<p>If you start out with €0, then the certainty equivalent of losing €2.5 with probability 1 is -€2.5.</p>
<p>Your exercise basically asks you to calculate what difference winning the lottery with a small probability makes. Given this utility function, not much.</p>

CC BY-SA 4.0
2020-10-05T10:00:01.663Z
2020-10-05T10:00:01.663Z
Giskard
<p>I am questioning if the CE of a lottery can be negative? For me it doesn't make much sense by definition.</p>
<p>I encountered this problem on the following exercise:</p>
<p>Imagine a case where we have a lottery(like the EuroMillions) with a price ticket of €2.5. By construction we win €13m with a probability of 1/139,838,160 and we loose with probability (1- 1/139,838,160).</p>
<p>Imagine that the utility function is <span class="math-container">$U(x)=-e^{(-x)}$</span>.</p>
<p>I compute the CE as you can find in the picture. Is this wrong? How can I interpret it?</p>
<blockquote>
<p>For the lottery with the prize of €13m, and let <span class="math-container">$n=139,838,160$</span>:
<span class="math-container">$$U(x) = \left(\frac{1}{n} \cdot e^{-(13,000,000-2.5)}\right) + \left(\frac{n-1}{n} \cdot e^{-(-2.5)}\right) = -12.182$$</span>
Therefore
<span class="math-container">\begin{align*}
U(CE)=-e^{-CE}&amp;=-12.182\\
\ln\left(e^{-CE}\right)&amp;=\ln(12.182)\\
CE&amp;=-2.49999
\end{align*}</span></p>
</blockquote>
<p>I hope it is clear.
Thanks a lot!</p>

<microeconomics><choice-theory><uncertainty>
Can the Certainty Equivalent be negative?
Gonçalo Gameiro
# Can the Certainty Equivalent be negative?

(Asked by: username_0 on 2020-10-05)

I am questioning if the CE of a lottery can be negative? For me it doesn't make much sense by definition.

I encountered this problem on the following exercise:

Imagine a case where we have a lottery(like the EuroMillions) with a price ticket of €2.5. By construction we win €13m with a probability of 1/139,838,160 and we loose with probability (1- 1/139,838,160).

Imagine that the utility function is $U(x)=-e^{(-x)}$.

I compute the CE as you can find in the picture. Is this wrong? How can I interpret it?

> For the lottery with the prize of €13m, and let $n=139,838,160$:
> $$U(x) = \left(\frac{1}{n} \cdot e^{-(13,000,000-2.5)}\right) + \left(\frac{n-1}{n} \cdot e^{-(-2.5)}\right) = -12.182$$
> Therefore
> \begin{align\*}
> U(CE)=-e^{-CE}&=-12.182\\
> \ln\left(e^{-CE}\right)&=\ln(12.182)\\
> CE&=-2.49999
> \end{align\*}

I hope it is clear.
Thanks a lot!

**Tags:** microeconomics, choice-theory, uncertainty


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## 1 answers

### Answer by username_1 on 2020-10-05. Score: 7 | Accepted answer

If you start out with €0, then the certainty equivalent of losing €2.5 with probability 1 is -€2.5.

Your exercise basically asks you to calculate what difference winning the lottery with a small probability makes. Given this utility function, not much.


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