<p>Suppose random variable $X\sim F$. </p>

<ol>
<li>Certainty equivalent of the lottery $X$ is the constant $c$ that solves: 

<blockquote>
  <p>$u(c) = \mathbb{E}(u(X))$</p>
</blockquote></li>
<li>Because $u(x) = \min(\sqrt{x}, \sqrt{M})$ is concave, by Jensen's inequality:

<blockquote>
  <p>$u(c) = \mathbb{E}(u(X)) \leq u(\mathbb{E}(X)) $</p>
</blockquote></li>
<li>Agent is risk averse because $u$ is concave.</li>
<li>If agent prefers lottery $X\sim F$ to lottery $Y\sim G$, it is not necessary that $F$ FOSD $G$. Consider the following two lotteries:</li>
</ol>

<blockquote>
  <p>$X = \frac{M}{2}$ with probability 1.</p>
  
  <p>$Y = M$ with probability $\frac{1}{2}$ and $Y = 0$ with probability $\frac{1}{2}$. </p>
</blockquote>

<p>Since $u$ is concave, lottery $X$ is preferred over lottery $Y$. This because $X$ gives the expected value of the lottery $Y$ with probability 1. Also, $F$ does not FOSD $G$ because $F\left(\frac{M}{2}\right) = 1 &gt; \frac{1}{2} = G\left(\frac{M}{2}\right) $.</p>

CC BY-SA 3.0
2017-01-26T12:16:28.747Z
2017-01-26T12:16:28.747Z
Amit
<p>I am solving old problems from various qualifiers from different universities to prepare myself for an upcoming test. I came across this and wanted to ask if anyone can confirm my answers? </p>

<p><a href="https://i.sstatic.net/U24sU.png" rel="noreferrer"><img src="https://i.sstatic.net/U24sU.png" alt="enter image description here"></a></p>

<p>My answers: </p>

<p>** I use $\succeq$ to denote "at least as good as".</p>

<p>(a) A certainty equivalence, in general, is the amount of money $c(F,u)$ so that: </p>

<p>$F \in \Delta(\mathbb{R})$, $\delta_{C(F,u)} \backsim F$ $\equiv u[C(F,u)] = U(F)$</p>

<p>Here, I've never seen something like this and so it is my best guess:</p>

<ul>
<li>$ \forall$  $x&lt;M $ , $C(F,\sqrt(x))= u^{-1}[\int\sqrt(x)dF$] </li>
<li>$ \forall$ $x\geq M$, $u[C(F,\sqrt(M))]= [\sqrt(M)] \implies C(F,u)=x$ </li>
</ul>

<p>Since $u()$ isn't 1-1, thus not invertible, over $x\geq M$ I eventually just decided my result above was true?  </p>

<ul>
<li>Or should it be something more like: </li>
</ul>

<p>$u[C(F,u)] = F(M)\sqrt(x) + [1-F(M)]*\sqrt(M)$</p>

<hr>

<p>(b) I know that the certainty equivalent is less than or equal to the expected value of $F$ iff an agent is risk averse.  </p>

<p>I think that is the same as saying $u[C(F,u)] \leq u(\int x dF)$ , $\forall F \in \Delta(\mathbb{R})$</p>

<hr>

<p>(c) An agent is risk averse if and only the agent's preferences are represented by a concave utility index $u(.)$  and so this agent is risk averse since: </p>

<ul>
<li>$x &lt; M$ $\implies u(x)=\sqrt(x)$, which is clearly concave.  </li>
<li>$x\geq M$ let ${x_1,x_2} \subset [M,\infty)$ and let $\alpha \in [0,1]$ Then $\alpha*x_1 + (1-\alpha)*x_2 \in [M,\infty)$ </li>
</ul>

<p>Now, note that $u(x_i)=\sqrt(M)$, for $ i=1,2,3$</p>

<p>and so $$u(x_3) \geq \alpha u(x_1) + (1-\alpha)u(x_2)$$ </p>

<p>$$\to u(\alpha x_1 + (1-\alpha)x_2) \geq  \alpha u(x_1) + (1-\alpha)u(x_2)$$</p>

<hr>

<p>(D) Again, I am not sure about this. All i know about F.O.S.D is that for two lotteries F,G, then for all EU money monotone preference:
 $$F \geq_{FOSD} G \iff F \succeq G $$</p>

<p>Any and all help is appreciated. </p>

<microeconomics><expected-utility><uncertainty>
Microeconomics - Expected Utility Theory - Piecewise utility index, certainty equivalence, etc.
justhereforhelp
# Microeconomics - Expected Utility Theory - Piecewise utility index, certainty equivalence, etc.

(Asked by: username_0 on 2016-07-26)

I am solving old problems from various qualifiers from different universities to prepare myself for an upcoming test. I came across this and wanted to ask if anyone can confirm my answers?

[![enter image description here](https://i.sstatic.net/U24sU.png)](https://i.sstatic.net/U24sU.png)

My answers:

\*\* I use $\succeq$ to denote "at least as good as".

(a) A certainty equivalence, in general, is the amount of money $c(F,u)$ so that:

$F \in \Delta(\mathbb{R})$, $\delta\_{C(F,u)} \backsim F$ $\equiv u[C(F,u)] = U(F)$

Here, I've never seen something like this and so it is my best guess:

* $ \forall$ $x* $ \forall$ $x\geq M$, $u[C(F,\sqrt(M))]= [\sqrt(M)] \implies C(F,u)=x$

Since $u()$ isn't 1-1, thus not invertible, over $x\geq M$ I eventually just decided my result above was true?

* Or should it be something more like:

$u[C(F,u)] = F(M)\sqrt(x) + [1-F(M)]\*\sqrt(M)$

---

(b) I know that the certainty equivalent is less than or equal to the expected value of $F$ iff an agent is risk averse.

I think that is the same as saying $u[C(F,u)] \leq u(\int x dF)$ , $\forall F \in \Delta(\mathbb{R})$

---

(c) An agent is risk averse if and only the agent's preferences are represented by a concave utility index $u(.)$ and so this agent is risk averse since:

* $x < M$ $\implies u(x)=\sqrt(x)$, which is clearly concave.
* $x\geq M$ let ${x\_1,x\_2} \subset [M,\infty)$ and let $\alpha \in [0,1]$ Then $\alpha\*x\_1 + (1-\alpha)\*x\_2 \in [M,\infty)$

Now, note that $u(x\_i)=\sqrt(M)$, for $ i=1,2,3$

and so $$u(x\_3) \geq \alpha u(x\_1) + (1-\alpha)u(x\_2)$$

$$\to u(\alpha x\_1 + (1-\alpha)x\_2) \geq \alpha u(x\_1) + (1-\alpha)u(x\_2)$$

---

(D) Again, I am not sure about this. All i know about F.O.S.D is that for two lotteries F,G, then for all EU money monotone preference:
$$F \geq\_{FOSD} G \iff F \succeq G $$

Any and all help is appreciated.

**Tags:** microeconomics, expected-utility, uncertainty

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## 1 answers

### Answer by username_1 on 2017-01-26. Score: 2

Suppose random variable $X\sim F$.

1. Certainty equivalent of the lottery $X$ is the constant $c$ that solves:
   > $u(c) = \mathbb{E}(u(X))$
2. Because $u(x) = \min(\sqrt{x}, \sqrt{M})$ is concave, by Jensen's inequality:
   > $u(c) = \mathbb{E}(u(X)) \leq u(\mathbb{E}(X)) $
3. Agent is risk averse because $u$ is concave.
4. If agent prefers lottery $X\sim F$ to lottery $Y\sim G$, it is not necessary that $F$ FOSD $G$. Consider the following two lotteries:

> $X = \frac{M}{2}$ with probability 1.
>
> $Y = M$ with probability $\frac{1}{2}$ and $Y = 0$ with probability $\frac{1}{2}$.

Since $u$ is concave, lottery $X$ is preferred over lottery $Y$. This because $X$ gives the expected value of the lottery $Y$ with probability 1. Also, $F$ does not FOSD $G$ because $F\left(\frac{M}{2}\right) = 1 > \frac{1}{2} = G\left(\frac{M}{2}\right) $.


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