高级微观经济理论_英文参考课件-lecture6.pdf

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1、IV Uncertainty 1.Preferences 2.Von Neumann-Morgenstern Utility 3.Risk aversion 4.ApplicationsConsumer Theory(6)(c)by Foxit Software Company,2004 Foxit PDF Editor-1.Preferences Basic definitions Preferences over gamblesConsumer Theory(6)(c)by Foxit Software Company,2004 Foxit PDF Editor-1.1 Basic def

2、initionsDenote the finite set of outcomes by A=a1,a2,.,an.Definition 1.simple gamble(lottery)A simple gamble(lottery),denoted by gs,is defined asgs=(p1a1,.,pnan|pi 0,nXi=1=1.pi:the probability that aiwill occur.ai:prizes,involving no uncertainties.examples?Consumer Theory(6)(c)by Foxit Software Comp

3、any,2004 Foxit PDF Editor-Basic definitionsDefinition pound gamble A compound gambleis a gamble whose prizes are gambles(simple or compound).Example 1.g=13g1,13g2,13g3,whereg1=(a1,0,0)g2=(14a1,38a2,38a3)g3=(14a1,38a2,38a3)Consumer Theory(6)(c)by Foxit Software Company,2004 Foxit PDF Editor-Basic def

4、initionsDefinition 3.reduced gamble A Reduced gamble of thecompound gamble:The simple gamble that lead to the sa-me probability distribution over outcomes as a compoundgamble.Example 2.Suppose that A=a1,a2.Consider the com-pound gamble yielding outcome a1with probability,andyielding a lottery ticket

5、 with probability 1 ,where thelottery ticket itself is a simple gamble.It yields the outcomea1with probability and the outcome a2with probability1 .What is the corresponding reduced gamble?Consumer Theory(6)(c)by Foxit Software Company,2004 Foxit PDF Editor-1.2 Preferences over gamblesLet G be the s

6、et of gambles,g G.Axiom 1:Completeness:For all g,g0 G,we have eitherg g0or g0 g or both.Axiom 2:Transitivity:For all g,g0,g00G,if gg0,g0 g00,then g g00.Axiom 1 and 2 Preferences are rational.Axiom 3:Continuity:For any gamble g G,suppose theoutcomes of the gamble is A=(a1,.,an)and a1a2 an.There is so

7、me probability,0,1 suchthat g (a1,(1 )an).Consumer Theory(6)(c)by Foxit Software Company,2004 Foxit PDF Editor-Axiom 4:Monotonicity.For all probabilities,0,1,(a1,(1 )an)(a1,(1 )an)if and only if .Axiom 5:Substitution.Consider any g=(p1g1,.,pkgk)and h=(p1h1,.,pkhk),where giand hiare gambles.If gi hif

8、or all i,then g h.Axiom 6:A compound gamble a its reduced gamble Ifgs=(p1a1,.,pnanis a reduced gamble of a compoundgamble g,then gs g.Consumer Theory(6)(c)by Foxit Software Company,2004 Foxit PDF Editor-2.Von Neumann-Morgenstern(VNM)Utility FunctionDefinition 4.Expected utility property The utility

9、func-tion has the expected utility property if,for every gambleg G,u(g)=nXi=1piu(ai).Definition 5.Von Neumann-Morgenstern Utility FunctionUtility functions possessing the expected utility property areknown as Von Neumann-Morgenstern utility functions.Consumer Theory(6)(c)by Foxit Software Company,20

10、04 Foxit PDF Editor-Existence of a VNM utility functionTheorem 1.Suppose preferences over gambles satisfyAxioms 1-6,then there exists a utility function u represen-ting such that u has the expected utility property.Consumer Theory(6)(c)by Foxit Software Company,2004 Foxit PDF Editor-Sketch of the Pr

11、oof Step(1).Construct a utility function:define u(g)to be thenumber satisfyingg (u(g)a1,(1 u(g)an).Step(2).Prove u represents,i.e.g g0a u(g)u(g0).Step(3).We must show u(g)has the expected utility pro-perty,i.e.u(g)=nXi=1piu(ai).Consumer Theory(6)(c)by Foxit Software Company,2004 Foxit PDF Editor-2.4

12、 Property of VNM utility functionTheorem 2.Suppose u(.)is VNM utility function,then itspositive affine transformation v(g)=+u(g),0 stillhas the expected utility property and represents the samepreferences.Proof:Step(1).Prove that v(g)=+u(g)has the expectedutility property if u has the expected utili

13、ty property.Step(2).Prove:if u(g)and v(g)have the expected utilityproperty and represent the same preferences,then theremust exist and 0 such that v(g)=+u(g).Consumer Theory(6)(c)by Foxit Software Company,2004 Foxit PDF Editor-Example:Expected utility and expected value ofgamblesExample 3.Suppose A=(10,4,2),10 (1 10,0 2),u(10)=14 (0.6 10,0.4 2),u(4)=.6 2 (0 10,1 2),u(2)=0.Consider two gamblesg1=(0.8 10,0.2 4,0 2)g2=(0.9 10,0.03 4,0.07 2)Consumer Theory(6)(c)by Foxit Software Company,2004 Foxit PDF Editor-