4. Consider an economy with two goods - xylitol (X) and yogurt (Y) -and two agents -Addy and Bill. Addy and Bill wish to trade with one another in order to maximize their individual utilities. We will consider how their trading decisions depend on the initial endowments of xylotol and Y and on their utility functions. Suppose Addy is endowed with one unit of xylitol and half a unit of yogurt and Bill is endowed with 1 unit of xylitol and 1.5 units of yogurt. Additionally, suppose their utility functions are given by:

UAddy(X, Y ) = XY

UBill(X, Y ) = Y + 2X

a) Find the set of Pareto Optimal Allocations in this economy and depict these in an Edgeworth box. What is this set of points called?

b) Find the equilibrium consumption of xylitol and yogurt by Addy and Bill in this economy and determine the price ratio that supports this equilibrium. Will this equilibrium allocation be Pareto Efficient?

c) Show that there are gains from trade in this setting. From what endowment point would there not be gains from trade?

d) Go back to the original preferences of the problem. Suppose the government decides that the competitive equilibrium is not a good allocation and they would prefer for Addy to consume(3/4, 3/2) and Bill to consume(5/4, 1/2) . Is this an attainable competitive equilibrium from the initial equilibrium?