Consider a pure-exchange economy with two goods {1, 2} and two consumers {A, B}. A and B wish to trade with one another to maximize their individual utilities. Suppose A is endowed with 1 unit of good 1 and half a unit of good 2, i.e., A = (1, 1 2 ). B is endowed with a half unit of good 1 and 1 unit of good 2, i.e., B = ( 1 2 , 1). In addition, suppose their utility functions are given by uA(x A) = (x A 1 )(x A 2 ) uB(x B) = (x B 1 )^1/5 (x B 2 )^4/5

1. Suppose the government decides that the competitive equilibrium is not a good allocation and they would prefer for A and B to consume respectively ( 5/4 , 5/6 ) and ( 1/4 , 2/3 ). Is this a competitive equilibrium for some endowments? Why or why not? Is it attainable from the initial endowments?
2. Suppose that the government can announce that p1 = 1 and p2 = 3. Would this achieve the government's desired allocations?
3. Could the government achieve its objective through lump sum redistribution (i.e., a change of the initial endowments such that there are still 3/2 units of each good in this economy)? If so, how could it redistribute to achieve its desired allocation?