Consider the optimal saving problem of an agent over three dates \(t \in\{0,1,2\}\), assuming that the agent is endowed with initial wealth \(w_{0}>0\), has a constant discount factor normalized to one and is characterized by a strictly increasing and strictly concave utility function \(u\) such that \(u^{\prime \prime \prime}>0\) (i.e., the agent is prudent). At the final date \(t=2\), the agent receives the random income \(\tilde{x}\). Consider the following optimal saving problem, where \(\alpha_{0}\) and \(\alpha_{1}\) denote saving at dates \(t=0\) and \(t=1\), respectively:

\[\begin{equation*}\max _{\left(\alpha_{0}, \alpha_{1}\right) \in \mathbb{R}^{2}}\left(u\left(w_{0}\alpha_{0}\right)+\mathbb{E}\left[u\left(\alpha_{0}\alpha_{1}\right)+u\left(\tilde{x}+\alpha_{1}\right)\right]\right) \tag{8.42}\end{equation*}\]

(i) Consider first the case where there is no early resolution of uncertainty, i.e., the random variable \(\tilde{x}\) is only observed at the final date \(t=2\). Show that in this case the optimal saving decision satisfies \(\alpha_{1}^{*}=2 \alpha_{0}^{*}-w_{0}\) (i.e., it is optimal to smooth consumption over the first two dates).
(ii) Consider then the case where the realization of the random variable \(\tilde{x}\) is observed at the intermediate date \(t=1\). Show that in this case the optimal saving \(\alpha_{0}^{\mathrm{i*}}\) (where the superscript \({ }^{\mathrm{i}}\) stands for informed) from date \(t=0\) to \(t=1\) satisfies the first order condition \[\begin{equation*}
u^{\prime}\left(w_{0}-\alpha_{0}^{\mathrm{i} *}\right)=\mathbb{E}\left[u^{\prime}\left(\frac{\alpha_{0}^{\mathrm{i} *}+\tilde{x}}{2}\right)\right] . \tag{8.43}
\end{equation*}\]
(iii) Deduce that \(\alpha_{0}^{*} \geq \alpha_{0}^{\mathrm{i} *}\), i.e., an early resolution of uncertainty reduces the optimal level of saving before the resolution of uncertainty.