Calvo model+New Keyesian model

Econ5140Macroeconomic AnalysisFall2014Jenny XuHKUSTCalvo Model and New Keyesian Framework for Monetary Policy Analysis1Introduction•In the early1970s,1980s,and early1990s, the standard model used for most mone-tary policy analysis combined the assump-tion of nominal rigidity with a simple struc-ture linking the quantity of money to ag-gregate spending.•This linkage was usually directly through a quantity theory equation in which nominal demand was equal to the nominal money supply,often with a random disturbance included,or through a traditional textbook IS-LM model.•While the theoretical foundations of these models were weak,the approach proved remarkably useful in addressing a wide range of monetary policy topics.2•More recently,attention has been placed on ensuring that the model structure is consistent with the underlying behavior of optimizing economic agents.•The standard approach tody builds a dy-namic,stochastic,general equilibrium frame-work based on optimizing behavior,com-bined with some form of nominal wage and /or price rigidity.•Early examples of models with these prop-erties include those of Yun(1996),Good-friend and King(1997),Rotemberg and Woodford(1995,1997),and McCallum and Nelson(1999).•This lecture shows how a basic MIU model, combined with the assumption of monop-olistically competitive goods markets and price stickiness,can form the basic for a simple linear macroeconomic model that is useful for policy analysis.3Calvo’s model•An alternative model of staggered price adjustment is due to Calvo(1983).He assumed thatfirms adjustment their price infrequently and that opportunities to ad-just arrived as an exogenous Poisson pro-cess.•Each period,there is a constant probabil-ity1−ωthatfirm can adjust its price;the expected time between price adjustment .is11−ω•Because these adjustment opportunities oc-cur randomly,the interval between price changes for an individualfirm is a random variables.4•Following Rotemberg(1987),suppose the representativefirm i set its price to min-imize a quadratic loss function that de-pends on the difference between thefirm’s actual price in period t,p it,and its optimal price,p∗t.This latter price might denote the profit maximization price forfirm i in the absence of any restrictions or costs as-sociated with price adjustment.•If thefirm can adjust at time t,it will set its price to minimize1 2E t∞∑j=0βj(p t+j−p∗t+j)2(1)subject to the assumed process for determining when thefirm will next be able to adjust.•If only the terms in(1)involving the price set at time t are written out,they are1 2E t∞∑j=0(ωβ)j(p it−p∗t+j)2(2)whereωj is the probability thatfirm has not ad-justed after j periods so that the price set at t still holds in t+j.5•Thefirst order condition for the optimal choice of p it requires thatp it∞∑j=0ωjβj−∞∑j=0ωjβj E t p∗t+j=0(3)•Rearranging,and letting x t denote the op-timal price set by allfirms adjusting their price,x t=(1−ωβ)∞∑j=0ωjβj E t p∗t+j(4)The price set by thefirm at time t is a weighted average of current and expected future value of the target price p∗.•Equation(4)can be rewritten asx t=(1−ωβ)p∗t+ωβE t x t+1(5) If the price target p∗depends on the ag-gregate price level and output.6•We can replace p∗t with p t+γy t+ϵt,where ϵt is a random disturbance to capture other determinants of p∗.In general,thefirm’s optimal price will be shown to be a func-tion of its marginal cost,which,in turn, can be related to a measure of output.•With a large number offirms,a fraction 1−ωwill actually adjust their prices each period,and the aggregate price level can be expressed as p t=(1−ω)x t+ωp t−1.•We then have the following two equations to describe the evolution of x t and p t:x t=(1−ωβ)(p t+γy t+ϵt)+ωβE t x t+1(6)p t=(1−ω)x t+ωp t−1(7)•The aggregate inflation can be derived asπt=βE tπt+1+(1−ω)(1−ωβ)ω(γy t+ϵt)(8)7Discussion and Comparison•This expression is quite similar to the one we derived from Taylor’s model.Currentinflation depends on expected inflation andthe current output.•One difference is that the coefficient in front of E tπt+1is nowβ,but not1.Thisis because in??,we did not discount thereal wage.•Another difference,noted by Kiley(2002), is that the Tylor-type staggered adjust-ment model display less persistence thanthe Calvo-type model when both are cali-brated to the same frequency of price changes.•In Taylor(1979.1980),after two periods, all wages are adjusted after two periods.Or,no wages arefixed for more than twoperiods.8•In Calvo(1983),if we assumeω+12,then the expected time between price changes is two periods.So on average,prices are adjustment every two periods.•However,many prices remainfixed for more than two periods.For exam,w3=0.125 of all prices remainfixed for at least three periods.In Calvo,there is a tail of distri-bution of prices which have remainedfixed for many periods.•One attractive aspect of Calvo’s model is that it show how the coefficient on output in the inflation equation depends on the frequency with which prices are adjusted.•A rise inω,which means that the average time between price changes for an individ-ualfirm increase.Output movement havea smaller impact on current inflation,hold-ing expected future inflation constant.9A New Keyesian Model for MonetaryPolicy Analysis•The model is consistent with general equi-librium model in which all agents face well-defined decision problems and behave op-timally.•Three key modifications will be made here.–First,endogenous variations of the cap-ital stock are ignored.This follows Mc-Callum and Nelson(1999),who arguethat little is lost for the purpose of short-run business cycle analysis by assumingan exogenous process for the capitalstock.At least for the US,there is little relationshipbetween the capital stock and output dynamicsat business cycle frequencies.Cogley and Nason(1995)show that the re-sponse of investment and the capital stock toproductivity shocks contributes little to dynam-ics in RBC models.10•–The second key modification is to in-corporate differentiated goods whose in-dividual prices are set by monopolis-tically competitivefirms facing Calvo-type price stickiness.–Third,monetary policy is represented by a rule for setting the nominal rateof interest.The nominal quantity ofmoney is then endogenously determinedto achieve the desired nominal interestrate.•The modification yield a framework,often referred to as new Keynesian.The result-ing version of the MIU model can be linked directly to the more traditional aggregate supply-demand(AS-IS-LM)model.The new Keynesian model will be then used in later lectures to explore a variety of mon-etary policy issues.11The basic model•The model consists of households that sup-ply labor,purchase goods for consump-tion,and hold money and bonds andfirms that hire labor and produce and sell differ-entiated products in monopolistically com-petitive goods markets.•Eachfirm sets the price of the good it pro-duces,but not allfirms reset their price in each period.Households andfirm be-have optimally;Households maximize the expected present value of utility,andfirms maximize profits.•There is also a central bank that controls the nominal rate of interest.The central bank,in contrast to households andfirms, is not assumed to behave optimally.12Households•The preference of the representative house-hold are defined over a composite con-sumption good C t,real money balance M tP t ,and leisure1−N t,where N t is the time de-voted to market employment.Households maximize the expected present discounted value of utility:E t∞∑t=0βt[C1−σt+l1−σ+γ1−b(M t+iP t+i)1−b−χN1+ηt+i1+η](9)•There is a continuum of suchfirms of measure1,andfirm j produces good c j.The composite consumption good that en-ters the household’s utility function is de-fined asC t=[∫1cθ−1θjtdj]θθ−1,θ>1(10)The parameterθgoverns the price elastic-ity of demand for the individual goods.13•The household’s decision problem can be dealt with in two stages.–First,regardless of the level of C t the household decide on,it will always beoptimal to purchase combination of in-dividual goods that minimizes the costof achieving this level of the compositegood.–Second,given the cost of achieving any given level of C t,the household choosesC t,N t,and M t optimally.•Dealingfirst with the problem of minimiz-ing the cost of buying C t,the household’s decision problem is tomin c jt ∫1p jt c jt dj(11)subject to[∫1cθ−1θjtdj]θθ−1≥C t(12)where p jt is the price of good j.14•Lettingψt be Lagrangian multiplier on the constraint,thefirst order condition for good j isp jt−ψt[∫1cθ−1θjtdj]1θ−1c−1θjt=0(13)Rearranging,c jt=(p jtψt )−θC t,From the def-inition of the composite level of consump-tion,we can solve forψt and c jt.–The lagrangian multiplier is the appro-priately aggregated price index for con-sumption.ψt=[∫1p jt1−θdj]11−θ≡P t(14)–The demand for good j can then be written asc jt=(p jtP t)−θC t(15)The price elasticity of demand for good j is equal toθ.Asθ−→∞,the indi-vidual goods become closer and closer substitutes,and as a consequence,in-dividualfirms have less market power.15•Given the definition of the aggregate price index P t,the budget constraint of the house-hold is,in real terms,C t+M tP t+B tP t=W tP tN t+M t−1P t+(1+i t−1)B t−1P t+Πt(16)where M t(B t)is the household’s nomi-nal holding of money(one-period bonds).Bonds pay a nominal rate of interest i.Real profits received fromfirms are equal toΠt.•In the second stage of the household’s de-cision problem,consumption,labor supply, money,and bond holding are chosen to maximize expected utility subject to the budget constraint.•This leads to the following conditions,which, in addition to the budget constraint,must hold in equilibrium:16•C−σt =β(1+i t)E tP tP t+1C−σt+1(17)γ(M tP t)−bC−σt=i t1+i t(18)χNηtC−σt=W tP t(19)These conditions represent–The Euler condition for the optimal in-tertermporal allocation of consumption,–The intratemporal optimality condition setting the MRS between money and consumption equal to the opportunity cost of holding money,–and the intratemporal optimality con-dition setting the MRS between leisure and consumption equal to the real wage.17Firms•Firms maximize profits,subject to three constraints.Thefirst is the production summarizing the avail-able technology.For simplicity,we have ignored capital,so output is a function solely of labor sup-ply input N jt and an aggregate productivity distur-bance Z t:c jt=Z t N jt,E(Z t)=1(20)where constant return to scale has been assumed.The second constraint on thefirm is the demand curve each faces.This is given by(15).The third constraint is that each period somefirms are not able to adjust their price.•The specific model of price stickiness we will use in due to Calvo(1983).Each period,thefirms that adjust their price are randomly selected,and a fraction1−ωof allfirms adjust while the remainingωfraction do not adjust.The parameterωis a measure of the degree of nominal rigidity;18•Before analyzing thefirm’s pricing decision,con-sider its cost minimization problem,which involves minimizing W t N jt subject to producing c jt=Z t N jt.This problem can be written,in real terms,asmin N j (W tP t)N jt+φt(c jt−Z t N jt)(21)whereφt is equal to thefirm’s real marginal cost. Thefirst order condition impliesφt=W tP t Z t(22)•Thefirm’s pricing decision problem then involves picking p jt to maximizeE t∞∑i=0ωi△i,t+i[(p jtP t+i)C jt+i−φt+i C jt+i](23)where the discount factor△i,t+i is given byβi(C t+iC t)−σ. Using the demand curve(15)to eliminate c jt,this objective function can be written asE t∞∑i=0ωi△i,t+i[(p jtP t+i)1−θ−φt+i(p jtP t+i)−θ]C t+i(24)while individualfirms produce differentiated prod-ucts,they all have the same production technology and face demand curves with constant and equal demand elasticities.19•All firms adjusting in period t face the same prob-lem,so all adjusting firms will set the same price.Let P ∗t be the optimal price chosen by all firms ad-justing at time t .The first order condition for the optimal choice of P ∗t isP ∗t P t =θθ−1E t ∑∞i =0ωi βi C 1−σt +i φt +i (P t +i P t )θE t ∑∞i =0ωi βi C 1−σt +i (P t +i P t)θ−1(25)•Consider the case in which all firms are able to adjust their prices every period (ω=0),it reduce toP ∗t P t =θθ−1φt =µφt (26)Each firm sets price P ∗t equal to a markup µ>1over its nominal marginal cost P t φt .Because price exceeds marginal cost,output will be inefficiently low.•When prices are flexible,all firms charge the same price.In this case,p ∗t =P t and φt =1µt .Using the definition of real marginal cost,this meansW t P t =Z t µ(27)in a flexible price equilibrium.20•However,the real wage must also equal the marginal rate of substitution between leisure and consump-tion to be consistent with household optimization.This conditions implies thatW t P t =Z tµ=χNηtC−σt(28)•Letˆx t denote the percentage deviation of a vari-able X t around its steady state and let the super-script f denote theflexible price equilibrium.Then approximating(28)around the steady state yields ηˆn f t+σˆc f t=ˆz t.From the production function,ˆy f t =ˆn ft+ˆz t,and because output is equal to con-sumption in equilibrium,ˆy ft =ˆc ft.•Combining these conditions,theflexible-price equi-librium outputˆy ftcan be expressed asˆy f t =1+ησ+ηˆz t(29)When prices are sticky,output can differ from the flexible price equilibrium level.•The average price in period t satisfiesP1−θt =(1−ω)(P∗t)1−θ+ωP1−θt−1(30)21•Equation(25)and(30)can be approximated arounda zero average inflation,steady-state equilibriumto obtain an expression for aggregate inflation of the formπt=βE tπt+1+˜κˆφt(31)where˜κ=(1−ω)(1−βω)ωis an increasing function ofthe fraction offirms able to adjust each period andˆφt is the real marginal cost,expressed as a percentage deviation around its steady state value.•Equation(31)is often referred to as the new Key-nesian Philips curve.Unlike more traditional Philips curve equations,the new Keynesian Philips curve implies that real marginal cost is the correct driving variables for the inflation process.It also implies that the inflation process is forward-looking,with current inflation a function of ex-pected future inflation.•Solving(31),πt=˜κ∞∑i=0βi E tˆφt+i(32)which shows that inflation is a function of the present discounted value of current and future real marginal costs.This derivation reveals how˜κ,the impact of real marginal cost on inflation,depends on the structural parametersβandω.22•We can show that inflation is also related to the an output gap measurement.We rewriteˆφt=γ(ˆy t−ˆy f t)(33) whereγ=σ+η.This implies that the inflation adjustment equation becomesπt=βE tπt+1+κx t(34)and x t= whereκ=γ˜κ=γ(1−ω)(1−βω)ωˆy t−ˆy f t is the gap between actual output andflexible price equilibrium output.•Equation(34)relates output,in the form of the deviation around the level of output that would occur in the absence of nomi-nal price rigidities,to inflation.It forms one of the two key components of an optimizing model that can be used for monetary analysis.The other component is a linearized version of the household’s Euler condition.23Equilibrium•We now have all the components of a simple general equilibrium model that isconsistent with optimizing behavior on thepart of households andfirms.Because consumption is equal to outputin this model,(17),(25),and(30)provideequilibrium conditions that determines out-put,the price set byfirms adjusting theirprice,and the aggregate price level oncethe behavior of the nominal rate of inter-est rate is specified.With the nominal interest rate treated asthe monetary policy instrument,(18)sim-ply determines the nominal quantity of money in equilibrium.•Equation(17)can be approximated around the zero-inflation steady state asˆy t=E tˆy t+1−1σ(ˆi t−E tπt+1)(35)24•Expressing this in terms of the output gap x t=ˆy t−ˆy f t,x t=E t x t+1−1σ(ˆi t−E tπt+1)+µt(36)whereµt=E tˆy ft+1−ˆy f t depends only onthe exogenous productivity disturbance.Combining(36)and(34)gives a simple two-equation,forward-looking,rational-expectations model for inflation and the output gap measure x t.For convenience,(34)is repeated here:πt=βE tπt+1+κx t(37)•This two equation model represents the equilibrium conditions for a well-specified general equilibrium model.Equation(36) and(35)represents the demand side of the economy(an expectational,forward-looking IS curve),while the new Keyne-sian Philips curve(37)corresponds to the supply side.25•The model can be closed by assuming that the central bank implements monetary pol-icy through control of the nominal interest rate.–The linearized version of(18)can then be used tofind the equilibrium nominal money supply.–Alternatively,if the central bank implements money policy by setting a path for the nominalsupply of money,(36)and(37)together withthe linearized version of(18),determines x t,πt,andˆi t.•If a policy rule for the nominal interest rule is added to the model,this must be done with care to ensure that the policy rule does not render the system unstable or introduce multiple equilibria.For example, suppose monetary policy is represented by the following rule forˆi t,ˆi t=ργˆi t−1+v t(38) With this policy rule,there exist multiple bounded equilibria and the equilibrium is locally indeterminate.Stationary sunspot equilibria are possible.26•Bullard and Mitra(2002)show that a unique stationary equilibrium exists as long asδ>1.Settingδ>1is referred to as the Tay-lor Principle because John Taylor was the first to stress the importance of interest rate rules that called for responding more than one for one to changes in inflation.•Suppose that,instead of reacting solely to inflation,the central bank respond to both inflation and the output gap according toˆi t=δππt+δx x t+v t(39)This type of policy rule is called a Taylor rule(Tay-lor1993),and variants of it have been shown to provide a reasonable empirical description of the policy behavior of many central banks(Clarida, Gali,and Gertler2000).With this policy rule,stability now depends on both the policy parametersδπandδx.27The monetary transmission mechanism•The model consisting of(36)and(37)as-sumes that the impact of monetary policy on output and inflation operates through the real rate of interest.•As long as the central bank is able to affect the real interest rate through its control of nominal interest rate,monetary policy can affect real output.•The basic interest rate transmission mech-anism for monetary policy could be ex-tended to include effects on investment spending if capital were reintroduced into the model(Christiano,Eichenbaum,and Evans2001).Increases the real interest rate would reduce the demand for capital and lead to a fall in investment spending.28•In addition to these interest rate channels, monetary policy is often thought to af-fect the economy either indirectly through credit or directly through the quantity of money.–Since measure of money and bank credit move together.This is called the creditchannel of money transmission process.–It is sometimes argued that changes in the money supply have direct effectson aggregate demand that are indepen-dent of the interest rate channels thatoperates on consumption.Real money holding represent part ofhousehold wealth;an increase in realbalance should induce an increase inconsumption spending through a wealtheffect.This channel is often called thePigou effect.29。