GSDE-新凯恩斯模型-ch3

Lectures on Monetary Policy,In‡ation and the Business CycleThe Basic New Keynesian ModelbyJordi GalíFebruary2007Motivation and OutlineEvidence on Money,Output,and Prices:The Long RunShort Run E¤ects of Monetary Policy Shocks(i)persistent e¤ects on real variables(ii)slow adjustment of aggregate price level(iii)liquidity e¤ectMicro Evidence on Price-setting Behavior:signi…cant price and wage rigiditiesFailure of Classical Monetary ModelsA Baseline Model with Nominal Rigiditiesmonopolistic competitionsticky prices(staggered price setting)competitive labor markets,closed economy,no capital accumulationHouseholdsRepresentative household solves1X t=0 t U(C t;N t)max E0whereC t Z10C t(i)1 1 di 1 subject to Z10P t(i)C t(i)di+Q t B t B t 1+W t N t T tfor t=0;1;2;:::plus solvency constraint.Optimality conditions1.Optimal allocation of expendituresC t(i)= P t(i)P t C t implying Z10P t(i)C t(i)di=P t C t whereP t Z10P t(i)1 di 11 2.Other optimality conditionsU n;t Uc;t =W tP tQ t= E t U c;t+1U c;t P t P t+1Speci…cation of utility:U(C t;N t)=C1t1N1+'t1+'implied log-linear optimality conditions(aggregate variables)w t p t= c t+'n tc t=E t f c t+1g 1 (i t E t f t+1g )where i t log Q t is the nominal interest rate and log is the discount rate.Ad-hoc money demandm t p t=y t i tFirmsContinuum of…rms,indexed by i2[0;1]Each…rm produces a di¤erentiated goodIdentical technologyY t(i)=A t N t(i)1Probability of being able to reset price in any given period:1 , independent across…rms(Calvo(1983)).2[0;1]:index of price stickinessImplied average price duration11Aggregate Price DynamicsP t= (P t 1)1 +(1 )(P t)1 11Dividing by P t 1:= +(1 ) P t P t 1 11tLog-linearization around zero in‡ation steady statet=(1 )(p t p t 1)(1) or,equivalentlyp t= p t 1+(1 )p tOptimal Price Setting1X k=0 k E t Q t;t+k P t Y t+k j t t+k(Y t+k j t) maxP tsubject toY t+k j t=(P t=P t+k) C t+kfor k=0;1;2;:::whereQ t;t+k k C t+k C t P t P t+kOptimality condition:1X k=0 k E t Q t;t+k Y t+k j t P t M t+k j t =0 where t+k j t 0t+k(Y t+k j t)and M 1Equivalently,1X k=0 k E t Q t;t+k Y t+k j t P t P t 1 M MC t+k j t t 1;t+k =0 where MC t+k j t t+k j t=P t+k and t 1;t+k P t+k=P t 1Perfect Foresight,Zero In‡ation Steady State:P t P t 1=1; t 1;t+k=1;Y t+k j t=Y;Q t;t+k= k;MC=1MLog-linearization around zero in‡ation steady state:1X k=0( )k E t f c mc t+k j t+p t+k p t 1g p t p t 1=(1 )Equivalently,1X k=0( )k E t f mc t+k+p t+k gp t= +(1 )Flexible prices( =0):p t= +mc t+p t=)mc t= (symmetric equilibrium)Particular Case: =0(constant returns)=)MC t+k j t=MC t+kLog-linearization of optimality condition around zero in‡ation steady state: p t p t 1=1X k=0( )k E t f(1 )c mc t+k+ t+k g(2)= E t f p t+1 p t g+(1 )c mc t+ t(3) where cmc t mc t mc=mc t+ ,and log 1.Combining(1)and(3):t= E t f t+1g+ c mc twhere(1 )(1 )Generalization to 2(0;1)(decreasing returns)De…nemc t (w t p t ) mpn t(w t p t ) 11(a t y t ) log(1 )Using mc t +k j t =(w t +k p t +k ) 11 (a t +k y t +k j t ) log(1 ),mc t +k j t=mc t +k +1 (y t +k j t y t +k )=mc t +k1(p t p t +k )(4)Implied in‡ation dynamicst = E t f t +1g + c mc t(5)where(1 )(1 )1 1 +EquilibriumGoods markets clearingY t(i)=C t(i)for all i2[0;1]and all t.Letting Y t R10Y t(i)1 1 di 1,Y t=C tfor all bined with the consumer’s Euler equation:y t=E t f y t+1g 1 (i t E t f t+1g )(6)Labor market clearingN t=Z10N t(i)di=Z10 Y t(i)A t 11 di= Y t A t 11 Z10 P t(i)P t 1 di T aking logs,(1 )n t=y t a t+d t where d t (1 )log R10(P t(i)=P t) 1 di(second order). Up to a…rst order approximation:y t=a t+(1 )n tMarginal Cost and Outputmc t=(w t p t) mpn t=( y t+'n t) (y t n t) log(1 )= +'+ 1 y t 1+'1 a t log(1 )(7) Under‡exible pricesmc= +'+ 1 y n t 1+'1 a t log(1 )(8)=)y n t= y+ya a twhere y ( log(1 ))(1 )+'+ (1 )>0and ya 1+'+'+ (1 ).=)cmc t= +'+ 1 (y t y n t)(9) where y t y n t e y t is the output gapNew Keynesian Phillips Curvet= E t f t+1g+ e y t(10) where +'+ 1 .Dynamic IS equatione y t=E tf e y t+1g 1 (i t E t f t+1g r n t)(11) where r n t is the natural rate of interest,given byr n t + E t f y n t+1g= + ya E t f a t+1gMissing block:description of monetary policy(determination of i t).Equilibrium under a Simple Interest Rate Rulei t= + t+ y e y t+v t(12) where v t is exogenous(possibly stochastic)with zero mean. Equilibrium Dynamics:combining(10),(11),and(12)e y t t =A T E tf e y t+1g E t f t+1g +B T(b r n t v t)(13) whereA T 1+ ( + y) ;B T 1and 1+ y+Uniqueness()A T has both eigenvalues within the unit circle Given 0and y 0,(Bullard and Mitra(2002)):( 1)+(1 ) y>0is necessary and su¢cient.E¤ects of a Monetary Policy ShockSet b r n t=0(no real shocks).Let v t follow an AR(1)processv t= v v t 1+"v tCalibration:v=0:5, =1:5, y=0:5=4, =0:99, ='=1, =2=3, =4. Dynamic e¤ects of an exogenous increase in the nominal rate(Figure1): Exercise:analytical solutionE¤ects of a Technology ShockSet v t=0(no monetary shocks).T echnology process:a t= a a t 1+"a t:Implied natural rate:b r n t= ya(1 a)a t Dynamic e¤ects of a technology shock( a=0:9)(Figure2) Exercise:AR(1)process for a tEquilibrium under an Exogenous Money Growth Processm t= m m t 1+"m t(14) Money market clearingb l t=b y t b i t(15)=e y t+b y n t b i t(16) where l t m t p t denotes(log)real money balances.Substituting(15)into(11):(1+ )e y t= E t f e y t+1g+b l t+ E t f t+1g+ b r n t b y n t(17) Furthermore,we haveb l t 1=b l t+ t m t(18)Equilibrium dynamicsA M;024e y t t b l t 135=A M;124E t f e y t+1g E t f t+1gb l t 135+B M24b r n tb y n t m t35(19)whereA M;0 241+ 00100 1135;A M;124 10 000135;B M24 1000000 135Uniqueness()A M A 1M;0A M;1has two eigenvalues inside and one outside the unit circle.E¤ects of a Monetary Policy ShockSet b r n t=y n t=0(no real shocks).Money growth processm t= m m t 1+"m t(20) where m2[0;1)Figure3(based on m=0:5)E¤ects of a Technology ShockSet m t=0(no monetary shocks).T echnology process:a t= a a t 1+"a t:Figure4(based on a=0:9).Empirical EvidenceTechnical AppendixOptimal Allocation of Consumption ExpendituresMaximization of C t for any given expenditure level R10P t(i)C t(i)di Z t can be formalized by means of the LagrangeanL= Z10C t(i)1 1 di 1 Z10P t(i)C t(i)di Z tThe associated…rst order conditions are:C t(i) 1 C t1 = P t(i)for all i2[0;1].Thus,for any two goods(i;j)we have:C t(i)=C t(j) P t(i)P t(j)which can be plugged into the expression for consumption expenditures to yieldC t(i)= P t(i)P t Z t P tfor all i2[0;1].The latter condition can then be substituted into the de…nition of C t,yieldingZ10P t(i)C t(i)di=P t C tCombining the two previous equations we obtain the demand schedule:C t(i)= P t(i)P t C tLog-Linearized Euler EquationWe can rewrite the Euler equation as1=E t f exp(i t c t+1 t+1 )g(21) In a perfect foresight steady state with constant in‡ation and constant growth we must have:i= + +with the steady state real rate being given byr i= +A…rst order Taylor expansion of exp(i t c t+1 t+1 )around that steady state yields:exp(i t c t+1 t+1 )'1+(i t i) ( c t+1 ) ( t+1 )=1+i t c t+1 t+1which can be used in(21)to obtain,after some rearrangement of terms,the log-linearized Euler equationc t=E t f c t+1g 1 (i t E t f t+1g )Aggregate Price Level DynamicsLet S(t) [0;1]denote the set of…rms which do not re-optimize their posted price in period t.The aggregate price level evolves according toP t= Z S(t)P t 1(i)1 di+(1 )(P t)1 11= (P t 1)1 +(1 )(P t)1 11where the second equality follows from the fact that the distribution of prices among…rms not adjusting in period t corresponds to the distribution of e¤ective prices in period t 1,with total mass reduced to .Equivalently,dividing both sides by P t 1:= +(1 ) P t P t 1 1 (22)1twhere t P t P t 1.Notice that in a steady state with zero in‡ation P t=P t 1:Log-linearization around a zero in‡ation( =1)steady state implies:t=(1 )(p t p t 1)(23)Price DispersionFrom the de…nition of the price index:1=Z10 P t(i)P t 1 "di=Z10exp f(1 )(p t(i) p t)g di'1+(1 )Z10(p t(i) p t)di+(1 )22Z10(p t(i) p t)2dithus implying the second order approximationp t'E i f p t(i)g+(1 )2Z10(p t(i) p t)2diwhere E i f p t (i )gR 10p t (i )di is the cross-sectional mean of (log)prices.In addition,Z 10 P t (i )P t 1 di =Z 10exp1 (p t (i ) p t ) di '1 1 Z 10(p t (i ) p t )di +12 1 2Z 10(p t (i ) p t )2di '1+12 (1 )1 Z 10(p t (i ) p t )2di +12 1 2Z 10(p t (i ) p t )2di =1+12 1 1 Z 10(p t (i ) p t )2di '1+12 1 1 var i f p t (i )g >1where 1 1 + ,and where the last equality follows from the observation that,up to second order,Z 10(p t (i ) p t )2di 'Z 10(p t (i ) E i f p t (i )g )2di var i f p t (i )gFinally,using the de…nition of d t we obtain d t (1 )log Z 10 P t (i )P t 1 di '12 var i f p t (i )g。