"gradient vectors" b. of u.( •} with strictly positive coefficients l

(L - 1/2) .(L - 1/2} q^.. Hence the upper contour set {y. € iR^:

u.(yj 2: u.(x.)> is strictly supported by p at x^. The utility maximization condition is thus satisfied at x. under p, implying that zAp) = x. - =

i T I i £ £ i p.(a ) . Moreover, since b, ■ (x. + e } - b. -x. - b. *e = 1 for all £, the i 11 ill wealth expansion path of consumer i under p must be a straight line parallel to e (locally around x.). Hence D x.(ptp-u.) - (l/p.)e . Since the upper i contour set (y. e [rS u.(y.),£ u.(xj) is strictly supported bv p at x., i + i- i, i' i ' r I

S.{p,p-w.) = 0. Hence Dz.(p) - - D x.(p,pmu.)zAp) = ea. 1.1 i w. i r r i i r

17.E.4 Here is the offer curve of a function zM that is continuous, homogeneous of degree zero, satisfies Walras* law, and cannot be generated from a rational preference. The last property follows from the fact that the weak axiom of revealed preference is not satisfied at p and p\ On the other hand, it is easy to check that if the offer curve does not go through the initial endowment point as in Figure 17,E.2, then the weak axiom always holds.

Figure I7JB.4

17.E,5 We shall prove the assertion by contradiction. Suppose that there is an economy that generates the given function z(-), whose initial endowment w satisfies 0 £ w ^ 1 and 0 ^ ^ 1, and whose consumers have the consumption 2

sets R . Let x.(-) be the demand function of consumer i, z.(0 be his excess l

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