1.  Consider an economy that is populated by two types of consumers.  The first has
preferences over a private good, x, and a non-excludable public good, G, that are given
by:
u = log x + log G.
The second group of consumers has similar preferences given by
u = log x + βlog G
for 0 =< β < 1. The argument G in both types’ utility function is G = Σi gi  where the sum
is over all consumers of both types.  Assume that there are N consumers of each type in
the economy.  Type 1 consumers have an initial endowment of one unit of the private
good, while type 2 consumers have an initial endowment of 0.5 units of the private good.
One unit of the private good can be costlessly transformed into one unit of the public
good (and vice versa),

a) Find the decision rule that each type 1 and each type 2 consumer will use in trying to
decide how much to spend on the public good.  In a symmetric Nash equilibrium in
which all type 1 consumers contribute the same amount to the public good, and all type 2
consumers contribute the same amount to the public good, how much will the economy
spend on public goods?
b) Now assume that there is a social planner who has access to a poll tax (a lump sum tax
that is levied at the same rate on everyone in the economy).  The revenue from the tax
can be used to fund spending on the public good.   If the social planner seeks to maximize
the sum of utilities in the economy, find the optimal level of the poll tax.  How does the
level of the public good compare with the level in the private provision equilibrium of
(a)?