哈佛大学：《中级微观经济学》（英文版）Lecture 3 Comparative Statics

Properties of the indirect utility function MWG Proposition 3. D. 3: Suppose that u(is a continuous utility function representing a locally non-satiated preference relation z defined on the consumption set R+. The indirect utility function v(p w)IS (a homogeneous of degree zero (b) strictly increasing in w and non-Increasing In p, (c)quasiconvex; that is, the set ip, w): v(p, w)<y is convex for any 2, and (d)continuous in p and w

Properties of the indirect utility function. MWG Proposition 3.D.3: Suppose that u. is a continuous utility function representing a locally non-satiated preference relation  defined on the consumption set  L. The indirect utility function vp,w is: (a) homogeneous of degree zero, (b) strictly increasing in w and non-increasing in p, (c) quasiconvex; that is, the set p,w : vp,w  v is convex for any v, and (d) continuous in p and w

We have already proven that Marshallian demand is homogeneous of degree zero, this implies that indirect utility is homogeneous of degree zero Nonincreasing in p follows from the fact that if p falls you can always buy the old bundle and thus be no worse off Increasing in w uses that fact plus local non-satiation with an increase in w you can always buy the old bundle plus a little bit more of the good that you are not satiated with Continuity I leave up to you

We have already proven that Marshallian demand is homogeneous of degree zero, this implies that indirect utility is homogeneous of degree zero. Nonincreasing in p follows from the fact that if p falls you can always buy the old bundle, and thus be no worse off. Increasing in w uses that fact plus local non-satiation: with an increase in w you can always buy the old bundle plus a little bit more of the good that you are not satiated with. Continuity I leave up to you

To show quasi-convexity, assume that v(p,w)≤ v and v(p’,w′)≤y. For any a E [O, 1] consider the price wealth pair )=(ap+(1-a)p,a+(1-a)n) Assume that v(p, w)is not quasi convex L e. there exists an x, such that ap·x+(1-a)p·x≤1+(1-a) but u(x)>y If u(x)>y, then x must not have been affordable at the old budget sets (otherwise it would have been chosen and would have yielded higher utility), which implies that p·x> w and p'°x>w

To show quasi-convexity, assume that vp,w  v and vp ,w   v . For any   0, 1 consider the price wealth pair p ,w  p  1  p ,w  1  w  Assume that vp,w is not quasi convex, i.e. there exists an x, such that p  x  1  p  x  w  1  w , but ux  v. If ux  v, then x must not have been affordable at the old budget sets (otherwise it would have been chosen and would have yielded higher utility), which implies that p  x  w and p  x  w

But these together imply that ap·x+(1-a)·x>a+(1-a) Which is a contradiction

But these together imply that p  x  1  p  x  w  1  w which is a contradiction