<p>You want the household to be <em>ex-ante</em> indifferent between taking the lottery and paying some $p$ and voiding it.</p>

<p>With van-Neumann-Morgenstein utility, the utility of taking the lottery is given by $$U_L(w) \equiv 0.5 U(w-1000) + 0.5U(w+1000)$$</p>

<p>Now, you're looking for a payment $p(w)$ such that $u(w - p(w)) = U_L(w)$. </p>

CC BY-SA 3.0
2015-05-31T15:54:06.500Z
2017-08-21T07:54:52.350Z
2017-08-21T07:54:52.350Z
FooBar
<blockquote>
  <p>Given a consumer with a utility function, $u(w)$, and a wealth of $w&gt;1000$. Assuming that the consumers relative risk aversion is constant and equal to 1, that is $R_r(w)=1$ for $w&gt;0$, the consumer is facing a lottery that gives him 50% chance to win 1000 and 50% chance to lose 1000.</p>
</blockquote>

<p>My question is, how much is the consumer willing to pay to avoid this lottery, and how does it depend on $w$? I can't seem to figure it out, and I really have no idea where to start.</p>

CC BY-SA 3.0
2015-05-31T13:21:40.453Z
Start [here](http://en.wikipedia.org/wiki/Risk_aversion#Relative_risk_aversion). Than graph your `u(w)`. Make sure that w>1000 is a correct condition. It seems strange. May be, u(w) is defined on w>0 and his/her current wealth is a point in a range w>1000?
garej
CC BY-SA 3.0
2015-05-31T13:23:19.953Z
$u(w)$ is defined for $w>0$ - his current wealth is $w>1000$ when he is facing the lottery. Sorry I didn't make that clear :)

I'll have a look at your link, thanks.
Phillip Bredahl
CC BY-SA 3.0
2015-05-31T13:26:21.797Z
You should edit your question, and my comment will be redundant. And do not forget to carefully read site rules =))
garej
<microeconomics><risk><risk-aversion>
Relative risk aversion and lotteries exercise
Phillip Bredahl
# Relative risk aversion and lotteries exercise

(Asked by: username_0 on 2015-05-31)

> Given a consumer with a utility function, $u(w)$, and a wealth of $w>1000$. Assuming that the consumers relative risk aversion is constant and equal to 1, that is $R\_r(w)=1$ for $w>0$, the consumer is facing a lottery that gives him 50% chance to win 1000 and 50% chance to lose 1000.

My question is, how much is the consumer willing to pay to avoid this lottery, and how does it depend on $w$? I can't seem to figure it out, and I really have no idea where to start.

**Tags:** microeconomics, risk, risk-aversion

> **Comments:**
> **username_1**: Start [here](http://en.wikipedia.org/wiki/Risk_aversion#Relative_risk_aversion). Than graph your `u(w)`. Make sure that w>1000 is a correct condition. It seems strange. May be, u(w) is defined on w>0 and his/her current wealth is a point in a range w>1000?  
> **username_0**: $u(w)$ is defined for $w>0$ - his current wealth is $w>1000$ when he is facing the lottery. Sorry I didn't make that clear :)  I'll have a look at your link, thanks.  
> **username_1**: You should edit your question, and my comment will be redundant. And do not forget to carefully read site rules =))  

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## 1 answers

### Answer by username_2 on 2015-05-31. Score: 2

You want the household to be *ex-ante* indifferent between taking the lottery and paying some $p$ and voiding it.

With van-Neumann-Morgenstein utility, the utility of taking the lottery is given by $$U\_L(w) \equiv 0.5 U(w-1000) + 0.5U(w+1000)$$

Now, you're looking for a payment $p(w)$ such that $u(w - p(w)) = U\_L(w)$.


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