Suppose we have a 2 person economy, with endowments (w/1,1, w1,2) and (w/2,1, /2,2). You may assume utility functions are monotone and quasi concave. Prove the following two claims: . Given a number a E R, if (71,1, 21,2) = arg max ful (1,1, $1,2) : u2(12,1, 12,2) 2 1, T1,1 + 12,1 = W1,1 + W/2,1, 21,2 + 12,2 = W1,2 + w/2,2} then ($1,1, 21,2, 12,1, 12,2) is Pareto efficient. In words, if an allocation maximizes the utility of person 1 subject to the feasibility constraint and subject to person 2 getting a minimum utility u, then such an allocation is P.E. . If ($1,1, 21,2, 12,1, 12,2) is a Pareto efficient allocation, then there is a value u such that (21,1, 21,2, 12,1, 12,2) solves the problem in the bullet point above. HINT: For the first bullet point, proceed by contradiction: suppose a Pareto im- provement exists. Does the Pareto improvement satisfy the utility constraint? If yes, what do we know about the utility person 1 gets from the Pareto improvement? Is this compatible with the claim that (21,1, 21,2) maximizes the utility of person 1 subject to all the constraints? HINT: For a you may use the following conjecture: u = 12(12,1, 12,2). These two claims put together prove that the set of Pareto efficient allocations may be calculated as the set of allocations that maximize the utility of one person subject to two constraints: the other person receives a minimal utility level a, and the feasibility constraint. The first bullet point says that any solution to such a problem is PE, the second bullet point says that we are not "missing out" on any PE allocations. The role of i is that each PE allocation corresponds to a different u value