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Question 3 In this question you will argue that the set of competitive equilibrium prices of a com- petitive economy has essentially no structure other than closedness. Consider a two-commodity world, let prices be normalized to the sphere S = {p ( R4 + l llpll = 1), fix & > 0, and denote Se = {p ES | p, > E and p2 > {}. Fix an arbitrary set E C S, and suppose that it is closed. Define the function Z : S - R? as follows: (i) for commodity 1, Zi(p) = min {llp - pll : p e E); (1) (ii) and for commodity 2, Zz(p) = -"i Zi(p). (2) P2 With this construction: (a) Argue that Z is defined for all pe S. (b) Argue that Z is continuous and satisfies Walras's law.! (c) Argue that there exists an exchange economy , (u', when) where each u' : R. - R is continuous, locally non-satiated and quasi-concave and such that for all p E Se, [ilx'(p) - w'] = Z(p), where x'(p) = argmax, {u'(x) : p . x

d, then D, - being the greedy one - will get nothing, while D, will be rewarded with two times what he asked for, up to $200M; that is, D, will get min (2d,, 200M} . Symmetrically, if d,