formal machinery for modelling preferences: preference relations, the axioms they satisfy, and indifference curves that represent them graphically. Without these tools, you cannot solve for the optimal bundle, which is where the course is heading. Make sure you are comfortable with the budget constraint material first.
Consumer preferences rank bundles of goods. Three axioms (completeness, transitivity, more is better) make preferences well-behaved enough to work with. Indifference curves are the graphical representation of preferences: each curve connects all bundles that give the consumer the same level of satisfaction. These curves slope downward, never cross, and are typically bowed inward (strictly convex).
Strict preference (A ≻ B)
The consumer prefers bundle A to bundle B. Given a free choice between the two, they would always pick A.
Indifference (A ~ B)
The consumer is equally satisfied with bundle A and bundle B. They have no reason to choose one over the other.
Weak preference (A ≿ B)
Bundle A is at least as good as bundle B. This means either A ≻ B or A ~ B. Think of it as "A is not worse than B."
Completeness
For any two bundles A and B, the consumer can always state a preference: either A ≿ B, or B ≿ A, or both (in which case A ~ B). No pair of bundles is incomparable.
Transitivity
If A ≿ B and B ≿ C, then A ≿ C. Preferences are logically consistent. Without transitivity, a consumer could be led around in circles.
More is better (monotonicity)
If bundle A has at least as much of every good as bundle B, and strictly more of at least one good, then A ≻ B. In plain terms, getting more stuff (of goods you like) is always preferred.
Indifference curve (IC)
A curve showing all bundles that give the consumer the same level of satisfaction. Every point on a given IC is equally preferred. Think of it as a contour line on a satisfaction map.
Strictly convex (indifference curves)
If you pick any two bundles on the same IC and draw a straight line between them, that line lies entirely above the IC (in the preferred region). This reflects the idea that consumers prefer balanced mixtures to extremes.
Symbol | Meaning |
|---|---|
A ≻ B | Consumer strictly prefers A to B |
B ≻ A | Consumer strictly prefers B to A |
A ~ B | Consumer is indifferent between A and B |
A ≿ B | A is at least as good as B (weak preference) |
Weak preference (≿) is the building block. Strict preference and indifference are derived from it: A ≻ B means A ≿ B but not B ≿ A. A ~ B means both A ≿ B and B ≿ A.
1. Completeness. For any two bundles A and B, the consumer can rank them. There is no "I don't know" option. Either A ≿ B, B ≿ A, or both.
2. Transitivity. For any three bundles A, B, and C: if A ≿ B and B ≿ C, then A ≿ C. This rules out circular preferences.
3. More is better. If bundle A has at least as much of every good as B, and more of at least one good, then A ≻ B.
An indifference curve shows the set of all bundles that make the consumer equally well off. Bundles on a higher IC (further from the origin) are preferred to bundles on a lower IC.
If you have four ICs labelled IC₁ through IC₄, with IC₄ being the furthest from the origin, then IC₄ ≻ IC₃ ≻ IC₂ ≻ IC₁. Higher up means more beneficial for the consumer.
ICs slope downward when "more is better" holds. If both X and Y are "goods" (the consumer wants more of each), then getting more X while staying on the same IC requires giving up some Y. The curve must slope downward.
ICs cannot cross. Proof by contradiction: suppose two ICs cross at bundle A. Let B be on IC₁ only and C be on IC₂ only. Since A and B are on the same IC, A ~ B. Since A and C are on the same IC, A ~ C. By transitivity, B ~ C. But if C has more of both goods than B, then by "more is better," C ≻ B. Contradiction. Therefore ICs cannot cross.
ICs are strictly convex. For standard preferences, any straight line connecting two points on an IC lies above the IC. This captures the intuition that consumers prefer mixtures to extremes.
Both X and Y are goods: ICs slope downward (standard case).
One good, one bad (e.g., Y = food, X = rubbish): ICs slope upward. To stay equally happy while receiving more rubbish, the consumer needs more food to compensate. The preferred direction is toward more Y and less X, so ICs closer to the Y-axis are better.
One good, one neutral (e.g., X is a good, Y is neutral): ICs are vertical lines. Only X matters, so the consumer is indifferent across any amount of Y at a given level of X. Moving rightward (more X) reaches higher ICs.
The consumer's optimal bundle is the point on the budget constraint that reaches the highest possible indifference curve. At this point, the slope of the IC equals the slope of the budget constraint. This will be developed fully in the next set of notes.
Indifference curves show up whenever someone faces a trade-off between two things they value. A commuter choosing between travel time and travel cost is navigating indifference curves, each representing a constant level of overall satisfaction. Strict convexity captures why most people prefer a moderate commute at a moderate price over either extreme.
Students often believe indifference curves can cross "if the consumer changes their mind." They cannot. The proof by contradiction relies on the axioms holding at a single point in time. If preferences change, you have a different set of ICs entirely.
Students sometimes think higher utility means a curve further to the right. It means further from the origin in the direction of more of both goods, which is typically up and to the right.
Students confuse the slope of the IC with the slope of the budget constraint. The IC slope is the MRS (based on preferences). The budget constraint slope is the MRT (based on prices). They happen to be equal only at the optimal bundle.
Students occasionally draw ICs that touch the axes. Standard strictly convex ICs for Cobb-Douglas preferences approach but do not touch the axes.
⚠️ The proof that indifference curves cannot cross is a classic exam question. Memorise the logic: assume they cross, find two bundles that must be indifferent by transitivity, show that "more is better" creates a contradiction.
⚠️ Know the three axioms by name and be able to state each precisely. Exams often present a scenario and ask which axiom is violated.
⚠️ Be able to draw ICs for goods-goods, good-bad, and good-neutral cases, and explain the direction of increasing preference in each.
⚠️ Strict convexity is frequently tested. Understand what it means graphically (the line between two points on an IC lies above the IC) and intuitively (consumers prefer mixtures).
True or false: If A ~ B and B ~ C, then A ~ C. (True, by transitivity.)
Fill in the blank: Indifference curves slope downward when both goods satisfy the _______ assumption. (Answer: more is better / monotonicity)
True or false: Two indifference curves from the same preference map can cross at exactly one point. (False: ICs can never cross.)
True or false: If Good X is a "bad," indifference curves will have a positive slope. (True: the consumer needs compensation in the form of more of the good to accept more of the bad.)
Fill in the blank: Strictly convex ICs reflect a consumer's preference for _______ over extremes. (Answer: mixtures / balanced bundles)
Q: State the completeness axiom and give an example of a situation where it might be violated.
A: Completeness requires that for any two bundles A and B, the consumer can say A ≿ B, B ≿ A, or both. It might be violated if a consumer faces two completely unfamiliar goods and cannot form any ranking between them.
Q: Prove that indifference curves cannot cross, using the axioms of consumer preferences.
A: Suppose two ICs cross at bundle A. Let B be another point on IC₁ and C another point on IC₂. Then A ~ B (both on IC₁) and A ~ C (both on IC₂). By transitivity, B ~ C. But if C lies to the northeast of B (more of both goods), then by "more is better," C ≻ B. This contradicts B ~ C. Therefore ICs cannot cross.
Q: A consumer has strictly convex indifference curves. Bundle A = (2, 8) and Bundle B = (8, 2) lie on the same IC. Is the bundle (5, 5) preferred, less preferred, or indifferent to A and B?
A: Preferred. Strict convexity means any mixture of two bundles on the same IC lies above the IC, placing it on a higher indifference curve. Since (5, 5) is the midpoint of A and B, it is strictly preferred to both.
Q: Draw and label indifference curves for the case where X is a "good" and Y is a "neutral." Which direction represents increasing preference?
A: The ICs are vertical lines (parallel to the Y-axis). Since only X matters, moving to the right (increasing X) reaches higher ICs. The amount of Y is irrelevant to the consumer's ranking.
Preferences and indifference curves combine with the budget constraint to determine the optimal bundle (covered in the Utility Functions and Optimal Bundle notes). The shape of ICs, whether strictly convex, L-shaped, or linear, depends on the utility function type: Cobb-Douglas, perfect complements, or perfect substitutes. The MRS (slope of the IC) will be formally derived once marginal utility is introduced.
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