Consider an economy with trade in $K$ assets at market prices $(q_1,$dots,$q_K).$ Assets pay returns $r_s=(r_s^1,$dots,$r_s^K)$ in each of $S$ states that occur with probabilities $(p_1,$dots,$p_S).$

$($a$)$ Write down the optimization problem for the choice of a portfolio that minimizes the variance subject to the constraint that it satisfies the budget constraint and that it achieves a certain level of mean return.

$($b$)$ Derive the first$-$order conditions for the variance$-$minimizing portfolio. Are these conditions sufficient as well?

$($c$)$ Assume that asset $K$ is riskless with return $R$ and price $q_K-=1.$ Show that the first$-$order conditions for the $K-1$ risky assets can be written in the following form:

$[{}_k-q_k*R]=\frac{(k,a)}{{}^2(a)}*[(a)-R*{\displaystyle \underset{k=1}{\overset{K}{}}}{q}_k*a_k]$