Questions: Strong Axiom of Revealed Preference (SARP)

Question 1

A researcher observes three choices: at prices p₁, the consumer picks bundle A when B was affordable; at prices p₂, she picks B when C was affordable; at prices p₃, she picks C when A was affordable. Each individual pair satisfies WARP. Does this consumer satisfy SARP?

• A. Yes — since every pairwise comparison satisfies WARP, SARP is automatically satisfied
• B. No — the chain A ≻ B ≻ C ≻ A forms a revealed preference cycle, which SARP prohibits
• C. Yes — SARP only applies when a consumer is observed more than five times
• D. Indeterminate — we need to know the actual budget sets to determine SARP violations

Answer: (B)

This is precisely the scenario SARP was designed to catch. WARP only checks direct pairwise comparisons, so each pair satisfying WARP tells us nothing about the transitive chain. SARP requires that if A is revealed preferred to B through *any* chain — direct or indirect — then B cannot be revealed preferred to A through any chain. A ≻ B, B ≻ C, and C ≻ A creates a cycle of length three that SARP rules out, even though no single pairwise comparison violates WARP.

Question 2

An economist wants to know whether a consumer's choices over 50 shopping trips are consistent with utility maximization. What is the minimal condition she needs to check?

• A. Whether demand curves slope downward in every observed price-quantity pair
• B. Whether the consumer spent their entire budget in each period
• C. Whether the revealed preference relation derived from all observations contains any cycle
• D. Whether the consumer chose the cheapest bundle in every period

Answer: (C)

SARP is both necessary and sufficient for consistency with utility maximization: the choices can be rationalized by some utility function if and only if the revealed preference relation is acyclic. The economist needs to build the revealed preference graph (A ≻ B if A was chosen when B was affordable) and check for cycles — no cycles means a rationalizing utility function exists. Checking demand slopes or budget exhaustion is neither necessary nor sufficient.

Question 3

Satisfying SARP is necessary and sufficient for the existence of a utility function that rationalizes all observed choices.

• T. True
• F. False

Answer: True

This is the central result: SARP precisely characterizes utility-maximizing behavior in the revealed preference framework. 'Necessary' means any utility maximizer must satisfy SARP — if a utility function exists, you can never have a revealed preference cycle, because a utility function assigns real numbers and real numbers cannot form a cycle (you can't have u(A) > u(B) > u(C) > u(A)). 'Sufficient' means that SARP compliance guarantees some utility function rationalizes the choices.

Question 4

A consumer's choices satisfy WARP in most pairwise comparison. This is sufficient to conclude that some utility function rationalizes their behavior.

• T. True
• F. False

Answer: False

WARP is weaker than SARP. WARP only rules out direct pairwise contradictions (if A is directly revealed preferred to B, don't directly reveal B preferred to A). It does not rule out indirect cycles through chains of three or more observations. Such cycles are incompatible with utility representation, so WARP compliance alone does not guarantee the existence of a utility function. SARP, which extends the no-cycle condition to all indirect chains, is the correct condition for utility rationalizability.

Question 5

Why is a cycle in the revealed preference relation incompatible with the existence of a utility function rationalizing the consumer's choices?

Model answer: A utility function assigns a real number to every bundle, and the function rationalizes choices only if every chosen bundle gets a higher utility number than any affordable alternative. If A is revealed preferred to B is revealed preferred to C is revealed preferred to A, we need u(A) > u(B) > u(C) > u(A) — but real numbers cannot form a strict cycle. No such assignment exists, so no utility function can rationalize a choice pattern with a revealed preference cycle.

This is the deep connection between acyclicity and numerical representability. The existence of a utility function is equivalent to the existence of a complete, transitive, consistent ordering of all bundles. A cycle directly violates transitivity (A is preferred to B is preferred to C, yet C is preferred to A). Since utility functions must represent complete transitive preferences, cycles are the exact obstruction to utility representation.