Reference no: EM13887981

There are I consumers and one good, which may be used for public or private consumption. Each consumer is endowed with one unit of the good and divides this one unit between public and private consumption. That is, each consumer i, for i = 1, . . . , I , chooses  xi , where 0 ≤ xi ≤ 1, and the total amount available for public consumption is g= x1 + x2 + . . . + xI . For all i, the utility function of consumer n is ui(1 - xi , g), where ui is continuous, strictly increasing, and strictly concave. In equilibrium, each consumer knows the choice of xi for all other consumers and chooses his personal consumption and contribution to public consumption so as to maximize his or her own utility.

(a) Describe the equilibrium formally.

(b) Prove that an equilibrium exists.

(c) Either prove that the equilibrium is Pareto optimal or show that it is not by means of a counter example.