Questions: Weak Axiom of Revealed Preference (WARP)

Question 1

At prices p¹, a consumer chooses bundle x¹ = (4, 2) when bundle x² = (2, 4) was also affordable. At prices p², the consumer chooses x² = (2, 4), and x¹ = (4, 2) is also affordable at p². Does this violate WARP?

• A. No — the consumer simply has different preferences at different times, which is not a WARP violation
• B. Yes — x¹ was directly revealed preferred to x², but then x² was chosen when x¹ was still affordable, which is a direct reversal
• C. No — WARP only applies when the two bundles cost exactly the same under both price vectors
• D. Yes — any change in the chosen bundle between two observations violates WARP

Answer: (B)

WARP states: if A is directly revealed preferred to B (A was chosen when B was affordable), then B cannot be directly revealed preferred to A (B cannot be chosen when A is affordable). Here, x¹ was chosen when x² was affordable at p¹, so x¹ is revealed preferred to x². Then x² was chosen at p² when x¹ was still affordable — this is a direct reversal and violates WARP. Option A confuses 'revealed preference' with 'preferences changing over time.' WARP treats observed choices as stable preference signals; if they contradict each other, the behavior is inconsistent.

Question 2

A consumer's choice data is consistent with WARP but reveals the pattern: A is revealed preferred to B, B is revealed preferred to C, and C is revealed preferred to A. Has WARP been violated?

• A. Yes — any cycle in revealed preferences violates WARP
• B. No — WARP only prohibits direct pairwise reversals (e.g., A preferred to B then B preferred to A); a three-way cycle does not involve a direct reversal of any single pair
• C. Yes — WARP is equivalent to transitivity, so three-way cycles are forbidden by WARP
• D. No — but only because the consumer is indifferent among A, B, and C

Answer: (B)

WARP checks pairwise consistency only: if A is ever chosen over B, then B can never be chosen over A. A three-way cycle (A ≻ B, B ≻ C, C ≻ A) does not involve any single pair being reversed — each comparison is observed only once in one direction — so WARP is not violated. This is exactly why WARP is weaker than transitivity: transitivity would forbid this cycle (if A ≻ B and B ≻ C, then A ≻ C, contradicting C ≻ A), but WARP does not examine chains of three or more choices.

Question 3

WARP is equivalent to transitivity of preferences — both impose the same consistency requirements on choice behavior.

• T. True
• F. False

Answer: False

WARP is strictly weaker than transitivity. WARP rules out direct pairwise reversals: if A is revealed preferred to B, B cannot be revealed preferred to A. Transitivity rules out cycles of any length: if A ≻ B and B ≻ C, then A ≻ C. A consumer can satisfy WARP while exhibiting intransitive cycles (A ≻ B ≻ C ≻ A), since each of those three pairwise comparisons involves a different pair and no single pair is reversed. The Strong Axiom of Revealed Preference (SARP) closes this gap by prohibiting cycles of any length.

Question 4

A violation of WARP implies that the consumer's behavior cannot be rationalized by any well-behaved utility function.

• T. True
• F. False

Answer: True

True. A well-behaved utility function produces choices that are always consistent with revealed preference: if A is chosen over B when B is affordable, the utility of A is higher, so B will never be chosen over A when A is affordable. WARP is a necessary condition for rationalizability — any utility-maximizing consumer must satisfy WARP. Conversely, if WARP is violated in observed choice data, no utility function (however constructed) can explain those choices. This is what makes WARP an empirically testable prediction that requires no functional form assumptions.

Question 5

What is the key difference between WARP and transitivity, and why does this make WARP a weaker rationality condition?

Model answer: Transitivity requires consistency across chains of comparisons: if A ≻ B and B ≻ C, then A ≻ C must hold. WARP only requires consistency within pairwise comparisons: if A is chosen over B in one observation, B cannot be chosen over A in another. WARP is weaker because it only looks at each pair in isolation — it permits a consumer to exhibit the three-way cycle A ≻ B ≻ C ≻ A without violating any pairwise constraint. Transitivity would forbid this cycle. WARP is the minimum consistency condition for two-observation revealed preference; SARP (Strong Axiom) imposes transitivity across the full revealed preference relation and is equivalent to utility rationalizability in general.

The difference matters practically: WARP can be tested with just two choice observations (two price-budget combinations), making it a minimal and empirically tractable consistency test. Detecting intransitive cycles requires observing choices over three or more situations. The weakness of WARP is both a theoretical limitation and a practical feature.