Question

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=1/2​ if it is liquidated early in period 1 . At t=1 a fraction ω people learn that they only care about consumption at t=1, while a fraction (1−ω) 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 ωu(C₁​)+(1−ω)u(C₂​) where the utility function is u(C)=ln(C). For all questions state any assumptions you are making in deriving your answers. If ω=21​ what is the expected utility of an optimal bank contract if the possibility of a bank run, πˢ, is zero i.e. if it is known with certainty that there won't be a bank run?

Answer 1

In this model, agents with different preferences at different time periods consider the expected utility of an optimal bank contract.

Step-by-step explanation:

In this model, agents have an endowment of $1 at t=0 and can invest in a storage asset or an illiquid asset. At t=1, a fraction ω of agents care only about consumption at t=1, while a fraction (1-ω) care only about consumption at t=2. The expected utility of an agent at t=0 is ωu(C1)+(1-ω)u(C2), where u(C)=ln(C) is the utility function.

If ω=1/2, and the possibility of a bank run, πʲ, is zero, the expected utility of an optimal bank contract can be calculated using the given utility function and the probabilities. Plug in the values for C1 and C2 to calculate the expected utility.