# 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

• 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
• 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