<p>Whenever you need to make someone indifferent between $x$ and $y$, it means that</p>

<p>$$U(x) = U(y)$$</p>

<p><strong>a</strong>:</p>

<p>Denote amount he would pay by $z$. Paying $z$ to avoid the lottery gives him "certain utility" $u(100.000 - z)$. With Von Neumann-Morgenstern utility as given,<br>
we can denote the lottery utilities as 
$$prob*U(10.000 + x) + (1-Prob)*U(10.000-x)$$ (Understand why we can do that!) </p>

<p><strong>b</strong></p>

<p>Without uncertainty, he will have $u(w)$. Then, you need to compute the utility of the lottery as the function of $p, x,y$.</p>

<p>That is, you need to solve</p>

<p>$$U(100000) = prob*U(w+y) + (1-prob)*U(w-x)$$</p>

<p>Now, I'm not sure whether you were meant to say <em>loses x or wins y</em>, I suppose it is <em>loses x or wins y</em>. Anyhow, the value of the lottery is on the right hand side. Can it ever dominate the left hand side, given the additional information for $p$ that you were told in that subquestion?</p>

CC BY-SA 3.0
2015-02-04T19:07:18.830Z
Thanks this is great! I have one question though..how come in part a) if I take U=2w^(1/2) or U= ln(2w) which are both valid for p=0.5, I get that the answer for U=ln(2w) is twice as much?
User38
CC BY-SA 3.0
2015-02-04T16:49:03.390Z
2015-02-04T16:49:03.390Z
FooBar
<p>The question:</p>

<p>Consider a person with constant relative risk aversion $p$.</p>

<p>(a) Suppose the person has wealth of $100,000$ and faces a gamble in which he wins or loses $x$ with equal probabilities. Calculate the amount he would pay to avoid the gamble, for various values of p (say, between $0.5$ and $40$), and for some values of $x$. For large gambles, do large values of p seem reasonable? What about small gambles?</p>

<p>(b) Suppose $p &gt; 1$ and the person has wealth $w$. Suppose he is offered a gamble in which he loses x or wins y with equal probabilities. Show that he will reject the gamble no matter how large $y$ is if $p &gt;= (log(0.5)+log(1-x/w))/log(1-x/w)$.</p>

<p>I'm not sure where to start with this. Am I solving for the risk premium and multiplying by $w$?</p>

<p>I know that for someone with CRRA utility $u(w)= (1/(1-p))w^(1-p)$ and that an individual will pay $\pi(w)$ to avoid the gamble if $u((1-\pi)w)=E[u(1+\epsilon \tilde)w)]$. But I'm not sure how to apply this information to solve the question.</p>

CC BY-SA 3.0
2015-02-04T16:52:03.807Z
Welcome to Econ.SE. I formatted your question, notice that you can use dollar signs to add latex-like mathematic formulas. I was unsure about the last part, where you had *tilde*.
FooBar
<utility><risk><expected-utility><risk-aversion>
Question about constant relative risk aversion
User38
# Question about constant relative risk aversion

(Asked by: username_0 on 2015-02-04)

The question:

Consider a person with constant relative risk aversion $p$.

(a) Suppose the person has wealth of $100,000$ and faces a gamble in which he wins or loses $x$ with equal probabilities. Calculate the amount he would pay to avoid the gamble, for various values of p (say, between $0.5$ and $40$), and for some values of $x$. For large gambles, do large values of p seem reasonable? What about small gambles?

(b) Suppose $p > 1$ and the person has wealth $w$. Suppose he is offered a gamble in which he loses x or wins y with equal probabilities. Show that he will reject the gamble no matter how large $y$ is if $p >= (log(0.5)+log(1-x/w))/log(1-x/w)$.

I'm not sure where to start with this. Am I solving for the risk premium and multiplying by $w$?

I know that for someone with CRRA utility $u(w)= (1/(1-p))w^(1-p)$ and that an individual will pay $\pi(w)$ to avoid the gamble if $u((1-\pi)w)=E[u(1+\epsilon \tilde)w)]$. But I'm not sure how to apply this information to solve the question.

**Tags:** utility, risk, expected-utility, risk-aversion

> **Comments:**
> **username_1**: Welcome to Econ.SE. I formatted your question, notice that you can use dollar signs to add latex-like mathematic formulas. I was unsure about the last part, where you had *tilde*.  

---

## 1 answers

### Answer by username_1 on 2015-02-04. Score: 1 | Accepted answer

Whenever you need to make someone indifferent between $x$ and $y$, it means that

$$U(x) = U(y)$$

**a**:

Denote amount he would pay by $z$. Paying $z$ to avoid the lottery gives him "certain utility" $u(100.000 - z)$. With Von Neumann-Morgenstern utility as given,  
we can denote the lottery utilities as
$$prob\*U(10.000 + x) + (1-Prob)\*U(10.000-x)$$ (Understand why we can do that!)

**b**

Without uncertainty, he will have $u(w)$. Then, you need to compute the utility of the lottery as the function of $p, x,y$.

That is, you need to solve

$$U(100000) = prob\*U(w+y) + (1-prob)\*U(w-x)$$

Now, I'm not sure whether you were meant to say *loses x or wins y*, I suppose it is *loses x or wins y*. Anyhow, the value of the lottery is on the right hand side. Can it ever dominate the left hand side, given the additional information for $p$ that you were told in that subquestion?

> **Comments:**
> **username_0**: Thanks this is great! I have one question though..how come in part a) if I take U=2w^(1/2) or U= ln(2w) which are both valid for p=0.5, I get that the answer for U=ln(2w) is twice as much?  

---

