QUESTION $(5)$ Assume that there is an economy with two future states and two present assets with the following payoffs:

vec$(V{)}_{T1}=\left[\begin{array}{cc}{V}_{T,1,1}& V_{T,1,2}\end{array}\right]=\left[\begin{array}{cc}1& 1\end{array}\right],$vec$(V{)}_{T2}=\left[\begin{array}{cc}{V}_{T,2,1}& V_{T,2,2}\end{array}\right]=\left[\begin{array}{cc}0& 2\end{array}\right].$

The prices of the assets are:

$\left[\begin{array}{cc}{q}_{0,1}& q_{0,2}\end{array}\right]=\left[\begin{array}{cc}1& 1\end{array}\right].$

$($a$.$i$)$ What are the state prices $($or prices of Arrow$-$Debreu assets$)$ if they exist?

$($a$.$ii$)$ Find the risk free rate, $r_f,$ and the risk neutral probabilities $\left[\begin{array}{cc}{}_{T,1}& {}_{T,2}\end{array}\right]$ if they exist.

$($a$.$iii$)$ Is the market arbitrage free and complete?

$($b$.$i$)$ Assume that the asset endowment holdings of an individual are $\left[\begin{array}{cc}{e}_{T,1}& e_{T,2}\end{array}\right]=\left[\begin{array}{cc}\frac{?}{\text{b}\text{a}\text{r}}(e{)}_1& \frac{?}{\text{b}\text{a}\text{r}}(e{)}_2\end{array}\right].$ Determine the "budget plan" or "trading possibilities".

$($b$.$ii$)$ Using the probabilities of the states $\left[\begin{array}{cc}{}_{T,1}& {}_{T,2}\end{array}\right]$ you found in $($a$.$ii$),$ and that the individual's Bernoulli utility function is $.$ Find the individual's optimal state consumption levels.

$($c$)$ An investor desires to have state contingent wealth $($for contingent future consumption $,$ expenditures$),\left[\begin{array}{cc}{W}_{T,1}& W_{T,2}\end{array}\right]=\left[\begin{array}{cc}2& 8\end{array}\right].$ What amount of shares assets, vec$({)}_0=\left[\begin{array}{cc}{}_{0,1}& {}_{0,2}\end{array}\right],$ should the investor buy? As a financial consultant, would you advice the investor to buy the portfolio $($considering the price and the payoffs of the portfolio vec$({)}_0)?$

$($d$.$i$)$ Assume that there is a third asset with the following payoffs: vec$(V{)}_{T,3}=\left[\begin{array}{cc}{V}_{T,3,1}& V_{T,3,2}\end{array}\right]=\left[\begin{array}{cc}2& 4\end{array}\right].$ What is the price of this asset, $q_{0,3},$ in case of no arbitrage opportunity?