<p>Yes. Indeed, the book mentions that every continuous preference relation on <span class="math-container">$\mathfrak{R}_+^K$</span> 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.</p>
<p>There are too many examples of non-monotone preferences that admit a utility representation. Think of constant or decreasing utility functions.</p>
<p>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.</p>

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2023-02-22T15:30:53.190Z
2023-02-22T15:30:53.190Z
Michael Greinecker
<p>The book I am working with (Microeconomics Theory by A. Rubinstein) states that:</p>
<p>&quot;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&quot;.</p>
<p>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?</p>

<microeconomics><utility><preferences>
Can a preference relation not satisfy monotonicity and still be represented by an Utility function?
aliosha karamazov
# Can a preference relation not satisfy monotonicity and still be represented by an Utility function?

(Asked by: username_0 on 2023-02-22)

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?

**Tags:** microeconomics, utility, preferences

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## 1 answers

### Answer by username_1 on 2023-02-22. Score: 3 | Accepted answer

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.


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