宏觀經(jīng)濟(jì)學(xué)(英文版)課件Chapter04-IM1e

Chapter4

DeterminingAggregateProduction

BriefChapterSummary

4.1TheAggregateProductionFunction(pages106–116)

LearningObjective1

Describetheaggregateproductionfunction.

?Long-runincreasesinrealGDPpercapitacanbeexplainedbyanaggregateprodufunction,whichisanequationthatshowstherelationshipbetweeninputsemployedfirmsandthemaximumoutputtheinputscanproduce.TheCobb-Douglasproductio

functiondoesagoodjobofaccountingforU.S.dataonthegrowthofrealGDP.

4.2AModelofRealGDPintheLongRun(pages116–120)

LearningObjective2

ExplainhowrealGDPisdeterminedinthelongrun.

?Amodelofasingle,representativefirmcanbeusedtoexplainhowallfirmschoose

inputsinordertoproducetheirprofit-maximizingamountsofoutputs.Theequilibrium

quantitiesofinputscanbecombinedwiththeaggregateproductionfunctiontodeterminethelong-runequilibriumlevelofrealGDP.

4.3AccountingforGrowthinRealGDP(pages120–126)

LearningObjective3

Discussthecontributionsofcapital,labor,andefficiencytothegrowthrateofreal

GDP.

?Growthaccountingallowsustodeterminehowmuchofthegrowthrateofrealduetolabor,capital,andtotalfactorproductivity.Themostimportantdeterminantseconomicgrowtharefactorssuchaschangesintechnology,increasesinlaborskills,improvementsinmanagerialtechniques.

4.4GDPperHourWorkedAmongCountries(pages127–131)

LearningObjective4

Understandthatdifferencesinefficiencyexplaindifferencesinlaborproductivity

acrosscountries.

?DifferencesintotalfactorproductivityhelptoexplainthedifferencesinGDPperworkedacrosscountries.Totalfactorproductivityincludestheeffectofknowledge,

governmentinstitutions,macroeconomicpolicies,geography,andthequalityofthelaborforce.

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CHAPTER4|DeterminingAggregateProduction53

4ATheCobb–DouglasProductionFunctionandConstantReturnsto

Scale(pages140–142)

LearningObjective4A

DeriveimportantpropertiesoftheCobb–Douglasproductionfunction.

?TheCobb-Douglasproductionfunctionhasseveralusefulcharacteristics,includingconstantreturnstoscale.

ListofKeyTerms

Aggregateproductionfunct,ionp.107.AnequationthatshowstherelationshipbetweentheinputsemployedbyfirmsandthemaximumoutputfirmscanproducewiththoseCapital-laborratop.123.Thedollarvalueofcapitalgoodsperunitoflabor;measureeitherthedollarvalueofcapitaldividedbythetotalnumberofhoursworked,orasdollarvalueofcapitaldividedbythetotalnumberofworkers.

Cobb-Douglasproductionfunct,ionp.108.Aproductionfunctionthattakestheform

Y=AKaL1a.

Constantreturnsto,scapl.e107.Apropertyofaproductionfunctionsuchthatifallinincreasebythesamepercentage,realGDPincreasesbythesamepercentage.

MarginalproductofcapMitKl),(p.109.Theextraoutputafirmreceivesfromaddingonemoreunitofcapital,holdingallotherinputsandefficiencyconstant.

Marginalproductofla),(p.111.Theextraoutputafirmreceivesfromaddingonmoreunitoflabor,holdingallotherinputsandefficiencyconstant.

Profit,p.116.Totalrevenueminustotalcost.

TotalfactorproductiviyFP)p.108.AnindexoftheoveralllevelofefficiencyoftransformingcapitalandlaborintorealGDP.

ChapterOutline

THESURPRISINGECONOMICRISEOFINDIA

Between1990and2010,realGDPinIndiagrewatanaverageannualrateof4.7%,well

abovetheaveragegrowthrateof1.4%experiencedbytheUnitedStates.Aftercenturipoverty,Indiahasbeguntoclosethegapbetweenitsstandardoflivingandthestalivinginhigh-incomecountriessuchastheUnitedStatesandthecountriesofWestern

Europe.ButIndiaremainsapoorcountry,withapopulationover1.2billion,morethofwhomareemployedinagriculture.Still,therapidgrowththatbeganinthe1990s

provideshopeforabetterlife.Explainingeconomicgrowthcanbecomplex.ThischafocusesonafewkeyideasthatexplainwhatdeterminesthelevelofGDPinthelon

TeachingTips

KeyIssueR:ealGDPhasincreasedsubstantialvyertimeiheUnitedStatesandother

developedcountries.

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KeyQuestio:nWhatarethemainfactorsthatdeterminethegrowthrateofrealGDP?

4.1TheAggregateProductionFunction(pages106–116)

LearningObjective:Describetheaggregateproductionfunction.

Thelong-runincreaseinrealGDPpercapitaisthebestmeasureofthequantity

andservicesavailableperperson.ThechapterbeginstodevelopamodeltoexplainrealGDPandthenadjuststhemodeltoexplainrealGDPpercapita.

From1949to2010,theaverageannualgrowthrateofrealGDPintheUnitedSt

3.3percent.The’sfiprrmoductionfunctionistherelationshipbetweentheinputsemployedbyafirmandthemaximumoutputthefirmcanproducewiththoseinputs.Afirm’s

technologyistheprocessesitusestoturninputsintooutputsofgoodsandservices.

Onamacroeconomiclevelwecansaythattheaggregateproductionfunctiiosnanequationthatshowstherelationshipbetweentheinputsemployedbyfirmsandthemaximumoutputfirmscanproducewiththoseinputs.Atthemacroeconomiclevel,we

measureoutputasrealGDPandincludeonlylaborandcapitalasinputs.Ageneral

oftheaggregateproductioncanbewrittenas:

Y(K,L),oF(K,L),

where

Y=realGDP

K=quantityofcapitalgoodsavailabletocafpitma,sorckthe

L=quantityoflabor

A=indexofhowefficientlytheeconomytransformscapitalandlaborintorealTechnology,governmentregulationsandinstitutions,thequalityofthelaborforce,andteffectofan’tiogneographyaresomeofthefactorsthatcanafe.ctthevalueof

A.TheCobb–DouglasProductionFunction

Theeconomicgrowthmodelinthischapterassumesthattheproductionfunctionhas

constantreturnstoslnes.tantreturnstoscaeapropertyofaproductionfunctionsuchthatifallinputsincreasebythesamepercentage,realGDPincreasesbythesamepercentage.MacroeconomistsoftenworkwithaparticularproductionCft–ion.TheDouglasproductionfunctionisaproductionfunctionthattakehKeαL1,rm

where:

α=capistal’shareinnationalincome

1–α=laor’shareinnationalincome

ForaCobb–Douglasproductionfunctionwithconstantreturnstoscale,theexponentsof

thecapitalandlaborvariablessumtoone.ThisfunctiondoesagoodjobofaccouU.S.dataonthegrowthofrealGDP.Since1949,theshareofincomegoingtothecapitalintheU.S.hasbeenaboutone-thirdandtheaverageshareofincomegoinghasbeenabouttwo-thirds.Equation(4.1)i–sDotuhgelasCooductionfunctionwithα

havingavalueofone-third:

YK?L?(4.1)

Inthisfunctini,scalledthetotalfactorproductivity.fchtrproductivity

(TFP)isanindexoftheoveralllevelofefficiencyoftransformingcapitalandlabor

GDP.Ifcapital,labor,ortotalfactorincrease,realGDPalsoincreases.Graphsofthe

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CHAPTER4|DeterminingAggregateProduction55

productionfunctioncanbedrawnwith)ealonGDte(verticalaxisandeitherthevalue

ofcapitalK)(orlabnthehorizontalaxis.Byshowingtheproductionfunctionthis

way,totalfactorproductiAvtyan(deitherlabororcapitalareheldconstant.

B.TheMarginalProductsofCapitalandLabor

Therearethreeimportantfeaturesoftheproductionfunctiongraphthatholdstotalfaproductivityandlaborconstant.First,theproductionfunctionslopesupward.The

marginalproductofcapMitlK)stheextraoutputafirmreceivesfromaddingone

moreunitofcapital,holdingallotherinputsandefficiencyconstant.FortheCobb-DouproductionfunctionofEquation(4.1),themarginalproductofcapitalis:

MPK=|3||K|.(4.2)

(1(Y

Second,theproductionfunctionflattensoutasthecapitalstockincreases.Thatis,

slopeoftheproductionfunctiondecreasesasthecapitalstockincreases.Thisflattening

occursbecauseofdiminishingmarginalreturns.Holdinglaborandefficiencyconstant,thmarginalproductofcapitaldecreasesasthecapitalstockincreases.Thethirdfeatureo

productionfunctionistherelationshipbetweenthemarginalproductofcapitalandtheslopeoftheproductionfunction.Thisslopeis:

Change爄n爎eal燝DP

.

Change爄n燾apital爏tock

Thisisanotherexpressionforthemarginalproductofcapital.Duetodiminishing

marginalreturns,themarginalproductofcapitalandtheslopeoftheproductionfunc

decreaseastheeconomyaccumulatesmorecapitalgoods.

Everythingthatappliesofcapitalalsoappliestanoductoflabor

(MPL)istheextraoutputafirmreceivesfromaddingonemoreunitoflabor,ho

otherinputsandefficiencyconstant.FortheCobb-DouglasproductionfunctionshowninEquation(4.1),themarginalproductoflaboris:

MPL=|3||L|.(4.3)

(2(Y

Theshapeoftheaggregateproductionfunctionflattensoutasthequantityoflaborincreases.Therefore,themarginalproductoflabordecreasesasmorelaborisadded,solaborexperiencesdiminishingmarginalreturns.Allcountrieshavedownward-slopingmarginalproductofcapitalandmarginalproductoflaborcurvesbecausediminishingmarginalreturnsapplytoalltypesofcapitalandlabor.

C.CalculatingTotalFactorProductivity

TheCobb-DouglasproductionfunctioninEquation(4.1)showsrealGDPasafunctionoftotalfactorproductivity,capital,andlabor.Itisnotpossible“ovteorallobeleotfhe

efficiency”tocalculatetotalfactorproductivitydirectly.BecausewecanobserverealG

capital,laborandlabor’sshareofincome,andcapital’sshareofincome,wecanuseEquation(4.1)fortheaggregateproductionfunctionAtoasc:alculate

A牋=Y.(4.4)

KL

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D.ChangesinCapital,Labor,andTotalFactorProductivity

Ifeitherlaborortotalfactorproductivityincreases,theproductionfunctionshiftsupwaandrealGDPincreases.Increasesinthecapitalstockwouldalsoshifttheproduction

functionupwardandincreaserealGDP.Increasesinlabororthecapitalstockaresubdiminishingmarginalreturns.Increasesintotalfactorproductivityarenotsubjectto

diminishingmarginalreturns.

TeachingTips

MakingtheConnectiononpage114discussesforeigndirectinvestmentandrealGDP

growthinChina.TheWorldBankhasmoreinformationanddataonthesetopicsfor

andothercountriesat.

Table4.1onpage115visuallysummarizesthechangestoanaggregateproductionfunction.

4.2AModelofRealGDPintheLongRun(pages116–120)

LearningObjective:ExplainhowrealGDPisdeterminedinthelongrun.

ToexplainhowrealGDPisdetermined,weneedtoexplainhowfirmschoosetheqofcapitalgoodstopurchaseandlabortohire.Weassumethatfirmsaremotivateddesiretomaximizeprofit,whichisequaltototalrevenueminustotalcost.Economists

assumethat(1)firmspurchasecapitalandhirelaboronlyifdoingsomaximizesproffirmsoperatepirfectlycompetitivemark,etsoeachfirmpriaker,(3)firmstake

thepricesofcapitalgoodsandlaborasgiven,and(4)firmsdecidehowmuchcaplabortohireandhowmuchoutputtoproduceusingtheavailabletechnologybasedpricesofoutputandinputs.

Wecanuseasimplemolrepresentthebehaviorofasinglerepresentativefirm.Thefirmproducesoup,ut,andsellsitattheperfectlycompePt.itivTeheprfiicrem,takesthenominalwagerateW,,andtnhoeminalrentalcostof,cRa,pitagiven.The’sfitroalrevenueiPsY.ThefmtotalcostWLisK.Profitsare:

Profit=Reven–ueCost

ProfitY–W(LK)

ProfitY–WL–RK

A.TheMarketsforCapitalandLabor

Thedemandandsupplymodelcanbeusedtoexplainthequantitiesoapitalandlaborfirmshire.Wetreatthesupplyofcapitalandlaborasfixedandunresponsivetomarket

prices.Weassumethatthefirmdoingthehiringissmallrelativetothemarket.Thefromhiringonemoreworkerequalsthepriceofoutputmultipliedbytheextraoutpu

hiringtheworker.Thechangeinprofitfromhiringonemoreworkeristhedifferencebetweentheadditionalrevenueearnedandtheadditionalcostpaid:

?ProfitPL–W.

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CHAPTER4|DeterminingAggregateProduction57

ThefirmwillhireadditionalworkerssoPnLgs.Thealwag,ew,isthe

nominalwagedividedbythepriceofoutput,sothefirmwillhireadditionalwork

asthemarginalproductoflaborisgreaterthantherealwage:

W

MPL>

P

.

Themarginalproductoflaborwilldecreaseasthefirmhiresmoreworkers—duetodiminishingmarginalreturns.Thefirmwillcontinuetohiremorelaboruntil

W

MPL=

P

.

Iftherealwagedecreases,thefirmwillhireadditionalworkersuntilthemarginal

productoflaborfallsenoughtorestoretheaboveequality.Iftherealwagedecreasfirmwilllayoffworkersuntilthemarginalproductoflaborrisesenoughtorestorethe

equality.Therefore,themarginalproductoflaborcurveisthedemandforlaborcursimilarargumentcanbemadeaboutafirmhiringcapital.Thechangeinprofitfrom

purchasingonemorecapitalgood,holdingthequantityoflaborconstant,is:

?ProfitPK–R.

Thefirmwillpurchasecapitalgoodswheneverthechangeinprofitisgreaterthan

orwheneverP×MPK>R.Therealrentalpriceofcapitatlh,e,ominalrentalprice

dividedbythepriceofoutput.Thefirmmaximizesprofit,sothefirmwillpurchase

goodswhenever:

R

MPK>

P

.

Themarginalproductwilldecreaseasthefirmpurchasesmorecapitalgoodsduetodiminishingmarginalreturns.Thefirmwillcontinuetopurchasecapitaluntil:

R

MPK=

P

.

Iftherentalcostdecreases,thefirmwillpurchasemorecapitalgoodsuntilthemaproductofcapitalfallsenoughtorestoretheaboveequality,andiftherentalpricerfirmwillletitscapitalstockwearoutuntilthemarginalproductofcapitalrisesenrestoretheequality.Therefore,themarginalproductofcapitalcurveisthedemandcforcapitalgoods.Whatistrueaboutasinglefirmistrueforeveryfirmintheeconomy.

Economistsoftenspeakoftheaggregatelabormarkaentdtheaggregatecapitalmar.ketThedemandcurvesslopedownward,reflectingdiminishingmarginalreturnstocapitalanlabor.Thesupplycurvesareverticallines,reflectingtheassumptionthatthequantity

supplieddoesnotrespondtochangesinprice.Theintersectionsofthedemandandcurvesdeterminetheequilibriumquantitiesofcapitalandlabor.

B.CombiningtheFactorMarketswiththeAggregateProductionFunction

Wecancombinetheequilibriumquantitiesofcapitalandlaborwiththeaggregateproductionfunctiontodeterminethelong-runequilibriumlevelofrealGDP.

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C.WhatHasHappenedtotheRealWageandtheRealRentalCostofCapitalOverTime?

Therealrentalcostofcapital,calculatedasthemarginalproductofcapital,hasave$0.116ofrealGDPperdollarofcapitalbetween1949and2009andhasneverbeenfar

fromthisaverage.Therealwage,calculatedasthemarginalproductoflabor,hasincdramaticallysince1949.Therealwagewas$9.67perhourin1949,androseto$3hourin2009,foranaverageannualgrowthrateof2.2%.

D.Aggregation

Inreality,workersandfirmsarenotinterchangeable.Forexample,manyfinancialfirmsoffworkersduring2007–2009duetothefinancialcrisisandthefallingdemandfor

financialservices.Theseworkershaveaspecificsetofskillsthatdonotreadilytranothersectors.Likewise,capitalgoods,suchassoftware,thatareusefulinthefinancial

sector,arenotnecessarilyusefulinhighwayconstruction.Thesameistrueforthe

aggregateproductionfunction.Inreality,everyfirmhasitsownproductionfunction.

4.3AccountingforGrowthinRealGDP(pages120–126)

LearningObjective:Discussthecontributionsofcapital,labor,andefficiencytothe

growthrateofrealGDP.

ThecapitatockfortheUnitedStatwes7.4timeslargerin2009thanitwasin1949,

whilelaborandtotalfactorproductivitywerebothabouttwiceaslargein2009asRobertSolowofMIThasdevelopedaprocedureknownasgrowthaccountingtodeterhowmuchofthegrowthrateinrealGDPisduetoeachofthethreecauses.

A.AccountingforRealGDPGrowth

Weusethefollowingsymbols:

gY=growthrateofrealGDP

gK=growthrateofthecapitalstock

gL=growthrateoflaborhours

gA=growthrateoffactorproductivity

For194–92009,thesegrowthrateswere:

gY=3.29%

gK=3.39%

gL=1.09%

gA=1.41%

TheproductionfunctionofEquation(4.1)showshowcapital,labor,andtotalfactor

productivitydeterminetlelofrealGDP.Thefocusinthissecgtothisraotthe

ofrealGDPbyusingthefollowingequation:

g=g+(|1|g+(|2|g

YA3K3L.(4.5)

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CHAPTER4|DeterminingAggregateProduction59

Thisisthebasicgrowthaccountingequation.Itallowsustodeterminethecontributi

eachcomponentoftheproductionfunctiontothegrowthofrealGDP.UsingthegrowthratesaboveandEquation(4.5):

ContributionoftotalfactorprodutAivit=y:1.41%

3K3

Contributionoflab:|g=(|2|(1.09)=0.73%

Contributionofcapit(|a:|g=(|1|(3.39)=1.13%

3L3

TherelativesharetothegrowthrateofrealGDPis’sthoniuptntdividedby

thegrowthrateofrealGDP.Tofindtherelativeshareofcapital,laborandtotalfactorproductivity,wecandivideEquationg(Y4:.5)by

g3g3g

1=(|gA|+(|牋1|(|gK|+(|2|(|gL|

YYY.

(4.6)

SubstitutingtheabovevaluesintoEquation(4.6),therelativeshareoftotalfactorproductivitygrowthis:

g3.29

(|gA|=(|1.41|=0.429,r42.9%.

Y

Therelativeshareofcapitalgrowthis:

3g?3.29

(|1|(|gK|=(|1|(|3.39|=0.343,r34.3%

Y

.

Therelativeshareoflaborgrowthis:

3g?3.29

(|2|(|gL|=(|2|(|1.09|=0.221,r22.1%

Y

.

ThetwomostimportantdeterminantsofthegrowthrateofrealGDParethegrowthcapitalgoodsandthegrowthrateoftotalfactorproductivity.Totalfactorproductivity

growthaloneaccountsfornearly43percentofthegrowthinU.S.realGDP.Thatmechangesintechnology,theskilllevelofthelaborforce,improvementsinorganizationotherintangiblefactorshavebeenthemostimportantdeterminantsofeconomicgrowth.

B.AccountingforLaborProductivityGrowth

Toexplainthegrowthoflaborproductivity,wecanuseamodifiedversionoftheCobb-

Douglasproductionfunction.Thefunctionhasconstantreturnstoscaleobymultiplyin

andLby/wehave:

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LLLL

Y=AF(|K,|=AF(|K|

Outputperhourworked,orlaborproductivityis:

Y

y=

L

.

Thecapital-laborratioisthedollarvalueofcapitalgoodsperunitoflabor,measuredaseitherthedollarvalueofcapitaldividedbythetotalnumberofhoursworkedorasthe

dollarvalueofcapitaldividedbythetotalnumberofworkers.Thecapital-laborratioc

writtenas:

K

k=

L

.

Ignoringthe1:

yf(k).

Becausetheshareofcapitalinincomeisone-third,wehave:

y=Ak13

(4.7)

Solaborproductivitydependsontotalfactorproductivityandthecapital-laborratio.

IntheappendixtothechapterthefollowingequationsfortheabsoluteandrelativecontributionstothegrowthrateofrealGDPperhourworkedarederived:

yA3K,

g=g+(|1|g

and

1=||g+|3|||g.

(g(1(g

C.TotalFactorProductivityastheUltimateSourceofGrowth

(4.8)

(4.9)

Capitalaccumulationissubjecttodiminishingmarginalreturns.Foreconomicgrowthtosustainable,itmustbedrivenbyincreasesintotalfactorproductivity.IntheUnitedentrepreneursandmanagersareundercompetitivepressurefromotherfirms.Thedrivforprofitprovidesanincentivefortechnologicalchange.Inmarketeconomies,peoplewhavetheirownmoneyonthelinemakedecisionsaboutwhichinvestmentstomakeand

whichtechnol