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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.

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  • $\begingroup$ You are using a lot of undefined notation. (Also some undefined concepts, but those are minor.) $\endgroup$
    – Giskard
    Commented Nov 2, 2019 at 15:29
  • $\begingroup$ 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? $\endgroup$
    – Giskard
    Commented Nov 2, 2019 at 23:27
  • $\begingroup$ $E(u(x)) = U(F)$ ($U(F)$ is Neumann-Morgenstern expected utility of the lottery) $\endgroup$
    – Aqqqq
    Commented Nov 3, 2019 at 8:42
  • $\begingroup$ $F$ is a lottery. $x$ is one of the sure outcome of the lottery. $\endgroup$
    – Aqqqq
    Commented Nov 3, 2019 at 8:49
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    $\begingroup$ 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? $\endgroup$
    – Giskard
    Commented Nov 3, 2019 at 11:53

1 Answer 1

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

Now, from monotonicity:

$$CE<E(x)$$

Second, assume $CE<E(x)$. By monotonicity and the definition of $CE$:

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

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  • $\begingroup$ The part after "second" is not necessary. Or is there any reason for putting it in? $\endgroup$
    – Aqqqq
    Commented Nov 5, 2019 at 9:59
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    $\begingroup$ Not really, you can make it work also with "iff" statements. $\endgroup$ Commented Nov 5, 2019 at 14:40
  • $\begingroup$ 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. $\endgroup$
    – Giskard
    Commented Nov 9, 2019 at 21:04

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