# What additional axiom to GARP do we need to generate a differentiable or smooth utility function

After researching for a while, I find this:

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

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

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