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An incentive fee above a hurdle changes a manager’s utility function into $F(y)=U(\epsilon y+\alpha (y-w_0{)}_{+})$. For constant absolute risk aversion utility, each branch is strictly concave, but the marginal reward jumps upward at the hurdle. Thus the effective terminal-wealth utility is not concave, and concavification of incentive utility can reveal optimal risk-taking lotteries.

Let risky excess payoff have nonzero mean vector $b$ and positive definite covariance matrix $V$. With unrestricted holdings, the minimum-variance portfolio achieving target excess expected wealth $c$ isA Lagrange multiplier or the Cauchy-Schwarz inequality in the $V$ inner product proves this: every component orthogonal to $V^{-1}b$ adds variance without changing $b^T\theta =c$. The minimum variance is $c^2/(b^T{V}^{-1}b)$.

For constant absolute risk aversion utility with coefficient $\gamma >0$ and Gaussian excess payoffs, expected-utility maximization gives $\theta ={\gamma }^{-1}{V}^{-1}b$. This coincides with the target-mean portfolio when $c>0$ and $\gamma =1/\lambda$. A negative target has $\lambda <0$ and cannot correspond to a positive risk-aversion parameter. If $b=0$, only target $c=0$ is feasible, and the zero portfolio uniquely minimizes variance.

For a competitive equilibrium with one productive asset, the primitives are the probability law and information filtration of total output $d$, each agent's utility function and discount factor, the initial ownership shares ${\theta }_0^j$ summing to one, and the permitted consumption and trading strategies. Prices are not primitives of the equilibrium problem. The derived objects are the ex-dividend price $S$, each agent's consumption $C^j$ and holdings ${\theta }^j$, and their marginal-utility pricing processes. At a common price, every agent solves the individual problem, and resource and share clearing requireThe individual discrete dividend budget equations, optimality Euler equations and terminal conditions from part (i), together with these clearing equations, define equilibrium even when the market is incomplete. Summing the individual budget constraints is consistent with total fruit consumption.

For the given constant absolute risk aversion utility, $U_j^{\prime }(c)=e^{-{\gamma }_{j}c}$. Hence each interior optimum satisfiesThe same $S$ must satisfy all these equations, but their conditional expectation identities do not by themselves identify the different agents' pricing densities pointwise.

To illustrate explicit exponential-utility risk sharing, first consider a complete or otherwise implementable interior allocation. With positive constant welfare weights $a_j$, a pointwise resource multiplier ${\nu }_t$ satisfiesWriteSolving the marginal equations and summing the consumptions givesThe output risk is shared in proportion to risk tolerance ${\gamma }_j^{-1}$; differences in patience produce the time-dependent transfer term. The common normalized state-price density is ${\zeta }_t={\nu }_t/{\nu }_0$. The weights, up to a common scale, are determined by the initial wealth budgetsand, for a fundamental price,If consumption is required to be nonnegative and the interior formula produces a negative allocation, replace it byFor $d_t>0$ this determines the resource multiplier uniquely. At zero total output all constrained consumptions are zero and the corresponding marginal inequalities apply.

A single traded tree need not span the contingent transfers in the general risk-sharing formula. A genuine equilibrium example within the literal one-tree market is obtained by taking all ${\beta }_j=\beta$ and initial shares ${\theta }_0^j=\Gamma /{\gamma }_j$. ThenEvery agent simply keeps the initial shares, all budget constraints and clearing conditions hold, and their discounted marginal utilities coincide. The fundamental price above satisfies every Euler equation. For example, nonnegative bounded output ensures the required price and utility integrability and the terminal terms vanish; the supporting-line argument from part (i) then verifies individual optimality. For heterogeneous discount factors or arbitrary initial ownership, the displayed coupled Euler-budget-clearing system remains the correct one-tree formulation; using the unconstrained Pareto allocation as a traded equilibrium requires an additional financing check.

Here $\alpha >0$ and ${\gamma }_j>0$. The asset payoff has a centered Laplace distribution. Its moment-generating function is $\mathbb{E}{e}^{tX}={\alpha }^2/({\alpha }^2-t^2)$ for $|t|<\alpha$, and diverges otherwise. Expected constant absolute risk aversion utility for a holding $\theta$ isTo maximize it, minimize the logarithm of the positive factor on $|{\gamma }_j\theta |<\alpha$. Its derivative is ${\gamma }_{j}p+2{\gamma }_j^2\theta /({\alpha }^2-{\gamma }_j^2{\theta }^2)$, and its second derivative is strictly positive. It diverges at the endpoints, so the unique critical point is the global optimizer:At $p=0$ use the continuous limit ${\theta }_j=0$. The other quadratic root is outside the finite-utility domain.

Let ${\Gamma }^{-1}={\sum }_j{\gamma }_j^{-1}$. Aggregate demand is $D(p)=-(\sqrt{1+p^2{\alpha }^2}-1)/(\Gamma p)$. Unit positive supply requires $p<0$ and $\sqrt{1+p^2{\alpha }^2}=1-\Gamma p$. Squaring and excluding $p=0$ givesThis negative price satisfies the unsquared equation and every individual finite-utility constraint. When $\alpha \le \Gamma$, there is no finite clearing price. Indeed $D(p)<\alpha /\Gamma$ for every finite negative $p$, with limit $\alpha /\Gamma$ as $p\to -\infty$. If $\alpha <\Gamma$ this maximum limiting demand is below supply; if equal, demand only approaches one at an infinitely negative price. A formal positive solution obtained by squaring when $\alpha <\Gamma$ violates the original demand equation.

Because $S_1$ is Gaussian, the terminal wealth $W_1$ is normal. For constant absolute risk aversion utilitythe moment-generating function of a normal variable givesMaximizing this expected utility is equivalent to maximizingafter dropping a constant. The unique first-order condition isand strict concavity givesIt agrees with part (a) exactly whenFor the usual risk-averse convention $\gamma >0$, the asserted correspondence assumes $w_1>(1+r)w_0$; under that natural target-return condition the choice is unique.