The next question differs in one important respect from the example in the text. In the example, the economy was endowed with units of public good which could be converted into a private good. In this question, the economy is endowed with private good which can be converted into a public good. How does this difference alter the equations that characterize the solutions?
Consider an economy composed of two people, George and Harriet, who have the utility functions

Here, z is the quantity of public goods, and bG and bH are, respectively, George's and Harriet's consumption of bread. The economy is endowed with 120 loaves of bread, and 20 loaves of bread are needed to produce each unit of public good.

a) Find the Pareto optimal allocation in which George and Harriet have the same utility.

b) Suppose that 70 loaves of the economy's endowment are initially held by George, and that 50 loaves are initially held by Harriet. Find the allocation attained when the public good is funded through voluntary contributions.