拉姆齐模型
合浦一中-钢材涨价
2023年2月16日发(作者：张富明)LectureNotes3:
TheNeoclassicalGrowthModel-Part2
JennyXu,Econ5140,Fall2014
1
MainFeatures
•Endogenizessavingratebyintroducinghouse-
holdsandfirmsoptimization.
•DemonstratesthatpredictionsoftheSolow
modeldonothinderontheassumptionof
afixedsavingrate.
•Allowswelfareanalysis.
•Itisarepresentativehousehold,infinitehori-
zonmodel.
•Providesausefulframeworkforstudying
manyothereconomicquestions.
2
Assumptions
•Aclosedeconomywithasinglegood(used
asbothconsumptiongoodandinvestment
good).
•Perfectlycompetitiveoutput,capitaland
labourmarkets.
•Informationstructure:Perfectforesight.
3
DynamicEfficiencyintheRamseyModel
•Aneconomyissaidtobedynamicallyin-
efficientifitoversavesalongitsbalanced
seconomythepathof
steadystateconsumptionliesbelowfeasi-
blealternativepathsatallpointsintime.
•WeseeintheSolowmodelandOLGmodel
dynamicsinefficiencymayexistandthere
aboutRamseymodel?
•Atthesteadystate,
k
∗
=(
α
1
β
−1+δ
)
1
1−α(1)
c
∗
=k
∗α
−δk
∗
(2)
4
DynamicEfficiencyintheRamsey
Model:II
•Sotomaximizethesteadystateconsump-
tion,weneedtohave
∂c
∗
∂k
∗
=0⇒α(k
G
)
α−1
=δ(3)
•Fromk∗
=(
α
1
β
−1+δ
)
1
1−α,wehave
α(k
∗
)
α−1
=
1
β
−1+δ(4)
•Sowemusthavek∗
G ,since 1 β −1+δ> δ! •DynamicinefficiencyisnotpossibleinRam- seyModel! 5 ImpossibilityofDynamicInefficiencyin theRamseyModel •Thedynamicinefficiencycannotoccurin otbeanequi- libriumfortheeconomytofollowapath wherehigherconsumptioncanbeattained conomywereon suchapath,householdwouldreducetheir savingandtakeadvantageofthisoppor- tunity. •Thesteadystatevalueofcapitalperworker, k ∗ ,towhichtheeconomyconvergesto,is knownasthemodifiedgolden-rulecapital stock. •Thisresultalsoappliedtothemodelifwe introducethetechnologicalprogressand thepopulationgrowth. 6 Theimpactofachangeinthediscount factor •Wedefineβ=1 1+ρ . •Supposetheeconomyisatthesteadystate. Considerapermanentunexpecteddecline inρ. •Recallthemeaningofρ!Isthefuturecon- sumptionmoreorlessvaluabletohouse- holds? •Howis△ct =0-locusaffected? •Howis△kt =0-locusaffected? •Whatisthedynamics? 7 TheSteadyState(BGP) •Wecanextendthebenchmarkmodeltoa modelwithtechnologyprogressandpopu- lationgrowth. •Exogenoustechnologicalprogress:A t A t−1 = 1+g,whereg>=0. •- nouslaborgrowthrate: L t L t−1 =1+n,where n>=0. •Similarly,wecansolveforthesteadystate wherethecapitalpereffectivelaborand consumptionpereffectivelaborwillbecon- stant. 8 •Atthesteadystate,(goandcheckyour- self) k ∗ =( α 1 β −1+δ ) 1 1−α(5) c ∗ =k ∗α −(n+g+δ)k ∗ (6) •Variablesinunitsofeffectivelabour,c,k, yareconstant. •Variablesinpercapitaunits,K L , Y L , C L growattheexogenousrateoftechnologi- calprogress1+g. •Variablesinlevels,K,Y,C,growatthe rate(1+n)(1+g). TransitionalDynamics •Themodelpredictsthattheeconomycon- vergestoitssteadystatealongaunique saddlepath. •Givenk0 ,therewillbeuniquec 0 thatplaces theeconomytowardsthesteadystate. •Thedynamicsisgivenby ∆c t c t−1 =[β(1+αkα−1 t −δ)]σ−1(7) ∆k t =kα t −ct−δkt (8) or ˆ k t+1 =a 1 ˆ k t ˆc t =a 2 ˆ k t •Thesteadystateisalsoknownasthemod- ifiedgoldenrule. 9 Behaviorofsavingratealongthepath: •Savingrateisdefinedas: s t = y t−ct y t (9) •IntheRamseymodel,withoptimizingagents, thebehaviourofthesavingratealongthe transitionpathisambiguous. •Itdependsontherelativeimportanceofan incomeandasubstitutioneffects. •Asubstitutioneffect:ask↑,returnonsav- ing↓,thesavingrate↓astheeconomyde- velops. •Anincomeeffect:inapooreconomy,the actualincomeislowerthanthelong-runor 10 onsumption smoothing,thesavingrateislowwhenk ↑,thegapbetweencurrent incomeandpermanentincomefalls,the savingrate↑astheeconomydevelops. •Atthesteadystate(BGP).savingrateis constant. ComparisonwithSolowModel •SimilaritieswiththeSolowModel –Inthesteadystate,thesavingrateis constant. –Qualitativeresponsesofcandkinthe Ramseymodeltoadecreaseinthedis- countrateareidenticaltoresponsesin theSolowmodeltoanincreaseinthe savingrate. •DifferenceswiththeSolowModel –Savingrateisgenerallynotconstantalong fferencehas asubstantialimpactontherateofcon- vergence. –Nopossibilityofdynamicinefficiency. 11