<p>None if the axiom is to be on preferences, as any smooth utility representation can be monotone transformed into a non-smooth utility function.</p>

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2020-09-20T23:28:50.523Z
Well, if it is on the binary relation $\succsim$, then I believe Debreu has an axiom: for any $y$, set $\{x|x\sim y\}$ is of class $\mathcal C^k$.
High GPA
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2020-09-21T06:03:52.990Z
@HighGPA Not sure why you are referring to this, seems like it is about the indifference curves, not the utility representation.
Giskard
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2020-09-21T06:18:39.117Z
To my limited understanding Debreu proved that having $C^k$ indifference hypersurface is equivalent to k-differentiable utility when $k=2$.
High GPA
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2020-09-21T06:27:03.217Z
Debreu defined that a preference is of class $C^2$ if the set $\{x|x\sim y\}$ is of class $C^2$. Along with other axioms such as monotonicity, continuity, completeness and transitivity, the preference relation is represented with a second differentiable utility
High GPA
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2020-09-21T06:48:07.000Z
@HighGPA I am afraid there is a slight difference. This preference relation *can* be represented by a $C^2$ utility function. It can also be represented by other utility functions. So I guess your question is not when the utility function is *guaranteed* to be smooth, but when a smooth utility function that represents the preferences *exists*.
Giskard
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2020-09-22T02:47:34.987Z
Well, Debreu's conditions on binary relations is both sufficient and necessary.  Regarding my original question on finite demand data sets: yes, the existence of smooth preference is enough.
High GPA
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2020-09-20T22:38:50.463Z
2020-09-20T22:50:52.453Z
2020-09-20T22:50:52.453Z
Giskard
<p>After researching for a while, I find this:</p>
<p><a href="https://www.jstor.org/stable/1913607?seq=2#metadata_info_tab_contents" rel="nofollow noreferrer">https://www.jstor.org/stable/1913607?seq=2#metadata_info_tab_contents</a></p>
<p>They come up with an axiom called SSARP that generates a preference with smooth demand function. But what about smooth utility function? What axiom can guarantee that?</p>
<hr />
<p>The axiom should be best on the demand data set just like GARP</p>

<microeconomics><game-theory><decision-theory><theory><demand>
What additional axiom to GARP do we need to generate a differentiable or smooth utility function
High GPA
# What additional axiom to GARP do we need to generate a differentiable or smooth utility function

(Asked by: username_0 on 2020-09-20)

After researching for a while, I find this:

<https://www.jstor.org/stable/1913607?seq=2#metadata_info_tab_contents>

They come up with an axiom called SSARP that generates a preference with smooth demand function. But what about smooth utility function? What axiom can guarantee that?

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The axiom should be best on the demand data set just like GARP

**Tags:** microeconomics, game-theory, decision-theory, theory, demand

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## 1 answers

### Answer by username_1 on 2020-09-20. Score: 3 | Accepted answer

None if the axiom is to be on preferences, as any smooth utility representation can be monotone transformed into a non-smooth utility function.

> **Comments:**
> **username_0**: Well, if it is on the binary relation $\succsim$, then I believe Debreu has an axiom: for any $y$, set $\{x|x\sim y\}$ is of class $\mathcal C^k$.  
> **username_1**: @HighGPA Not sure why you are referring to this, seems like it is about the indifference curves, not the utility representation.  
> **username_0**: To my limited understanding Debreu proved that having $C^k$ indifference hypersurface is equivalent to k-differentiable utility when $k=2$.  
> **username_0**: Debreu defined that a preference is of class $C^2$ if the set $\{x|x\sim y\}$ is of class $C^2$. Along with other axioms such as monotonicity, continuity, completeness and transitivity, the preference relation is represented with a second differentiable utility  
> **username_1**: @HighGPA I am afraid there is a slight difference. This preference relation *can* be represented by a $C^2$ utility function. It can also be represented by other utility functions. So I guess your question is not when the utility function is *guaranteed* to be smooth, but when a smooth utility function that represents the preferences *exists*.  
> **username_0**: Well, Debreu's conditions on binary relations is both sufficient and necessary.  Regarding my original question on finite demand data sets: yes, the existence of smooth preference is enough.  

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