Consider an exchange economy with two people, A and B, and two goods, 1 and 2. There are ten units of each good in the economy. In the initial allocation, person A has 4 units of good 1 and 5 units of good 2; person B has 6 units of good 1 and 5 units of good 2. These consumers can trade according to the prices of the two goods, p1 and p2. That is, each agent decides how many units of each good to supply and demand (trade), given the prices.
(a) Suppose person A's utility function is UA = x1x2, where xi is A's consumption of good i ∈ {1, 2}. Compute A's preferred final allocation (the allocation that A wants to trade to) as a function of the prices.
(b) Suppose person B's utility function is UB = y1 +y2 where yi is B's consumption of good i ∈ {1, 2}. Using your answer to part (a), what is the equilibrium price ratio and the equilibrium final allocation?