An agent with utility $U(x)=-\exp (-\gamma x)$, where $\gamma>0$ is a constant, may select at time 0 a portfolio of $n$ assets, which he then holds to time 1 . The values $X=\left(X_{1}, \ldots, X_{n}\right)^{T}$ of the assets at time 1 have a multivariate normal distribution with mean $\mu$ and nonsingular covariance matrix $V$. Prove that the agent will prefer portfolio $\psi \in \mathbb{R}^{n}$ to portfolio $\theta \in \mathbb{R}^{n}$ if and only if $q(\psi)>q(\theta)$, where

$$q(x)=x \cdot \mu-\frac{\gamma}{2} x \cdot V x$$

Determine his optimal portfolio.

The agent initially holds portfolio $\theta$, which he may change to portfolio $\theta+z$ at cost $\varepsilon \sum_{i=1}^{n}\left|z_{i}\right|$, where $\varepsilon$ is some positive transaction cost. By considering the function $t \mapsto q(\theta+t z)$ for $0 \leqslant t \leqslant 1$, or otherwise, prove that the agent will have no reason to change his initial portfolio $\theta$ if and only if, for every $i=1, \ldots, n$,

$$\left|\mu_{i}-\gamma(V \theta)_{i}\right| \leqslant \varepsilon$$