Let (pi=left(pi_{1}, ldots, pi_{S} ight)) be a vector of probabilities, i.e., (pi in Delta_{S}), where (Delta_{S}) is

Let \(\pi=\left(\pi_{1}, \ldots, \pi_{S}\right)\) be a vector of probabilities, i.e., \(\pi \in \Delta_{S}\), where \(\Delta_{S}\) is the simplex

\[\Delta_{S}:=\left\{\pi \in \mathbb{R}_{+}^{S}: \sum_{s=1}^{S} \pi_{s}=1\right\}\]

Consider the optimal choice problem (8.2) and suppose that there exists an optimal solution \(\left(x_{1}^{*}(\pi), \ldots, x_{S}^{*}(\pi)\right) \in \mathbb{R}_{S}^{+}\), for each \(\pi \in \Delta_{S}\), and define

\[U^{*}(\pi):=\sum_{s=1}^{S} \pi_{s} u\left(x_{s}^{*}(\pi)\right)\]

Show that the map \(\pi \mapsto U^{*}(\pi)\) from \(\Delta_{S}\) onto \(\mathbb{R}\) is convex in the vector of probabilities.