= Certainty equivalent and willingness to pay
{title2=$u(c_{\rm CE})=\mathbb E u(X),\quad\mathbb E u(w-c_{\rm buy}+X)=u(w)$}
A <certainty equivalent> exchanges a <lottery> for a sure outcome on the same <utility function> scale. A maximum purchase price compares wealth after paying with wealth without purchasing; its <utility indifference price> therefore depends on the initial wealth and on the admissible monetary domain. These two numbers are not generally interchangeable when the <utility function> is nonlinear.
For example, suppose $u(x)=x^2$ is applied only on nonnegative wealth, $X\ge0$, $\mathbb EX=\mu$ and $\operatorname{Var}(X)=\sigma^2$. The <certainty equivalent> is $\sqrt{\mu^2+\sigma^2}$. A purchase price satisfying the wealth comparison instead obeys
$$
(w-c+\mu)^2+\sigma^2=w^2,
\qquad c=w+\mu-\sqrt{w^2-\sigma^2},
$$
provided the square root exists and every resulting wealth is in the increasing domain. The sufficient condition $w\ge\sqrt{\mu^2+\sigma^2}$ ensures $w-c\ge0$. The formulas differ even though both use the same <expected utility> representation. Applying the square function indiscriminately to negative wealth would remove the usual preference for more money and invalidate that monotonic price interpretation.
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