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

Economics 2010a Fa2003 Edward L. Glaeser Lecture 3

Economics 2010a Fall 2003 Edward L. Glaeser Lecture 3

3. Comparative Statics a. Indirect Utility Functions b. Expenditure Functions and duality C. Expenditure Function and Price Indices d. Slutsky via Utility Functions e. Slutsky via Preferences

3. Comparative Statics a. Indirect Utility Functions b. Expenditure Functions and Duality c. Expenditure Function and Price Indices d. Slutsky via Utility Functions e. Slutsky via Preferences

f. Composite Commodity Theorem g. Application: Labor Supply

f. Composite Commodity Theorem g. Application: Labor Supply

Indirect Utility Functions ndirect Utility functions represent the level of utility as a function of prices and wages and we write v(p, w) It is useful many times, to have utility solely as a function of "exogenous" parameters Define the indirect utility function as v(p, w)=u(x(p, w)) where x(p, w) is the Marshallian demand function which solves the consumers problem to maximize u(x)subject to ≥p·x

Indirect Utility Functions Indirect Utility functions represent the level of utility as a function of prices and wages, and we write vp,w. It is useful many times, to have utility solely as a function of "exogenous" parameters. Define the indirect utility function as vp,w  uxp,w where xp,w is the Marshallian demand function, which solves the consumers problem to maximize ux subject to w  p  x

As an aside, remember that utility or indirect utility units have no meaning. The same preferences are u(x)andf(u(x))if f() is a strictly monotonic function

As an aside, remember that utility or indirect utility units have no meaning. The same preferences are ux and fux if f. is a strictly monotonic function

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