2
$\begingroup$

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!

Improve this question
$\endgroup$
4
$\begingroup$

There are many such utility functions. Most commonly:

\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.

Improve this answer
$\endgroup$
4
  • $\begingroup$ 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? $\endgroup$ – Rory Nov 24 '17 at 19:07
  • $\begingroup$ @Rory you can obtain $U'$ by simply adding $1$ to $U$ $\endgroup$ – Herr K. Nov 24 '17 at 21:00
  • 1
    $\begingroup$ @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. $\endgroup$ – Giskard Nov 24 '17 at 23:44
  • $\begingroup$ @denesp: I didn't realize that was what OP was asking. In that case I'd agree with you. $\endgroup$ – Herr K. Nov 25 '17 at 2:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.