Suppose u is a utility function that represents a consumer's preferences. Then u must be complete and transitive.

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The statement asks about the relationship between a utility function u and the consumer's underlying preferences. The key idea is that a utility function is a numerical representation of a preference relation.
When we say u represents a consumer's preferences, we mean the induced ordinal ranking given by u matches the ranking of the preferences: x ≽ y if and only if u(x) ≥ u(y) (up to monotonic transformations that preserve order).
For a utility representation to exist, the preference relation must be complete: for any two bundles x and y, the consumer either prefers x to y, y to x, or is indifferent (both x ≽ y and y ≽ x). If this condition fails, no single utility function can consistently rank all bundles to match the preferences.
Similarly, the preference relation must be transitive: if x ≽ y and y ≽ z, then x ≽ z. Without transitivity, you could have cyclical preferences (x ≽ y ≽ z ≽ x), which cannot be captured by any real-valued utility function that preserves the order.
Thus, when a utility function u represents preferences, the underlying preference relation is complete and transitive, ensuring that u can faithfully encode the ordering of bundles. In short, the statement that u must be complete and transitive (in the sense of representing a complete and transitive preference relation) is correct because those are the foundational rationality conditions that permit a utility representation.
If someone argued that a utility function could exist without these properties, they would be conflating the existence of a function with the necessity that its representing relation be rational; the standard framework requires completeness and transitivity for a faithful representation, so the claim aligns with core microeconomic theory.

ECON6001/ECON6701 Microeconomic Analysis Quiz 1

Suppose u is a utility function that represents a consumer's preferences. Then u must be complete and transitive.

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