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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?


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

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

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  • $\begingroup$ 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 $\mathcal C^k$. $\endgroup$ – High GPA Sep 20 at 23:28
  • $\begingroup$ @HighGPA Not sure why you are referring to this, seems like it is about the indifference curves, not the utility representation. $\endgroup$ – Giskard Sep 21 at 6:03
  • $\begingroup$ To my limited understanding Debreu proved that having $C^k$ indifference hypersurface is equivalent to k-differentiable utility when $k=2$. $\endgroup$ – High GPA Sep 21 at 6:18
  • $\begingroup$ 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 $\endgroup$ – High GPA Sep 21 at 6:27
  • $\begingroup$ @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. $\endgroup$ – Giskard Sep 21 at 6:48

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