Consider an economy with \(I\) agents characterized by power utility functions \(u^{i}(x)=\frac{1}{\alpha} x^{\alpha}\), with common risk aversion parameter \(0 eq \alpha<1\), and endowment \(\left(e_{1}^{i}, \ldots, e_{S}^{i}\right) \in\) \(\mathbb{R}_{+}^{S}\), for all \(i=1, \ldots, I\). Assume that at \(t=0\) there are \(S\) markets for the \(S\) Arrow securities paying in correspondence of the \(S\) states of the world, with prices \(\left(p_{1}^{*}, \ldots, p_{S}^{*}\right)\). Show that in correspondence of an equilibrium of this economy the optimal expected utility of each agent is concave with respect to the vector of probabilities of the \(S\) states of the world. Deduce that additional information will decrease the ex-ante expected utility of each agent as long as the individual endowments are not proportional to the aggregate endowment, which we denote by \(\left(e_{1}, \ldots, e_{S}\right)\).