Consider an economy consisting of two people. The utility of person i (where i is either 1 or 2) is
where s is the quantity of public good, and ci is person i's contribution to the provision of the public good (measured in private goods). In this economy, one unit of private goods can always be transformed into one unit of public goods, so s must satisfy the "adding-up condition":
a) Define the marginal rate of substitution as the amount of private goods which exactly compensates an individual for the loss of one unit of public good. Then
Find each person's marginal rate of substitution.
b) Find the Samuelson condition for this economy.
c) An efficient allocation in this economy is a triplet (s,c1,c2) which satisfies both the adding-up condition and the Samuelson condition. Show that there are two efficient allocations associated with every s greater than 4.
d) Show that, if s is greater than 4, the two people's contributions are not equal. Show also that the contributions made by the agents become more inequitable as s rises.