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

Question

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:

$u[C(F,u)] = F(M)\sqrt(x) + [1-F(M)]*\sqrt(M)$

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(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 $u[C(F,u)] \leq u(\int x dF)$ , $\forall F \in \Delta(\mathbb{R})$

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(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 $$u(x_3) \geq \alpha u(x_1) + (1-\alpha)u(x_2)$$

$$\to u(\alpha x_1 + (1-\alpha)x_2) \geq \alpha u(x_1) + (1-\alpha)u(x_2)$$

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(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.

Answer 1

Suppose random variable $X\sim F$.

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

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

2. 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)) $

3. Agent is risk averse because $u$ is concave.
4. 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 > \frac{1}{2} = G\left(\frac{M}{2}\right) $.