Question

Consider the Diamond-Dybvig model. There are three dates (T=0, 1, 2) and a unit mass of ex-ante identical consumers. Each of the consumers has an endowment of y = 2 to invest at date T = 0. The type of each consumer is revealed at date T = 1. A fraction t = 0.25 are impatient and consume only at T = 1. The remaining fraction are patient and indifferent between consuming at either T = 1 or T = 2. An individual's type is private information. Consumers have two choices how to invest. First, they can invest in the productive technology (illiquid asset) where one invested unit earns a gross return R = 1.5, if invested for two periods. One unit invested for only one period earn a gross return of 1. Second, consumers can store their endowment privately. Endowment in storage will not lose its value over time. Each consumer has the utility function U (c) = ln(c). The discount rate p=0.8. Denote the consumption allocation of as consumption of consumer type t in period T.

a. Set up the optimization problem and find the efficient consumption allocation c]*, cz* to impatient and patient consumers respectively.

Answer 1

The Diamond-Dybvig model optimization problem involves maximizing the utility function U(c) = ln(c) based on the individual consumer's type and investment choices. The efficient consumption allocation is found by considering the intertemporal budget constraint and the different return rates for immediate or later consumption.

Step-by-step explanation:

The Diamond-Dybvig model can be used to determine the optimal consumption allocation for different types of consumers depending on whether they are impatient or patient. In this scenario, individuals have an endowment of y = 2 and can invest in either a productive technology with a return R = 1.5 over two periods or store it privately with no appreciation. The goal is to maximize the utility function U(c) = ln(c) for each consumer based on their type that is revealed at T = 1 and their time preferences.

Impatient consumers, which are a fraction t = 0.25, will consume at T = 1, whereas patient consumers will consume at T = 1 or T = 2. Since the utility is logarithmic and the discount rate is p=0.8, we need to set up the optimization problem considering the intertemporal budget constraint, which indicates how the consumption today and in the future affects utility. The efficient consumption allocation (c1*, c2*) can be found by taking into account the utility maximization subject to the budget constraint.

To approach this optimization problem, we assume that investments in the productive technology will be made unless immediate consumption is needed. This allows us to use the endowment and return figures to calculate the budget sets across both periods and apply the utility function to determine the optimal allocation that accounts for the fraction of impatient and patient consumers and their respective consumption preferences.