Consider a world with N consumers, with N equally probable states, and with one good, money. The consumers all have identical von Neumann-Morgenstern utility functions, with the utility for $x being ln(x). In staten, for n = 1, . . . , N , the endowment of consumer n is $0, whereas the endowment of every other consumer is $1.
(a) Compute the Arrow-Debreu equilibrium.
(b) In the equilibrium, what insurance premium do consumers pay when they have an endowment of $1? What insurance bene?t do they receive when their endowment is $0?
Suppose that in this world, a public agency can transfer wealth from those with money to the person without any. There is leakage, however, in that if $x is taken from a wealthy consumer, only $x/2 is given to the needy one. Suppose that the social objective is to maximize the expected utility of a typical consumer.
(c) If there are complete Arrow-Debreu insurance markets, what is the socially optimal transfer by the public agency?
(d) If there is no insurance, what is the socially optimal transfer by the public agency?
Suppose now that it is possible for the government agency to determine and to announce, before the trade in insurance contracts occurs, who is going to have an endowment of $0.
(e) What would be the Arrow-Debreu equilibrium if the announce- ment were made?
(f) Would the expected utility of consumers be improved by the announcement, where the expectation is calculated from the point of view of a moment before the announcement was made?