Decision Theory Question: Existence and uniqueness of the certainty equivalent of p

Question

Let $X = (x_*,x^*)$ be an interval in the real line and denote by $\Delta(X)$ the set of simple probability distributions on $X$. Consider a preference relation $\succcurlyeq$ on $\Delta(X)$ that satisfies the axioms of expected utility theory. If $\succcurlyeq$ displays monotonicity with respect to first order stochastic dominance and risk aversion then for all $p \in \Delta(X)$ the certainty equivalent of $p$ exists and is unique.

Sketch of the proof:

1) We define the certainty equivalent of lottery $p$: $CE_p \sim \int_{X}u(x)p(x)dx$

2) We know that if $p$ FOSD $q$ then $p\succcurlyeq q$

3) Because of risk aversion: $\int_{X}u(x)p(x)dx \leq u(\int_{X}xp(x)dx) =u(p)$

4) We want to show that $\exists$ a lottery, call it $s$ such that $CE_p \sim s$; in this respect (from 3 above) we know that $p \succcurlyeq s$; We then pick a particular lottery $q$ such that $p \succcurlyeq q$

5) Since by assumption $\succcurlyeq$ on $\Delta(X)$ satisfies the axioms of expected utility theorem (in particular continuity) there exists a unique $\alpha$ such that $\alpha p + (1-\alpha) q \sim s$ ( see MWG p. 177). We therefore proved that $\exists$ a compound lottery $s$ that is the certainty equivalent of lottery $p$.

QUESTION: Am I missing any detail in the proof?

Answer 1

Your notation is a bit misleading: it would be better to write $\mathbb{E}u(p)$ or $U(p)$ for the expected-utility associated with $p$ instead of $u(p)$, and $u(\mathbb{E}p)$ for the utility of the expected value of $p$. Formally $u$ is defined on $X$ and not on $\Delta(X)$.

Regarding your proof, it seems to me that: $(i)$ you don't explain how to find $s$; $(ii)$ you don't find a certainty equivalent, because $s$ is a lottery. What you want to find is a sure monetary prize, i.e. a degenerate lottery, that the decision-maker values equally as the lottery $p$.

For instance, you can notice that, by the monotonicity of $u$, \begin{equation} u(x_{*})\leq \int_{X}{u(x)p(x)dx} \leq u(x^{*}) \end{equation}

In addition, the function $u:x \rightarrow u(x)$ is continuous. Therefore, by the intermediate value theorem, there exists $CE_p \in [x_{*},x^{*}]$ such that $u(CE_p) = \int_{X}{u(x)p(x)dx}$.

For the uniqueness, imagine that $CE'_p$ is another certainty equivalent of $p$, i.e. that $u(CE'_p)=u(CE_p)$. Since $u$ is strictly increasing (which can be seen as a consequence of the monotonicity with respect to first-order stochastic dominance), we obtain $CE'_p=CE_p$.

Notice that you don't need risk aversion to prove the result, but further implies that $CE_p \leq \mathbb{E}p$.