F 0 if z

Then F(l 2) p > 0 Gil 2) and JxdF(x) 2(1 - p) > I JxdG(x). Hence F( ) does not first-order stochastically dominate G(0, but the mean of F( ) is larger than that of G(0. 6.D.3 Any elementary increase in risk from a distribution F( -) is a mean-preserving spread of F(0. in Example 6.D.2, we saw that any mean-preserving spread of F( -) is second-order stochastically dominated by F(0. Hence the assertion follows. 6.D.4 Let L (p p p ) and L' - be tW0 *otteries' (a) By a direct calculation, the...

T

Quasiconcave, by the supporting hyperplane theorem Theorem M.G.3 , there exist p R p e R. , w gt '0, and w gt 0 such that x xfp.w and x x p w' . Then uix v p,w and u x' v p w' , Hence vlp.w v pT,w' . Thus, by the homogeneity, v p,aw vlp'.aw' . But as we saw above, x p,aw ax and r p aw gt ax'. Hence v p,aw u ax and v p aw' u ax'L Thus u ax u ax' . Therefore u x is homogeneous of degree one. 3-D.4 a Let e. 1,0 05 e We shall prove that for every p e w Rt a R, ana x - co gt m x R if x x p,w , then...

A A

Note that v-S p,w v 0 for every v e. iRw. Now let v e K3 Note that v v - v p3 p and the third coordinate of v - v p p is equal to zero. So denote its first two coordinates by v IR . Then, by Proposition 3, v-S p,w v v-S p,w v s 0. 2.F.17 a Yes. In fact, x apfaw ccw ap w pp c p,w . b Yes. In fact, p-x p,w IkPkxk P gt w IkPk w- c Suppose that p - gt p,w w' and p-xip' w' w. The first inequality implies that P w C p w that is, w p s w' Q p' . Tne second inequality implies similarly...

L 1112 122 lo2

iii The equilibrium allocation of b is Pareto optima if and oniy if iv The ecuiiibrium allocation of c is Pareto ootima if and onlv if u Thus the information in c and d are socially valuable. The important fact here is that the equilibrium prices are not changed by the introduction of information or contingent commodities, and consumer 1 attains the same utility level at every state. Nothing as in Example 19.H.1 happens in this 20.H.7 First printing errata 1 he utility function of consumer 2...

Lexicographic Indifference Curve

For example, x y but y gt x. 3.C.1 Let v be a lexicographic ordering. To prove the completeness, suppose that we do not have x gt y. Then y 2 x and Xj y 0r y gt Hence either uy gt x ' or y x and y gt Thus y gt x. To prove the transitivity, suppose that x gt y and y gt - z. Then x amp y and y z., Hence x. If x. gt z,, then x gt - z. If x. z then x, - yT z Thus X- 2 y and y_ z Hence z_. Thus x gt z. To show that the strong monotonicity, suppose that x 2 y and...

Info

It follows from this construction that u 0 is continuos at every x 1,1 . The preference gt is convex and monotone. But, whatever the choice of the value of u U is, it cannot be continuous at 1,1 . In fact, 1 - 1 n, 1 - 1 n 1,1 and 1 l n, 1 1 n 1,1 and u 1 n, 1 I n I 1 n 2 3. Hence, if 2 lt u Irl , then 2,0 gt 1 - 1 n, 1 - i n but 1,1 gt - 2,0 if u l,l lt 3, then 1 1 n, 1 1 n gt 2,1 but 2,1 gt - 1,1 , If u l,l 3, then all upper contour sets of gt are closed if u l,l 2, then all lower contour...