Consider an economy with two possible states of the world, with associated probabilities of occurrence \((1 / 2,1 / 2)\), and a representative agent with expected logarithmic utility function of the form

\[u\left(x_{0}, x_{1}, x_{2}\right)=\log \left(x_{0}\right)+\frac{1}{2} \log \left(x_{1}\right)+\frac{1}{2} \log \left(x_{2}\right) .\]

In the economy it is possible to trade the two Arrow securities and a risk free asset with constant unitary payoff. Determine the equilibrium prices \(\left(q_{1}^{*}, q_{2}^{*}\right)\) of the two Arrow securities and the return \(r_{f}\) of the risk free asset in the representative agent economy when the aggregate endowment is given by \(\left(e_{0}, e_{1}, e_{2}\right)=(1,3,1)\).