In a two-person, two-product economy, both products are labeled like X and Y and the two consumers are called A and B. Let XA, for example, denote the consumption amount of good X per person A. Each of the two individuals has initial endowments of one unit of each asset. The preferences of a are given by the utility function

UA(XA,YA) = 1/3 log XA + 2/3 log YA

B's preferences are a little stranger than this. They are given by

UB (XB, YB, XA) = log XB + log YB + log (2-XA)

That is, B obtains "disutility" from A's consumption of the first good.

a) What is the Walrasian equilibrium (or equilibria) for this economy? (This is an economy with externalities). You will find things to be fairly simple if you normalize the prices of the two goods to add up to one. Let p be the price of good x, then 1 - p is the price of good y.

b) Is the equilibrium allocation optimal in part (a) of Pareto? If so, why? (That is, how do you know it is so?) If not, how do you know?