Continue with the social planner's problem in the Allen-Gale model, Section 2.3. Given the planner's optimal choice on \(c_{1} / c_{2}\) after \(R\) is revealed \(t=1\), solve problem (2.11) for the planner's optimal choice of \(\alpha\) with
- Logarithm-form utility function, i.e., \(u(\cdot)=\ln (\cdot)\);
- Distribution of \(R\) following
\[
f(R)=\left\{\begin{array}{cl}
\frac{1}{3} & \text { for } 0 \leq R \leq 3 \\
0 & \text { otherwise. }
\end{array}\right.
\]
Compute \(\hat{R}\) under the optimal \(\alpha\), and delineate \(c_{1}\) and \(c_{2}\) as functions of \(R\) in a \(c-R\) space.

Transcribed Image Text:

(1 - p)c = R(1-) (2.3)