Consider an exchange economy (ui,ei)i€I in which each ui is continuous and quasiconcave on Rn+. Suppose that Xbar=(x1bar, x2bar,...,xIbar)>>0 is pareto efficient, that each ui is differentiable at Xibar, and that gradient(ui(Xibar)>>0. follow the steps below to derive another version of the second welfare theorem:

a) show that for any two consumers i and j, the gradient vectors gradient(ui(xibar) and gradient (uj(xjbar) must be proportional. that is, there must exist some a>0 (which may depend on i and j) such that gradient(ui(xibar)=a*gradient (uj(xjbar). interpret this condition in the case of the edgeworth box economy.

b) define pbar=gradient(u1(X1bar)>>0. show that for every consumer i, there exists a lambdai>0 such that gradient(ui(xibar)=lamdai*pbar.

c) use theorem 1.4 to argue that for every consumer i,xibar solves

max ui(xi) s.t. pbar*xi <= pbar*xibar

theorem 1.4: sufficiency of consumers first order conditions under the conditions of the beginning of this problem

a) show that for any two consumers i and j, the gradient vectors gradient(ui(xibar) and gradient (uj(xjbar) must be proportional. that is, there must exist some a>0 (which may depend on i and j) such that gradient(ui(xibar)=a*gradient (uj(xjbar). interpret this condition in the case of the edgeworth box economy.

b) define pbar=gradient(u1(X1bar)>>0. show that for every consumer i, there exists a lambdai>0 such that gradient(ui(xibar)=lamdai*pbar.

c) use theorem 1.4 to argue that for every consumer i,xibar solves

max ui(xi) s.t. pbar*xi <= pbar*xibar

theorem 1.4: sufficiency of consumers first order conditions under the conditions of the beginning of this problem

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