Certainty equivalent and willingness to pay ($u(c_{\mathrm{CE}})=\mathbb{E}u(X),\mathbb{E}u(w-c_{\mathrm{buy}}+X)=u(w)$)

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, $X\ge 0$, $\mathbb{E}X=\mu$ and $\mathrm{Var}(X)={\sigma }^2$. The certainty equivalent is $\sqrt{{\mu }^2+{\sigma }^2}$. A purchase price satisfying the wealth comparison instead obeysprovided the square root exists and every resulting wealth is in the increasing domain. The sufficient condition $w\ge \sqrt{{\mu }^2+{\sigma }^2}$ ensures $w-c\ge 0$. 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.