Consider a representative agent economy where the representative agent's utility function is given by \(\mathbf{u}(x)=\log (x)\) and the aggregate endowment process is \(e=\left\{e_{0}, e\left(A_{t}\right) ; A_{t} \in \mathscr{F}_{t}, t=1, \ldots, T\right\}\). Show that, in correspondence of a no-trade equilibrium in the representative agent economy, the value at the initial date \(t=0\) of the aggregate endowment process \(e\) is given by \(e_{0}\left(1-\delta^{T+1}\right) /(1-\delta)\) (normalizing \(q_{0}=1\), as at the end of Sect. 6.2).