(i) An investor in a single-period market with time- 0 wealth $w_{0}$ may generate any time-1 wealth $w_{1}$ of the form $w_{1}=w_{0}+X$, where $X$ is any element of a vector space $V$ of random variables. The investor's objective is to maximize $E\left[U\left(w_{1}\right)\right]$, where $U$ is strictly increasing, concave and $C^{2}$. Define the utility indifference price $\pi(Y)$ of a random variable $Y$.

Prove that the map $Y \mapsto \pi(Y)$ is concave. [You may assume that any supremum is attained.]

(ii) Agent $j$ has utility $U_{j}(x)=-\exp \left(-\gamma_{j} x\right), j=1, \ldots, J$. The agents may buy for time- 0 price $p$ a risky asset which will be worth $X$ at time 1 , where $X$ is random and has density

$$f(x)=\frac{1}{2} \alpha e^{-\alpha|x|}, \quad-\infty<x<\infty .$$

Assuming zero interest, prove that agent $j$ will optimally choose to buy

$$\theta_{j}=-\frac{\sqrt{1+p^{2} \alpha^{2}}-1}{\gamma_{j} p}$$

units of the risky asset at time 0 .

If the asset is in unit net supply, if $\Gamma^{-1} \equiv \sum_{j} \gamma_{j}^{-1}$, and if $\alpha>\Gamma$, prove that the market for the risky asset will clear at price

$$p=-\frac{2 \Gamma}{\alpha^{2}-\Gamma^{2}}$$

What happens if $\alpha \leqslant \Gamma ?$