Consider the Allen and Gale model of contagion, where the storage asset allows the costless transfer of one unit from on

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Consider the Allen and Gale model of contagion, where the
storage asset allows the costless transfer of one unit from one
time period to another and the illiquid asset pays out R = 2 in
period 2 but only 1 4 if it is liquidated early in period 1. There
are four regions A, B, C and D, where in each region there is a
continuum of investors measure 1 with a Prob. ω of being early
consumers who only to consume in period 1, C1 and Prob. 1 − ω of
being late consumers who only consume in period 2, C2 and so the
expected utility function is given by ω ln(c1) + (1 − ω) ln(c2)
Suppose there are two states of the world, S1 and S2, that occur
with equal probability, where the pattern of regional liquidity
shocks in each state of the world is described by the following
table
Table of Regional Liquidity Shocks
A
B

C
D
S1 ωH = 3
/10 ωL = 1/10
ωH = 3/10 ωL =
1/10
S2 ωL =
1/10 ωH = 3/10
ωL = 1/10
ωH = 3/10
For all questions state any assumptions you are making in
deriving your answers.
(a) What is the optimal bank contract for agents in this
economy? What is the level of investment by Banks in the illiquid
asset, I, under this contract?
(b) What is the level of interbank deposits that will allow
banks to offer the optimal contract with no shortage of
liquidity.
(c) Consider the complete markets interbank network structure
where banks hold deposits from both regions with negatively
correlated liquidity shocks, so that e.g. banks in region A hold
deposits from banks in region B and region D.Describe the asset
flows between regions A and B in the case of shock S1
(d) Considering the complete markets interbank network structure
of part c). Analyze the asset flows between regions in the case of
a perturbation state S˜, where liquidity demand in period 1 is 0.2
in regions B, C and D and where liquidity demand is 0.2 + ε, in
region A where ε = 0.1. Are there bank runs in any region?
(e) Now suppose that the illiquid asset pays out R = 2 in period
2 but only 1 8 if it is liquidated early in period 1. How does the
effect your answer to part d)?