<p>There are many such utility functions. Most commonly: </p>

<p>\begin{equation}
u(x)=L-\mathrm e^{-ax},\tag{1}
\end{equation}
where typically $a&gt;0$. Or 
\begin{equation}
u(x)=L-(x-a)^2.\tag{2}
\end{equation}</p>

<p>Both of these are uniformly bounded below $L\in\mathbb R$ and both tend to $L$ as the functions approach their respective maximum.</p>

CC BY-SA 3.0
2017-11-24T19:07:52.963Z
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
CC BY-SA 3.0
2017-11-24T21:00:16.517Z
@Rory you can obtain $U'$ by simply adding $1$ to $U$
Herr K.
CC BY-SA 3.0
2017-11-24T23:44:31.030Z
@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
CC BY-SA 3.0
2017-11-25T02:16:39.527Z
@denesp: I didn't realize that was what OP was asking. In that case I'd agree with you.
Herr K.
CC BY-SA 3.0
2017-11-24T16:53:32.197Z
2017-11-24T18:26:43.297Z
2017-11-24T18:26:43.297Z
Herr K.
<p>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.</p>

<p>I'd really appreciate any help offered! </p>

<utility><expected-utility>
Modelling Bounded Utility Functions
Rory
# Modelling Bounded Utility Functions

(Asked by: username_0 on 2017-11-24)

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!

**Tags:** utility, expected-utility

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## 1 answers

### Answer by username_1 on 2017-11-24. Score: 4

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.

> **Comments:**
> **username_0**: 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?  
> **username_1**: @Rory you can obtain $U'$ by simply adding $1$ to $U$  
> **username_2**: @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.  
> **username_1**: @denesp: I didn't realize that was what OP was asking. In that case I'd agree with you.  

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