范微观济学现代观点课后习题详解

第1 市125个消费者租用商品房的保留价格为500第1 市125个消费者租用商品房的保留价格为500美元，第Supposethattherewere25peoplewhohadareservationpriceof$500,andthepersonhadareservationpriceof$200.WhatwouldthedemandcurvelookpQ200p50p 2124套住房可以出租，均衡价格是多少?26Intheaboveexample,whatwouldtheequilibriumpricebeiftherewere24torent?Whatiftherewere26apartmentstorent?Whatiftherewere25apartmentsto1-2A点所示，特1-21-1由同（1），如图1-2C点所示。1 Ifpeoplehavedifferentreservationprices,whydoesthemarketdemandcurvePP越高，保留价PPP的消费者逐渐增多，P下降时，市场需求会不断增加。当消费者足够多，互相Inthetextweassumedthatthecondominiumpurchaserscamefromtheinner-ringpeople—peoplewhowerealreadyrentingapartments.WhatwouldhappentothePriceofinner-ringapartmentsifallofthecondominiumpurchaserswereouter-ringpeople—thepeoplewhowerenotcurrentlyrentingapartmentsintheinnerring?来的需求曲线D11-3 2SupposenowthatSupposenowthatthecondominiumpurchaserswereallinner-ringpeople,butthateachcondominiumwasconstructedfromtwoapartments.Whatwouldhappentothepriceofapartments?供给曲线由S向左移动至S′。新的均衡价格P′高于原来的均衡价格P。 Whatdoyousupposetheeffectofataxwouldbeonthenumberofapartmentsthatwouldbebuiltinthelongrun? 7．假设需求曲线是D(p)=100-2p，如果垄断者有60套住房，他会定什么价格?他可以租出多少套?如果他有40套住房，他会定什么价格?他可以租出多少套?SupposethedemandcurveisD(p)=100–2p.Whatpricewouldthemonopolistsetifhehad60apartments?Howmanywouldherent?Whatpricewouldhesetifhehad40apartments?Howmanywouldherent?3maxq(5012qmaxq(5012qq＝506050 Ifourmodelofrentcontrolallowedforunrestrictedsubletting,whowouldendupgettingapartmentsintheinnercircle?WouldtheoutcomebeParetoefficient?第2 预算约1p1x1p2x2=m112的价格提高了7倍，收入增加了3倍。根据原来的价格和收入写出新预算线的方程。Originallytheconsumerfacesthebudgetlineplxl+p2x2=m.Thenthepriceofgooddoubles,thepriceofgood2becomes8timeslarger,andincomebecomes4timeslarger.Writedownanequationforthenewbudgetlineintermsoftheoriginalpricesandincome.答：p1x1+p2x2=m，则新的预算线方程是2p1x18p2x24m'21p'8p；m'4m212以新的预算线为p'pxm，即2pxx1 2124Whathappenstothebudgetlineifthepriceofgood2increases,butthepriceofgoodandincomeremain答：预算线会变得更为平坦，横截距仍然不变。理由如下：假设原来的预算线为mpxWhathappenstothebudgetlineifthepriceofgood2increases,butthepriceofgoodandincomeremain答：预算线会变得更为平坦，横截距仍然不变。理由如下：假设原来的预算线为mpxpxm，将预算线重新整理为xx21 221 31122倍，预算线是变平了还是变Ifthepriceofgood1doublesandthepriceofgood2triples,doesthebudgetbecomeflatterorm答：预算线变得更为平坦。理由如下：由预算线xx221 211倍，那么预算线的斜率将从p1变为2p1p3pWhatisthedefinitionofanumeraire绘出同一个预算集。因此这条预算线为：p1x1+p2x2=m，就与下列的预算线是相同的 mx2x1+x2 ppm22。入限定为5假设政府对每加仑汽油征税157假设政府对每加仑汽油征税157Supposethatthegovernmentputsataxof15centsagallonongasolineandthendecidestoputasubsidyongasolineatarateof7centsagallon.Whatnettaxisthiscombinationequivalentto?p1x1+p2x2=mu1Supposethatabudgetequationisgivenbyp1x1+p2x2=m.Thegovernmentdecidestoimposealump-sumtaxofu,aquantitytaxongood1oft,andaquantitysubsidyongood2s.Whatistheformulaforthenewbudget答：(p1t)x1p2s)x2muIftheincomeoftheconsumerincreasesandoneofthepricesdecreasesatthesametime,willtheconsumernecessarilybeatleastaswell-off? 第3 l．如果在(x1，x2)和(y1，y2)同时可以得到的情况下，消费者选择(x1x2)能否得出x1x2)﹥（y1，y2）的结论Ifweobserveaconsumerchoosing(x1,x2)when(y1,y2)isavailableonetime,arewejustifiedinconcludingthat(x1,x2)﹥(y1,y2)?6答：不能得出(x答：不能得出(x1x2)﹥（y1y2）的结论。理由如下：从最优化原理(下选择了(x1x2)，那么就可以得出结论(x1x2)>（y1y2。但就一般的情费者的最优商品束并不一定是惟一的，消费者在(y1y2)可得的情况下选择了(x1，x2)时消费者也可能对二者是无差异的。因此对于一般的情况只能得出结论(x1，x2（y1，y2ConsideragroupofpeopleA,B,Candtherelation“atleastastallas,”asin“AisatleastastallasB.”Isthisrelationtransitive?Isitcomplete?说A不会比B矮，BC矮，所以AC矮，即AC至少一样高。Takethesamegroupofpeopleandconsidertherelation“strictlytallerthan.”Isthisrelationtransitive?Isitreflexive?Isitcomplete?喜爱Y消费束胜过Z，那么可以认为消费者喜爱X消费束胜过。一个。这种偏好关系是传递的吗?完备的吗AcollegefootballcoachsaysthatgivenanytwolinemenAandB,healwaysprefersonewhoisbiggerandfaster.Isthispreferencerelationtransitive?Isit无差异曲线自身能相交吗?3-1能绘成一条单独的无差异曲线吗Cananindifferencecurvecrossitself?Forexample,couldFigure3-9depictaindifference7 CouldFigure3-1beasingleindifferencecurveifpreferencesare图 Ifbothpepperoniandanchoviesarebads,willtheindifferencecurvehaveapositiveanegative情况下，无差异曲线离原点越近，代表的效用就越高。如图3－3所示。8图3－3 图3－3 Explainwhyconvexpreferencesmeansthat“averagesarepreferredtoextremes.”3－4所示，在凸状的偏好中，X、Y位于同一无差异曲线上，ZXY的加权平均，由3－4Z位于比X、Y更高的无差异曲线上，即Z比X、Y更受消费者的偏爱。XZ是X和YY图 915美元钞票的边际替代率是多少Whatisyourmarginalrateofsubstitutionof$1billsfor$5美元的钞票，那么需要5张1美元的钞票来补偿，因此x55x11151Ifgood1isa“neutral”,whatisitsmarginalrateofsubstitutionforgood03－59图 图 Thinkofsomeothergoodsforwhichyourpreferencesmightbe第4 效1.一个效用函数自乘奇数次是单调变换。那么该效用函数自乘偶数次还是单调变换吗?(提示：考虑f(u)=u2这种情况)Thetextsaidthatraisinganumbertoanoddpowerwasamonotonictransformation.Whataboutraisinganumbertoanevenpower?Isthisamonotonictransformation?(Hint:considerthecasef(u)=u2.)（1）u2（2）u1/（3）u1/（4）u（6）u（7）u2,（8）u2,Whichofthefollowingaremonotonictransformations?(1)u=2v－13;(2)u=－l/v2;(3)u=l/v2;(4)u=lnv;(5)u=－e－v;(6)u=v2;(7)u=v2forv>0;(8)u=v2forv<0.3．如果是单调偏好的话，经过原点的射线就只会同每条无差异曲线相交一次。请严格）Weclaimedinthetextthatifpreferencesweremonotonic,thenadiagonallinethroughtheoriginwouldintersecteachindifferencecurveexactlyonce.3．如果是单调偏好的话，经过原点的射线就只会同每条无差异曲线相交一次。请严格）Weclaimedinthetextthatifpreferencesweremonotonic,thenadiagonallinethroughtheoriginwouldintersecteachindifferencecurveexactlyonce.Canyouprovethisrigorously?(Hint:whatwouldhappenifitintersectedsomeindifferencecurvetwice?)x1y1x2y2x1y1x2y2x1y1差异曲线相交一次，如图4－1所示。4．哪种偏好可用形如u(x1x2v(x1x213x113x2x1x2Whatkindofpreferencesarerepresentedbyautilityfunctionofthe?Whatabouttheutilityfunctionv(xl,答：（1）完全替代的偏好可用形如u(x1,x2) x1x2的效用函数表示。理由如下u(x1x2x1x2做单调变换f(u)u2u'(x,x)x 2代表的偏好的类型，所以，u(x1,x2) x1x2也代表完全替代的偏好（2）v(x1x213x113x21v(x,x)13x13x做单调变换f(v) v，得v'(x,x)xx和（1）的理由 12 同，可知v(x1x213x113x2这种形式的效用函数表示?效用函数5．哪种偏好可用形如u(x1x2x11v(x,x)13x13x做单调变换f(v) v，得v'(x,x)xx和（1）的理由 12 同，可知v(x1x213x113x2这种形式的效用函数表示?效用函数5．哪种偏好可用形如u(x1x2x1v(x,x) x2x2是u(x1x22 11Whatkindofpreferencesarerepresentedbyautilityfunctionoftheformu(x1,x2)?Istheutilityfunctionv(x1,x2)=x21+u(x1,+x2amonotonictransformation答：（1）线性偏好可以用形如u(x1x2x1 （2）效用函数v(x,x) 2x1x2x2是u(x1,x2)的单调变换。理由如下：2 16．考虑效用函数u(x1x2)x1x2。它表示哪种偏好函数v(xx)x2是212)x 是u(xx的单调变u(xxw(x22 2122Considertheutilityfunctionu(x1,x2).Whatkindofpreferencesdoesrepresent?Isthefunctionv(x1,x2)=x21x2amonotonictransformationofu(xl,x2)?Isfunctionw(x1,x2)=x2lx22amonotonictransformationofu(x1,答：（1）用函数u(x1x2v(xx)xx不是2u(xx（2）函数 调变换不改变它代表的偏好，所以如果v(xxxx是u(xx2的单调变换，那么这2 1个效用函数代表相同的偏好，从而它们的边际替代率相同，但是计算发现，效用函数xu(x,x)xx的边际替代率MRSv(xxx2x的边际替代率 1 21MRS)x2x肯定不是u(xx，这两个边际替代率不相等，所以v(x x1（3）函数w(x,x)x2 是u(x,x)的单调变换。理由如下：对效用函2 2u(x,x)xxf(uu4w(xxx2 1 7．请解释为什么效用函数的单调变换不会改变边际Canyouexplainwhytakingamonotonictransformationofautilityfunctionchange7．请解释为什么效用函数的单调变换不会改变边际Canyouexplainwhytakingamonotonictransformationofautilityfunctionchangethemarginalrateof假设对一个效用函数作单调变换，比如说，v(x1x2f(u(x1x2f/u/MRS=v/ f/uu/ 222第5 选Iftwogoodsareperfectsubstitutes,whatisthedemandfunctionforgoodp20xppm0到m/pp 1 pm/1 maxx1x10,x2s.t.p1x1p2x2mp1，2p2mp1x00xm/p211p2/mp1(1p1mmp1x00xm/p211p2/mp1(1p1mmax11ppp0x1m/222p20xppm0到m/pp 1 pm/1 图 Supposethatindifferencecurvesaredescribedbystraightlineswithaslopeof-Givenarbitrarypricesandmoneyincomepl,p2,andm,whatwilltheconsumer’soptimalchoiceslooklike?答：如图5－2所示，比较预算线与无差异曲线的斜率可知， b时，消费者 b时5－2糖Supposethataconsumeralwaysconsumes2spoonsofsugarwitheachcupofcoffee.thepriceofsugar5－2糖Supposethataconsumeralwaysconsumes2spoonsofsugarwitheachcupofcoffee.thepriceofsugarisplperspoonfulandthepriceofcoffeeisp2percupandtheconsumerhasmdollarstospendoncoffeeandsugar,howmuchwillheorshewanttopurchase?m和。分析如下x12p1 2p1mpxpxm；。1 2122p2p 4p1，p2并有m美元可支出。那么他的最优消费束是Supposethatyouhavehighlynonconvexpreferencesforicecreamandolives,likethosegiveninthetext,andthatyoufacepricespl,p2andhavemdollarstospend.Listthefortheoptimalconsumption ）或者为（m,0。这是因为：在凹性偏好的m最优选择为只消费两种商品中的一种商品。即消费者的最优消费束为）（m,045．如果一个消费者有效用函数u(xxxx，她会将她收入的多少比例花在商品 1上Ifaconsumerhasautilityfunctionu(xl,x2)=x1x42,whatfractionofherincomewillspendongood斯效用函数，对其进行单调变换，可得u(x,x) x/。关于柯布—道格拉斯效用函1/54spendongood斯效用函数，对其进行单调变换，可得u(x,x) x/。关于柯布—道格拉斯效用函1/54 xx10,x21s.t.p1x1p2x2由拉格朗日乘子法有：maxxx1(mpxpx11 2x1(1)xx 112p px111 。所以消费者会把 )的收入花费在商品则p p (1)x122145Forwhatkindofpreferenceswilltheconsumerbejustaswell-offfacingaquantitytaxasanincome5－3所示，在征税后，无论在哪种税收条件下，消费者的最优选择都不变，5－3第6 需Iftheconsumerisconsumingexactlytwogoods,andsheisalwaysspendingallofhermoney,canbothoftheminferiorx1(p1,p2,图 px1(p1,p2,图 p1x1p2x2m，因为消费者总是花光她的全部收入，所以此式恒成立。预算线两边对收入求导得：dx1dx2pp12Showthatperfectsubstitutesareanexampleofhomothetic果消费者对(x1x2的偏好甚于y1y2，则对任意的t0，该消费者对(tx1tx2的偏于(ty1ty2下面来证明完全替代是相似偏好的一个特例。由于完全替代偏好的效用函数是ux1x2x1x2，如果ux1x2uy1y2x1x2y1y2的正值t，都会有tx1tx2ty1ty2，可得utx1tx2uy1y2，所以完全替代是相似偏3．证明柯ShowthatCobb-Douglaspreferencesarehomothetic答：关于相似偏好的概念见第2题。由于柯布道格拉斯效用函数为：uxxx utxtxtxtuxxaatxa 2 1 ux1x2ux1x2uy1y2 ，那么对于任何正 ，就一定utx1tx2tux1x2tuy1y2uty1ty2成立，从而得到utx1tx2uty1ty2AstheincomeoffercurveistotheEngel,whichcurveisthepriceoffercurve6－26-3图 Ifthepreferencesareconcavewilltheconsumereverconsumebothofthegoods图 Arehamburgersandbunscomplements图 ArehamburgersandbunscomplementsorWhatistheformoftheinversedemandfunctionforgood1inthecaseofm p2。理由如下：首先x1完全互补品的效用函数是：min{x1x2更一般的这种类型的效用函数是：min{ax1,，但这种特殊形式并不影响分析)，在预算约束下，消费者选择效用最大x10,x2min{x1,s.t.p1x2p2x2x10,x2min{x1,s.t.p1x2p2x2少 mmp1x1p2x1mx1，pp m进而得到商品1的反需求函数为p1 p2x18．如果需求函数是x1p1，那么反需求函数就是x p1Trueorfalse?Ifthedemandfunctionisx1=－p1,thentheinversedemandfunctionis第7 显示偏1．当价格是p1p2=（1，2）时，消费者需求是(x1x2=(1，2)，当价格(q1q21)时，消费者需求是)=(2，1)。这种行为同最大化的行为模型相一致y1,Ifpricesare(pl,p2)=(1,2),thecombinationofconsumerdemandis(x1,x2)=(1,2),andifpricesare(ql,q2)=(2,1),thecombinationofconsumerdemandis(y1,y2)=(2,1).Isthisbehaviorconsistentwiththemodelofmaximizingbehavior?答：这种行为同最大化的行为模型不一致。p1x1p2x25p1y1p2y24(x1x2被显示偏好于y1y2。同样的，由于q1x1q2x24q1y1q2y25从中可知y1y1被显示偏好于x1x2所以这种行为违背了显示偏好弱公理，与最大化的行2．当价格是(p1p2=(2，1)时，消费者需求是(x1x2=(1，2)，当价格是(q1q22)时，消费者需求是(y1,y2)=(2，1)。这种行为同最大化行为模型相一致吗Ifpricesare(pl,p2)=(2,1),thecombinationofconsumerdemandis(x1,x2)=(1,2),andpricesare(ql,q2)=(1,2),thecombinationofconsumerdemandis(y1,y2)=(2,1).Ispricesare(ql,q2)=(1,2),thecombinationofconsumerdemandis(y1,y2)=(2,1).Isthisbehaviorconsistentwiththemodelofmaximizingbehavior?(x1,x2)(y1,y2)O32xIntheprecedingexercise,whichbundleispreferredbytheconsumer,thex-bundleor答：xWesawthattheSocialSecurityadjustmentforchangingpriceswouldtypicallyrecipientsatleastaswell-offastheywereatthebaseyear.Whatkindofpricechangesleavethemjustaswell-off,nomatterwhatkindofpreferencesthey这是因为，对于指数化的保险金，当价格从（pbpb）变为(qq qxqxqxb1 2 1 2(q,q)t(p,p pbxpbxpbpb1 2 1 2InthesameInthesameframeworkastheabovequestion,whatkindofpreferenceswouldleavetheconsumerjustaswell-offaswasinthebaseyear,forallprice7－2所示。7－2中，AB第8 斯勒茨基方SupposeaconsumerhaspreferencesbetweentwogoodsthatareperfectWillthethingthattheentiredemandresponseequalstothesubstitutioneffecthappenifpricesSupposethatpreferencesareconcave.Isitstillthecasethatthesubstitutioneffectis8－18－1等于A-8－2x向消费者退税，情况会发生什么变化Inthecaseofthegasolinetax,whatwouldhappeniftherebatetotheconsumersbasedontheiroriginalconsumptionofgasoline,x,ratherthanontheirfinalconsumptiongasoline,pxymp代表石油价格，x表示花在其他商品上的货币，m表示总收入，从而消费者的最优选择为x* ，特别的px*y*px*y*ptxym图形分析见图8－3。图 43题中，政府支出比它得到的税收更多还是更少Inthecasedescribedintheprecedingquestion,wouldthegovernmentbepayingoutmoreorlessthanitreceivedin_8－3政府的税收补贴，所以政府支出比它得到的税收Inthiscasewouldtheconsumersbebetterofforworseoffifthetaxwithrebatebasedonoriginalconsumptionwereineffect?第9 购买和销Ifaconsumer’snetdemandsare(5,-3)andherendowmentis(4,4),whatareher（9，1（9，当商品价格是(pl，p2)=(2，3)时，消费者的消费是(x1，x2)=(4，4)买卖是没有成本的。那么消费者一定偏好商品束y1,y2=(3，5)吗?品束(y1，y2)吗Thepricesare品束(y1，y2)吗Thepricesare(pl,p2)=(2,3),andtheconsumeriscurrentlyconsuming(x1,x2)=(4,Thereisaperfectmarketforthetwogoodsinwhichtheycanbeboughtandsoldcostlessly.Willtheconsumernecessarilypreferconsumingthebundle(y1,y2)=(3,5)?Willshenecessarilypreferhavingthebundle(y1,y2)?（2）消费者肯定选择拥有商品束(y1y2)=(3，5)时，消费者出售禀赋的收入是费者的更多的收入也即更大的预算空间，因此他一定会偏好拥有商品束(y1,)，但是在有商品束(y1,y2)的情况下，消费者的最优选择却未必还是该商品束，因此他未必偏好3．当商品价格为p1p2(2,3时，,消费者的消费是(x1x2(4,4)为(q1q22,4)，那么在新的价格下，消费者的境况会变得更好一些吗Thepricesare(pl,p2)=(2,3),andtheconsumeriscurrentlyconsuming(x1,x2)=(4,4).Nowthepriceschangeto(q1,q2)=(2,4).Couldtheconsumerbebetteroffunderthesenewprices?更好（如图9－1中的无差异曲线S1所示），也有可能变的更差（如图9－1中的无差异曲线S2所示。图 图 TheU.S.currentlyimportsabouthalfofthepetroleumthatituses.Therestofitsaremetbydomesticproduction.CouldthepriceofoilrisesomuchthattheU.S.wouldbemadebetteroff?__pxmpw__p，预算线是最优选择位于ABAB段位于油价上涨前的预算集内，这就是说，油价上涨前，美国在消费得起ABB点对应的消费束，由显示偏好原理，BAB段对应的消费束，所以，如果油价上涨后，美国的选择位于AB段上，那么他的境况必然变差。但是如果美国在石油价格上升后从石油进口国转变为石油出口国，即新的最优选择位于BC段，那么美国的情况有可能变好（如图S2）也有可能变差（9－2中的S1）这要视具体的效用函数来决定图 2430时(碰巧发生在考试周的前夕)，这将Supposethatbysomemiraclethenumberofhoursinthedayincreasedfrom24tohours(withluckthiswouldhappenshortlybeforeexamweek).Howwouldthisaffectthebudgetconstraint?2430时，预算线会平行地向外移动。Ifleisureisaninferiorgood,whatcanyousayabouttheslopeofthelabor替代效应＋（RR)(RR)项恒为正替代效应＋（RR)(RR)项恒为正，从而等式右边的后一项，也即收入效应恒为负。由于替代效应恒为负第10 跨时期选1．如果利率是20%，那么20年后交付的100万美元在今天的价值是多少Howmuchis$1milliontobedelivered20yearsinthefutureworthtodayiftherateis20就有100万0.03310-1不同利率和到期时间下，12．当利率提高时，跨时期的预算线是变得更陡峭还是变得更平Astheinterestraterises,doestheintertemporalbudgetconstraintsteeperorcwc(1r)cw1r)w，所以112 111的斜率为(1r)，当利率提高时，预算线的斜率的绝对值增大，因此预算线会变得更加3．在研究跨期的食品购买时，商品完全替代的假设是否有WouldtheassumptionthatgoodsareperfectsubstitutesbevalidinastudyintertemporalfoodAconsumer,whoisinitiallyalender,remainsalenderevenafteradeclineininterestrates.Isthisconsumerbetterofforworseoffafterthechangeininterestrates?Iftheconsumerbecomesaborrowerafterthechangeishebetterofforworseoff?答：（1）假设在利率下降前，某消费者是一个贷款者，如果利率下降后，他仍然是一此时的最优选择变为B由于B在原来的预算线之下，这就意味着：在利率变化前，消费者可以选择B点的时候却放弃了它，转而选择A那么由显示偏好原理可知，消费者一定偏10－2好还是变坏。分析如图10-3所示，假设利率降低后，消费者的最优选择点为C点，此时消10－55．如果利率是10%，一年后10－55．如果利率是10%，一年后100美元的现值是多少？如果利率为5%，它的现值又是thepresentvalueiftheinterestrateis5%?第11 资产市假定资产A在下个时期能卖11美元。如果与A相似的资产的报酬率是10%，请问资产A的现值是多少?SupposeassetAcanbesoldfor$11nextperiod.IfassetssimilartoAarepayingarateofreturnof10%,whatmustbeassetA’scurrentprice?答：资产Ap=11/1.1=10111万美元的价格出售的住宅，现在可以卖到10万美元。这幢房子的报酬率是多少？Ahouse,whichyoucouldrentfor$10,000ayearandsellfor$110,000ayearfromcanbepurchasedfor$100,000.Whatistherateofreturnonthis10%利率，人们面临的边际税率是40%，不征税债券的报酬率应是多Thepaymentsofcertaintypesofbonds(e.g.municipalbonds)arenottaxable.Iftaxablebondsarepaying10%andeveryonefacesamarginaltaxrateof40%,whatratereturnmustthenontaxablebonds报酬率应该等于征税债券的税收调整后的报酬率。在边际税率时40%10%的报酬率经税收调整后为10%(140%)＝6%，因此不征税的债券的报酬率为6%410美元的价格才可得到，如果利率是10%，该稀缺资源今天的价格应是多少Supposethatascarceresource,facingaconstantdemand,willbeexhaustedin10years.IfalternativeSupposethatascarceresource,facingaconstantdemand,willbeexhaustedin10years.Ifalternativeresourcewillbeavailableatapriceof$40andiftheinterestrateis10%,mustthepriceofthescarceresourcebe答：15.42美元。分析如下：10年后，稀缺资源的价格应(1第12 不确定1．图12－1是消费者在不确定的环境中购买保险时的预算线，请解释预算线上初始禀HowcanonereachtheconsumptionpointstotheleftoftheendowmentinFigure12-12－1(1)u(c1c212a(1c12c2（a>0(2)u(c,c,,)cc 1 2(3)u(c1,c2,1,2)1lnc12lnc2Whichofthefollowingutilityfunctionshavetheexpectedutilityproperty?(a)21,π2)=a(π1c1+π2c2),π1c1+π2c2,=如果效用函数u(x如果效用函数u(x)满足u(c1c21,2)a[1v(c12v(c2b，就说u(x)具（1（2，3．一个厌恶风险的人要在以下两种情形之间进行选择：他有25%的概率得到1000元，有75％的概率得到100美元；或者，他可以得325美元。他会怎样选Arisk-averseindividualisofferedachoicebetweenagamblethatpays$1000withprobabilityof25%and$100withaprobabilityof75%;orapaymentof$325.Whichwouldhechoose?答：此人会选择第二种情形。因为第一种选择给消费者带来的预期收入为12－2风险厌恶者的效用函数4．如果消费者可以确定地得到320美元，他会怎样选择?Whatifthepaymentwas$320?答：3320Drawautilityfunctionthatexhibitsrisk-lovingbehaviorforsmallgamblesandrisk-aversebehaviorforlargergambles.6．为什么相对于火灾来说，在洪灾的情况下，6．为什么相对于火灾来说，在洪灾的情况下，相邻而居的人们更难以相互提供Whymightaneighborhoodgrouphaveahardertimeselfinsuringforflooddamageversusfiredamage?第13 风险资1．如果无风险资产的报酬率是6%，某种风险资产的报酬率为9%3%，那么，如果一个人愿意接受的标准差水平是2%，则他能够实现的最大报酬率是多少Iftherisk-freerateofreturnis6%,andifariskyassetisavailablewithareturnof9%andastandarddeviationof3%,whatisthemaximumrateofreturnyoucanachieveifyouarewillingtoacceptastandarddeviationof2%?Whatpercentageofyourwealthwouldhavetobeinvestedintheriskyasset?答：（1）此人能够实现的最大报8%。理由如下：假设此人的总投x比例3x2%x23rpxrs1x)rf39%368%，这里rsrf分别是风险资产和无风险资产的报酬率，rp是投资组合的报酬率。21题中的风险价格是多少Whatisthepriceofriskintheabove21rp0.081Ifastockhasaβof1.5,thereturnonIfastockhasaβof1.5,thereturnonthemarketis10%,andtherisk-freerateofreturnis5%,whatexpectedrateofreturnshouldthisstockofferaccordingtotheCapitalAssetPricingModel?Iftheexpectedvalueofthestockis$100,whatpriceshouldthestockbesellingfor答：（1）该股票的预期收益率是12.5%。分析如下：根据资本资产定价模型rirfi(rmrf)rf5%rm10%rp5%1.510%512.5%（2）如果该股票的预期值为100美元，那么今天出售的价格应该为： 1 14章消费者剩余1美元的从量税，这种税收的额外净损失是多少？Agoodcanbeproducedinacompetitiveindustryatacostof$10perunit.Thereare100consumersareeachwillingtopay$12eachtoconsumeasingleunitofthegood(additionalunitshavenovaluetothem.)Whatistheequilibriumpriceandquantitysold?Thegovernmentimposesataxof$1onthegood.Whatisthedeadweightlossofthistax?以所有的消费者都会购买这种产品，即厂商的销量为100。此时消费者剩余为（12-10）14-1供给等于需求时，市场均衡为11。此时消费者剩余为（12-11）×100=100，生产者剩余为（11-1-10）×100=01×100=100100＋100＝20014-214-2供给等于需求时，市场均衡2．假设需求曲线为Dp)10p。那么消费者消费614-2供给等于需求时，市场均衡2．假设需求曲线为Dp)10p。那么消费者消费6SupposethatthedemandcurveisgivenbyD(p)=10-p.Whatisthegrossbenefitfromconsuming6unitsofthegood?6的最高价格之和，也就是CS(10q)dq6018423246Intheaboveexample,ifthepricechangesfrom4to6,whatisthechangein464CSCS2CS1(10q)dq(10q)dq(10q)dq1000bSupposethataconsumerisconsuming10unitsofadiscretegoodandthepriceincreasesfrom$5perunitto$6.However,afterthepricechangetheconsumercontinuestoconsume10unitsofthediscretegood.Whatisthelossintheconsumer’ssurplusfromthispricechange?不影响结论），当5元时，消费者消费10单位商品，消费者剩余SADG，当格提高到6元后，消费者剩余为：SACBSCEG，这里之所以要减去三角形CEG是因为费者为多消费的商品愿意支付的价格低于实际购买价格6CSSCEGSCSSCEGSAGD10图14- 第15 市场需IfthemarketdemandcurveisD(p)=100-0.5p,whatistheinversedemandAnaddict’sdemandfunctionforadrugmaybeveryinelastic,butthemarketfunctionmightbequiteelastic.Howcanthis图 图 吸毒者2的需求图15－3吸毒者3的需图 图 )IfD(p)=12-2p,whatpricewillmaximizeRp124p0p3)SupposethatthedemandcurveforagoodisgivenbyD(p)=100/p.Whatpricewillmaximizerevenue?Trueorfalse?Inatwogoodmodelifonegoodisaninferiorgoodtheothergoodbealuxuryp1x1(p1,p2,m)p2x2(p1,p2,m)上式两边关于m求导，得到pp21s11s22s11s2222求收入弹性。所以如果一种商品是低档品，不妨假设为商品1，那么必有11s1s21，所以必有212第16 均Whatistheeffectofasubsidyinamarketwithahorizontalsupplycurve?Withaverticalsupplycurve?t的补贴，所以补贴的好处全部由消费者得到图16- 图16- Supposethatthedemandcurveisverticalwhilethesupplycurveslopesupward.Iftaxisimposedinthismarketwhoendsuppaying图16-图16- 斜，令红、蓝铅笔的价格分别为pr和pb。如果政府只对红铅笔征税，会出现什么结果？Supposethatallconsumersviewredpencilsandbluepencilsasperfectsubstitutes.Supposethatthesupplycurveforredpencilsisupwardsloping.Letthepriceofredpencilsandbluepencilsbeprandpb.Whatwouldhappenifthegovernmentputataxonlyonred美国近一半的汽油是进口的。假设石油生产者愿意以25oilproducersarewillingtosupplyasmuchoilastheUnitedStateswantsataconstantpriceof$25abarrel.Whatwouldhappentothepriceofdomesticoilifataxof$5abarrelwereplacedonforeignoil?00内的石油与国外的石油对消费者而言是完全替代的，所以消费者会选择价格便宜的石油消00图16- Suppose图16- Supposethatthesupplycurveisvertical.Whatisthedeadweightlossofataxinthis图16- Considerthetaxtreatmentofborrowingandlendingdescribedinthetext.Howrevenuedoesthistaxsystemraiseifborrowersandlendersareinthesametax1题可知，对借款者和贷款者征税，市场达到均衡的条件是：rb(q)rl1 1 者处于同一纳税等级，即tbtl，则由（1）07．当tltb7．当tltb1Doessuchataxsystemraiseapositiveornegativeamountofrevenuewhentl<第17 Consideranauctionofantiquequiltstocollectors.Isthisaprivate-valueorcommon-value如果竞价增量是l美元。那么，在这个利润最大化的英国式拍卖中，保留价格应该是多少?（编者注：拍卖者并不清楚每个竞价者的评价，只知道对某个特定的竞价者而言，他是高）incrementof$1.Whatshouldthereservationpricebeinaprofit-maximizing答：在这个利润最大化的英国式拍卖中，保留价格0（88（810（10，8）合出现的概率都是0.25。令r表示拍卖者的保留价格，那么：（8，9，9，100.2580.2590.2590.2510r9r（09，9，100.2500.2590.2590.25109792（0，10，10，100.2500.25100.25100.2510107.5102.50时，拍卖者可以获得最大利润。SupposethatwehavetwoCopiesofIntermediateMicroeconomicsSupposethatwehavetwoCopiesofIntermediateMicroeconomicstosellto(enthusiastic)students.Howcanweuseasealed-bidauctionthatwillguaranteethatthebidderswiththetwohighestvaluesgetthebooks?的期望收益是（其中“Prob”代表“概率”：Prob(b1min{b2,b3})[v1min{b2,1项是投标人l2l胜出时，他所享有的消费者剩余。如果b1min{b2b31获得的消费者剩余为零，从而也就没有必要考虑包含Prob(b1min{b2b3})假定v1min{b2b3}，那么投标人1就会使他胜出的概率尽可能大，他可以通过设实现这一点。另一方面，假定v1min{b2b3，那么投标人1就会使他胜出的概4．考虑原书中Ucom公司的例子，这种拍卖设计是有效率的吗?它使利润实现了最大ConsidertheUcomexampleinthetext.Wastheauctiondesignefficient?Diditmaximize：（1）政府将经营许可证授予对此项经营评价最高的企业，从这层意义上讲，这种（2）它没有使拍卖者实现最大利润，因为成交价格AgametheoristfillsajarwithpenniesandauctionsitoffonthefirstdayofclassusinganEnglishauction.Isthisaprivate-valueoracommon-valueauction?Doyouthinkthewinningbidderusuallymakesaprofit?答：（1）是一个共同价值拍卖，因为奖品的价值对所有竞标者都是相同的（2）第18 技1．假定生产函数为f(xxxx2 1Considertheproductionfunctionf(x1,x2)=x21x22.Doesthisexhibitconstant,ordecreasingreturnstof(tx,tx)Considertheproductionfunctionf(x1,x2)=x21x22.Doesthisexhibitconstant,ordecreasingreturnstof(tx,tx)t2x2t2x2tfx,xtfx,x4 12 12．假定一生产函数为f(xx)x2 1Considertheproductionordecreasingreturnsto.Doesthisexhibitconstant,111 f(tx,tx)4txt t6f(x1,x2)tf(x1,x2223 3．柯布一道格拉斯生产函数为f(x1x2)Axaxb。其规模报酬的情况取决于a+b1a+b值分别是多少TheCobb-Douglasproductionfunctionisgivenbyf(x1,x2)=Axa1xb2.Itturnsoutthetypeofreturnstoscaleofthisfunctionwilldependonthemagnitudeofa+b.Whichvaluesofa+bwillbeassociatedwiththedifferentkindsofreturnstoscale?答：利用2题的方法，给定t>1，可得f(tx,tx) axtabf(x1x2 13x2Thetechnicalrateofsubstitutionbetweenfactorsx2andx1is-4.Ifyoudesiretoproducethesameamountofoutputbutcutyouruseofx1by3units,howmanymoreofx2willyou答：12xMRSdx221Trueorfalse?Ifthelawofdiminishingmarginalproductdidnothold,theworld’ssupplycouldbegrowninaInaproductionprocessisitpossibletohavedecreasingmarginalproductinanandyetincreasingreturnstoInaproductionprocessisitpossibletohavedecreasingmarginalproductinanandyetincreasingreturnsto时规模报酬仍然可能出现递增比如柯布—道格拉斯生产函数fx,x 1/22/ 22123/22/1/24/ 0 11xx4912出递减，同时，对任意的t1f(txtx12x23)t7/6x1/2x2/ 第19 利润最大Intheshortrun,ifthepriceofthefixedfactorisincreased,whatwillhappentomaxpf(x，zwxrzmaxpf(x，zwxrzrz中。所以r的提高会引起利润的下降。Ifafirmhadeverywhereincreasingreturnstoscale,whatwouldhappentoitsprofitspricesremainedfixedandifitdoubleditsscaleoff(txtf(xx，其利润为pf(xwx企业生产规模扩大一倍后，其利润是pf(2x2wx2[pf(xwxIfafirmhaddecreasingreturnstoscaleatalllevelsofoutputanditdividedupintotwoequal-sizesmallerfirms,whatwouldhappentoitsoverallprofits？pwmaxpf(xwxpf(x*wx*pwmaxpf(xwxpf(x*wx*x*12[pf ) wx]pf(x)*** Agardenerexclaims:“Foronly$1inseedsI’vegrownover$20inproduce!”Besidesfactthatmostoftheproduceisintheformofzucchini,whatotherobservationswouldacynicaleconomistmakeaboutthissituation?Ismaximizingafirm’sprofitsalwaysidenticaltomaximizingthefirm’sstock（1）在一个具有完全确定性的环境中，企业的未来利润流是已知的，那么这些利润现值就是企业的价值，用公式表示，就是V(1r)6pMP1w11？IfpMP1>w1,thenshouldthefirmincreaseordecreasetheamountoffactor1inorderincreasemaxpf(x)pMPw 1果x2的价格下降，那么果x2的价格下降，那么x1的使用量会发生什么变化？该厂商的利润水平会有什么变化？Supposeafirmismaximizingprofitsintheshortrunwithvariablefactorx1andfixedfactorx2.Ifthepriceofx2goesdown,whathappenstothefirm’suseofx1?Whathappenstothefirm’slevelofprofits?的表达式可知w2的下降不会影响x1的使用量，但会提高它的利润Aprofit-maximizingcompetitivefirmthatismakingpositiveprofitsinlong-runequilibrium(may/maynot)haveatechnologywithconstantreturnstoscale.利润pf(xLwxL0xL而言，会选择t第20 成本最小Provethataprofit-maximizingfirmwillalwaysminimizecosts.maxpf(x)wxpf(x*)w对于给定的qf(x*)minws.t.f(x)pf(x')wx'pf(x*)wx*Ifafirmisproducingwherethesame2,whatcanitdotoreducecostsbutMP1/1MP2/MP1/1MP2/Supposethatacost-minimizingfirmusestwoinputsthatareperfectsubstitutes.Ifthetwoinputsarepricedthesame,whatdotheconditionalfactordemandslooklikefortheinputs?产量是q，那么该厂商的条件要素需求函数为：x10qx2qminx1x10,x2x1由于约束条件和目标函数式相同，所以不论x1x2的取值如何，厂商都可以达到最小成x1=0到qx2qx1。Thepriceofpaperusedbyacost-minimizingfirmincreases.Thefirmrespondstothischangebychangingitsdemandforcertaininputs,butitkeepsitsoutputconstant.Whathappenstothefirm'suseofpaper?wx0，其中x(x1,......xn)，w(w1 wn)不等式wx0依然成立，从而企业的产量可以维持不变。如果除纸以外的其他某些要素需求不变，即xiwi0，i=2,3……n有x1w10，由于w10x10变化(i)和要素需求变化xi，显示的成本最小化理论会导出什么不等式Ifafirmusesninputs(n>2),whatinequalitydoesthetheoryofrevealedIfafirmusesninputs(n>2),whatinequalitydoesthetheoryofrevealedcostminimizationimplyaboutchangesinfactorprices(△ωi)andthechangesinfactordemands(△xi)foragivenlevelofoutput?假设在q的产量，在q的产由（1）式-（2）式就得到21Whichofthefollowingaretrue?(1)Averagefixedcostsneverincreasewithoutput;averagetotalcostsarealwaysgreaterthanorequaltoaveragevariablecosts;(3)averagecostcanneverrisewhilemarginalcostsaredeclining.F ，所以AFC随着Q的增加而减少QACAVCAFCQQ2当02Q2TC(Q)3Q 2QQ2当02Q2TC(Q)3Q 2 3Q6当Q 121Q0Q1Q2QQ1 1 MC(Q)AC(Q)33Q3 Q36Q则当2Q3时，MCACAfirmproducesidenticaloutputsattwodifferentplants.Ifthemargin