ECO359: Assignment 3w
1. Consider the following generalization of the Diamond-Dybvig model discussed in class. Suppose that the consumers’ utility is given by
for the parameter η > 1. This utility function is commonly used in economics and is called CRRA (stands for constant relative risk aversion). Note that U (c1 , c2 ) is negative (which shouldn’t confuse you) and is increasing in both c1 and c2 (which ultimately the property of the utility function that we want and like). Suppose that l = 0 and 1 ≤ r2 < R. The rest of the model is as in the lecture.
(a) Suppose, as in the lecture, the bank collects money from consumers in the form of deposits and invests in the short-term technology and in the long-term technology. Hence, Find that maximizes the consumers’ expected utility, i.e., solves
Find corresponding . Hint: write the first–order condition and solve for the optimal . It's pretty straightforward, just be careful in derivations.
(b) Prove that 1 < < < R. Hint: use the fact that η > 1 and R > 1.
(c) Parameter η captures the risk-aversion of consumers with higher η corresponding to a higher degree of risk-aversion. There is a general consumption-smoothing principle that roughly states that risk-averse consumers want to smoothen their consumption across different states (in this model, “patient” and “impatient” states). For λ = 1/2 and R = 1.5 plot numerically and for a range of η’s greater than one. Does your plot confirm the consumption-smoothing principle or contradicts it.
(d) Find the limits of and as η → ∞ . How does your answer relate to the consumption-smoothing principle.
2. Consider the following variation of the Diamond-Dybvig model with l = r2 = 1 < R. The bank’s liabilities consist of a mass γ of consumers’ deposits and a mass 1 - γ of equity capital. Consumers have CRRA preferences with η = 2:
The equity capital is provided by patient equity holders of the bank with utility
Equity holders cannot withdraw their funds from the bank at t = 1. All agents have one unit of endowment at t = 0 that they contribute to the bank as a deposit or equity capital, respectively. The bank invests all funds into the long-term technology. A depositor can withdraw at t = 1 or t = 2. The equity holders get whatever is left over after paying all depositors.
(a) Let ^λ ≥ λ be the fraction of consumers that “run” on the bank at t = 1. Compute the payoff of a patient consumer from withdrawing at t = 1 (as a function of ^λ). Hint1: you need to consider separately two cases γ ≤ 1 and γ > 1. In the former case, the bank always has enough money at t = 1 after liquidating part of its investment to pay all consumers , while in the latter case, the bank might not have sufficient funds to cover all running consumers. Hint2: Note the difference with the lecture: there are at most γ mass of consumers who run on the bank, because 1 - γ of equity holders cannot withdraw at t = 1 .
(b) Compute the payoff of a patient consumer from staying put and withdrawing at t = 2 (as a function of^λ). Hint1: Note that if a fraction ^λ of consumers “runs” at t = 1, then the total mass of agents who run is ^λγ. Hint2: First, compute the amount of investment in the LT technology that remains after paying all consumers who “run” at t = 1 . Then, compute the return on this investment and divide it by the mass of patient consumers who do not run. If this quantity, call it z, is greater than , then patient consumers get simply . Otherwise, they get z.
(c) Consider parameters R = 1.2 and λ = 1/2. Draw the patient consumer’s payoff as a function of ^λ ∈ [λ, 1] from “running” and from “staying put” for two values of γ = 0.95 and γ = 0.85. How many stable equilibria are there in each case? Discuss how bank capitalization is helpful in preventing bank runs. Explain the economic mechanism.
(d) What is the minimal capital requirement (that is, the minimal value of 1 - γ) that guarantees that no bank run occurs in equilibrium. Hint: The previous part should give you a hint. If it doesn’t, try a grid of γ ∈ [0.85, 0.95] and see how the equilibrium changes.
(e) Consider parameters R = 1.2, λ = 1/2, and γ = 0.95. Suppose there is a risk of a bank run and the bank can raise additional equity from patient investors. How much equity would you raise to prevent the bank run? Would patient investors be willing to invest in the bank’s equity? Would patient investors be willing to invest in the bank’s equity if they do not believe that the bank will be able to raise sufficient equity capital to prevent the bank run? Hint: To solve the last two questions, you need to compare the payoff of new equity investors from investing in the equity and from keeping their money and investing them in the short–term technology.