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<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:
- $x < M$ $\implies u(x)=\sqrt(x)$, 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.