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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