Revealed preference: WARP, SARP and GARP

Revealed preference theory was proposed by Paul Samuelson (CUET 2019). Its direction is the opposite of the earlier lessons: from observed prices, incomes and choices it derives what the preferences must be (JAM 2025). The data are a list of observations, each a price vector and the bundle bought at those prices.

What does it assume? Only that she behaves consistently: that her choices come from a stable ranking, that the ranking is transitive, and that more is preferred (non-satiation, so that she can be induced to buy any bundle if it is made cheap enough). It does not assume a cardinal utility function or any measure of satisfaction (UOH 2014, 2016).

Suppose at prices p she buys bundle x, and another bundle y costs no more than x at those prices. She could have bought y and did not. We say x is revealed preferred to y. In symbols, with p·x the cost of x at prices p:

Two statements from CUET 2023 go together. If one bundle is chosen when another could have been chosen, the first is revealed preferred to the second. And if the consumer chooses the most preferred bundle she can afford, then any bundle y that costs no more than her chosen x must be one she likes less: x is preferred to y. Both are true. The first is the observation; the second is the reason the observation tells us something.

If x is revealed preferred to y, she must never, at another set of prices, choose y when x was affordable. In words: if X is chosen when Y is available, then when Y is chosen X must not be available (CUET 2024).

To test it, compute a table. For two observations 0 and 1 with chosen bundles x⁰ and x¹ at prices p⁰ and p¹, find what each bundle costs at each set of prices.

JNU 2021 asks which of three conditions are inconsistent with WARP. A is “x⁰ affordable under budget 1, but x¹ not under budget 0”: only one direction, so fine. C is the same in the other order, also fine. B is “each affordable under the other”: both directions at once, a violation. So the answer is B only.

Two numerical questions of the same kind. Prices (1, 2) with choice (1, 2), and prices (2, 1) with choice (2, 1) (UOH 2018). Each bundle costs 5 at its own prices and 4 at the other prices, so each was affordable when the other was bought: a violation. And a household with prices P₂ = 1: in situation A income 40, P₁ = 1, buys 20 units of X₁ (so 20 of X₂); in situation B income 61, P₁ = 2, buys 15 of X₁ (so 31 of X₂) (UOH 2016). At A’s prices, B’s bundle costs 15 + 31 = 46, above the 40 she spent, so it was out of reach. No violation: the consumer is consistent.

Pick a scenario, then move the second situation: change its price, its income and where on the line she buys. The panel on the right says which bundle was affordable when. Slide the second choice until the verdict flips; it changes exactly when one of the bundles crosses the other’s budget line.

WARP compares two choices at a time. But x can be revealed preferred to y, and y to z, so x is preferred to z indirectly, through a chain. The stronger axioms look along chains.

• SARP (strong): no two different bundles can each be revealed preferred to the other, even through chains. Equivalently, no cycle of revealed preferences.
• GARP (generalised): if x is revealed preferred to y through a chain, then y is never strictly directly revealed preferred to x (that is, y never cost more than x at y’s own prices). It allows ties, so it is the right condition for data that came from a utility maximiser with indifference.

Data that satisfy GARP can be explained by some well-behaved utility function (this is Afriat’s theorem); data that violate it cannot. A maximiser’s choices always satisfy all three axioms.

With two goods, WARP implies SARP (Rose, 1958). To find data that satisfy WARP but break SARP you need three goods and three observations. Here is one, built for this lesson, where she spends exactly Rs 1 each time:

No two observations contradict each other, so WARP holds. But 1 is revealed preferred to 2, 2 to 3 and 3 to 1: a cycle, so SARP fails. Since each step is only a tie in cost, GARP allows it (she may be indifferent between all three bundles). A strict version: prices (1, 1, 2) with (0, 2, 0), prices (1, 2, 1) with (2, 0, 0), and prices (2, 1, 1) with (1, 0, 1). At the third prices she spends Rs 3, but (0, 2, 0) would have cost only Rs 2, so the last step is strict, and GARP fails too.

Revealed preference also explains why the substitution effect of Lesson 12 is never positive, with no calculus. Suppose the prices change from p⁰ to p¹ and she is given Slutsky compensation, so her new income just buys the old bundle: p¹·x⁰ = p¹·x^c, where x^c is what she buys at the new prices after compensation. If x^c differs from x⁰, then x⁰ was affordable when x^c was chosen, so x^c is revealed preferred to x⁰. WARP then says x^c was not affordable at the old prices: