Consider an economy with two possible states of the world and two agents \(a\) and \(b\). Assume that both agents are characterized by a logarithmic utility function \(u(x)=\log (x)\) defined on consumption at time \(t=1\). Let \(p_{1}\) and \(p_{2}\) denote the equilibrium prices of the two Arrow securities and suppose that the initial endowments of the agents are given by \(\left(e_{1}^{i}, e_{2}^{i}\right)=\alpha^{i}\left(e_{1}, e_{2}\right)\), for \(i \in\{a, b\}\), for some couple \(\left(\alpha^{a}, \alpha^{b}\right)\) satisfying \(\alpha^{a}+\alpha^{b}=1\) and where \(\left(e_{1}, e_{2}\right)\) denotes the economy's aggregate endowment. Show that, if the two agents have homogeneous beliefs \((\pi, 1-\pi)\) about the possible realization of the state of the world, then they will not trade in equilibrium.