Question

Let 𝑥 and 𝑦 be two consumption bundles, assumed to be non-negative and perfectly divisible. Further, the assumptions of completeness, transitivity, reflexivity, non-satiation, continuity, and strict convexity are satisfied. Then, which of the following statements is NOT CORRECT?

• A. Either $𝑥 ≥ 𝑦, or 𝑦 ≥ 𝑥$, or both
• B. $𝑦 > 𝑥$ if 𝑦 contains more of at least one good and no less of any other
• C. 𝑥 is not indifferent to itself
• D. For 𝑥 (or 𝑦), its better set is strictly convex

Answer 1

The Correct Option is C

Fundamental Assumptions of Consumer Preference Theory

The fundamental assumptions of consumer preference theory include the properties of completeness, transitivity, reflexivity, non-satiation, continuity, and strict convexity. Let’s analyze each option with respect to these properties:

1. Option (A): Either x ⪰ y or y ⪰ x or both: This option follows from the completeness assumption. Completeness states that for any two bundles x and y, a consumer can compare them and determine either x ⪰ y (x is at least as good as y) or y ⪰ x (y is at least as good as x), or both (indifference). Thus, this statement is correct.
2. Option (B): y ≻ x if y contains more of at least one good and no less of any other: This option is consistent with the non-satiation assumption. Non-satiation states that “more is better,” meaning a consumer will always prefer a bundle that contains more of at least one good and no less of any other. Thus, this statement is correct.
3. Option (C): x is not indifferent to itself: This statement is incorrect. The reflexivity assumption in preference theory states that every bundle is at least as good as itself. In other words, x ∼ x, meaning a bundle x is always indifferent to itself. Therefore, the statement x is not indifferent to itself directly violates the reflexivity assumption. Hence, this is the correct answer to the question, as it is the only statement that is NOT correct.
4. Option (D): For x (or y), its better set is strictly convex: This option follows from the strict convexity assumption. Strict convexity implies that if a consumer prefers two bundles x and y, they will also strictly prefer any convex combination (weighted average) of x and y over either bundle. This means the better set (set of bundles preferred to x) will be strictly convex. Thus, this statement is correct.

Explanation of the Correct Answer:

Option (C) violates the fundamental assumption of reflexivity, which states that any bundle x is always at least as good as itself (x ∼ x). Therefore, the statement “x is not indifferent to itself” is incorrect.