北京大学CCER-中级微观经济学-尼克尔森教材课后习题答案(2-21章)

1CHAPTERCHAPTERCHAPTERCHAPTER2222THETHETHETHEMATHEMATICSMATHEMATICSMATHEMATICSMATHEMATICSOFOFOFOFOPTIMIZATIONOPTIMIZATIONOPTIMIZATIONOPTIMIZATIONTheproblemsinthischapterareprimarilymathematical.TheyareintendedtogivestudentssomepracticewithtakingderivativesandusingtheLagrangiantechniques,buttheproblemsinthemselvesofferfeweconomicinsights.Consequently,nocommentaryisprovided.Alloftheproblemsarerelativelysimpleandinstructorsmightchoosefromamongthemonthebasisofhowtheywishtoapproachtheteachingoftheoptimizationmethodsinclass.SolutionsSolutionsSolutionsSolutions2.122(,)43Uxyxya.86UU=x,=yxyb.8,12c.86UUdUdx+dy=xdx+ydyxyd.for0860dydUxdxydydx8463dyxx==dxyye.1,2413416xyUf.4(1)2/33(2)dydxg.U=16contourlineisanellipsecenteredattheorigin.Withequation,slopeofthelineat(x,y)is.224316xy43dyxdxy2.2a.Profitsaregivenby2240100RCqq*44010dqqdq2*2(1040(10)100100)b.soprofitsaremaximized224ddqc.702dRMRqdq230dCMCqdq2222SolutionsSolutionsSolutionsSolutionsManualManualManualManualsoq*=10obeysMR=MC=50.2.3Substitution:21soyxfxyxx120fxx0.50.5,0.25x=,y=f=Note:.Thisisalocalandglobalmaximum.20fLagrangianMethod:?1)xyxy£=y=0x£=x=0yso,x=y.usingtheconstraintgives0.5,0.25xyxy2.4SettinguptheLagrangian:.?0.25)xyxy£1£1yxxySo,x=y.Usingtheconstraintgives.20.25,0.5xyxxy2.5a.2()0.540ftgtt.*40400,dfgttdtgb.Substitutingfort*,.*2()0.5(40)40(40)800ftgggg.*2()800ftggc.dependsongbecauset*dependsong.2*1()2ftgso.*222408000.5()0.5()ftgggd.,areductionof.08.Noticethat8003225,80032.124.92soa0.1increaseingcouldbepredictedtoreduce22800800320.8gheightby0.08fromtheenvelopetheorem.ChapterChapterChapterChapter2/The2/The2/The2/TheMathematicsMathematicsMathematicsMathematicsofofofofOptimizationOptimizationOptimizationOptimization33332.6a.Thisisthevolumeofarectangularsolidmadefromapieceofmetalwhichisxby3xwiththedefinedcornersquaresremoved.b..Applyingthequadraticformulatothisexpressionyields22316120Vxxttt.Todeterminetrue22162561441610.60.225,1.112424xxxxxtxxmaximummustlookatsecondderivative--whichisnegativeonly221624Vxttforthefirstsolution.c.IfsoVincreaseswithoutlimit.33330.225,0.67.04.050.68txVxxxxd.ThiswouldrequireasolutionusingtheLagrangianmethod.Theoptimalsolutionrequiressolvingthreenon-linearsimultaneousequations—atasknotundertakenhere.Butitseemsclearthatthesolutionwouldinvolveadifferentrelationshipbetweentandxthaninpartsa-c.2.7a.SetupLagrangianyieldsthefirstorderconditions:1212?ln()xxkxx12212£10?0£0xxxkxxHence,.Withk=10,optimalsolutionis2215or5xx125.xxb.Withk=4,solvingthefirstorderconditionsyields215,1.xxc.OptimalsolutionisAnypositivevalueforx1reducesy.120,4,5ln4.xxyd.Ifk=20,optimalsolutionisBecausex2providesadiminishing1215,5.xxmarginalincrementtoywhereasx1doesnot,alloptimalsolutionsrequirethat,oncex2reaches5,anyextraamountsbedevotedentirelytox1.2.8Theproofismosteasilyaccomplishedthroughtheuseofthematrixalgebraofquadraticforms.See,forexample,MasColelletal.,pp.937–939.Intuitively,becauseconcavefunctionsliebelowanytangentplane,theirlevelcurvesmustalsobeconvex.Buttheconverseisnottrue.Quasi-concavefunctionsmayexhibit“increasingreturnstoscale”;eventhoughtheirlevelcurvesareconvex,theymayriseabovethetangentplanewhenallvariablesareincreasedtogether.2.9a.11210.fxx11220f.xx4444SolutionsSolutionsSolutionsSolutionsManualManualManualManual21111(1)0.fxx21222(1)0.fxx111212210.ffxxClearly,allthetermsinEquation2.114arenegative.b.If12ycxxsinceα,β0,x2isaconvexfunctionofx1./1/21xcxc.Usingequation2.98,222222222221122111222(1)()(1)fffxxxx=whichisnegativeforα+β1.222212(1)xx2.10a.Since,thefunctionisconcave.0,0yyb.Because,and,Equation2.98issatisfiedandthe1122,0ff12210fffunctionisconcave.c.yisquasi-concaveasis.Butisnotconcaveforγ1.Alloftheseresultsyycanbeshownbyapplyingthevariousdefinitionstothepartialderivativesofy.5CHAPTERCHAPTERCHAPTERCHAPTER3333PREFERENCESPREFERENCESPREFERENCESPREFERENCESANDANDANDANDUTILITYUTILITYUTILITYUTILITYTheseproblemsprovidesomepracticeinexaminingutilityfunctionsbylookingatindifferencecurvemaps.TheprimaryfocusisonillustratingthenotionofadiminishingMRSinvariouscontexts.Theconceptsofthebudgetconstraintandutilitymaximizationarenotuseduntilthenextchapter.CommentsCommentsCommentsCommentsononononProblemsProblemsProblemsProblems3.1Thisproblemrequiresstudentstographindifferencecurvesforavarietyoffunctions,someofwhichdonotexhibitadiminishingMRS.3.2Introducestheformaldefinitionofquasi-concavity(fromChapter2)tobeappliedtothefunctionsinProblem3.1.3.3ThisproblemshowsthatdiminishingmarginalutilityisnotrequiredtoobtainadiminishingMRS.Allofthefunctionsaremonotonictransformationsofoneanother,sothisproblemillustratesthatdiminishingMRSispreservedbymonotonictransformations,butdiminishingmarginalutilityisnot.3.4Thisproblemfocusesonwhethersomesimpleutilityfunctionsexhibitconvexindifferencecurves.3.5Thisprob