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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 1
5x p,w Sw 0. 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...
Info Lsi
2.F.I3 First printing errata In the last part of condition of b , the inequality p-x gt w should be p'-x gt w Also, in the last part of c t the relation x' p,w should be x' p,w J a We say that a Walrasian demand correspondence satisfies the weak axiom if the following condition is satisfied For any p,w and p w' , if x e xip.w , x' e x p w' , p' x w and p x' w, then xJ lt e jdp.w . Or equivaiently, for any p,w and p' 1 , if x e x p,w , x' xtp w' , p-x' w, and x' e x p,w , then p'-x gt w b If x e...
Info Qpb
if e gt 0 is sufficiently small. Then S p,w is not negative semidefimte and hence the demand function in Exercise 2.E.1 does not satisfy the Weak Axiom. 2.F,11 By Proposition 2.F.3, S p,w p 0 and hence s p,w - p p2 snlp.wl Also p S p,w 0 and hence s Cp.w - pj p s p,w . We saw this in the answer for Exercise 2.F.9 as well. Thus s p,w -s2I P w . 2-F.12 By applying Proposition LD.l to the Walrasian choice structure, we know that x p,w satisfies the weak axiom in the sense of Definition LC.L By...
Info Gor
3.D.7 a Since p X lt w and x the weak axiom implies p -xx gt w . Thus x has to be on the bold line in the following figure. In the following four question, we assume the given preference can be a differentiate utility function u . b If the preference is quasilinear with respect to the first good, then we can take a utility function u so that du x dx 1 for every x Exercise 3.C.5 b . Hence the first-order condition implies ouCx Sx - p2 -Di ' r i 0.0 1,1 , t 0 1 1 , eacn t 0,1. Since P 1 lt anc IS...
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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...
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lt EVlp ,p w . Also, by the auasilinearity, CV p ,p2,w 2 K p ,p2, v Exercise 3.L5 . Hence CV p ,pltv lt CV p ,p2,w . It is worthwhile to remark that, although EVlp0,pl,w l' p ,p2,w , we can obtain the strict reverse inequality Y p ,p w gt EVlp0 2,w f while preserving CV p0,p w lt CF p ,p2,w , by decreasing p2 only slightly. 3.1.5 According to Exercise 3.E.7, we can write the expenditure function e p2, ,pL u - elp ,- ,pL u - e p2 gt 1pL u - e p2, ,pL u 3,1.6 Let u v p0,w . If E.CTUp p wJ 0,...
S 1
19.C.3 Write x. x, x_ e R and define x. t 71 .x j it .x . 3 . Then, bv the ccncavitv of the Bernoulli utilitv function, x, x Hence bv the strong monotonicity, which implies the local nonsatiation , p-x. p'x But here D-x. - o-x. y d K 7i .x j - y p x . y ii .DX . - y D X , Hence y in .o - d x . 0. s si' s si By dividing both sides of this inequality by p, we obtain 7i . - p p x 2 0. A possible interpretation is thus that the equilibrium consumption x. is biased towards the states s whose...
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x I01 2 - 1 - 21 2 2 . is a core allocation and does not have the equal treatment property. In fact, the quasilinearity implies that, for each feasible allocation x and for each coalition S, if WW 5 Wli ttS iieSW2i 4iS l 2 - 2i S ' then S cannot improve upon x this can be shown by applying the argument in the answer to Exercise 15.B.8 to coalition S. Denote the value on the right-hand side by v S . This is the same notation as in Appendix A by the quasilinearity, the exchange economy give rise...
Info Azb
this implies that p e IR v p,w v is convex for every v and w. c For the linear indifference curves, we have w Pi A, 1 - A A 10,1 if Pl p2 x p,w w pj P2 1J v p,w w p p , As for the limit argument with respect to p. First consider the case with p lt 1 and p 1. Then 6 p tp - 1 - as p - 1. Case L p1 lt p Since P2 Pj gt we ave Thus - . w p lim pI w p p2J 1 im - w p . Since p, p0 lt 1, we have pT p0 Thus 5-1.6 , . p2 lim p w pi p2 I im -1- 0. Thus the CES Walrasian demands converge to the Walrasian...
Info Psu
this implies that p e IR v p,w v is convex for every v and w. c For the linear indifference curves, we have w Pi A, 1 - A A 10,1 if Pl p2 As for the limit argument with respect to p. First consider the case with p lt 1 and p 1. Then 6 p Cp - 1 - as p - 1. Since p2 pj gt we ave Thus - . w p lim pI w p p j 1 im - w p . Since p, p0 lt 1, we have p, p0 Thus 5-1.6 , . p2 lim p w pi p2 I im -1- 0. Thus the CES Walrasian demands converge to the Walrasian demand of the linear indifference curves. As...
Lexicographic Indifference Curve
relation that is not monotone in R . 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...
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3.G.I2 Note first that, according to Exercise 3.G.1I, the wealth expansion path is linear in the direction of V b p and intercept - l b p V a p . If the underlying preference is homothetic, then - l b p V a p 0. Hence a p must be a constant function. If the underlying utility function is homogeneous of degree one in w, then v p,w must be homogeneous of degree on in w by Exercise 3.D.3 a , Hence alp 0 for every p gt gt 0. If the preference is quasilinear in good 1, then please first go back to...
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gradient vectors b. of u. with strictly positive coefficients L - 1 2 . L - 1 2 q Hence the upper contour set y. iR u. yj 2 u. x. gt is strictly supported by p at x . The utility maximization condition is thus satisfied at x. under p, implying that zAp x. - 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 contour set y. e rS u....
Info Dol
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...