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The book I am working with (Microeconomics Theory by A. Rubinstein) states that:
"In the case that preferences are represented by a utility function, preferences satisfying monotonicity (or strong monotonicity) can be represented by an increasing (or strongly increasing) utility function".
Can it be the case that a preference relation does not satisfy monotonocity (Can inferior goods be such a case?) and if yes, how can it be represented by a Utility function?
Yes. Indeed, the book mentions that every continuous preference relation on $\mathfrak{R}_+^K$ can be represented by a continuous utility function by a theorem of Debreu (it actually follows from an earlier theorem of Eilenberg). Rubinstein then proceeds to give a simpler proof under the additional (but unnecessary) assumption that preferences satisfy monotonicity.
There are too many examples of non-monotone preferences that admit a utility representation. Think of constant or decreasing utility functions.
Inferior goods refer to demand behavior and not directly to a utility function or preferences. In general, strictly monotone preferences can give rise to inferior goods, so there is no natural connection between these concepts.
