曼昆宏观经济学第七版英文课件第七章

MACROECONOMICS©2010WorthPublishers,allrightsreservedSEVENTHEDITIONPowerPoint®SlidesbyRonCronovichN.GregoryMankiwCHAPTEREconomicGrowthI:

CapitalAccumulationandPopulationGrowth7ModifiedforEC204byBobMurphyMACROECONOMICS©2010WorthPubInthischapter,youwilllearn:theclosedeconomySolowmodelhowacountry’sstandardoflivingdependsonitssavingandpopulationgrowthrateshowtousethe“GoldenRule”tofindtheoptimalsavingrateandcapitalstockInthischapter,youwilllear3CHAPTER7

EconomicGrowthIWhygrowthmattersDataoninfantmortalityrates:20%inthepoorest1/5ofallcountries0.4%intherichest1/5InPakistan,85%ofpeopleliveonlessthan$2/day.One-fourthofthepoorestcountrieshavehadfaminesduringthepast3decades.Povertyisassociatedwithoppressionofwomenandminorities.

Economicgrowthraiseslivingstandardsandreducespoverty….3CHAPTER7EconomicGrowthIncomeandpovertyintheworld

selectedcountries,2000MadagascarIndiaBangladeshNepalBotswanaMexicoChileS.KoreaBrazilRussianFederationThailandPeruChinaKenyaIncomeandpovertyintheworllinkstopreparedgraphs@Gnotes:circlesizeisproportionaltopopulationsize,

colorofcircleindicatescontinent,press“play”onbottomtoseethecrosssectiongraphevolveovertime,clickhereforoneinstructionguideIncomepercapitaandLifeexpectancyInfantmortalityMalariadeathsper100,000AdultliteracyCellphoneusersper100,000linkstopreparedgraphs@Gap6CHAPTER7

EconomicGrowthIWhygrowthmattersAnythingthateffectsthelong-runrateofeconomicgrowth–evenbyatinyamount–willhavehugeeffectsonlivingstandardsinthelongrun.1,081.4%243.7%85.4%624.5%169.2%64.0%2.5%2.0%…100years…50years…25yearspercentageincreasein

standardoflivingafter…annualgrowthrateofincomepercapita6CHAPTER7EconomicGrowth7CHAPTER7

EconomicGrowthIWhygrowthmattersIftheannualgrowthrateofU.S.realGDPpercapitahadbeenjustone-tenthofonepercenthigherduringthe1990s,theU.S.wouldhavegeneratedanadditional$496billionofincome

duringthatdecade.7CHAPTER7EconomicGrowth8CHAPTER7

EconomicGrowthIThelessonsofgrowththeory…canmakeapositivedifferenceinthelivesofhundredsofmillionsofpeople.Theselessonshelpusunderstandwhypoorcountriesarepoordesignpoliciesthat

canhelpthemgrowlearnhowourowngrowthrateisaffectedbyshocksandourgovernment’spolicies8CHAPTER7EconomicGrowth9CHAPTER7

EconomicGrowthITheSolowmodelduetoRobertSolow,

wonNobelPrizeforcontributionsto

thestudyofeconomicgrowthamajorparadigm:widelyusedinpolicymakingbenchmarkagainstwhichmost

recentgrowththeoriesarecomparedlooksatthedeterminantsofeconomicgrowthandthestandardoflivinginthelongrun9CHAPTER7EconomicGrowth10CHAPTER7

EconomicGrowthIHowSolowmodelisdifferentfromChapter3’smodel1.

Kisnolongerfixed:

investmentcausesittogrow,

depreciationcausesittoshrink2.

Lisnolongerfixed:

populationgrowthcausesittogrow3. theconsumptionfunctionissimpler10CHAPTER7EconomicGrowth11CHAPTER7

EconomicGrowthIHowSolowmodelisdifferentfromChapter3’smodel4.

noGorT

(onlytosimplifypresentation;

wecanstilldofiscalpolicyexperiments)5.

cosmeticdifferences11CHAPTER7EconomicGrowth12CHAPTER7

EconomicGrowthITheproductionfunctionInaggregateterms:Y=F(K,L)Define:y=Y/L=outputperworker

k=K/L=capitalperworkerAssumeconstantreturnstoscale:

zY=F(zK,zL)foranyz>0Pickz=1/L.ThenY/L=F(K/L,1)y=F(k,1)y=f(k) wheref(k)=F(k,1)12CHAPTER7EconomicGrowth13CHAPTER7

EconomicGrowthITheproductionfunctionOutputper

worker,y

Capitalper

worker,k

f(k)Note:thisproductionfunctionexhibitsdiminishingMPK.1MPK=f(k+1)–f(k)13CHAPTER7EconomicGrowth14CHAPTER7

EconomicGrowthIThenationalincomeidentityY=C+I (remember,noG)In“perworker”terms:

y=c+i

wherec=C/Landi=I

/L

14CHAPTER7EconomicGrowth15CHAPTER7

EconomicGrowthITheconsumptionfunctions=thesavingrate,

thefractionofincomethatissaved(sisanexogenousparameter) Note:sistheonlylowercasevariablethat

isnotequalto

itsuppercaseversiondividedbyLConsumptionfunction:c=(1–s)y

(perworker)15CHAPTER7EconomicGrowth16CHAPTER7

EconomicGrowthISavingandinvestmentsaving(perworker) =y–c

=y–(1–s)y

=syNationalincomeidentityisy=c+i Rearrangetoget:i=y–c=sy

(investment=saving,likeinchap.3!)Usingtheresultsabove,

i=sy=sf(k)16CHAPTER7EconomicGrowth17CHAPTER7

EconomicGrowthIOutput,consumption,andinvestmentOutputper

worker,y

Capitalper

worker,k

f(k)sf(k)k1

y1

i1

c1

17CHAPTER7EconomicGrowth18CHAPTER7

EconomicGrowthIDepreciationDepreciationperworker,δk

Capitalper

worker,k

δkδ=therateofdepreciation=thefractionofthecapitalstockthatwearsouteachperiod1δ18CHAPTER7EconomicGrowth19CHAPTER7

EconomicGrowthICapitalaccumulationChangeincapitalstock =investment–depreciation

Δk =i

–δk Sincei=sf(k),thisbecomes:Δk=s

f(k)

–δk

Thebasicidea:Investmentincreasesthecapitalstock,depreciationreducesit.19CHAPTER7EconomicGrowth20CHAPTER7

EconomicGrowthITheequationofmotionforkTheSolowmodel’scentralequationDeterminesbehaviorofcapitalovertime……which,inturn,determinesbehaviorof

alloftheotherendogenousvariables

becausetheyalldependonk.E.g., incomeperperson: y=f(k) consumptionperperson: c=(1–s)

f(k)

Δk=s

f(k)

–δk

20CHAPTER7EconomicGrowth21CHAPTER7

EconomicGrowthIThesteadystateIfinvestmentisjustenoughtocoverdepreciation

[sf(k)

=δk],thencapitalperworkerwillremainconstant:

Δk=0.Thisoccursatonevalueofk,denotedk*,

calledthesteadystatecapitalstock.Δk=s

f(k)

–δk

21CHAPTER7EconomicGrowth22CHAPTER7

EconomicGrowthIThesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)δkk*

22CHAPTER7EconomicGrowth23CHAPTER7

EconomicGrowthIMovingtowardthesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)δkk*

Δk=sf(k)−

δkdepreciationΔkk1investment23CHAPTER7EconomicGrowth24CHAPTER7

EconomicGrowthIMovingtowardthesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)δkk*

k1Δk=sf(k)−

δkΔkk224CHAPTER7EconomicGrowth25CHAPTER7

EconomicGrowthIMovingtowardthesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)δkk*

Δk=sf(k)−

δkk2investmentdepreciationΔk25CHAPTER7EconomicGrowth26CHAPTER7

EconomicGrowthIMovingtowardthesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)δkk*

Δk=sf(k)−

δkΔkk226CHAPTER7EconomicGrowth27CHAPTER7

EconomicGrowthIMovingtowardthesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)δkk*

Δk=sf(k)−

δkk2Δkk327CHAPTER7EconomicGrowth28CHAPTER7

EconomicGrowthIMovingtowardthesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)δkk*

Δk=sf(k)−

δkk3Summary:

Aslongask<k*,investmentwillexceeddepreciation,

andkwillcontinuetogrowtowardk*.28CHAPTER7EconomicGrowthNOWYOUTRY:

Approachingk*fromaboveDrawtheSolowmodeldiagram,

labelingthesteadystatek*.Onthehorizontalaxis,pickavaluegreaterthank*fortheeconomy’sinitialcapitalstock.Labelitk1.Showwhathappenstokovertime.

Doeskmovetowardthesteadystateor

awayfromit?NOWYOUTRY:

Approachingk*30CHAPTER7

EconomicGrowthIAnumericalexampleProductionfunction(aggregate):Toderivetheper-workerproductionfunction,dividethroughbyL:Thensubstitutey=Y/Landk=K/Ltoget30CHAPTER7EconomicGrowth31CHAPTER7

EconomicGrowthIAnumericalexample,cont.Assume:s=0.3δ=0.1initialvalueofk=4.031CHAPTER7EconomicGrowth32CHAPTER7

EconomicGrowthIApproachingthesteadystate:

AnumericalexampleYear

k

y

c

i

k

Dk1 4.000 2.000 1.400 0.600 0.400 0.2002 4.200 2.049 1.435 0.615 0.420 0.1953 4.395 2.096 1.467 0.629 0.440 0.1894 4.584 2.141 1.499 0.642 0.458 0.184…10 5.602 2.367 1.657 0.710 0.560 0.150…25 7.351 2.706 1.894 0.812 0.732 0.080…100 8.962 2.994 2.096 0.898 0.896 0.002…

9.000 3.000 2.100 0.900 0.900 0.00032CHAPTER7EconomicGrowthNOWYOUTRY:

SolvefortheSteadyStateContinuetoassume

s=0.3,δ=0.1,andy=k1/2Usetheequationofmotion

Δk=sf(k)

−

δk

tosolveforthesteady-statevaluesofk,y,andc.NOWYOUTRY:

SolvefortheSANSWERS:

SolvefortheSteadyStateANSWERS:

SolvefortheStead35CHAPTER7

EconomicGrowthIAnincreaseinthesavingrateInvestment

and

depreciationkdks1f(k)Anincreaseinthesavingrateraisesinvestment……causingktogrowtowardanewsteadystate:s2f(k)35CHAPTER7EconomicGrowth36CHAPTER7

EconomicGrowthIPrediction:Highers

⇒

higherk*.Andsincey=f(k),

higherk*⇒highery*.Thus,theSolowmodelpredictsthatcountrieswithhigherratesofsavingandinvestment

willhavehigherlevelsofcapitalandincomeperworkerinthelongrun.36CHAPTER7EconomicGrowthInternationalevidenceoninvestmentratesandincomeperpersonIncomeperpersonin2003

(logscale)Investmentaspercentageofoutput

(average1960-2003)Internationalevidenceoninve38CHAPTER7

EconomicGrowthITheGoldenRule:IntroductionDifferentvaluesofsleadtodifferentsteadystates.

Howdoweknowwhichisthe“best”steadystate?The“best”steadystatehasthehighestpossible

consumptionperperson:c*=(1–s)f(k*).Anincreaseins

leadstohigherk*andy*,whichraisesc*

reducesconsumption’sshareofincome(1–s),

whichlowersc*.

So,howdowefindthesandk*thatmaximizec*?38CHAPTER7EconomicGrowth39CHAPTER7

EconomicGrowthITheGoldenRulecapitalstocktheGoldenRulelevelofcapital,

thesteadystatevalueofk

thatmaximizesconsumption.Tofindit,firstexpressc*intermsofk*: c* =y*

−

i* =f

(k*)

−

i*

=f

(k*)

−

δk*

Inthesteadystate: i*

=

δk*becauseΔk=0.39CHAPTER7EconomicGrowth40CHAPTER7

EconomicGrowthIThen,graph

f(k*)andδk*,

lookforthe

pointwhere

thegapbetweenthemisbiggest.TheGoldenRulecapitalstocksteadystateoutputanddepreciationsteady-statecapitalper

worker,k*

f(k*)δ

k*40CHAPTER7EconomicGrowth41CHAPTER7

EconomicGrowthITheGoldenRulecapitalstockc*=f(k*)−

δk*

isbiggestwheretheslopeoftheproductionfunction

equals

theslopeofthedepreciationline:steady-statecapitalper

worker,k*

f(k*)δ

k*MPK=δ

41CHAPTER7EconomicGrowth42CHAPTER7

EconomicGrowthIThetransitiontothe

GoldenRulesteadystateTheeconomydoesNOThaveatendencytomovetowardtheGoldenRulesteadystate.AchievingtheGoldenRulerequiresthatpolicymakersadjusts.Thisadjustmentleadstoanewsteadystatewithhigherconsumption.Butwhathappenstoconsumption

duringthetransitiontotheGoldenRule?42CHAPTER7EconomicGrowth43CHAPTER7

EconomicGrowthIStartingwithtoomuchcapitalthenincreasingc*requiresafallins.InthetransitiontotheGoldenRule,consumptionishigheratallpointsintime.timet0ciy43CHAPTER7EconomicGrowth44CHAPTER7

EconomicGrowthIStartingwithtoolittlecapitalthenincreasingc*requiresan

increaseins.Futuregenerations

enjoyhigherconsumption,

butthecurrent

oneexperiences

aninitialdrop

inconsumption.timet0ciy44CHAPTER7EconomicGrowth45CHAPTER7

EconomicGrowthIPopulationgrowthAssumethepopulationandlaborforcegrow

atraten(exogenous):EX:SupposeL=1,000inyear1andthepopulationisgrowingat2%peryear(n=0.02).ThenΔL=n

L=0.02

×

1,000=20,

soL=1,020inyear2.45CHAPTER7EconomicGrowth46CHAPTER7

EconomicGrowthIBreak-eveninvestment(δ

+

n)k=break-eveninvestment,

theamountofinvestmentnecessary

tokeepkconstant.Break-eveninvestmentincludes:δ

k

toreplacecapitalasitwearsoutn

k

toequipnewworkerswithcapital (Otherwise,kwouldfallastheexistingcapitalstockisspreadmorethinlyoveralargerpopulationofworkers.)46CHAPTER7EconomicGrowth47CHAPTER7

EconomicGrowthITheequationofmotionforkWithpopulationgrowth,

theequationofmotionforkis:break-eveninvestmentactualinvestmentΔk=s

f(k)

−

(δ

+

n)

k47CHAPTER7EconomicGrowth48CHAPTER7

EconomicGrowthITheSolowmodeldiagramInvestment,break-eveninvestmentCapitalper

worker,k

sf(k)(δ

+

n

)

kk*

Δk

=

sf(k)

−

(δ

+n)k48CHAPTER7EconomicGrowth49CHAPTER7

EconomicGrowthITheimpactofpopulationgrowthInvestment,break-eveninvestmentCapitalper

worker,k

sf(k)(δ

+n1)

kk1*

(δ

+n2)

kk2*

Anincreaseinncausesanincreaseinbreak-eveninvestment,leadingtoalowersteady-statelevelofk.49CHAPTER7EconomicGrowth50CHAPTER7

EconomicGrowthIPrediction:Highern

⇒lowerk*.Andsincey=f(k),

lowerk*

⇒lowery*.Thus,theSolowmodelpredictsthatcountrieswithhigherpopulationgrowthrateswillhavelowerlevelsofcapitalandincomeperworkerinthelongrun.50CHAPTER7EconomicGrowthInternationalevidenceonpopulationgrowthandincomeperpersonIncomeperpersonin2003

(logscale)Populationgrowth

(percentperyear,average1960-2003)Internationalevidenceonpopu52CHAPTER7

EconomicGrowthITheGoldenRulewithpopulationgrowthTofindtheGoldenRulecapitalstock,

expressc*intermsofk*: c* =y*

−

i* =f

(k*)

−

(δ

+n)k*

c*ismaximizedwhen

MPK=δ

+n

orequivalently,

MPK−

δ

=nIntheGolden

Rulesteadystate,

themarginalproduct

ofcapitalnetofdepreciationequalsthepopulation

growthrate.52CHAPTER7EconomicGrowth53CHAPTER7

EconomicGrowthIAlternativeperspectivesonpopulationgrowthTheMalthusianModel(1798)Predictspopulationgrowthwilloutstripthe

Earth’sabilitytoproducefood,leadingtotheimpoverishmentofhumanity.SinceMalthus,worldpopulationhasincreasedsixfold,yetlivingstandardsarehigherthanever.Malthusneglectedtheeffectsoftechnologicalprogress.53CHAPTER7EconomicGrowth54CHAPTER7

EconomicGrowthIAlternativeperspectivesonpopulationgrowthTheKremerianModel(1993)Positsthatpopulationgrowthcontributestoeconomicgrowth.Morepeople=moregeniuses,scientists&engineers,sofastertechnologicalprogress.Evidence,fromverylonghistoricalperiods:Asworldpop.growthrateincreased,sodidrateofgrowthinlivingstandardsHistorically,regionswithlargerpopulationshaveenjoyedfastergrowth.54CHAPTER7EconomicGrowthChapterSummary1.

TheSolowgrowthmodelshowsthat,

inthelongrun,acountry’sstandardoflivingdepends:positivelyonitssavingratenegativelyonitspopulationgrowthrate2.

Anincreaseinthesavingrateleadsto:higheroutputinthelongrunfastergrowthtemporarilybutnotfastersteadystategrowthChapterSummary1. TheSolowgrChapterSummary3.

IftheeconomyhasmorecapitalthantheGoldenRulelevel,thenreducingsavingwillincreaseconsumptionatallpointsintime,makingallgenerationsbetteroff. IftheeconomyhaslesscapitalthantheGoldenRulelevel,thenincreasingsavingwillincreaseconsumptionforfuturegenerations,butreduceconsumptionforthepresentgeneration.ChapterSummary3. IftheeconoMACROECONOMICS©2010WorthPublishers,allrightsreservedSEVENTHEDITIONPowerPoint®SlidesbyRonCronovichN.GregoryMankiwCHAPTEREconomicGrowthI:

CapitalAccumulationandPopulationGrowth7ModifiedforEC204byBobMurphyMACROECONOMICS©2010WorthPubInthischapter,youwilllearn:theclosedeconomySolowmodelhowacountry’sstandardoflivingdependsonitssavingandpopulationgrowthrateshowtousethe“GoldenRule”tofindtheoptimalsavingrateandcapitalstockInthischapter,youwilllear59CHAPTER7

EconomicGrowthIWhygrowthmattersDataoninfantmortalityrates:20%inthepoorest1/5ofallcountries0.4%intherichest1/5InPakistan,85%ofpeopleliveonlessthan$2/day.One-fourthofthepoorestcountrieshavehadfaminesduringthepast3decades.Povertyisassociatedwithoppressionofwomenandminorities.

Economicgrowthraiseslivingstandardsandreducespoverty….3CHAPTER7EconomicGrowthIncomeandpovertyintheworld

selectedcountries,2000MadagascarIndiaBangladeshNepalBotswanaMexicoChileS.KoreaBrazilRussianFederationThailandPeruChinaKenyaIncomeandpovertyintheworllinkstopreparedgraphs@Gnotes:circlesizeisproportionaltopopulationsize,

colorofcircleindicatescontinent,press“play”onbottomtoseethecrosssectiongraphevolveovertime,clickhereforoneinstructionguideIncomepercapitaandLifeexpectancyInfantmortalityMalariadeathsper100,000AdultliteracyCellphoneusersper100,000linkstopreparedgraphs@Gap62CHAPTER7

EconomicGrowthIWhygrowthmattersAnythingthateffectsthelong-runrateofeconomicgrowth–evenbyatinyamount–willhavehugeeffectsonlivingstandardsinthelongrun.1,081.4%243.7%85.4%624.5%169.2%64.0%2.5%2.0%…100years…50years…25yearspercentageincreasein

standardoflivingafter…annualgrowthrateofincomepercapita6CHAPTER7EconomicGrowth63CHAPTER7

EconomicGrowthIWhygrowthmattersIftheannualgrowthrateofU.S.realGDPpercapitahadbeenjustone-tenthofonepercenthigherduringthe1990s,theU.S.wouldhavegeneratedanadditional$496billionofincome

duringthatdecade.7CHAPTER7EconomicGrowth64CHAPTER7

EconomicGrowthIThelessonsofgrowththeory…canmakeapositivedifferenceinthelivesofhundredsofmillionsofpeople.Theselessonshelpusunderstandwhypoorcountriesarepoordesignpoliciesthat

canhelpthemgrowlearnhowourowngrowthrateisaffectedbyshocksandourgovernment’spolicies8CHAPTER7EconomicGrowth65CHAPTER7

EconomicGrowthITheSolowmodelduetoRobertSolow,

wonNobelPrizeforcontributionsto

thestudyofeconomicgrowthamajorparadigm:widelyusedinpolicymakingbenchmarkagainstwhichmost

recentgrowththeoriesarecomparedlooksatthedeterminantsofeconomicgrowthandthestandardoflivinginthelongrun9CHAPTER7EconomicGrowth66CHAPTER7

EconomicGrowthIHowSolowmodelisdifferentfromChapter3’smodel1.

Kisnolongerfixed:

investmentcausesittogrow,

depreciationcausesittoshrink2.

Lisnolongerfixed:

populationgrowthcausesittogrow3. theconsumptionfunctionissimpler10CHAPTER7EconomicGrowth67CHAPTER7

EconomicGrowthIHowSolowmodelisdifferentfromChapter3’smodel4.

noGorT

(onlytosimplifypresentation;

wecanstilldofiscalpolicyexperiments)5.

cosmeticdifferences11CHAPTER7EconomicGrowth68CHAPTER7

EconomicGrowthITheproductionfunctionInaggregateterms:Y=F(K,L)Define:y=Y/L=outputperworker

k=K/L=capitalperworkerAssumeconstantreturnstoscale:

zY=F(zK,zL)foranyz>0Pickz=1/L.ThenY/L=F(K/L,1)y=F(k,1)y=f(k) wheref(k)=F(k,1)12CHAPTER7EconomicGrowth69CHAPTER7

EconomicGrowthITheproductionfunctionOutputper

worker,y

Capitalper

worker,k

f(k)Note:thisproductionfunctionexhibitsdiminishingMPK.1MPK=f(k+1)–f(k)13CHAPTER7EconomicGrowth70CHAPTER7

EconomicGrowthIThenationalincomeidentityY=C+I (remember,noG)In“perworker”terms:

y=c+i

wherec=C/Landi=I

/L

14CHAPTER7EconomicGrowth71CHAPTER7

EconomicGrowthITheconsumptionfunctions=thesavingrate,

thefractionofincomethatissaved(sisanexogenousparameter) Note:sistheonlylowercasevariablethat

isnotequalto

itsuppercaseversiondividedbyLConsumptionfunction:c=(1–s)y

(perworker)15CHAPTER7EconomicGrowth72CHAPTER7

EconomicGrowthISavingandinvestmentsaving(perworker) =y–c

=y–(1–s)y

=syNationalincomeidentityisy=c+i Rearrangetoget:i=y–c=sy

(investment=saving,likeinchap.3!)Usingtheresultsabove,

i=sy=sf(k)16CHAPTER7EconomicGrowth73CHAPTER7

EconomicGrowthIOutput,consumption,andinvestmentOutputper

worker,y

Capitalper

worker,k

f(k)sf(k)k1

y1

i1

c1

17CHAPTER7EconomicGrowth74CHAPTER7

EconomicGrowthIDepreciationDepreciationperworker,δk

Capitalper

worker,k

δkδ=therateofdepreciation=thefractionofthecapitalstockthatwearsouteachperiod1δ18CHAPTER7EconomicGrowth75CHAPTER7

EconomicGrowthICapitalaccumulationChangeincapitalstock =investment–depreciation

Δk =i

–δk Sincei=sf(k),thisbecomes:Δk=s

f(k)

–δk

Thebasicidea:Investmentincreasesthecapitalstock,depreciationreducesit.19CHAPTER7EconomicGrowth76CHAPTER7

EconomicGrowthITheequationofmotionforkTheSolowmodel’scentralequationDeterminesbehaviorofcapitalovertime……which,inturn,determinesbehaviorof

alloftheotherendogenousvariables

becausetheyalldependonk.E.g., incomeperperson: y=f(k) consumptionperperson: c=(1–s)

f(k)

Δk=s

f(k)

–δk

20CHAPTER7EconomicGrowth77CHAPTER7

EconomicGrowthIThesteadystateIfinvestmentisjustenoughtocoverdepreciation

[sf(k)

=δk],thencapitalperworkerwillremainconstant:

Δk=0.Thisoccursatonevalueofk,denotedk*,

calledthesteadystatecapitalstock.Δk=s

f(k)

–δk

21CHAPTER7EconomicGrowth78CHAPTER7

EconomicGrowthIThesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)δkk*

22CHAPTER7EconomicGrowth79CHAPTER7

EconomicGrowthIMovingtowardthesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)δkk*

Δk=sf(k)−

δkdepreciationΔkk1investment23CHAPTER7EconomicGrowth80CHAPTER7

EconomicGrowthIMovingtowardthesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)δkk*

k1Δk=sf(k)−

δkΔkk224CHAPTER7EconomicGrowth81CHAPTER7

EconomicGrowthIMovingtowardthesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)δkk*

Δk=sf(k)−

δkk2investmentdepreciationΔk25CHAPTER7EconomicGrowth82CHAPTER7

EconomicGrowthIMovingtowardthesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)δkk*

Δk=sf(k)−

δkΔkk226CHAPTER7EconomicGrowth83CHAPTER7

EconomicGrowthIMovingtowardthesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)δkk*

Δk=sf(k)−

δkk2Δkk327CHAPTER7EconomicGrowth84CHAPTER7

EconomicGrowthIMovingtowardthesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)δkk*

Δk=sf(k)−

δkk3Summary:

Aslongask<k*,investmentwillexceeddepreciation,

andkwillcontinuetogrowtowardk*.28CHAPTER7EconomicGrowthNOWYOUTRY:

Approachingk*fromaboveDrawtheSolowmodeldiagram,

labelingthesteadystatek*.Onthehorizontalaxis,pickavaluegreaterthank*fortheeconomy’sinitialcapitalstock.Labelitk1.Showwhathappenstokovertime.

Doeskmovetowardthesteadystateor

awayfromit?NOWYOUTRY:

Approachingk*86CHAPTER7

EconomicGrowthIAnumericalexampleProductionfunction(aggregate):Toderivetheper-workerproductionfunction,dividethroughbyL:Thensubstitutey=Y/Landk=K/Ltoget30CHAPTER7EconomicGrowth87CHAPTER7

EconomicGrowthIAnumericalexample,cont.Assume:s=0.3δ=0.1initialvalueofk=4.031CHAPTER7EconomicGrowth88CHAPTER7

EconomicGrowthIApproachingthesteadystate:

AnumericalexampleYear

k

y

c

i

k

Dk1 4.000 2.000 1.400 0.600 0.400 0.2002 4.200 2.049 1.435 0.615 0.420 0.1953 4.395 2.096 1.467 0.629 0.440 0.1894 4.584 2.141 1.499 0.642 0.458 0.184…10 5.602 2.367 1.657 0.710 0.560 0.150…25 7.351 2.706 1.894 0.812 0.732 0.080…100 8.962 2.994 2.096 0.898 0.896 0.002…

9.000 3.000 2.100 0.900 0.900 0.00032CHAPTER7EconomicGrowthNOWYOUTRY:

SolvefortheSteadyStateContinuetoassume

s=0.3,δ=0.1,andy=k1/2