Preferences and indifference curves

A bundle is a list of quantities, one for each good: say 3 kg of rice and 2 litres of milk. Take two bundles, A and B. Ask her which she likes better. Her answer is one of three things: she prefers A (write A ≻ B), she prefers B, or she is indifferent (A ∼ B), meaning they are equally good to her. Write A ≿ B for “A is at least as good as B”: it covers both A ≻ B and A ∼ B.

That is all a preference is: her answers to every such question. No numbers and no “units of happiness” are needed yet. The next lesson shows how a number (utility) can be attached afterwards, and how little of it means anything.

An economist calls a ranking rational if it satisfies two requirements (and a trivial third, reflexivity: every bundle is as good as itself).

• Completeness. Any two bundles can be compared: A ≿ B or B ≿ A (or both). She is never stuck saying “I cannot say”.
• Transitivity. If A ≿ B and B ≿ C, then A ≿ C. Her rankings fit together in a chain.

Try the lab. Choose a ready-made ranking, or build your own. Break completeness by leaving a pair unranked. Break transitivity with a cycle, and read the money pump.

The next assumption says more is good, and textbooks state it in three strengths, from strongest to weakest. Writing y ≥ x for “y has at least as much of every good as x”, and y ≫ x for “y has strictly more of every good”:

• Strong monotonicity. If y ≥ x and y ≠ x, then y ≻ x. More of even one good, and no less of any other, is strictly better. (JNU calls this strict monotonicity.)
• Monotonicity. If y ≫ x, then y ≻ x. Only strictly more of every good is guaranteed to be better.
• Local non-satiation. Whatever bundle she holds, however small the distance, there is a bundle that close which she likes better. She is never fully satisfied, but she may dislike some goods.

Each implies the next, and not the other way round: strong monotonicity ⇒ monotonicity ⇒ local non-satiation. So strict monotonicity implies local non-satiation, and a preference with a bliss point (a best bundle, with worse bundles all round it) violates all three. That is JNU 2016: of local non-satiation and strict monotonicity, strict monotonicity implies local non-satiation, and not the reverse.

Non-satiation is what makes the budget line the right place to look for a best bundle. If she is never satisfied, she spends all her income; a bundle inside the budget set always has a better one near it.

With two goods, every bundle is a point. Join all the bundles that are exactly as good as a given one and you get its indifference curve. Through every point there is one curve (this is where completeness and transitivity are used), and each curve is a whole set of bundles of equal value to her. Bundles on a higher curve are better.

With goods that she likes (strong monotonicity), three properties follow. Each is a favourite examiner’s question.

1. Curves slope downward. If she gets more x, she must lose some y to stay on the same curve, because otherwise she would have more of both and be strictly better off. The slope is negative because more is better (JNU 2016), not because the marginal rate of substitution diminishes. That is a different property (convexity), met in the next lesson.
2. Higher curves are better. A curve farther from the origin has more of the goods she likes.
3. Curves never cross. Suppose two curves cross at P. Pick A on the first curve and B on the second so that A has more of both goods than B (possible, because the curves separate on either side of P). Then A ≻ B by “more is better”. But A ∼ P because they share a curve, and P ∼ B because they share the other, so transitivity forces A ∼ B. Contradiction. Crossing curves break transitivity and more-is-better together (Hyderabad 2019).

“Slopes down” assumed both things are goods. Change that and the picture changes. A bad is something she dislikes (pollution, effort, noise). A neutral is something she does not care about. Try each case in the gallery, and read the direction in which utility rises.

Two questions we left out, and why

UGC NET December 2023 Q63 asks which statements about monotonicity are correct, and its printed key is “marks to all”. The reason is in the chapter above: only the statement defining strong monotonicity is correct, and the options never list it alone. (The statement that calls that definition “monotone” mixes up the two names, and the one saying local non-satiation implies monotonicity has the arrow the wrong way.) We do not serve it.

Hyderabad 2021 Q46 gives u(x, y) = 1000/(10 + (x−15)² + (y−20)²) and keys two options as correct: the function is continuous and strictly quasi-concave. It has a bliss point at (15, 20), so it is not monotone and does not satisfy non-satiation. The key is right, but the question is stored with a single-answer format, so it cannot be marked, and it is left out.