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1、第七章显示偏好显示偏好显示偏好分析u 假设我们察看到了一个消费者在不同预假设我们察看到了一个消费者在不同预算约束下的需求作出的消费选择。算约束下的需求作出的消费选择。这会显示关于消费者偏好的一些信息。这会显示关于消费者偏好的一些信息。我们可以利用这些信息我们可以利用这些信息 .显示偏好分析检验消费者总是选择买得起的最受偏检验消费者总是选择买得起的最受偏爱的消费束这一假说。爱的消费束这一假说。提示消费者的偏好关系提示消费者的偏好关系.偏好假设u偏好偏好u在选择数据的搜集期间内没有改动。在选择数据的搜集期间内没有改动。u严厉凸性严厉凸性.u单调性单调性.u凸性和单调性阐明最优的、负担得起的凸性和单
2、调性阐明最优的、负担得起的消费束是独一的。消费束是独一的。偏好假设x2x1x1*x2*假设偏好是凸的和单调的假设偏好是凸的和单调的(性状良性状良好好)。那么最优的、负担得起的。那么最优的、负担得起的消费束是独一的。消费束是独一的。直接显示偏好u设消费束设消费束x*是消费束是消费束y 可以负担得起的可以负担得起的情况下所选择的消费束。那么情况下所选择的消费束。那么x* 是是 y的的直接显示偏好直接显示偏好 (否那么否那么 y 将被选择将被选择).直接显示偏好x2x1x*y被选择的消费束被选择的消费束 x*需求束需求束是消费束是消费束y和和z的直接显示偏好的直接显示偏好z直接显示偏好ux 是是 y
3、 的直接显示偏好可以被写成的直接显示偏好可以被写成 u x y. Dpp间接显示偏好u假设假设 x 是是y的直接显示偏好，并且的直接显示偏好，并且y是是 z的直接显示偏好的直接显示偏好. 那么基于传送性那么基于传送性, x 是是 z的间接显示偏好的间接显示偏好. 可以被写成可以被写成 x z x y and y z x z.DppDppIppIpp间接显示偏好x2x1x*z选择选择 x*的时候的时候 z 买不起买不起.选择选择 y* 的时候，的时候， x* 买不起。买不起。间接显示偏好x2x1x*y*z选择选择 x* 的时候，的时候， z买不起。买不起。选择选择 y* 的时候，的时候， x*
4、买不起。买不起。间接显示偏好x2x1x*y*z选择选择 x* 的时候，的时候， z买不起。买不起。选择选择 y* 的时候，的时候， x* 买不起。买不起。 所以所以 x* 和和 z 不能直接比较不能直接比较间接显示偏好x2x1x*y*z选择选择 x* 的时候，的时候， z买不起。买不起。选择选择 y* 的时候，的时候， x* 买不起。买不起。 所以所以 x* 和和 z 不能直接比较不能直接比较间接显示偏好x2x1x*y*z但是但是 x*x* y*Dpp选择选择 x* 的时候，的时候， z买不起。买不起。选择选择 y* 的时候，的时候， x* 买不起。买不起。 所以所以 x* 和和 z 不能直接
5、比较不能直接比较间接显示偏好x2x1x*y*z但是但是 x*x* y*并且并且 y* zDppDpp选择选择 x* 的时候，的时候， z买不起。买不起。选择选择 y* 的时候，的时候， x* 买不起。买不起。 所以所以 x* 和和 z 不能直接比较不能直接比较间接显示偏好x2x1x*y*z但是但是 x*x* y*并且并且 y* z 所以所以 x* z.DppDppIpp显示偏好的两大公理u假假设要运用要运用显示偏好来分析示偏好来分析, 消消费者者选择必需符合两个必需符合两个规范范 显示偏好的弱公理和示偏好的弱公理和显示偏好的示偏好的强公理公理显示偏好的弱公理 (WARP)u假设消费束假设消费束
6、x是消费束是消费束y的直接显示偏好，的直接显示偏好，那么消费束那么消费束y绝不能够是消费束绝不能够是消费束x的直接的直接显示偏好显示偏好u x y not (y x).DppDpp显示偏好的弱公理 (WARP)u违反违反 WARP 的消费者选择数据无法同经济的消费者选择数据无法同经济理性相容理性相容.u运用经济理性分析所察看到的消费者选择运用经济理性分析所察看到的消费者选择的时候，的时候， WARP 是一个必要条件。是一个必要条件。显示偏好的弱公理 (WARP)u什么样的消费者选择数据违背了什么样的消费者选择数据违背了 WARP呢呢?显示偏好的弱公理 (WARP)x2x1xy显示偏好的弱公理
7、(WARP)x2x1xy选择选择x 的时候，的时候，y 买得起买得起所以所以 x y.Dpp显示偏好的弱公理 (WARP)x2x1xy选择选择x 的时候，的时候，y 买得起买得起所以所以 x y.选择选择y 的时候，的时候，x买得起买得起所以所以y x.DppDpp显示偏好的弱公理 (WARP)x2x1xy这种情况是相互矛盾的这种情况是相互矛盾的.选择选择x 的时候，的时候，y 买得起买得起所以所以 x y.选择选择y 的时候，的时候，x买得起买得起所以所以y x.DppDpp检验数据能否违背了 WARPu某一消费者所出了如下选择某一消费者所出了如下选择:u在价钱为在价钱为 (p1,p2)=(
8、$2,$2) 时选择了时选择了 (x1,x2) = (10,1).u在价钱为在价钱为 (p1,p2)=($2,$1)时选择了时选择了 (x1,x2) = (5,5).u在价钱为在价钱为 (p1,p2)=($1,$2) 时选择了时选择了 (x1,x2) = (5,4).u这些数据能否违背了这些数据能否违背了 WARP 呢呢?检验数据能否违背了 WARP检验数据能否违背了 WARP红色数据是被选择的消费束的本钱红色数据是被选择的消费束的本钱.检验数据能否违背了 WARP带圈的数据是负担得起，但未被选择的消费束。带圈的数据是负担得起，但未被选择的消费束。检验数据能否违背了 WARPCircles s
9、urround affordable bundles thatwere not chosen.检验数据能否违背了 WARP带圈的数据是负担得起，但未被选择的消费束。带圈的数据是负担得起，但未被选择的消费束。检验数据能否违背了 WARP检验数据能否违背了 WARP检验数据能否违背了 WARP(10,1) 是是 (5,4)的直接的直接显示偏好显示偏好,但但 (5,4)同时同时也是也是 (10,1)的直接显示的直接显示偏好。偏好。所以，以上数据违背了所以，以上数据违背了WARP。检验数据能否违背了 WARP(5,4) (10,1)(10,1) (5,4)x1x2DppDpp显示偏好的强公理 (SAR
10、P)u假假设消消费束束 x 是消是消费束束 y 的的显示偏好示偏好 (直接的或者直接的或者间接的接的) 并且并且 x y, 那么消那么消费束束 y 决不会是消决不会是消费束束 x的的显示偏好示偏好 (直接直接的或者的或者间接的接的) u x y or x ynot ( y x or y x ).DppDppIppIpp显示偏好的强公理u什么样的数据满足显示偏好的弱公理什么样的数据满足显示偏好的弱公理(WARP) ，但却违背了显示偏好的强公理，但却违背了显示偏好的强公理 (SARP)呢呢?显示偏好的强公理u思索以下数据思索以下数据:A: (p1,p2,p3) = (1,3,10) & (x1,x
11、2,x3) = (3,1,4)B: (p1,p2,p3) = (4,3,6) & (x1,x2,x3) = (2,5,3)C: (p1,p2,p3) = (1,1,5) & (x1,x2,x3) = (4,4,3)显示偏好的强公理A: ($1,$3,$10) (3,1,4).B: ($4,$3,$6) (2,5,3).C: ($1,$1,$5) (4,4,3).显示偏好的强公理显示偏好的强公理情况情况 A的条件下的条件下,消费束消费束 A 是消费是消费束束C的直接显示偏的直接显示偏好；好； A C.Dpp显示偏好的强公理Dpp情况情况 B的条件下的条件下,消费束消费束 B 是消费是消费束束A的
12、直接显示偏的直接显示偏好好; B A.显示偏好的强公理Dpp情况情况 C的条件下的条件下,消费束消费束 C 是消费是消费束束B的直接显示偏的直接显示偏好；好； C B.显示偏好的强公理显示偏好的强公理数据没有违背数据没有违背 WARP.显示偏好的强公理数据没有违背数据没有违背 WARP 但是但是 .我们有我们有A C, B A 并且并且 C B所以所以, 基于传送性基于传送性,A B, B C 并且并且 C A.DppDppDppIppIppIpp显示偏好的强公理数据没有违背数据没有违背 WARP 但是但是 .我们有我们有A C, B A 并且并且 C B所以所以, 基于传送性基于传送性,A
13、B, B C 并且并且 C A.DppDppDppIppIppIppIII显示偏好的强公理DppIppIIIB A 与与A B相互矛盾相互矛盾数据没有违背数据没有违背 WARP 但是但是 .显示偏好的强公理DppIppIIIA C 与与C A相互矛盾相互矛盾数据没有违背数据没有违背 WARP 但是但是 .显示偏好的强公理DppIppIII数据没有违背数据没有违背 WARP 但是但是 .C B 与与B C相互矛盾相互矛盾显示偏好的强公理III数据没有违背数据没有违背 WARP 但是但是却有却有3处违背了处违背了SARP.显示偏好的强公理u观测到的消费者选择数据满足观测到的消费者选择数据满足SAR
14、P是是性状良好的偏好关系的充要条件，或者性状良好的偏好关系的充要条件，或者说这是理性的数据说这是理性的数据.u对于性状良好的偏好关系来说，上述对于性状良好的偏好关系来说，上述3组组数据是非理性的。数据是非理性的。重建无差别曲线u假设我们具有满足假设我们具有满足 SARP的消费者选择的消费者选择数据数据.u那么我们就可以充分接近地重建消费者那么我们就可以充分接近地重建消费者的无差别曲线。的无差别曲线。u怎样做怎样做?重建无差别曲线u察看值察看值:A: (p1,p2) = ($1,$1) & (x1,x2) = (15,15)B: (p1,p2) = ($2,$1) & (x1,x2) = (10
15、,20)C: (p1,p2) = ($1,$2) & (x1,x2) = (20,10)D: (p1,p2) = ($2,$5) & (x1,x2) = (30,12)E: (p1,p2) = ($5,$2) & (x1,x2) = (12,30).u穿过消费束穿过消费束 A = (15,15)的无差别曲线在的无差别曲线在什么地方呢什么地方呢?重建无差别曲线x2x1ABECDA: (p1,p2)=(1,1); (x1,x2)=(15,15)B: (p1,p2)=(2,1); (x1,x2)=(10,20)C: (p1,p2)=(1,2); (x1,x2)=(20,10)D: (p1,p2)=(
16、2,5); (x1,x2)=(30,12)E: (p1,p2)=(5,2); (x1,x2)=(12,30).重建无差别曲线x2x1ABECDA: (p1,p2)=(1,1); (x1,x2)=(15,15)B: (p1,p2)=(2,1); (x1,x2)=(10,20)C: (p1,p2)=(1,2); (x1,x2)=(20,10)D: (p1,p2)=(2,5); (x1,x2)=(30,12)E: (p1,p2)=(5,2); (x1,x2)=(12,30).Begin with bundles revealedto be less preferred than bundle A.R
17、ecovering Indifference Curvesx2x1AA: (p1,p2)=(1,1); (x1,x2)=(15,15).Recovering Indifference Curvesx2x1AA: (p1,p2)=(1,1); (x1,x2)=(15,15).Recovering Indifference Curvesx2x1AA: (p1,p2)=(1,1); (x1,x2)=(15,15).A is directly revealed preferredto any bundle inRecovering Indifference Curvesx2x1ABECDA: (p1,
18、p2)=(1,1); (x1,x2)=(15,15)B: (p1,p2)=(2,1); (x1,x2)=(10,20).Recovering Indifference Curvesx2x1ABA: (p1,p2)=(1,1); (x1,x2)=(15,15)B: (p1,p2)=(2,1); (x1,x2)=(10,20).Recovering Indifference Curvesx2x1ABA is directly revealed preferredto B and Recovering Indifference Curvesx2x1BB is directly revealed pr
19、eferredto all bundles inRecovering Indifference Curvesx2x1Bso, by transitivity, A is indirectlyrevealed preferred to all bundles inRecovering Indifference Curvesx2x1Bso A is now revealed preferredto all bundles in the union.ARecovering Indifference Curvesx2x1ABECDA: (p1,p2)=(1,1); (x1,x2)=(15,15)C:
20、(p1,p2)=(1,2); (x1,x2)=(20,10).Recovering Indifference Curvesx2x1ACA: (p1,p2)=(1,1); (x1,x2)=(15,15)C: (p1,p2)=(1,2); (x1,x2)=(20,10).Recovering Indifference Curvesx2x1ACA is directly revealedpreferred to C and .Recovering Indifference Curvesx2x1CC is directly revealed preferredto all bundles inReco
21、vering Indifference Curvesx2x1Cso, by transitivity, A isindirectly revealed preferredto all bundles inRecovering Indifference Curvesx2x1Cso A is now revealed preferredto all bundles in the union.BARecovering Indifference Curvesx2x1Cso A is now revealed preferredto all bundles in the union.BATherefor
22、e the indifferencecurve containing A must lie everywhere else above this shaded set.Recovering Indifference CurvesuNow, what about the bundles revealed as more preferred than A?Recovering Indifference Curvesx2x1ABECDA: (p1,p2)=(1,1); (x1,x2)=(15,15)B: (p1,p2)=(2,1); (x1,x2)=(10,20)C: (p1,p2)=(1,2);
23、(x1,x2)=(20,10)D: (p1,p2)=(2,5); (x1,x2)=(30,12)E: (p1,p2)=(5,2); (x1,x2)=(12,30).ARecovering Indifference Curvesx2x1DA: (p1,p2)=(1,1); (x1,x2)=(15,15)D: (p1,p2)=(2,5); (x1,x2)=(30,12).ARecovering Indifference Curvesx2x1DD is directly revealed preferredto A.ARecovering Indifference Curvesx2x1DD is d
24、irectly revealed preferredto A.Well-behaved preferences areconvexARecovering Indifference Curvesx2x1DD is directly revealed preferredto A.Well-behaved preferences areconvex so all bundles on the line between A and D are preferred to A also.ARecovering Indifference Curvesx2x1DD is directly revealed p
25、referredto A.Well-behaved preferences areconvex so all bundles on the line between A and D are preferred to A also.AAs well, .Recovering Indifference Curvesx2x1Dall bundles containing thesame amount of commodity 2and more of commodity 1 thanD are preferred to D and therefore are preferred to A also.
26、ARecovering Indifference Curvesx2x1DAbundles revealed to be strictly preferred to ARecovering Indifference Curvesx2x1ABECDA: (p1,p2)=(1,1); (x1,x2)=(15,15)B: (p1,p2)=(2,1); (x1,x2)=(10,20)C: (p1,p2)=(1,2); (x1,x2)=(20,10)D: (p1,p2)=(2,5); (x1,x2)=(30,12)E: (p1,p2)=(5,2); (x1,x2)=(12,30).ARecovering
27、Indifference Curvesx2x1AEA: (p1,p2)=(1,1); (x1,x2)=(15,15)E: (p1,p2)=(5,2); (x1,x2)=(12,30).Recovering Indifference Curvesx2x1AEE is directly revealed preferredto A.Recovering Indifference Curvesx2x1AEE is directly revealed preferredto A.Well-behaved preferences areconvexRecovering Indifference Curv
28、esx2x1AEE is directly revealed preferredto A.Well-behaved preferences areconvex so all bundles on the line between A and E are preferred to A also.Recovering Indifference Curvesx2x1AEE is directly revealed preferredto A.Well-behaved preferences areconvex so all bundles on the line between A and E ar
29、e preferred to A also.As well, .Recovering Indifference Curvesx2x1AEall bundles containing thesame amount of commodity 1and more of commodity 2 thanE are preferred to E and therefore are preferred to A also.Recovering Indifference Curvesx2x1AEMore bundles revealed to be strictly preferred to ARecove
30、ring Indifference Curvesx2x1ABCEDBundles revealedearlier as preferredto ARecovering Indifference Curvesx2x1BCEDAll bundles revealedto be preferred to AARecovering Indifference CurvesuNow we have upper and lower bounds on where the indifference curve containing bundle A may lie.Recovering Indifferenc
31、e Curvesx2x1All bundles revealedto be preferred to AAAll bundles revealed to be less preferred to ARecovering Indifference Curvesx2x1All bundles revealedto be preferred to AAAll bundles revealed to be less preferred to ARecovering Indifference Curvesx2x1The region in which the indifference curve con
32、taining bundle A must lie.AIndex NumbersuOver time, many prices change. Are consumers better or worse off “overall as a consequence?uIndex numbers give approximate answers to such questions.Index NumbersuTwo basic types of indicesuprice indices, anduquantity indicesuEach index compares expenditures
33、in a base period and in a current period by taking the ratio of expenditures.Quantity Index NumbersuA quantity index is a price-weighted average of quantities demanded; i.e. u(p1,p2) can be base period prices (p1b,p2b) or current period prices (p1t,p2t).Quantity Index NumbersuIf (p1,p2) = (p1b,p2b)
34、then we have the Laspeyres quantity index; Quantity Index NumbersuIf (p1,p2) = (p1t,p2t) then we have the Paasche quantity index; Quantity Index NumbersuHow can quantity indices be used to make statements about changes in welfare?Quantity Index NumbersuIf thenso consumers overall were better off in
35、the base period than they are now in the current period.Quantity Index NumbersuIf thenso consumers overall are better off in the current period than in the base period.Price Index NumbersuA price index is a quantity-weighted average of prices; i.e. u(x1,x2) can be the base period bundle (x1b,x2b) or
36、 else the current period bundle (x1t,x2t).Price Index NumbersuIf (x1,x2) = (x1b,x2b) then we have the Laspeyres price index; Price Index NumbersuIf (x1,x2) = (x1t,x2t) then we have the Paasche price index; Price Index NumbersuHow can price indices be used to make statements about changes in welfare?
37、uDefine the expenditure ratioPrice Index NumbersuIfthenso consumers overall are better off in the current period.Price Index NumbersuBut, ifthenso consumers overall were better off in the base period.Full Indexation?uChanges in price indices are sometimes used to adjust wage rates or transfer paymen
38、ts. This is called “indexation.u“Full indexation occurs when the wages or payments are increased at the same rate as the price index being used to measure the aggregate inflation rate. Full Indexation?uSince prices do not all increase at the same rate, relative prices change along with the “general
39、price level.uA common proposal is to index fully Social Security payments, with the intention of preserving for the elderly the “purchasing power of these payments.Full Indexation?uThe usual price index proposed for indexation is the Paasche quantity index (the Consumers Price Index).uWhat will be t
40、he consequence?Full Indexation?Notice that this index uses currentperiod prices to weight both base andcurrent period consumptions.Full Indexation?x2x1x2bx1bBase period budget constraintBase period choiceFull Indexation?x2x1x2bx1bBase period budget constraintBase period choiceCurrent period budgetco
41、nstraint before indexationFull Indexation?x2x1x2bx1bBase period budget constraintBase period choiceCurrent period budgetconstraint after full indexationFull Indexation?x2x1x2bx1bBase period budget constraintBase period choiceCurrent period choiceafter indexationCurrent period budgetconstraint after
42、indexationFull Indexation?x2x1x2bx1bBase period budget constraintBase period choiceCurrent period choiceafter indexationCurrent period budgetconstraint after indexationx2tx1tFull Indexation?x2x1x2bx1bx2tx1t(x1t,x2t) is revealed preferred to(x1b,x2b) so full indexation makesthe recipient strictly bet
43、ter off ifrelative prices change betweenthe base and current periods.Full Indexation?uSo how large is this “bias in the US CPI?uA table of recent estimates of the bias is given in the Journal of Economic Perspectives, Volume 10, No. 4, p. 160 (1996). Some of this list of point and interval estimates
44、 are as follows: Full Indexation?Full Indexation?uSo suppose a social security recipient gained by 1% per year for 20 years.uQ: How large would the bias have become at the end of the period?Full Indexation?uSo suppose a social security recipient gained by 1% per year for 20 years.uQ: How large would the bias have become at the end of the period?uA: so after 20 years social security payments would be about 22% “too large.
