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1、Assignment(5)231.SupposetheproductionfunctionisCobb-Doug1.asand/(x1,x2)=x12x22(1) Writeanexpressionforthemargina1.productofxatthepoint(x,X2)m12=1.Jy2x,2(2) Wi1.1.themargina1.productofxincrease,decreaseorremainconstantforsma1.1.increasesinx,ho1.dingxfixed?MP.Because=0,themargina1.productOfxiwi1.1.inc
2、reaseinx2.x24(5) Ca1.cu1.atethetechnica1.rateofsubstitutionbetweenxandx.TRSMP1MP23v12v 1/22 x x2(6) Doesthistechno1.ogyhavediminishingtechnica1.rateOfsubstitution?TRS1_2八Yes.Because-=xix2(,(xpx2)(t1.)So,thistechno1.ogydemonstratesincreasingreturnstosca1.e.accordingtoaproductionfunctionf(x)=4x,wherex
3、isthenumberofunitsofinput.Thecommodityse1.1.sfor$100perunit.Theinputcosts$50perunit.(1) Writedownafunctionthatstatesthefirm,sprofitasafunctionoftheamountofinput.=py-cx=400Vx-50X(2) Whatistheprofit-maximizingamountofinputandoutput?Howmanyprofitsdoesitmakewhenitmaximizesprofits?Theprofit-maximizingpro
4、b1.emis:max=4(X)x-5()xThefirst-orderconditionforitis:ddx=200/Jx-50=0=x=16x=6)=4x=16,%-S)=400-50x=800(3) Supposethatthefirmistaxed$20perunitofitsoutputandthepriceofitsinputissubsidizedby$10.Whatisitsnewinput1.eve1.?Whatisitsnewoutput1.eve1.?Howmuchprotdoesitmakenow?(Hint:Agoodwaytoso1.vethisistowrite
5、anexpressionforthefirm,sprofitasafunctionofitsinputandso1.vefortheprofit-maximizingamountofinput.)Theprofitfunctionis:乃=(-20)y-(c-1.()x=320yx-4()xTomaximizetheprofit,thefirst-orderconditionmustbesatisfied.Jd=1607-40=0=x=16几=16)=4=16,%*=i6)=320-40x=640(4) Supposethatinsteadofthesetaxesandsubsidies,th
6、efirmistaxedat50%ofitsprofits.Writedownitsafter-taxprofitsasafunctionoftheamountofinput.VVhatistheprofit-maximizingamountofoutput?Howmuchprofitdoesitmakeaftertaxes?Theprofitfunctionis:Tr=(Py-cx)(1.50%)=2()0Vx-25xThefirst-orderconditiontomaximizethisprofitis:ddx=1()()7-25=OnR=I6EX=I6)=4=16,mg)=200-25
7、x=4003. Comp1.etethetab1.eYTCFCVCAFCAVCACMC11000100900100900100090021800100170050850900800325001002400100/38002500/37004330010032002580082580054500100440020880900120066000100590025/32950/310001500780001007900100/77900/78000/720008104001001030012.51287.51300240091400010013900100/913900/914000/936004.
8、 PrimoFungi,sproductionfunctionisQ=8JK1.(PrimoFungi,sproductismushroom).Thefirmmustpayinputpricesof$3and$27perunit,forcapita1.and1.abor,respective1.y.P1.easeanswerthefo1.1.owingquestions:(1) VVhatisPrimoFungi,s1.ong-runcostfunction?Inthe1.ong-run,a1.1.inputfactorsarevariab1.e,sothecostfunctionis:1.C
9、=vK+w1.Costminimizationprob1.emcanbedepictedas:max1.C=3K+271.s.t.Q=SyK1.=ySo1.vingtheprob1.emneedsthiscondition:MP2XT,z21.,z21.v3znrMP1.2K1.,2Uy2Kw278屈=y248Sothe1.ong-runcostfunctionis:C1.(v,w,y)=Ct(3,27,y)=3K*+27=.(2) VVhatisPrimoFungi,sshort-runcostfunctionifitsinputofcapita1.isfixedatK=4?Bydefini
10、tionandthegivenassumptions,intheshortrun,capita1.isfixedatK=4.Thenshort-runproductionfunctionrequiresthatthe1.aborinputsmustsatisfiesy=84=161.Thustoproduceyunitsofproducts,The1.aborinputsmustbe1.*=(y16)3Further,theshort-runcostfunctionisG(3,27,y,N)=27+3K=12+-(3) ShowthatPrimoFungi,s1.ong-runsupp1.yc
11、urveismoree1.asticthanitsshort-runsupp1.ycurvewhenitsinputofcapita1.isfixedatK=4.Inthe1.ong-run,1.C=9/41.MC=9/4isa1.waysexisted,soits1.ong-runsupp1.ycurveisP=1.MC=9/4.Thenwecanca1.cu1.ateitse1.asticityr=.dPQIntheshort-run,SMC=27y/128AVC=27y/256fora1.1.y0,soshort-runsupp1.ycurveisP=SMC=27y/128.Ca1.cu
12、1.ateitse1.asticityandwegetdQP_dy_ P_ dP QP y128Pry12821y 1= 1 0So,1.ong-runsupp1.ycurveismoree1.asticthanshort-runcurve.(4) Showthatevenifthemarketofmushroomisperfect1.ycompetitive,PrinioF1.1.ngiiswi1.1.ingtosupp1.ymushroomwhenitsinputofcapita1.isfixedatK=4.BecauseSMC=27y128AVC=27y256isexistedfora1.1.y0,PrimoFungicangetsometprofit*todiminishitsfixedcost,whichisbetterthantoproducenothing.Evenifthemarketofmushroomisperfect1.ycompetitive,PrimoFungiiswi1.1.ingtosupp1.y.
