斯坦福大学经济数学讲义

UsefulMathforMicroeconomics¤JonathanLevinAntonioRangelSeptember20011IntroductionMosteconomicmodelsarebasedonthesolutionofoptimizationproblems.Thesenotesoutlinesomeofthebasictoolsneededtosolvetheseproblems.Itiswortspendingsometimebecomingcomfortablewiththem—youwillusethemalot!Wewillconsiderparametricconstrainedoptimizationproblems(PCOP)oftheformmaxx2D(µ)f(x;µ):Herefistheobjectivefunction(e.g.profits,utility),xisachoicevariable(e.g.howmanywidgetstoproduce,howmuchbeertobuy),D(µ)isthesetofavailablechoices,andµisanexogeneousparameterthatmayaffectboththeobjectivefunctionandthechoiceset(thepriceofwidgetsorbeer,orthenumberofdollarsinone’swallet).Eachparameterµdefinesaspecificproblem(e.g.howmuchbeertobuygiventhatIhave$20andbeercosts$4abottle).IfweletΘdenotethesetofallpossibleparametervalues,thenΘisassociatedwithawholeclassofoptimizationproblems.Instudyingoptimizationproblems,wetypicallycareabouttwoobjects:1.Thesolutionsetx¤(µ)´argmaxx2D(µ)f(x;µ);¤ThesenotesareintendedforstudentsinEconomics202,StanfordUniversity.TheywereoriginallywrittenbyAntonioinFall2000,andrevisedbyJoninFall2001.LeoRezendeprovidedtremendoushelpontheoriginalnotes.Section5drawsonanexcellentcomparativestaticshandoutpreparedbyIlyaSegal.1thatgivesthesolution(s)foranyparameterµ2Θ.(Iftheproblemhasmultiplesolutions,thenx¤(µ)isasetwithmultipleelements).2.ThevaluefunctionV(µ)´maxx2D(µ)f(x;µ)thatgivesthevalueofthefunctionatthesolutionforanyparameterµ2Θ(V(µ)=f(y;µ)foranyy2x¤(µ):)Ineconomicmodels,severalquestionstypicallyareofinterest:1.Doesasolutiontothemaximizationproblemexistforeachµ?2.Dothesolutionsetandthevaluefunctionchangecontinuouslywiththeparameters?Inotherwords,isitthecasethatasmallchangeintheparametersoftheproblemproducesonlyasmallchangeinthesolution?3.Howcanwecomputethesolutiontotheproblem?4.Howdothesolutionsetandthevaluefunctionchangewiththeparam-eters?Youshouldkeepinmindthatanyresultwederiveforamaximizationproblemalsocanbeusedinaminimizationproblem.Thisfollowsfromthesimplefactthatx¤(µ)=argminx2D(µ)f(x;µ)()x¤(µ)=argmaxx2D(µ)¡f(x;µ)andV(µ)=minx2D(µ)f(x;µ)()V(µ)=¡maxx2D(µ)¡f(x;µ):2NotionsofContinuityBeforestartingonoptimization,wefirsttakeasmalldetourtotalkaboutcontinuity.Theideaofcontinuityisprettystraightforward:afunctionhiscontinuousif“small”changesinxproduce“small”changesinh(x).Wejustneedtobecarefulabout(a)whatexactlywemeanby“small,”and(b)whathappensifhisnotafunction,butacorrespondence.22.1ContinuityforfunctionsConsiderafunctionhthatmapseveryelementinXtoanelementinY,whereXisthedomainofthefunctionandYistherange.Thisisdenotedbyh:X!Y.WewilllimitourselvestofunctionsthatmapRnintoRm,soXµRnandYµRm.Recallthatforanyx;y2Rk,kx¡yk=sXi=1;:::;k(xi¡yi)2denotestheEuclideandistancebetweenxandy.Usingthisnotionofdistancewecanformallydefinecontinuity,usingeitheroffollowingtwoequivalentdefinitions:Definition1Afunctionh:X!Yiscontinuousatxifforevery0thereexists±0suchthatkx¡yk±andy2X)kh(x)¡h(y)k.Definition2Afunctionh:X!YiscontinuousatxifforeverysequencexninXconvergingtox,thesequenceh(xn)convergestof(x).Youcanthinkaboutthesetwodefinitionsasteststhatoneappliestoafunctiontoseeifitiscontinuous.Afunctioniscontinuousifitpassesthecontinuitytestateachpointinitsdomain.Definition3Afunctionh:X!Yiscontinuousifitiscontinuousateveryx2X.Figure1showsafunctionthatisnotcontinuous.Considerthetoppic-ture,andthepointx.Takeanintervalcenteredaroundh(x)thathasa“radius”.Ifissmall,eachpointintheintervalwillbelessthanA.Tosatisfycontinuity,wemustfindadistance±suchthat,aslongaswestaywithinadistance±ofx,thefunctionstayswithinofh(x).Butwecannotdothis.Asmallmovementtotherightofx,regardlessofhowsmall,takesthefunctionabovethepointA.Thus,thefunctionfailsthecontinuitytestatxandisnotcontinuous.Thebottomfigureillustratestheseconddefinitionofcontinuity.Tomeetthisrequirementatthepointx,itmustbethecasethatforeverysequencexnconvergingtox,thesequenceh(xn)convergestoh(x).Butconsider36-qaqx2±qh(x)2qA6-qaqxqh(x)qAqzn¾qyn-q³)qh(zn)?q©*qh(yn)6Figure1:TestingforContinuity.4thesequenceznthatconvergestoxfromtheright.Thesequenceh(zn)convergestothepointAfromabove.SinceAh(x),thetestfailsandhisnotcontinuous.Weshouldemphasizethatthetestmustbesatisfiedforeverysequence.Inthisexample,thetestissatisfiedforthesequenceynthatconvergestoxfromtheright.Ingeneral,toshowthatafunctioniscontinuous,youneedtoarguethatoneofthetwocontinuitytestsissatisfiedateverypointinthedomain.Ifyouusethefirstdefinition,thetypicalproofhastwosteps:²Step1:Pickanyxinthedomainandany0.²Step2:Showthatthereisa±x()0suchthatkh(x)¡h(y)kwheneverkx¡yk±x().Toshowthisyouhavetogiveaformulafor±x(¢)thatguaranteesthis.Theproblemsattheendshouldgiveyousomepracticeatthis.2.2ContinuityforcorrespondencesAcorrespondenceÁmapspointsxinthedomainXµRnintosetsintherangeYµRm.Thatis,Á(x)µYforeveryx.ThisisdenotedbyÁ:X¶Y.Figure2providesacoupleofexamples.Wesaythatacorrespondenceis:²non-empty-valuedifÁ(x)isnon-emptyforallxinthedomain.²convexifÁ(x)isaconvexsetforallxinthedomain.²compactifÁ(x)isacompactsetforallxinthedomain.Fortherestofthesenotesweassume,unlessotherwisenoted,thatcorre-spondencesarenon-e