<p>This is perhaps not the best way to prove the statement. But notice that a CARA utility is equivalent to having the general functional form:
<span class="math-container">\begin{equation}
u(x)=-\alpha \mathrm e^{-ax-b}+\beta,\quad\text{where $a,\alpha&gt;0$ and $b,\beta\in\mathbb R$}.
\end{equation}</span> 
Knowing this, the rest should be straightforward. </p>

CC BY-SA 4.0
2019-02-11T21:16:46.463Z
This actually worked! Thanks! And if I had to work out the inequality related to NIARA (non-increasing), should I proceed in a different way?
PostDocing
CC BY-SA 4.0
2019-02-11T22:02:51.233Z
@PhDing: Probably. It's hard to pin down the general functional form of a non-CARA utility function. So my method will most likely be of little use.
Herr K.
CC BY-SA 4.0
2019-02-10T22:37:00.333Z
2019-02-10T22:37:00.333Z
Herr K.
<p>I need to prove that Constant Absolute Risk Aversion (CARA) is equivalent to 
<span class="math-container">\begin{gather}
\int u'(x)dF(x) = u'(c(F,u))
\end{gather}</span></p>

<p>where <span class="math-container">$u(x)$</span> is a Bernoulli utility function, <span class="math-container">$F$</span> is the distribution of the lottery and <span class="math-container">$c(F,u)$</span> is the certainty equivalent. </p>

<p>I started from the fact that CARA is defined as <span class="math-container">$-\frac{u''(x)}{u'(x)}=a$</span>, where <span class="math-container">$a$</span> is a constant and that the certainty equivalent is defined as <span class="math-container">$u(c(F,u))=\int u(x)dF(x)$</span>. </p>

<p>I tried to mix up the two definitions but I am a bit lost with the interaction of integration and differentiation.</p>

<p>Do you have any hint?</p>

<microeconomics><risk-aversion>
Constant absolute risk aversion and certainty equivalent
PostDocing
# Constant absolute risk aversion and certainty equivalent

(Asked by: username_0 on 2019-02-09)

I need to prove that Constant Absolute Risk Aversion (CARA) is equivalent to
\begin{gather}
\int u'(x)dF(x) = u'(c(F,u))
\end{gather}

where $u(x)$ is a Bernoulli utility function, $F$ is the distribution of the lottery and $c(F,u)$ is the certainty equivalent.

I started from the fact that CARA is defined as $-\frac{u''(x)}{u'(x)}=a$, where $a$ is a constant and that the certainty equivalent is defined as $u(c(F,u))=\int u(x)dF(x)$.

I tried to mix up the two definitions but I am a bit lost with the interaction of integration and differentiation.

Do you have any hint?

**Tags:** microeconomics, risk-aversion

---

## 1 answers

### Answer by username_1 on 2019-02-10. Score: 2 | Accepted answer

This is perhaps not the best way to prove the statement. But notice that a CARA utility is equivalent to having the general functional form:
\begin{equation}
u(x)=-\alpha \mathrm e^{-ax-b}+\beta,\quad\text{where $a,\alpha>0$ and $b,\beta\in\mathbb R$}.
\end{equation}
Knowing this, the rest should be straightforward.

> **Comments:**
> **username_0**: This actually worked! Thanks! And if I had to work out the inequality related to NIARA (non-increasing), should I proceed in a different way?  
> **username_1**: @PhDing: Probably. It's hard to pin down the general functional form of a non-CARA utility function. So my method will most likely be of little use.  

---

