Let (left{y^{i}: mathbb{R}_{+} ightarrow mathbb{R}_{+} ; i=1, ldots, I ight}) be a Pareto optimal sharing rule.

Let \(\left\{y^{i}: \mathbb{R}_{+} \rightarrow \mathbb{R}_{+} ; i=1, \ldots, I\right\}\) be a Pareto optimal sharing rule. Show that \(y^{i}\) is linear if and only if

\[t_{u^{i}}^{\prime}\left(y^{i}(e)\right)=t_{u^{k}}^{\prime}\left(y^{k}(e)\right), \quad \text { for all } i, k=1, \ldots, I\]

where \(t_{u^{i}}^{\prime}\left(y^{i}(e)\right)\) denotes the first derivative of the risk tolerance of agent \(i\) computed in correspondence of the Pareto optimal consumption allocation \(y^{i}(e)\), where \(e\) denotes an arbitrary realization of the aggregate endowment.