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For each of the circuits shown in Figure P14.22, assume

Electrical Engineering: Principles & Applications | 6th Edition | ISBN: 9780133116649 | Authors: Allan R. Hambley ISBN: 9780133116649 150

Solution for problem P14.22 Chapter 14

Electrical Engineering: Principles & Applications | 6th Edition

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Electrical Engineering: Principles & Applications | 6th Edition | ISBN: 9780133116649 | Authors: Allan R. Hambley

Electrical Engineering: Principles & Applications | 6th Edition

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Problem P14.22

For each of the circuits shown in Figure P14.22, assume that the op amp is ideal and find the value of vo. Each of the circuits has negative feedback, and the summing-point constraint applies. (a) + 2 k 2 mA + vo (b) + 3 k + 5 V + 1 mA + vo 2 k (c) + 3 k +

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Session 16 Tuesday,March1,2016 10:27AM Table1 L Q MPL APL 1 2 2 2 2 7 5 3.5 3 14 7 4.6 4 20 6 5 5 25 5 5 6 29 4 4.8 7 32 3 4.57 8 34 2 4.25 9 35 1 3.88 10 35 0 3.5 11 34 -1 3.09 Theobjectiveistomaximizerent. Wageis$20perday. Thepriceofashirtis$10. • Profitmaximizationandfirm'semploymentdecision: o Anentrepreneurintendingtoproduceagoodhasalotofdecisionstomake. • Whattoproduce • Howmuchtoproduce • Wheretolocatetheirproductionfacility • Howlargeaspacetorent • Howmanymachines • Howmanyworkers • Whattopayforeachtypeoffactor • Atwhatpricetoselltheirgood o Thefirmisacompetitiveproducerinalargemarketforboththeproducts(shirts) andfactors(labor). o Disregardallexceptthreedecisions: • Howmanyworkerstoemploy • Howmuchtopayeachworker • Atwhatpricetoselltheirgood Table2 L P*Q$ VMPL$ VAPL$ LaborCost$ Rent$ 1 20 20 20 20 0 2 70 50 35 40 30 3 140 70 46 60 80 4 200 60 50 80 120 5 250 50 50 100 160 6 290 40 48 120 170 7 320 30 45.7 140 180 8 340 20 42.5 160 180 9 350 10 38.8 180 170 10 350 0 35 200 150 11 340 -10 30.9 210 120 1 1:Unitsoflabor 2:Revenue:pricexquantity 3:pricexmpl 4:pricexvapl 5:laborcost:costofemployment 6:Laborcost-Revenue=rent • Rentsarewhatafirmmakes • Accordingtotheobjectiveofrentmaximization,howmanyworkersdoesthisfirmhire o Therule:Hirelaboruptowherethewageequalsthevalueofthemarginalproduct oflabor. o Hires8laborers. o AtVMPL=$20,there's8workers • AtL=2theoutputad dedfromemployingthe2ndunitoflaboris$50. • Thatis,the2ndworkeradds$whiletheworkercosts$20. • Thererentsincreaseby$30whenthe2ndworkerisadded. • Addinganotherworker,itscontributionis$70,whileitcosts$20,sorentsincreaseby$ 50. • WecontinuewiththisexpansiontillwegetL=8.Herethisworkerwillad$20anditwill cost$20. • Forthe9thUnitoflaborthislaboradds$10totheoutputwhileitcosts$20.Willthe9th workerbeemployedatthisfirmWhy o No,becauseiftheyemploythe9thperson,therentswillfall,by$10.There'saloss. Derivingafirm'soptionalchoiceoflaboremployment(forexpositiononly) Q=quantityofoutput L=quantityoflaborusedtoproducetheoutput P=Pricepershirt W=wageperunitof labor • Firm'sdecision:chooseLsoastomaximizerents=PQ -WL. • WeknowthatQ=f(L) • So,chooseLsoastomaximizerents=Pf(L) -WL • TakethederivativewithrespecttoLandequatetozero.ThensolveforoptimalL: o Pdf(L)/dL-W=0orPMPL=worVMPL=W WhynotusetheAveragerule • TheValueofAverageProductofthe9thworkersis$38.88.Thatis,asanaverage,the9th workerproduces$38.88whilethisworkercostsonly$20. • Iftheemployeraddsthe9thworker,thisworkeradds$ 10tothetotalvalueofoutput. • Onaverage,theemployergets$38.88ofoutputoutofeachofthe9workersandpays each$20sobyemployingthe9thworker,theemployergetsalmost$19ofbenefit/rent. • However,thearrivalofthe9thworkerslowsdownt herestoftheworkers. • Thecostoftheslowdownoftheprevious8workersis($2.50 -$38.88)*8andisequalto $28.96. • Sotheemployer,byaddingthe9thworker,gets$19ofbenefitbutloses$29.Helosesin net$29-$19-$10.Thisisindeedthereduction inrents(from$180to$170). Derivingthedemandscheduleforlabor(byafirm): • Atwageof$20,wedecidedthattherentmaximizingfirmwouldhire 8workers. • Howmanyworkerswillthefirmhireifthewagegoesdownto$10perday o 9workers • Howmanyworkerswillthefirmhireifthewagegoesupto$40perday 20 10 Anexampleoftheuseofmarginalanalysisinproductionandemploymentoflabor: • JaneandJillaretwinsisters.TheyhavegraduatedwiththesameGPAandthesameskills fromthesamedepartment. • Assumethatthegoingwageforfreshgraduatesfromthisdepartsis$40Kayear. • Janegoesdowntowntoafurnituredesignfirmtoapplyforajob.Shecomeshomea nd tellsJillthatshegotthejob. • Nextday,Jillgoestothesamefirm(sheisasgoodasJane)andsheisturneddown. • Why • WhatisthevalueofemployingJanetothefirm(whatistheVMPLofemployJane) o $40K. • WhatisthevalueofemployingJillIs ithigherorlowerthan$40K o Lessthan$40K. • Animportantinsight:Whyisitthatourskilledgraduatesdonotreceive5millionjob offersupongraduation o ItisduetothefallingMPL.IftheMPLwasnotfallingsay,ifitwasincreasing,then firmswouldhaveinfinitedemandforworkers. Allocationofresourcesunderprivateproperty • Supposethereisapieceoflandownedbysomeone.Theownerhiresworkerstoworkon thelandtoproducefood.Thegoingwageis$20perworkerperday.Thepriceofa unitof foodis$10.Thetableoflaborinputsandoutputsisproducedbelow. L Output(Q) 1 2 2 7 3 14 4 20 5 25 6 29 7 32 8 34 9 35 10 35 11 34 12 32 13 30 14 28 15 25 Table4-Decisionofthefirmunderprivateproperty L P*Q($) VMPL($) VAPL($) LaborCost($) Rent($) 1 20 20 20 20 0 2 70 50 35 40 30 3 140 70 46 60 80 4 200 60 50 80 120 5 250 50 50 100 150 6 290 40 48 120 170 7 320 30 45.7 140 180 8 340 20 42.5 160 180 9 350 10 38.8 180 170 10 350 0 35 200 150 11 340 -10 30.9 220 120 12 320 -20 26.67 240 80 13 300 -20 25 260 40 14 280 -20 20 280 0 15 250 -30 16.67 300 -50 • Howmanyworkersdoesthisfirm -withprivatepropertyrights -hire o 8 • The$20wagemeansthata workerisabletoproduce$20worthofgoodsperday somewhereelseintheeconomy. • Thereforewhenthisindividualdecidestoworkfor$20atafarm,itmeansthat$20worth ofothergoodsdisappearelsewhereintheeconomy. • Underprivateproperty,thefa ctory/businessowneremploysworkersuptowherethe contributionofthelastworkeremployedinhisbusinessisnolessthantheywouldhave producedelsewhereintheeconomy(whichis$20perday). Considerthe6thworker: 6thworker: Shoes$20 WheatFarm$40 7thWorker: Shoes$20 WheatFarm$30 8thWorker Shoes$20 WheatFarm$20 9thWorker Shoes$20 WheatFarm$10 Session 17 Thursday,March3,2016 10:29AM Thebehavioroffirmsunderdifferentsystemsofpropertyrights:inallocatingresources,the systemofpropertyrightsmattersagreatdeal. • Underprivateproperty,rentsaremaximized.Howdoesthisprocessalsoensureefficient allocationofresourcesfortheeconomy o The$20wagemeansthataworkerisabletoproduce$20ofgoodsperday somewhereelseintheeconomy. o Whenthisindividualdecidestoworkfor$20atafarm,itmeansthat$20worthof othergoodsdisappearelsewhereintheeconomy. o Thewageraterepresentstheopportunitycostofthislabortothesociety. • Rentsrepresentthenetgainsinthevalueofoutput -requiringuseofscarceresources-in wheatproductionoverandabovethesameresourcesbeingusedelsewhere. ProductionandInputUseUnderCommonProperty • Consideragainapieceoflandonwhich foodisproduced.Now,assumethereisno "owner"andnoonecanbedeniedaccesstothisland.Eachworkergetsanequalshare - theValueofAverageProductofLabor.Goingwageis$20perworkerperday. • Priceis$10perbushelofwheat: L P*Q($) VMPL($) VAPL($) LaborCost($) Rent($) 1 20 20 20 20 0 2 70 50 35 40 30 3 140 70 46 60 80 4 200 60 50 80 120 5 250 50 50 100 150 6 290 40 48 120 170 7 320 30 45.7 140 180 8 340 20 42.5 160 180 9 350 10 38.8 180 170 10 350 0 35 200 150 11 340 -10 30.9 220 120 12 320 -20 26.67 240 80 13 300 -20 25 260 40 14 280 -20 20 280 0 15 250 -30 16.67 300 -50 • Undernoprivateproperty,howmanyworkerswillbeworkingonthislandWhatarethe rents o 14workers o Here,theVAPLis$20whichisequaltothewage. o Rentsare$0. • Whyarerents" dissipated"undercommonproperty o Excessoutputisgone. o The$20wagemeansthataworkerisabletoproduce$20worthofgoods somewhereelse. th • Forthe10 worker,thisworkerdoesnotaddanythingtooutputonthefarmandbybeing there,$20worthofothergoodsandservicesarenotbeingproduced.Sonetlossforthe 10 workeris$20. o AtL=11,VMPL=$-10,netlossis$30 o AtL=12,VMPL=$-20,netlossis$40 o AtL=13,VMPL=$-20,netlossis$40 o AtL=14,VMPL=$-20,netlossis$40 • Soaddingallthenetlosses:$180. • ThisisthemaximumrentthatcouldhavebeenearnedbychoosingworkerswhereVMPL =W.Thatis,inthesystemofcommonproperty,"dissipation"ofrentstakesplace.The rentrepresentsthenetgaininthevalueofoutput -requiringuseofscarceresources-in anactivityoverandabovethesameresourcesusedelsewhereintheeconomy.Thisnet gainisdissipatedbytheabsenceofpropertyrights. • TheMPLdiffersfromtheAPLbecauseMPLtakesintoaccounttheimpactof adding anotherworkeronpreviousworkers'averageproduct. • Ifthisimpactisignored,thenindividualsdonotownthefullconsequencesoftheiractions ,thisisthesourceoflostgainsfromtrade. • Anapplicationofthecommonpropertyoutcome:Whyis thatsalmonisarguedandfeared tobeonthebrinkofextinctionbutnotbeeforchicken ClickerQuestion:Undercommonproperty,workersaddthemselvesinuntilVAPL=W.Atthis levelofemploymentwagerentsfalltozerosinceworkersignoretheir pactonotherworkers' work. ProductionandInputUseunder"SocialistCooperatives":theworkers -whocollectivelyownthe land-decidewhocanjoinin.Eachworkergetsanequalshare. • P=$10 • W=$20 L P*Q($) VMPL($) VAPL($) LaborCost($) Rent($) 1 20 20 20 20 0 2 70 50 35 40 30 3 140 70 46 60 80 4 200 60 50 80 120 5 250 50 50 100 150 6 290 40 48 120 170 7 320 30 45.7 140 180 8 340 20 42.5 160 180 9 350 10 38.8 180 170 10 350 0 35 200 150 11 340 -10 30.9 220 120 12 320 -20 26.67 240 80 13 300 -20 25 260 40 14 280 -20 20 280 0 15 250 -30 16.67 300 -50 • Howmanyworkerswillbeworkingtheland o Only5workers o That'swherethevalueoftheaverageproductisthehighest o Therentsarenotmaximized. • Therentsare$150. • Butthe6thworkercouldadd $40onthislandandisonlyproducing$20elsewhere.The netgaintoconsumersof$20islostbyexcludingthisworker. • Thoseonthefarmwillget$50andotherworkersqueueuptogettherighttoworkthere andearnsomeofthe"rents" capturedbythemembersofcooperatives. • Sincecompetitionisfierce,somerationingtakesplace.Themembersofthecooperative havetodecidewhogetsinandwhodoesnot.Asextraresourcesarespenttogetin,a dissipationofrentstakesplace. ClickerQuestion:Whenworkers'socialistcooperativescontrolthelevelofemployment,they employworkerssuchthateachworkerispaidthehighestVAPL.Atthislevel,toofewworkers areemployed. Section7.4ApplicationsoftheLawofDiminishingMargin alProduct:Allocatingtwoormore variableinputsinproduction: • Withonlyoneinputchoice,welearnedthattherent(benefit)maximizingruleistohire laboruptowhere: o VMPL=WageoflabororP*MPL=W L • Now,supposethefirmalsoneedstomakedecisio nsaboutcapital(Machinery).INthis casetheruleformaximumbenefitfromemploymentofmachinesis:buyequipmentupto where o VMPK=WageofcapitalorP*MPK=W k • Theabovetworelationshipsimply: o P*MPL/W -P*MPK/W L K • Benefitsaremaximizedwhenemployme ntoflaborandcapitaltakeplaceuptowhere: themarginalbenefitofthelast$spentonlabor -themarginalbenefitofthelast$spent oncapital. • Forsimplicity,let'sassumewageforbothfactorsis$1.Thentheaboverulebecomes: employunitsoflaborandcapitaluptowhereMPL=MPK • INthissimplifiedcase,theruleistoemployLandKtillthemarginalproductivityoflabor employed=themarginalproductivityofcapitalemployed. • Ifresourcesareemployedsuchthat,MPL>MPK,itimpliesthat additiontooutputofone moreunitoflaboremployedarehigherthanthebenefitsofonemoreunitofcapital. Example • Supposeaclothingfirmcurrentlyemploys4.5hoursofdesigners/tailorstimeand4.5 hoursofadriver'stimeperday. • Thefirmhastotalof9hourofresourcestoemployperday. • Anadditionalhalfanhourofdesignerworkproduces$100worthofoutputwhilean additionalhalfhourofthedrivertimeproduces$150worthofoutput/service. • Shouldthisfirmchangetheallocationofitsr esources o Yes.Byreallocatinghalfanhourfromdesignertodriver,thefirmislosing$100but gaining$150,increasingitsrevenues. o Therefore,whereMPL>MPK,thefirmneedstomakeadjustmentsinitsemployment ofthesefactors. o Whereanemploymentof factorsaresuchthatMPL-MPK,nofurtheradjustmentis necessary(thisiswherethetotalbenefitsfromemploymentofbothfactorsarethe highest.) Example1: InM*A*S*HunitsduringtheKoreanwar,doctorsandnurseshadtodecidehowmuchattention togivetoeachwoundedsoldier.Thegoal(implicitly)istomaximizehumanlife. Theyspendtheirtimeandeffortonpatientssuchthat: Marginalbenefits(ofanhourofsurgery)onpatient1=Marginalbenefits(ofanhourofsurgery) onpatient2andsoon… Ifapatientistooseverelyinjuredandrequires10hoursofsurgerytosavehislife,thedoctors mayoptfornottreatinghim.Inthose10hours,theycouldsaveperhaps5lives.Inthismanner, themarginalbenefit(MB)ofspendingefforton theveryseverelywoundedislessthantheMB ofresourcesspentonthelessseverelywoundedpatients. Example2: Incasinos,itisthecasethatthe$machinesaremoreprofitablethannickelmachines.Thatis, theaverageprofitper$machineishighe rthantheaverageprofitpernickelmachine.Should thecasinoonlyinstall$machines SupposetheDollarmachineandNickelmachineseachcost$1000.Thetable1and2providethe marginalandaverageprofitsfor$andfornickelmachines. Supposethiscasinohasspacefor5machines. Shoulditinstallonly$machines,onlynickelmachines,oracombinationofthetwo

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Textbook: Electrical Engineering: Principles & Applications
Edition: 6
Author: Allan R. Hambley
ISBN: 9780133116649

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For each of the circuits shown in Figure P14.22, assume