Microeconomics - Expected Utility Theory - Piecewise utility index, certainty equivalence, etc. 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? 



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&lt;M $ , $C(F,\sqrt(x))= u^{-1}[\int\sqrt(x)dF$] 
$ \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: 


$x &lt; M$ $\implies u(x)=\sqrt(x)$, 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.
Suppose random variable $X\sim F$. 


Certainty equivalent of the lottery $X$ is the constant $c$ that solves: 


  $u(c) = \mathbb{E}(u(X))$

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)) $

Agent is risk averse because $u$ is concave.
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 &gt; \frac{1}{2} = G\left(\frac{M}{2}\right) $.