II$.$ Diamond$-$Dybvig Model of Bank Runs

Useful numerical calculations:

$($a$)\frac{[1-\mathrm{.}25(1\mathrm{.}28)]2}{\mathrm{.}75}=1\mathrm{.}813$

$($b$)\frac{[1-\mathrm{.}5(1\mathrm{.}28)]2}{\mathrm{.}5}=1\mathrm{.}44$

$($c$)\frac{[1-\mathrm{.}25(1\mathrm{.}28)](1\mathrm{.}4)}{\mathrm{.}75}=1\mathrm{.}27$

In the numerical example used by Diamond, $1$ unit of initial wealth invested in period $0$ has a return of $1$ in period $1($it$\text{'}$s basically like storage$)$ and a return of $R=2$ in period $2.$ The fraction of agents with early liquidity needs is $f=\frac{1}{4}.$

$(5$ points$)$ If agents invest by themselves $($autarky$),$ how much will they consume in period $1$ if they have an early liquidity need? How much will they consume in period $2$ if they don't have an early liquidity need? Explain how a bank can improve upon the autarky solution.