New Keynesian ModelJohn Du¤yEcon 2713 Notes Week #4John Du¤y () N ew Key nesian Model Econ 2713 Notes We ek #4 1 / 25New Keynesian ModelEmpirical work suggesting the impact of monetary policies on realactivity (see Clarida et al. (1999) for a list), has led to thedevelopment of a dynamic general equilibrium framework in whichmonetary policy can matter.The primary friction is that of nominal price rigidity, typically of theCalvo-pricing variety, (though the f ramework allows fully ‡exibleprices as a special case). Hence the name, “New Keynesian” model ofmonetary transmission.Another important dimension is that expectations of future variables,in‡ation and output play a key role. This leads to an importantdistinction between discretionary and commitment policies.Key references (used in this presentation): Clarida et. al. (1999),Woodford’s (2003) book Interest and Prices is the seminal statementof the model. Walsh’s (2003) bo ok Monetary Theory and Policy, 2ndEd. also has a nice presentation.John Du¤y () N ew Key nesian Model Econ 2713 Notes Week #4 2 / 25Modelling ConsiderationsAn optimizing model of households and …rms. Requires a productionfunction - labor only for now.Di¤erentiated goods and monopolistic competition – allows suppliersability to set prices.Sticky price adjustment - Calvo pricing assumption, only a certainfraction of …rms can adjust prices each period.Provide a mechanism for price level determination via interest raterules of the type used by central banks, as opposed to more typicaltheoretical approach relating changes in the price level to changes inthe supply of money (not an e¤ective policy target).John Du¤y () N ew Key nesian Model Econ 2713 Notes Week #4 3 / 25Household BehaviorThe Representative household consumes a basket of goods andsupplies labor to imperfectly competitive …rms.Its objective is tomax Et∞∑k =0βkU(Ct+k, Mt+k/Pt+k, Nt+k)subject to:PtCt+ Mt+ Bt= WtNt+ Mt1+ Rt1Bt1+ Πt+ Ttwhere Ctrepresents the real basket of goods consumed, Ptis theprice level, Mt/Pt, is real money balances, Ntis the labor supply, Wtis the nominal wage paid per unit of labor, Btis the time t quantityof nominal one period bonds paying gross nominal return Rtat timet + 1, Πtis nominal …rm pro…ts, and Ttis nominal lump-sumtaxes/transfers.John Du¤y () N ew Key nesian Model Econ 2713 Notes Week #4 4 / 25PreferencesFor tractability and speci…c results, it is useful to work with separablepreferences of the form:U(Ct, Mt/Pt, Nt)=C1σt1  σ+γ1  bMtPt1b χN1+ηt1 + ηWhether we include money in the utility function not (γ > 0, = 0),does not make much di¤erence.Policy will operate via interest rates rules to a¤ect the pricelevel/in‡ation.John Du¤y () N ew Key nesian Model Econ 2713 Notes Week #4 5 / 25Aggregate Consumption and the Price LevelThe basket of goods, (a continuum indexed by i) that is consumed bythe representative household is represented via a Dixit and Stiglitz(1977) CES aggregator:Ct=Z10c(θ1)/θitdiθ/(θ1)where θ > 0 is the elasticity of substitution among goods.The price index associated with this consumption basket is given by:Pt=Z∞0p1θitdi1/(1θ)Ptcan be interpreted as the minimum cost of buying a unit of theconsumption basket, as will become evident when we examine thehousehold’s optimal allocation of consumption expenditures acrossgoods available in the consumption basket.John Du¤y () N ew Key nesian Model Econ 2713 Notes Week #4 6 / 25Optimal Allocation of Consumption ExpendituresHouseholds seek to minimize the cost of buying the bundle Ct:mincitZ10pitcitdi subject to CtZ10c(θ1)/θitdiθ/(θ1)Set up the Lagrangian:Lt=Z10pitcitdi + λt CtZ10c(θ1)/θitdiθ/(θ1)!FOC with respect to good i:∂Lt/∂cit= pit λt c1/θitZ10c(θ1)/θitdi1/(θ1)!= 0.John Du¤y () N ew Key nesian Model Econ 2713 Notes Week #4 7 / 25Optimal Consumption Expenditures, continuedUsing the de…nition of Ctgiven above,pit= λtc1/θitC1/θtand rearranging we have:cit=pitλtθCt(1)John Du¤y () N ew Key nesian Model Econ 2713 Notes Week #4 8 / 25Firm i’s DemandUsing (1) in the de…nition of Ct, we have:Ct=24Z10(pitλtθCt)(θ1)/θdi35θ/(θ1)=1λtθZ∞0p1θitdiθ/(θ1)CtSolving for the Lagrange multiplier, we have:λt=Z∞0p1θitdi1/(1θ)= PtSo, λtis the price index representing the minimum cost of a unit ofthe consumption aggregate.John Du¤y () N ew Key nesian Model Econ 2713 Notes Week #4 9 / 25Firm i’s Demand, continuedWe can therefore rewrite the F.O.C. as:cit= CtpitPtθrepresenting the demand for the ith good, equivalently …rm i’sdemand. It follows that total expenditure equals PtCt, so we canwrite the budget constraint (as we have) in aggregate terms, withoutreference to i.John Du¤y () N ew Key nesian Model Econ 2713 Notes We ek #4 10 / 25Household’s Intertemporal Problem:max Et∞∑k =0βkC1σt+k1  σ+γ1  bMt+kPt+k1b χN1+ηt+k1 + ηsubject to:Ct+MtPt+BtPt=WtPtNt+Mt1Pt+ Rt1Bt1Pt+ΠtPt+TtPt.John Du¤y () N ew Key nesian Model Econ 2713 Notes We ek #4 11 / 25First Order ConditionsCan be rearranged to give:Euler equation for optimal intertemporal consumption:Cσt= βRtEtPtPt+1Cσt+1Optimal money holdings:γMtBtb=Rt 1RtCσtOptimal labor supply:χNηt=WtPtCσtJohn Du¤y () N ew Key nesian Model Econ 2713 Notes We ek #4 12 / 25Firm BehaviorPro…t maximizing …rms face three constraints:1The given production technology:yit= ZtNitwhich is linear in labor only (no capital). Ztis a productivitydisturbance. (More recently, models have added capital too).2The downward sloping demand curve:cit=pitPtθCt3Calvo staggered price adjustment. Each period, fraction ω of …rmsare unable to adjust their prices; fraction (1  ω) are able to adjusttheir prices. Probability of being able to change prices (1  ω) is iidacross …rms. So …rms set prices ta king this restriction into account.John Du¤y () N ew Key nesian Model Econ 2713 Notes We ek #4 13 / 25Cost MinimizationThe …rm chooses labor Nitto solve:minNitWtNit+ φ(cit ZtNit)N.B. Firms assumed to be wage-takers.F.O.C. implies that Wt= φZt, or φ = Wt/Zt.De…ne the real marginal cost mct= φ/Pt= Wt/(ZtPt)The …rm’s pricing problem is to set pitso as to maximize:Et∞∑k =0ωk∆k,t+ kpitPt+kcit +k φt+kcit +kor using