Let us consider an economy lasting for three dates, t=0, 1, 2, and inhabited by two types of utility maximizing consumers. There is a = 0.45 fraction of impatient and a (1 - ) = 0.55 fraction of patient consumers. At the time t=0, all consumers are endowed with 1 unit of consumption good (i.e., the numeraire) and are uncertain about their future liquidity needs. Put differently, until time t=1 consumers do not know their types. Agents' preferences over the consumption are represented by the generic utility function U(c1,c2). There are two types of assets. A short-term asset delivers 1 unit of consumption for each unit invested in the previous period or a long-term asset which delivers R=2.6 units of consumption at time t=2 for each unit invested at time t=0. The fraction of endowment investment in the short-term asset is denoted by , while (1- ) is the fraction invested in the long-term asset.

Question:

Let us assume the presence of financial intermediaries. Consumers' utility over consumption is represented by the following function U(c) = - 1/. Find the optimal investment that the bank makes in the short- and long-term assets.