<p>First, assume risk aversion. By the definition of the certainty equivalent and Jensen's:
<span class="math-container">$$u(CE(u,F))=E(u(x))&lt;u(E(x))$$</span></p>

<p>Now, from monotonicity:</p>

<p><span class="math-container">$$CE&lt;E(x)$$</span></p>

<p>Second, assume  <span class="math-container">$CE&lt;E(x)$</span>. By monotonicity and the definition of <span class="math-container">$CE$</span>:</p>

<p><span class="math-container">$$u(E(x))&gt;u(CE)=E(u(x))$$</span></p>

CC BY-SA 4.0
2019-11-05T09:59:04.617Z
The part after "second" is not necessary. Or is there any reason for putting it in?
Aqqqq
CC BY-SA 4.0
2019-11-05T14:40:53.753Z
Not really, you can make it work also with "iff" statements.
Weierstraß Ramirez
CC BY-SA 4.0
2019-11-09T21:04:46.027Z
You use strict monotonicity, but the question pointed out that "$u(x)$ *is monotonic and not strictly monotonic*". Luckily, OP was happy with the answer.
Giskard
CC BY-SA 4.0
2019-11-04T10:50:57.813Z
2019-11-04T10:50:57.813Z
Weierstraß Ramirez
<p>Utility function <span class="math-container">$u(x)$</span> is monotonic. I want to prove that <span class="math-container">$u(x)$</span> exhibits risk aversion if and only if for all lottery <span class="math-container">$F$</span>: <span class="math-container">$E(x) \geq CE(F,u)$</span> (CE is certainty equivalent).</p>

<p>(Definition of <span class="math-container">$CE$</span>: the certainty equivalent <span class="math-container">$CE(F,u)$</span> of lottery <span class="math-container">$F$</span> given Bernoulli utility <span class="math-container">$u(x)$</span> is such that <span class="math-container">$u(CE(F,u)) = E(u(x)) = U(F)$</span>)</p>

<p>I tried with </p>

<p><span class="math-container">$u(x)$</span> is risk averse <span class="math-container">$&lt;=&gt;$</span> <span class="math-container">$u(x)$</span> concave <span class="math-container">$&lt;=&gt;$</span> <span class="math-container">$E(u(x)) \leq u(E(x))$</span>   (Jensen's inequality)</p>

<p>I am not sure about how to proceed. Since <span class="math-container">$u(x)$</span> is monotonic and not strictly monotonic, I can not take inverse of <span class="math-container">$u(x)$</span> on both side of the inequality.</p>

CC BY-SA 4.0
2019-11-02T15:29:59.563Z
You are using a lot of undefined notation. (Also some undefined concepts, but those are minor.)
Giskard
CC BY-SA 4.0
2019-11-02T23:27:41.773Z
I see you have made some edits, but basically I would like you to explain the exact relationship between $x,F$ and $u$ in the inequality 
$$
E(x) \geq CE(F,u).
$$
Seems like $x$ and $F$ are the same thing?
Giskard
CC BY-SA 4.0
2019-11-03T08:42:44.723Z
$E(u(x)) = U(F)$ ($U(F)$ is Neumann-Morgenstern expected utility of the lottery)
Aqqqq
CC BY-SA 4.0
2019-11-03T08:49:20.557Z
$F$ is a lottery. $x$ is one of the sure outcome of the lottery.
Aqqqq
CC BY-SA 4.0
2019-11-03T08:55:51.563Z
"$x$ is one of the sure outcome of the lottery" What does this mean? Every lottery has *sure* outcomes? And more than one?
Giskard
CC BY-SA 4.0
2019-11-03T10:15:18.570Z
Sorry, just "outcome of the lottery" (just that the $u$ is defined on sure amounts of money without taking the probability of outcome into account.). E.g. if one lottery has 0.5 of probability of getting 100 dollar and 0.5 of probability of getting 0 dollar, then the 100 dollar and 0 dollar are the two outcomes of the lottery.
Aqqqq
CC BY-SA 4.0
2019-11-03T11:53:03.380Z
But then $x$ is just monetary values. $E(x)$ implies that $x$ is a stochastic variable, e.g. it has probabilities and assigned (in this case monetary) values. Just like $F$. So can you explain what the difference between $F$ and $x$ is? Perhaps you could link to and/or name your source?
Giskard
<microeconomics><utility><expected-utility><uncertainty>
How to prove the relationship between the expected value of a lottery and its certainty equivalent?
Aqqqq
# How to prove the relationship between the expected value of a lottery and its certainty equivalent?

(Asked by: username_0 on 2019-11-02)

Utility function $u(x)$ is monotonic. I want to prove that $u(x)$ exhibits risk aversion if and only if for all lottery $F$: $E(x) \geq CE(F,u)$ (CE is certainty equivalent).

(Definition of $CE$: the certainty equivalent $CE(F,u)$ of lottery $F$ given Bernoulli utility $u(x)$ is such that $u(CE(F,u)) = E(u(x)) = U(F)$)

I tried with

$u(x)$ is risk averse $<=>$ $u(x)$ concave $<=>$ $E(u(x)) \leq u(E(x))$ (Jensen's inequality)

I am not sure about how to proceed. Since $u(x)$ is monotonic and not strictly monotonic, I can not take inverse of $u(x)$ on both side of the inequality.

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

> **Comments:**
> **username_1**: You are using a lot of undefined notation. (Also some undefined concepts, but those are minor.)  
> **username_1**: I see you have made some edits, but basically I would like you to explain the exact relationship between $x,F$ and $u$ in the inequality  $$ E(x) \geq CE(F,u). $$ Seems like $x$ and $F$ are the same thing?  
> **username_0**: $E(u(x)) = U(F)$ ($U(F)$ is Neumann-Morgenstern expected utility of the lottery)  
> **username_0**: $F$ is a lottery. $x$ is one of the sure outcome of the lottery.  
> **username_1**: "$x$ is one of the sure outcome of the lottery" What does this mean? Every lottery has *sure* outcomes? And more than one?  
> **username_0**: Sorry, just "outcome of the lottery" (just that the $u$ is defined on sure amounts of money without taking the probability of outcome into account.). E.g. if one lottery has 0.5 of probability of getting 100 dollar and 0.5 of probability of getting 0 dollar, then the 100 dollar and 0 dollar are the two outcomes of the lottery.  
> **username_1**: But then $x$ is just monetary values. $E(x)$ implies that $x$ is a stochastic variable, e.g. it has probabilities and assigned (in this case monetary) values. Just like $F$. So can you explain what the difference between $F$ and $x$ is? Perhaps you could link to and/or name your source?  

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## 1 answers

### Answer by username_2 on 2019-11-04. Score: 2 | Accepted answer

First, assume risk aversion. By the definition of the certainty equivalent and Jensen's:
$$u(CE(u,F))=E(u(x))

Now, from monotonicity:

$$CE

Second, assume $CE. By monotonicity and the definition of $CE$:

$$u(E(x))>u(CE)=E(u(x))$$

> **Comments:**
> **username_0**: The part after "second" is not necessary. Or is there any reason for putting it in?  
> **username_2**: Not really, you can make it work also with "iff" statements.  
> **username_1**: You use strict monotonicity, but the question pointed out that "$u(x)$ *is monotonic and not strictly monotonic*". Luckily, OP was happy with the answer.  

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