2 Consider the following Diamond and Dybvig style model with 3 periods, t=0,1,2 where agents each have a $1 endowment at t=0 and care about consuming at t=1 or t=2. Their endowment can be invested in two assets; a storage asset that allows the costless transfer of one unit from one time period to another and an illiquid asset that pays out R=2 in period 2 but only pays out L=21 if it is liquidated early in period 1 . At t=1 a fraction people learn that they only care about consumption at t=1, while a fraction (1) people learn that they only care about consumption at t=2, and so the ex ante expected utility of an agent in t=0 is u(C1)+(1)u(C2) where the utility function is u(C)=ln(C). For all questions state any assumptions you are making in deriving your answers. (a) If =21 what is the expected utility of an optimal bank contract if the possibility of a bank run, S, is zero i.e. if it is known with certainty that there won't be a bank run? (b) Suppose that the probability of bank run is S=101, but all banks still offer the optimal contract as if there was no possibility of a bank run as in part (a). In the event of a bank run all a bank's assets are distributed equally between depositors. Now what is the expected utility of an agent in period 0 who deposits his/her endowment in a bank? (c) Suppose that the government implemented a weak narrow banking regulation that each bank must ensure that reserves and the liquidation of long term assets cover all potential withdrawals i.e. that 1x+xL>C1 where x denotes the per agent investment in the illiquid asset. What is the expected utility of an optimal bank contract under these regulations?.

1. 2 Consider the following Diamond and Dybvig style model with 3 periods, t=0,1,2 where agents
each have a $1 endowment at t=0 and care about consuming at t=1 or t=2. Their endowment can
be invested in two assets; a storage asset that allows the costless transfer of one unit from one
time period to another and an illiquid asset that pays out R=2 in period 2 but only pays out L=21
if it is liquidated early in period 1 . At t=1 a fraction people learn that they only care about
consumption at t=1, while a fraction (1) people learn that they only care about consumption at
t=2, and so the ex ante expected utility of an agent in t=0 is u(C1)+(1)u(C2) where the utility
function is u(C)=ln(C). For all questions state any assumptions you are making in deriving your
answers. (a) If =21 what is the expected utility of an optimal bank contract if the possibility of a
bank run, S, is zero i.e. if it is known with certainty that there won't be a bank run? (b) Suppose
that the probability of bank run is S=101, but all banks still offer the optimal contract as if there
was no possibility of a bank run as in part (a). In the event of a bank run all a bank's assets are
distributed equally between depositors. Now what is the expected utility of an agent in period 0
who deposits his/her endowment in a bank? (c) Suppose that the government implemented a
weak narrow banking regulation that each bank must ensure that reserves and the liquidation of
long term assets cover all potential withdrawals i.e. that 1x+xL>C1 where x denotes the per
agent investment in the illiquid asset. What is the expected utility of an optimal bank contract
under these regulations?