Econ60111 Lecture Notes

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1、ECON60111MacroeconomicTheoryLectureNotesRealBusinessCycles1AModelWithConstantEmploymentKeyingredients•Identical,infinitely-livedagents•Perfectlyfunctioningmarkets•ExogenousshocksBusinesscyclecharacteristics•Impulsesource:technologyshocks•Propagationmechanism:capitalaccumulationandintert

2、emporalsubstitutionSimplifyingassumptions•Constantpopulation(normalisedtoone)•Fulldepreciationofcapital•Logarithmicutilityfunction;Cobb-Douglasproductionfunction1.1TheRepresentativeAgentBasicfeatures•Eachagentisaconsumerandproducerofoutput•Eachagentproducesoutputusinglabourandcapital•Ea

3、chagentmaximiseshisexpectedlifetimeutilityPreferencesandtechnology•Intertemporalutilityfunction:∞−tVc=+∑t=0(1θψθ)[log()tt+log()],l,ψ>0(1)whereconsumption,leisurecl==tt•Productionfunction:αα1−yA=∈nk,(α0,1)(2)ttttwhereoutput,labour,capital,technologyynkA====shockttttConstraints•Timeconstr

4、aint:1=+ln(3)tt•Budget(resource)constraint:cky+=(4)tt+1tDecisionproblem•Maximisetheexpectedvalueof(1)subjectto(2),(3),(4)•TheLagrangianundernouncertainty:∞∞−−ttαα1−L=+∑∑tt==00(1θψ)[log()cntt+−log(1)]++(1θ)λt(Atntkt−ct−kt+1)whereLagrangeλ=multipliert•Thefirst-orderconditions:∂L−−tt1=+(1θ

5、θ)−+(1)λ=0t∂cctt1⇒=λ(5)tct∂L−−ttψαα−11−=−+(1θθ)++(1)λαAnk=0tttt∂−nn1ttψα−−11α⇒=λαAnk(6)tttt1−nt∂L−−tt+1αα−=−+(1θλ)++(1θλ)(1−α)Ank=0tt++11tt+1t+1∂kt+1−−1αα⇒=+λθ(1)λα(1−)Ank(7)tt++11tt+1t+1Equilibriumbehaviourunderuncertainty•Thestaticanddynamicoptimalityconditions[from(5),(6),(7)]:α−−11α

6、ψαAnkttt=(8)1−ncttαα−1−1Ankttt+++111=+(1θα)(1−)E(9)tcctt+1•Theresourceconstraint:α1−αckA+=nk(10)tt+1ttt1.2SteadyStateSolutionsforvariables∗∗∗•cckknnAA====,,,tttt•Equilibriumbehaviour[from(8),(9),(10)]:1−α∗−∗−αα11∗∗∗ψααAnkcAk1−nt=⇒=(11)∗∗∗∗∗1−ncnnnψ−α∗−∗1αα∗k1=+(1θα)(

7、1−)Ank⇒+=−1θα(1)A(12)∗n1−α∗∗∗∗∗∗∗αα1−ckkckA+=nk⇒+=A(13)∗∗∗nnn∗∗∗∗∗∗•knk=(,)θα[from(12)]⇒cnc=(,,,)θαψn[from(11)]⇒(,,)nN=θαψ∗∗[from(13)]⇒(,,)kK=θαψ⇒(,,)cC=θαψCalibration•Definition:calibrationisthetechniqueforassigningnumericalvaluestoparametersbasedonacombinationofempi