# How to prove the relationship between the expected value of a lottery and its certainty equivalent?

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.

• You are using a lot of undefined notation. (Also some undefined concepts, but those are minor.) – Giskard Nov 2 '19 at 15:29
• 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 Nov 2 '19 at 23:27
• $E(u(x)) = U(F)$ ($U(F)$ is Neumann-Morgenstern expected utility of the lottery) – Aqqqq Nov 3 '19 at 8:42
• $F$ is a lottery. $x$ is one of the sure outcome of the lottery. – Aqqqq Nov 3 '19 at 8:49
• 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 Nov 3 '19 at 11:53

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$$:

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

• The part after "second" is not necessary. Or is there any reason for putting it in? – Aqqqq Nov 5 '19 at 9:59
• Not really, you can make it work also with "iff" statements. – Weierstraß Ramirez Nov 5 '19 at 14:40
• 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 Nov 9 '19 at 21:04