Question

Intermed Microecon Theory

please help.

3.4 Problem 4 Suppose we have a 2 person economy, with endowments (w,u2), where is the endowment of personi. You may assume utility functions are monotone and represent concave preferences. Prove the following two claims: . Given a number ii є R, if (zi.r2) = arg max(m (zi) : "tr') 2 i, 팎 + 2 for each good n then (,2) is pareto efficient. In words, if an allocation amximizes the utiltiy 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 1 Remember: monotonicity says that more goods are always better. Convexity says that for every person i, ifr'-(r, zł) is indifferent to y'. (y, y ), then person i strictly prefers any average between z and y' to either or y 12 If (r,z2) is a pareto efficient allocation, then there is a value u such that 2) solves the problem in the bullet point above. 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 the other receiving a minimal utility level, and subject ot feasibility. The first bulet point says that any solution to such proble is PE, the second bullet point says that we are not "missig out" on any PE allocations