Hello!!! I have this exercise for the subject Banking Management and i'm a bit lost.

I will appreciate you help. Thank you!

This is it:

Assume a Diamond-Dybvig type economy, and a set of individuals who are endowed with

$1 each in t=0. In t=1, they are subject to a preference shock, and given this shock they can be of type-1 with probability p=0.5 or of type-2 with probability 1-p. Their utility function is:

U(C) = (C1 - ) / (1 - ) and =1.3

On the other hand, it is possible to invest in a technology that creates wealth, and for each

$1 invested in t=0, it generates R=$1.5 in t=2, but this investment can be liquidated prematurely and then yield k=$1 in t=1.

Furthermore, type 2 individuals can treasure their liquidity between t=1 and t=2 and rf=0

1.Calculate the ex-ante utility, when there are no banks in the economy

2.Assume now there is a bank that receives one unit in t=0. What is the optimal contract the bank can offer? Determine C1* and C*2

3.Assume C1*=$1.04675 and C2*=$1.4298, what is the expected utility of the intermediated solution? Is therefore the bank's existence justified?