Suppose that markets are complete and let [u^{i}left(x_{0}, x_{1}, ldots, x_{S} ight)=-gamma_{i} exp left(-x_{0} / gamma_{i} ight)-gamma_{i}

Suppose that markets are complete and let

\[u^{i}\left(x_{0}, x_{1}, \ldots, x_{S}\right)=-\gamma_{i} \exp \left(-x_{0} / \gamma_{i}\right)-\gamma_{i} \delta \sum_{s=1}^{S} \pi_{s} \exp \left(-x_{s} / \gamma_{i}\right)\]

for all \(i=1, \ldots, I\). Show that the equilibrium prices \(q^{*}=\left(q_{1}^{*}, \ldots, q_{S}^{*}\right)\) of the \(S\) Arrow securities are given by

\[q_{s}^{*}=\frac{\delta \pi_{s} \exp \left(-e_{s} / \sum_{i=1}^{I} \gamma_{i}\right)}{\exp \left(-e_{0} / \sum_{i=1}^{I} \gamma_{i}\right)}, \quad \text { for all } s=1, \ldots, S\]