高级微观经济理论_英文参考课件-lecture3.pdf

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1、IV Indirect Utility Function v(p,m)DefinitionProperties of indirect utility functionEnvelope TheoremConsumer Theory(3)DefinitionDef.:The indirect utility function is the maximum utility aconsumer can achieve given income and prices.v(p,m)=maxxu(x)s.t.px mv(p,m)=u(x(p,m).Consumer Theory(3)Basic prope

2、rties of indirect utility functionTheorem 1.Properties of v(p,m)If u(x)is continuousand strictly increasing in x,then v(p,m)is(1)continuous on(p,m),(2)homogeneous of degree zero in(p,m),(3)strictly increasing in m and decreasing in p,(4)quasiconvex in(p,m),(5)Roys identity:If x(p,m)is Marshallian de

3、mand func-tion,then xi(p,m)=v(p,m)piv(p,m)m,for i=1,2,.,n.Consumer Theory(3)(1).v(p,m)is continuous on(p,m)Proof(1):By the Theorem of Maximum.Theorem 1.Theorem of Maximum If the objective func-tion and the constraint are continuous,and if the domainis a compact set,then the maximum value function is

4、 conti-nuous.Consumer Theory(3)(2).v(p,m)is homogeneous of degree zero in(p,m)Definition 1.A function f(x1,x2,.,xn)is homogenousof degree r if t 0 we havef(tx1,tx2,.,txn)=trf(x1,x2,.,xn).Proof:(2)Consumer Theory(3)(3).v(p,m)is decreasing in p,strictly increasing in m.We need to prove:if p0 p,then v(

5、p,m)v(p0,m)Proof:By Envelope TheoremConsumer Theory(3)Envelope Theorem Case 1:Unconstrained optimization.Case 2:Constrained optimization.Consumer Theory(3)Envelope Theorem:unconstrained optimizationM(a)maxxf(x,a)M(a):maximum-value functionx:a variablea:a vector of m exogenous parametersx=x(a):the so

6、lution.M(a)=f(x(a),a).(1)Consumer Theory(3)Envelope Theorem:unconstrained optimization(cont.)Theorem 2.The Envelope Theorem 1 Assumethatfand x(a)are differentiable.ThenM(a)aj=faj.Proof:Meaning:Consumer Theory(3)Envelope Theorem:constrained optimizationM(a)=maxxf(x,a)(2)s.t.g(x,a)=0(3)Consider interi

7、or solution,x:a vector of n variablesa:a vector of m exogenous parametersL(x,a,)=f(x,a)+g(x,a).Consumer Theory(3)Envelope Theorem:constrained optimizationTheorem 3.Envelope Theorem 2 Suppose f,g and x(a)are differentiable w.r.t.a,thenM(a)aj=Laj|x(a),(a).Proof:Consumer Theory(3)Proof:(3)v(p,m)is stri

8、ctly increasing in m.Proof:By the first order condition and the Envelope Theo-rem.Consumer Theory(3)Proof:(4)v(p,m)is quasiconvex in(p,m)We need to show 0,1,v(p,m)max?v(p1,m1),v(p2,m2)?,whe-re pand mare convex combinations.LetB1=x|p1x m1B2=x|p2x m2B=x|px mWe only need to show:If x B,then x B1or x B2

9、.Consumer Theory(3)Proof:(5)Roys identityProof:By the Envelope Theorem.Consumer Theory(3)ExampleExample 1.Suppose u(x1,x2)=x1x12,what is the indi-rect utility function?How to prove that the indirect utility function you derived iscorrect?Consumer Theory(3)V.Expenditure Minimization and Expenditure f

10、unctionExpenditure minimization problem(EMP):minx0pxs.t.u(x)u.x 0Definition 2.Expenditure function Given market pricesp,the minimum expenditure that is necessary to achieve agiven level of utility,denoted by e(p,u).Consumer Theory(3)2.Solution to EMP:Hicksian demand xh(p,u)The solution to EMP:“Hicks

11、ian demand function”,xh(p,u).e(p,u)pxh(p,u).Hicksian demand function is also called“Compensated de-mand function”:Varying prices and income so as to keepthe consumer at the initial level of utility.How to derive Hicksian demand curve?Consumer Theory(3)Compensated law of demandTheorem 4.Suppose u is

12、strictly increasing and xh?p,u?is unique for all p.Then for any p0and p00,?p00 p0?.?xh?p00,u?xh?p0,u?0.Proof:By the definition of expenditure minimization.Note that if p00and p0differ only in the price of good i,thenthe result becomes?p00i p0i?xhi?p00,u?xhi?p0,u?0.Meaning:Consumer Theory(3)3.Propert

13、ies of expenditure functionTheorem 5.If u(.)is continuous and strictly increasing,then the expenditure function e(p,u)is(1)continuous on its domain(p,u).(2)increasing in p.(3)for p 0,strictly increasing in u.(4)homogeneous of degree 1 in p.(5)concave in p.(6)Shepards lemma:xhi(p,u)=e(p,u)pi.Consumer

14、 Theory(3)Proof(1)and(2)Proof(1):by the Theorem of Maximum.Proof(2):To show that e(p,u)is increasing in p,we only need to showpxh p0 xh0if p p0.Consumer Theory(3)Proof:(3)For p 0,e(p,u)is strictly increasing in uBy the Envelope Theorem.Consumer Theory(3)Proof:(4)e(p,u)is homogenous of degree 1 in p.

15、Proof:We need to show that if xhis the expenditure minimi-zing bundle at price p,then xhalso minimizes expenditureat tp,t 0.In this case,e(tp,u)=tpxh=te(p,u).This can be proven by contradiction.Consumer Theory(3)Proof:(5)e(p,u)is concave in p.Proof:This can be proved using the definition of concavef

16、unction and expenditure minimization.We need to prove that e(p+(1 )p0,u)e(p,u)+(1 )e(p0,u),0,1.Consumer Theory(3)Proof:(6)Shepards lemma:xhi(p,u)=e(p,u)pi.Proof:It follows directly from the Envelope Theorem.Consumer Theory(3)ExampleExample 2.Supposeaconsumersutilityfunctionisu(x1,x2)=x1x12,the prices of the goods are(p1,p2).What is the expenditure function?Solution:xh1(p1,p2,u)=p2(1 )p1#1uxh2(p1,p2,u)=p2(1 )p1#u.e(p1,p2,u)=(1 )1p1p12u.Consumer Theory(3)