Random withdrawal rate:-

Consider a three-date Diamond-Dybvig economy (t = 0, 1, 2). Consumers are ex ante identical; they save 1 at date 0. At date 1, consumers learn their preferences. A fraction λ has utility u(c₁) and a fraction (1 - λ) has utility u(c₂). At date 0, the consumers put their savings in a bank. They later cannot withdraw and invest in financial markets, so the Jacklin critique does not apply. That is, incentive compatibility issues are ignored in this exercise (a patient depositor cannot masquerade as an impatient one).

The bank invests the per-depositor savings into short- and long-term projects: i1+i2 = 1. The long-term technology yields (per unit of investment) R > 1 at date 2, but only l ₁ = r₂ = 1).

(i) • Show that the optimal allocation (c₁, c₂) satisfies

(ii) Suppose now that there is macroeconomic uncertainty, in that λ is unknown: λ = λ_L with probability β and λ = λ_Hwith probability 1 - β, where 0 _L H ω and z_ω denote the fraction of short-term investment that is not rolled over, and the fraction of long-term investment that is liquidated, respectively, in state of nature ω ∈ {L, H}). What does the solution look like for l = 0 and l close to 1? (Showoffs: characterize the solution for a general l!)