Consider an economy with \(S=N=3\), with the following dividend matrix:

\[D=\left[\begin{array}{lll}1 & 4 & 3 \\6 & 2 & 4 \\2 & 3 & 5\end{array}\right]\]

(i) Show that the market is complete.

(ii) Given the price vector \(p^{\top}=(2.15,2.7,3.35)\), determine the state price vector.

(iii) Determine the risk free rate of return \(r_{f}\) implicit in the couple \((p, D)\) and the portfolio which attains the riskless payoff \((1,1,1)\). Verify that the return of such a portfolio coincides with \(r_{f}\).

(iv) Determine the portfolio \(z^{c}\) which replicates the payoff \(c=(2,3,6)\). Verify that \(p^{\top} z^{c}=Q(c)=m^{\top} c\).

(v) Determine the portfolio \(z^{\mathbf{1}_{3}}\) which replicates the Arrow security \(\mathbf{1}_{3}\), with payoff \((0,0,1)\), and verify that \(p^{\top} z^{1_{3}}=m_{3}\).