# Modelling Bounded Utility Functions

I'm trying to work out how to model a utility function that is bounded below some level. More precisely, given a specified limit $L$, I want to work out how to ensure that the utility of any outcome $o$ asymptotically approaches $L$ as the value of $o$ increases.

I'd really appreciate any help offered!

\begin{equation} u(x)=L-\mathrm e^{-ax},\tag{1} \end{equation} where typically $a>0$. Or \begin{equation} u(x)=L-(x-a)^2.\tag{2} \end{equation}
Both of these are uniformly bounded below $L\in\mathbb R$ and both tend to $L$ as the functions approach their respective maximum.
• My interest is in ensuring that, given a stated, utility function $U(.)$, the transformation $U'$ of that utility function into a bounded utility function has the property: for every $x \in domU$, $U(x) <= U'(x)$. Are there any functions which additionally have this property? – Rory Nov 24 '17 at 19:07
• @Rory you can obtain $U'$ by simply adding $1$ to $U$ – Herr K. Nov 24 '17 at 21:00
• @HerrK. Yes but I believe the question is can you make $U'$ bounded and make it have this property at the same time. Unfortunately if $U$ has no upper bound then this property guarantees that $U'$ won't have one either. – Giskard Nov 24 '17 at 23:44