There are three dates t $=0,1,2$ a continuum of consumers with measure one, each endowed with wealth one. There are two types of these consumers, a fraction a is impatient, i$.$e$.,$ derive utility their utility is $1(,$ where is consumption at t$=1,$ and a fraction $1-$ a are patient whose utility is $1(,$ where is consumption at t$=2$ At t$=0$consumers don$$t know their type, i$.$e$.$ it is unknown whether the consumer wishes to consume at date t$=1$ or t$=2$ The investment in the long$-$term technology can be liquidated at t$=1,$ which then yields L Wealth can also be stored across pThere are three dates $t=0,1,2$ and a continuum of consumers with measure one, each endowed with wealth one. There are two types of

consumers: a fraction $$ is impatient, i$.$e$.$ their utility is $1-(\frac{1}{c_1}),$ where $c_1$ is consumption at $t=1,$ and a fraction $1-$ are patient whose

utility is $1-(\frac{1}{c_2}),$ where where $c_2$ is consumption at $t=2.$ At $t=0$ consumers don't know their type, i$.$e$.$ it is unknown whether the

consumer wishes to consume at date $t=1$ or $t=2.$ At $t=1$ all consumers observe their own type, but cannot observe others' types $($i$.$e$.$

type is private information$).$ There are two technologies to manage liquidity. Consumers can invest for one period only $($short term

technology$),$ in which case the investment will yield $r1.$ This technology is available at both $t=0$ and $t=1.$ Alternatively, wealth can be

invested for two periods $($long term technology with yield of $R)$ at date $t=0,$ which yields $0$ at $t=1,$ but $R>r^2$ at $t=2.$ The investment

in the long$-$term technology can be liquidated at $t=1,$ which then yields Wealth can also be stored across periods without cost.

a$)$ Suppose there is a bank to manage liquidity risk, all consumers deposit their wealth in the bank and the bank maximizes total

welfare of its depositors when it chooses $c_1^B,c_2^B,$ the promised payouts for early $(t=1)$ and late $(t=2)$ withdrawals, respectively.

I. Set up the bank's optimisation problem and derive the first order condition. What does the first order condition tell you about

the optimal allocation?

II$.$ Calculate the optimal investment in the long$-$term technology $I^B$ and the optimal allocation $c_1^B,c_2^B.$

III. Does the allocation $c_1^B,c_2^B$ always constitute an equilibrium? If not, derive the necessary condition for $c_1^B,c_2^B$ to be an

equilibrium and carefully explain your finding.

IV$.$ Now, suppose the condition that you derived in iii$)$ above does not hold. Derive the feasible $($incentive compatible$)$ allocation

$c_1^f,c_2^f$ that maximizes total welfare and can always be supported in an equilibrium.

b$)$ Now, suppose there is no financial intermediary to handle liquidity shocks. However, at $t=1$ a financial market for bonds

opens up and agents trade their wealth at $t=1$ for wealth at $t=2.$ Each bond pays $1$ at $t=2$ and its price is $p^M.$ Calculate the

consumer's optimal investment decision $I^M$ at $t=0,$ an expression for the price of the bond $p^M$ and the consumption levels in the two

states $c_1^M,c_2^M.$

c$)$ Is the bond market allocation efficient? Calculate the price of the bond widetilde$(p)$ that would ensure that the market delivers the

optimal allocation and discuss what would be needed for this price to clear the market.eriods without cost. ii$)$ Calculate the optimal investment in the long$-$term technology and the optimal allocation $,.$