高级微观经济理论_英文参考课件-lecture1.pdf

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1、Main TextbooksG.Jehle and P.Reny,“Advanced Microeconomic Theory”Other reading list:“Microeconomic Theory”by MasColell,A,M.D.Whinston,and J.Green,Oxford University Press,1995.“Microeconomic Analysis”by H.Varian W.W.Norton&Company,Inc.,1992.“Topics in Microeconomics:Industrial Organization,Auctions an

2、dIncentives,by E.Wolfstetter.(Chinese translation by Cuihong Fan,Shanghai University of Finance and Economics Press,2003).“Game Theory for Applied Economists”by Robert Gibbons,PrincetonUniversity Press,1992.Consumer Theory(1)Main topicsThis course will cover topics in consumer theory choice under un

3、certainty producer theory partial equilibrium analysis introductory game theory.Consumer Theory(1)Course GradesYour grade will be calculated as follows:1.Homework:15%2.Attendance:5%3.Midterm Exam:20%4.Final Exam:60%Consumer Theory(1)Chapter 1Consumer Theory I.Preference II.utility function III.Consu

4、mers problem IV.Indirect utility function V.Expenditure function VI.Properties of consumer demandSlutsky functionConsumer Theory(1)I.Preferences1.Basic definitions Consumption Set(or choice set),X=Rn+,is the set ofall consumption plans(or Consumption bundle)that theconsumer can conceive.Consumption

5、plan(x X,x=(x1,.,xn)repres-ents the quantity of good i in consumption plan x.Rn=R R R R:n-dimensional Cartesian Product.The element of Rnis an n-dimensional ordered vector.Consumer Theory(1)?Cartesian ProductDistinguishing between set and ordered vector:Set:the order of elements does not matter.Orde

6、red vector:the order of elements matters.In general,if S1has n elements and S2has m elements,wemay form m n ordered pairs.The Cartesian Product of two sets S1and S2,S1S2,is defi-ned to be the set of all ordered vectors(points)(x,y),wherex S1and y S2.Consumer Theory(1)Properties of consumption set XA

7、 set with no elements is an empty set.closed:A set is closed if it contains all of the points on itsboundary.convex:Set X is convex if for all x,y X,we have x+(1)y X,for all 0,1.Consumer Theory(1)2.Preference relations“Preference relation”is a binary relation on the set of al-ternatives X.Three type

8、s of preference relations:x y:“x is at least as good as y”.x?y a x ybuty 6 x x y a x yandy xConsumer Theory(1)3.Preference relation:Basic AssumptionsThe preference relation is rational if it satisfies Axiom 1and 2.Axiom 1:Completeness:For all x,y X,we have eitherx y or y x or both.Meaning?Axiom 2:Tr

9、ansitivity:For all x,y,z X,if x y,y z,then x z.Meaning?Consumer Theory(1)Preference relation:Basic properties(cont.)Axiom 3:Local nonsatiation:Given any x X and?0,there is a bundle y X with|yx|y1or x1=y1,x2 y2.Example 1.Consider the sequences of bundles xn=(1n,0)and yn=(0,1),are the lexicographic pr

10、eferences conti-nuous?Consumer Theory(1)II.Utility function1.Definition:A real-valued function u:X R is calleda utility function representing the preference if for anyconsumption bundles x,y X,x y if and only if(iff)u(x)u(y).Question:When can we represent a preference relation by autility function?C

11、onsumer Theory(1)2.Existence of a utility functionTheorem 1.If the preference relation is complete,transi-tive,continuous,and strictly monotonic,there exists a con-tinuous real-valued function,u that represents.Proof:(in four steps):Step 1:There exits a utility function u(x).Step 2:u(x)is unique.Ste

12、p 3:u(x)represents the preferences.Step 4:u(x)is continuous.Consumer Theory(1)Proof of Theorem 1 Step 1:Let e=(1,1,.,1)denote a unit consumptionbundle.For any consumption bundle x,letu(x)e x.Prove that u(x)exists for all x X.For every x X=Rn+,strict monotonicity implies thatx 0.By continuity,the set

13、s A t 0:te x,andB t 0:x te are both closed.Consumer Theory(1)Proof of Theorem 1(continued)Note:If t A B,then te x,let t=u(x)we haveu(x)e x.Thus,we need to show that t A B isnonempty.By strict monotonicity,t A implies t0 A for all t0 t.Therefore,A=t t,).Similarly,B=t 0,t.By completeness,t A B.Since A

14、 B=R+=0,t t,).We conclude that t t and therefore,t A Bis nonempty.Step 2:We show t 0 such that te x is unique.Supposet1e x and t2e x,transitivity implies t1e t2e and bystrict monotonicity,t1=t2.Consumer Theory(1)Proof of Theorem 1(continued)Step 3:Prove that this utility function represents the pre-

15、ferences.Consider two bundles x and y and the asso-ciated utility numbers u(x)and u(y).Then we havex ya u(x)e x y u(y)eby the definition of ua u(x)e u(y)eby transitivitya u(x)u(y)by strict monotonicityStep 4:we skip the proof for continuity.Consumer Theory(1)3.Ordinal property of utility functionA u

16、tility function that represents a preference relation isnot unique.Theorem 2.If u(x)is a utility function that represents,v(x)=f(u(x),where f:R R is strictly increasing inthe values of u(x),then v(x)also represents the same.Proof:Examples of strictly increasing monotonic transforms:1)2)3)Consumer Theory(1)4.Relationship between preferences and utility functionTheorem 3.Suppose u:X R is a utility function thatrepresents,then u(x)is strictly increasing iff is strictly monotonic.u(x)is quasiconcave iff is convex.u(x)is strictly quasiconcave iff is strictly convex.Consumer Theory(1)