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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.

The reservation bid price for a claim $Y$ is the fixed cash payment $p_i(Y)$ which leaves the agent indifferent: $E[U_i(X_i+Y-p_i(Y))]=E[U_i(X_i)]$. For Gaussian wealth $W$, exponential utility is $-\exp [-{\gamma }_{i}E(W)+{\gamma }_i^2\mathrm{Var}(W)/2]$, so its certainty equivalent is $E(W)-{\gamma }_i\mathrm{Var}(W)/2$. For $Y=\lambda X_0$, subtracting the original certainty equivalent givesThe marginal price is positive when the claim hedges the agent's endowment, negative when it adds positively correlated risk, and zero for an uncorrelated zero-mean infinitesimal position.

Assume $v_{00}>0$, so the claim is nontrivial. Buying $\theta$ units at per-unit price $p$ changes the certainty equivalent by $-(p+{\gamma }_i{v}_{0i})\theta -{\gamma }_i{v}_{00}{\theta }^2/2$. This strictly concave quadratic is maximized at ${\theta }_i=-(p+{\gamma }_i{v}_{0i})/({\gamma }_i{v}_{00})$. Market clearing ${\sum }_i{\theta }_i=0$ givesAt the optimum the certainty-equivalent gain isThe maximal expected utility equals the original negative utility multiplied by $e^{-{\gamma }_i{\Delta }_i}$, hence is improved. Paying a fixed amount ${\Delta }_i$ removes exactly that gain, proving it is the fixed-payment value of market access. If $v_{00}=0$, the Gaussian claim vanishes almost surely, all its covariances vanish, and the appropriate price and access value are zero; the printed formulas with a variance denominator require the nontrivial-claim assumption.

Let initial wealth be $w$ and let ${\theta }_i$ be the number of units held of risky asset $i$; the bank investment is $w-{\theta }^T{S}_0$. Put $b=\mu -(1+r)S_0$. Terminal wealth is normal, with mean $(1+r)w+{\theta }^{T}b$ and variance ${\theta }^{T}V\theta$. The expectation of the exponential utility isFor $\gamma >0$ and positive definite covariance matrix $V$, maximizing this is equivalent to maximizing the certainty equivalent ${\theta }^{T}b-\gamma {\theta }^{T}V\theta /2$. Thus the optimal investment isAll investors have proportional risky holdings. A market portfolio can therefore be represented by ${\theta }_M=V^{-1}b$, with any positive scalar multiple giving the same return. Suppose its initial value $P={\theta }_M^T{S}_0$ is positive and $b\ne 0$. Set $B=b^T{V}^{-1}b$. For returns $R_i=S_1^i/S_0^i-1$ and $R_M={\theta }_M^T{S}_1/P-1$,The beta of an asset is its covariance with the market return divided by the market variance. ConsequentlyThis is the capital asset pricing model. Since ${\beta }_i={\rho }_i{\sigma }_i/{\sigma }_M$, dividing the excess-return identity by ${\sigma }_i$ gives the Sharpe ratio identityThese ratio statements require nonzero variances and nonzero initial portfolio value. If $V$ is singular and $b$ is in its range, replace $V^{-1}$ by its pseudoinverse; any kernel holding can be added to an optimizer. If $b$ has a component in the kernel, it gives a deterministic excess gain that can be scaled without bound, so there is no finite optimal portfolio. If $b=0$, there is no distinguished risky market portfolio and its beta ratios are undefined.

Let $X$ denote the monetary reward. Its expected value and expected utility are different quantities:Thus indifference to a sure nonnegative reward requires $k^2=10.89$, giving $k=3.30$ dollars. This is the certainty equivalent, hence the lottery's monetary value in the question's comparison with a sure reward. The utility function $u(x)=x^2$ is increasing and strictly convex on nonnegative monetary amounts. Its certainty equivalent exceeds its expected value, $3.30>1.89$, demonstrating risk seeking; the same conclusion follows from the Jensen inequality.

The wording about paying an entry fee needs a qualification. A certainty equivalent compares a gross lottery with a sure gross reward; a literal utility indifference price also accounts for the fee's effect on wealth. If $u$ instead describes final wealth, with initial wealth $w$ and fee $c$, the comparison is $\mathbb{E}[(w-c+X{)}^2]=w^2$, on a domain where the utility function is increasing. Here $\mathrm{Var}(X)=7.3179$, so the nonnegative-wealth branch gives $c=w+1.89-\sqrt{w^2-7.3179}$ whenever that branch is admissible. An initial wealth and an admissible payment domain are not specified. Therefore $3.30$ is the determined certainty equivalent, while a literal wealth-dependent maximum fee is not uniquely determined. This is the distinction in certainty equivalent and willingness to pay. If negative rewards were allowed without requiring an increasing utility function, the equation for $k$ also has the algebraic root $-3.30$; the usual nonnegative-reward interpretation selects the positive root.