This question is about an Edgeworth Box economy (two goods, two consumers, and no firms) in which goods are not perfectly divisible. In particular, assume that goods can only be purchased in integer amounts. Formally, this means the set of feasible consumption vectors for consumer i is given by X; = {(zi1,%i2) : w;; N,z,5 N} where N denotes non-negative integers, and = A, B. Assume that consumer 4's initial endowment e; X,. Finally, assume that consumers' utility functions are linear and strictly increasing in both goods; that is u; () = Y;in + 0:%i 2 where v, > 0 and ; > 0. Assume that 4 # 72, A B 1. Write the conditions for (2%, z};, p*) to be an competitive equilibrium in this economy as well as the conditions for (#4,Zp) to be a Pareto optimal allocation. 2. Is it true that every Pareto-efficient allocation must be on the boundary of the Edgeworth Box? Explain your answer and illustrate with a diagram. 3. Assume that Z: B ;'g. Using a diagram, argue that the allocation where consumer A receives all of good 1 available and no good 2, while consumer B receives all good 2 available and no good 1 is Pareto optimal and (part of) a competitive equilibrium. 4. Argue that the economy where (v,4,d4) = (2,1), e4 = (1,1), (v5,d5) = (2,3), and e4 = (9,1) does not have a competitive equilibrium. 5. Does the First Welfare Theorem hold in the economy where (v4,04) = (1,1), ex = (2,1), (vp,08) = (1 +&,1) (where > 0 and small), and e4 = (1, 2)? 6. Now supposed good 1 is indivisible but good 2 is perfectly divisible as in the standard case. Formally, this is X; = {(z,1,%i2) : ;1 N,z;5 Ry }. Does the First Welfare Theorem hold in this economy? Provide a proof or a counterexample