高级微观经济理论_英文参考课件-lecture5.pdf

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1、Chapter 2Topics in Consumer Theory I.Duality II.Integrability III.Revealed preference IV.UncertaintyConsumer Theory(5)(c)by Foxit Software Company,2004 Foxit PDF Editor-I.DualityConsumers Problem:UMP or EMP.Duality between direct and indirect utility functions.We know:u(x)v(p,m).Now we learn v(p,m)u

2、(x).Consumer Theory(5)(c)by Foxit Software Company,2004 Foxit PDF Editor-1.Normalized indirect utility functionThe normalized indirect utility function isv(p)=maxxu(x)s.t.px=1Consumer Theory(5)Cuihong FAN3/?2.Solving for utility functionTheorem 1.Given the indirect function v(p),we can find the dire

3、ctutility function by solvingu(x)=minpv(p)s.t.px=1Proof?Consumer Theory(5)Cuihong FAN4/?ExampleExample 1.Suppose indirect utility function is v(p1,p2)=alnp1blnp2.What is the corresponding utility function?Consumer Theory(5)(c)by Foxit Software Company,2004 Foxit PDF Editor-II.IntegrabilityTwo ways o

4、f modeling consumer theory:Preference-based consumer theory Choice-based consumer theoryConsumer Theory(5)(c)by Foxit Software Company,2004 Foxit PDF Editor-1.The definition of IntegrabilityDefinition 1.IntegrabilityRecovering a consumers utility functionfrom his demand function.Why is it called Int

5、egrability?How to derive u(x)from x(p,m)?Why is the problem of Integrability of interest?Consumer Theory(5)(c)by Foxit Software Company,2004 Foxit PDF Editor-2.x(p,m)v(p,m)By Roys identityxi(p,m)=v(p,m)piv(p,m)m,i=1,.,k.How to find v(p,m)?Consumer Theory(5)(c)by Foxit Software Company,2004 Foxit PDF

6、 Editor-3.x(p,m)e(p,u)By shephards lemma,given utility u0and prices p,e(p,u0)pi=xhi(p,u0)=xi(p,e(p,u0)e(p0,u0)=p0 xh(p0,u0)initial condition.Is there a e(p,u)that solves the partial differential equations?Consumer Theory(5)(c)by Foxit Software Company,2004 Foxit PDF Editor-Integrability and Slutsky

7、matrixRemark 1.Slutsky matrix is symmetric and negative semidefinite isnecessary to solve the integrability problem.Why?Consumer Theory(5)(c)by Foxit Software Company,2004 Foxit PDF Editor-A necessary condition for the existence of a solution(1).A necessary condition for the existence of a solution

8、is that Sluts-ky matrix is symmetric.Theorem 2.Frobenius theoremA system of partial differentialequations of the formf(p)pi=gi(p),i=1,2,.,k.has a solution if and only ifgi(p)pj=gj(p)pii,j.How to apply Frobenius theorem?Consumer Theory(5)(c)by Foxit Software Company,2004 Foxit PDF Editor-Slutsky matr

9、ix must be negative semidefinite To assure that e(p,u)is an expenditure function,Slutsky matrixmust be negative semidefinite.Why?Consumer Theory(5)(c)by Foxit Software Company,2004 Foxit PDF Editor-Example:IntegrabilityExample 2.Given u and that xi(p,m)=impi,i=1,2,how to findu(x)?Consumer Theory(5)(

10、c)by Foxit Software Company,2004 Foxit PDF Editor-III.Revealed PreferencesSuppose we have some observations on a consumers behavior:(pt,xt)for t=1,2,.,T.We make some assumptions on preferences preferences do not change while we observe the behavior.are strictly convex are strictly monotone.Q:What ki

11、nd of choice behavior is rational?A:Consumer Theory(5)(c)by Foxit Software Company,2004 Foxit PDF Editor-1.Weak Axiom of Revealed Preferences(WARP)Definition 2.WARPA consumers choice behavior satisfies WARP iffor every distinct pair of bundles x0and x1,with x0chosen at pricesp0and x1chosen at prices

12、 p1,thenp0 x1 p0 x0 p1x0 p1x1.If the bundle x0is directly revealed preferred to the bundle x1then x1can never be directly revealed preferred to x0.Interpretation:Consumer Theory(5)(c)by Foxit Software Company,2004 Foxit PDF Editor-Checking WARP:examples 1 and 2Consumer Theory(5)(c)by Foxit Software

13、Company,2004 Foxit PDF Editor-Checking WARP:examples 3 and 4Consumer Theory(5)(c)by Foxit Software Company,2004 Foxit PDF Editor-Rule for checking WARPConsumer Theory(5)(c)by Foxit Software Company,2004 Foxit PDF Editor-ExampleExample 3.Suppose a consumers behavior is observed as follows:periodsp1p2

14、x1x21121222121Does his choice behavior satisfy WARP?Consumer Theory(5)(c)by Foxit Software Company,2004 Foxit PDF Editor-Another exampleExample 4.p1=(2,1,2),x1=(1,2,2)p2=(2,2,1),x2=(2,1,2)p3=(1,2,2),x3=(2,2,1)Show that WARP is satisfied but preferences are not rational.Consumer Theory(5)(c)by Foxit

15、Software Company,2004 Foxit PDF Editor-2.Relationship between WARP and traditionalconsumer theoryTheorem 3.WARP is equivalent to that Slutsky matrix is negative se-midefinite.This is proved in two steps:WARP is satisfied iffthe(Slutsky)compensated law of demand holds If the Slutsky compensated law o

16、f demand holds,then Slutsky matrixis negative semidefinite.Consumer Theory(5)(c)by Foxit Software Company,2004 Foxit PDF Editor-2.1 WARP and the(Slutsky)compensated law of demandDefinition 3.Slutsky income compensationAs prices change,howto change income to keep a consumers purchasing power constant

17、.Lemma 1.The Marshiallian demand function x(p,m)satisfies WARPif and only if the compensated law of demand holds.That is,for anycompensated price change from(p,m)to(p0,m0)=(p0,p0 x(p,m),we have(p0 p)x(p0,m0)x(p,m)0.Proof:Consumer Theory(5)(c)by Foxit Software Company,2004 Foxit PDF Editor-2.2 The co

18、mpensated law of demand and Slutsky matrixLemma 2.If the compensated law of demand holds,then Slutsky ma-trix is negative semidefinite.Proof:Rewrite the compensated law of demand as dpdx 0.dx=x(p,m)pdp+x(p,m)mdm=.Therefore,dpdx=dpx(p,m)p+x(p,m)mx(p,m)dp 0.Consumer Theory(5)(c)by Foxit Software Compa

19、ny,2004 Foxit PDF Editor-Weakness of WARP(1).(2).To obtain symmetry of Slutsky matrix and to assure that preferencesare rational,we need a stronger axiomSARP.Consumer Theory(5)(c)by Foxit Software Company,2004 Foxit PDF Editor-3.Strong Axiom of Revealed Preferences(SARP)Definition 4.SARPIf for every

20、 sequence of distinct consumptionbundles x0,x1,.,xk,where x0is revealed preferred to x1,x1is re-vealed preferred to x2,.,and xk1is revealed preferred to xk,it isnot the case that xkis revealed preferred to x0.Consumer Theory(5)(c)by Foxit Software Company,2004 Foxit PDF Editor-3.1 SARP and the traditional consumer theorySARP a preferences are rational.Slutsky matrix is symmetric and negative semidefinite.Proof:(See Houtthakker(1950)and Richter(1966).)Consumer Theory(5)(c)by Foxit Software Company,2004 Foxit PDF Editor-