(1) Consider the model of financial contagion of Allen and Gale.

a) Explain whether interbank deposits help provide liquidity when there is an aggregate excess demand for liquidity in the system.

b) Explain which financial networks are most susceptible to financial contagion.

(2)  (Past Exam 2016.  Question B.2) Consider the model of financial contagion of Allen and Gale. Specifically, consider an economy with three dates t = 0, 1, 2 and a single, all-purpose good at each date. There are four regions in the economy. In each region there is a continuum of identical banks. In each region there is a continuum of ex-ante identical agents of measure

1. Each agent has an endowment of one unit of the good at date t = 0 and nothing at dates t = 1, 2.  In order to provide for future consumption, each agent deposits his endowment in the representative bank of his region. The bank can invest the deposit in two assets, a short asset and a long asset. The short asset produces one unit of the good at date t + 1 for every unit invested at date t = 0, 1. The long asset produces R = 2 units of the good at date 2 for every unit invested at date 0, and produces the amount r = 0.3 at date 1).

At date 0 each agent is uncertain about his preferences over the timing of consumption. With some probability he expects to be an early consumer, who only values consumption at date 1.  With the complementary probability he expects to be a late consumer, who only values consumption at date 2. Note that he never values consumption at date 0.

The probability of being an early or late consumer depends on the state of nature that occurs. There are two, equally likely, states of nature, denoted by S1  and S2 .  The following table indicates the proportion of early consumers in each region depending on the state of nature (the letters A, B , C , D indicate the four regions).

Note that the average proportion of early (and late) consumers in the entire economy is 0 .5 in either state of nature.

Each agent has preferences represented by a Von Neumann-Morgenstern utility function, with

u(c) = ln c.

a) Determine the efficient solution (optimal risk sharing), that is, the investment in the long and in the short asset and the level of consumption for early and late consumers.  Is this solution incentive compatible?

b) Suppose now there is a market for interbank deposits. Show that in this case the first best can be achieved if the four representative banks are linked in a complete network.