New Keynesian model Marcin Kolasa Warsaw School of Economics Department of Quantitative Economics Marcin Kolasa (WSE) NK model 1 / 36 Flexible vs. sticky prices Central assumption in the (neo)classical economics: Prices (of goods and factor services) are fully flexible An increase in money supply immediately increases prices 1:1 Classical dichotomy: money is neutral and monetary policy has no real effects Consequences for models: we can abstract from money and nominal variables (New) Keynesian economics: Prices are sticky, i.e. they adjust sluggishly to macroeconomic shocks (including monetary shocks) Classical dichotomy does not hold: monetary policy has real effects Also, additional propagation channels for other shocks Consequences for models: money and nominal variables important Marcin Kolasa (WSE) NK model 2 / 36 Sticky prices: empirical evidence Price duration: US: average time between price changes is 2-4 quarters (Blinder et al., 1998; Bils and Klenow, 2004; Klenow and Kryvstov, 2005) Euro area: average time between price changes is 4-5 quarters (Rumler and Vilmunen, 2005; Altissimo et al., 2006) The higher inflation, the more frequently price changes occur Cross-industry heterogeneity Prices of tradables less sticky than those of nontradables Retail prices usually more sticky than producer prices Marcin Kolasa (WSE) NK model 3 / 36 Why are prices sticky? Lucas (1972): imperfect information Extensions: rational inattention (Sims, 2003; Mackowiak and Wiederholt, 2009), sticky information (Mankiw and Reis, 2007) Costs of changing prices (explicit or implicit): Menu costs Explicit contracts which are costly to renegotiate Long-term relationships with customers ’Good’ causes of price stickiness: in a stable economic environment agents trust in price stability Marcin Kolasa (WSE) NK model 4 / 36 New Keynesian model - basic features General equilibrium model Two stages of production, at one of them firms are monopolistically competitive - can set their prices Firms are not allowed to reoptimize their prices each period - prices are sticky Hence, monetary policy has real effects and so needs to be described within the model In a nutshell - the RBC model with: Sticky prices Monetary authority operating via an interest rate feedback rule Simplifications: No capital accumulation - labour is the only factor No trend productivity growth, only stationary stochastic shocks (hence, no need to normalize trending real variables) Marcin Kolasa (WSE) NK model 5 / 36 Households I Rent labour (the only production factor) to firms Own firms and so get their profits Divt Hold nominal bonds Bt , paying a nominal and risk-free (i.e. determined in the previous period) interest rate Rt−1 (expressed in gross terms) Make optimal consumption-savings (by adjusting bond holdings) and work-leasure decisions Intertemporal budget constraint: Wt Lt + Divt + Rt−1 Bt = Pt Ct + Bt+1 (1) where Pt is the price level of consumption and Wt is the nominal wage rate Marcin Kolasa (WSE) NK model 6 / 36 Households II Households maximize their expected lifetime utility : U0 = E0 ∞ X β t u(Ct , Lt ) (2) t=0 with instantaneous utility function u(Ct , Lt ) given by: u(Ct , Lt ) = L1+ϕ Ct1−θ −κ t 1−θ 1+ϕ The optimization is subject to the constraints: Budget constraint (1) Transversality condition: E0 lim t→∞ Marcin Kolasa (WSE) t Bt Y 1 Pt s=1 Rs NK model ! ≥0 (3) 7 / 36 Households’ optimization Lagrange function: LL = E0 ∞ X t=0 β t L1+ϕ Ct1−θ −κ t + λ̃t 1−θ 1+ϕ Wt Lt + Divt − Pt Ct +Rt−1 Bt − Bt+1 ! First order conditions: ∂LL = 0 =⇒ Ct−θ = λ̃t Pt ∂Ct ∂LL = 0 =⇒ κLϕ t = λ̃t Wt ∂Lt n o ∂LL = 0 =⇒ λ̃t = βEt λ̃t+1 Rt ∂Bt+1 Marcin Kolasa (WSE) NK model (4) (5) (6) 8 / 36 Firms Two stages of production: Final-goods firms produce output by combining intermediate goods Intermediate-goods firms produce using labour as the only input Contrary to the RBC model, final-goods production non-trivial since intermediate goods are not perfect substitutes. Therefore, the final output is not a simple sum of intermediate goods production. Marcin Kolasa (WSE) NK model 9 / 36 Final-goods firms I Final-goods firms produce according to the CES production function (Dixit-Stiglitz aggregator): φ φ−1 Z1 Yt (i) Yt = φ−1 φ di (7) 0 where: The continuum of intermediate-goods firms (indexed by i) is normalized to 1 Yt (i) is output produced by intermediate-goods firm i φ > 1 is elasticity of substitution between individual intermediate goods Note: When φ → ∞, Yt is a simple sum of intermediate products (like in the RBC model, where all producers are perfectly competitive) Marcin Kolasa (WSE) NK model 10 / 36 Final-goods firms II Maximization problem of final-goods firms: Z1 Pt Yt − Pt (i)Yt (i)di max Yt ,Yt (i) 0 subject to constraint (7) Final-goods firms are competitive, so they maximize their profits by chosing the inputs Yt (i), taking all prices (Pt (i) and Pt ) as given First order conditions: Yt (i) = Pt (i) Pt −φ Yt (8) Equation (8) defines the demand for intermediate input i Marcin Kolasa (WSE) NK model 11 / 36 Intermediate-goods firms I Linear production function in labour only: Yt (i) = Xt Lt (i) (9) Productivity Xt is common to all firms and follows the first-order autoregressive process: ln Xt = ρ ln Xt−1 + εt (10) where: 0 ≤ ρ < 1 and ε ∼ iid(0, σ 2 ) Labour inputs rented from households, technology available for free Prices are set according to the Calvo (1983) mechanism Marcin Kolasa (WSE) NK model 12 / 36 Price setting with flexible prices I Maximization problem of intermediate-goods firm i: max Pt (i),Yt (i),Lt (i) {Pt (i)Yt (i) − Wt Lt (i)} subject to the demand function (8) and the production function (9) Maximization problem rewritten using the demand and production function constraints: ( ) Wt Pt (i) −φ max Pt (i) − Yt Xt Pt Pt (i) Each intermediate-goods firms takes the economy-wide wage rate Wt and output Yt as given Marcin Kolasa (WSE) NK model 13 / 36 Price setting with flexible prices II First-order condition: Pt (i) = Note that Wt Xt φ Wt φ − 1 Xt (11) is marginal cost So, imperfectly competitive intermediate-goods firms set their prices as a (constant) mark-up over marginal costs, where the mark-up φ equals to φ−1 Note that since neither mark-ups nor marginal costs are firm-specific, all intermediate-goods firms choose the same prices Marcin Kolasa (WSE) NK model 14 / 36 Price setting with sticky prices I Calvo scheme: Each period a constant proportion 1 − γ (0 < γ < 1) of randomly selected intermediate-goods firms is allowed to reset their prices The remaining intermediate-goods firms have to keep their prices unchanged Firms allowed to reset their price take into account that they may not be allowed to do so in the future The probability that in period t + s the price of intermediate-goods firm i is still Pt (i) equals γ s The expected time of a price remaining fixed equals (1 − γ)−1 Marcin Kolasa (WSE) NK model 15 / 36 Price setting with sticky prices II Maximization problem of intermediate-goods firm i: ( ∞ ) X Wt+s Pt (i) −φ s s max Et λ̃t+s β γ Pt (i) − Yt+s Xt+s Pt+s Pt (i) s=0 Note: Profit maximization is dynamic: firms must take into account that they may not have a chance to reset their prices in the future Firms are owned by households, so they discount the utility value of their future profits by the discount factor β First-order condition: ∞ X s s Et λ̃t+s β γ Pt (i) − s=0 Marcin Kolasa (WSE) φ Wt+s φ − 1 Xt+s NK model Pt (i) Pt+s −φ Yt+s = 0 (12) 16 / 36 Price setting with sticky prices III First-order condition (12) is the same for each firm allowed to reset its price Therefore, all firms allowed to reoptimize at time t choose the same price, which we denote by P̃t The aggregate price level Pt is then: Pt 1 1 1−φ Z = Pt (i)1−φ di = 0 = 1−φ (1 − γ)P̃t1−φ + γPt−1 1 1−φ (13) where the first equality follows from (8) Marcin Kolasa (WSE) NK model 17 / 36 Monetary policy Prices are sticky, so monetary policy has real effects Monetary authorities set the short-term (one period) nominal interest rate according tho the Taylor-like feedback rule (see Taylor, 1993): Rt = R + aπ (πt − π̄) + ay (ln Yt − ln Y ) (14) where: R = π̄β is steady state nominal (gross) interest rate t π = PPt−1 is (gross) inflation and π̄ is the target (steady state) inflation Y is steady-state output aπ > 1, ay ≥ 0 Central bank can completely stabilize inflation by responding very aggresively to deviations of inflation from the target (i.e. by choosing a very large value for aπ ) Marcin Kolasa (WSE) NK model 18 / 36 General equilibrium Market clearing conditions: Output produced by firms must be equal to households’ total spending (consumption): Yt = Ct (15) Labour supplied by households must be equal to labour demanded by firms: −φ Z1 Z1 Z1 Pt (i) Yt Yt Yt (i) di = di = ∆t (16) Lt = Lt (i)di = Xt Pt Xt Xt 0 where: ∆t = 0 0 R1 Pt (i) −φ 0 Pt di ≥ 1 is a measure of price dispersion (∆t = 1 ⇔ ∀i : Pt (i) = Pt ) Proceeding similarly to (13) one can show that: !−φ P̃t Pt−1 −φ ∆t = (1 − γ) +γ ∆t−1 Pt Pt Marcin Kolasa (WSE) NK model (17) 19 / 36 Equilibrium dynamics Equilibrium dynamics can be summarized by 9 equations (4), (5), (6), (12), (13), (14), (15), (16) and (17), as well as the transversality condition (3) They describe the evolution of 9 endogenous variables: Ct , Lt , Wt , λ̃t , Yt , Lt , Pt , P̃t and Rt . This system can actually be reduced to just 5 endogenous variables: Yt , Pt , P̃t , ∆t and Rt The only exogenous driving force in the model is stochastic productivity Xt defined by (10) In principle, the New Keynesian model usually includes also other shocks Marcin Kolasa (WSE) NK model 20 / 36 Log-linear approximation of the model Due to nonlinearities and presence of expectations, the model does not have a closed-form solution Standard technique: log-linear approximation of the model equations around the (non-stochastic) steady-state Non-stochastic steady-state: no productivity shocks and all variables in the model constant Marcin Kolasa (WSE) NK model 21 / 36 New Keynesian Phillips curve Log-linearized (12) and (13) imply the following New Keynesian Phillips curve: πt = βEt πt+1 + (1 − γ)(1 − βγ) mct γ (18) where: mct is log devation of real marginal cost (i.e. XWt Pt t ) from its steady-state value we assumed for simplicity that the target (and so steady-state) inflation rate is zero Marcin Kolasa (WSE) NK model 22 / 36 New Keynesian IS curve Log-linearized (4), (6) and (15) imply the following New Keynesian IS curve: 1 yt = Et yt+1 − (Rt − R − Et πt+1 ) (19) θ where: yt = ln Yt − ln Y is log devation of output from its steady-state value we assumed for simplicity that the target (and so steady-state) inflation rate is zero Marcin Kolasa (WSE) NK model 23 / 36 Calibration Standard calibration: θ = 2 (more reasonable than 1 if there is no capital) β = 0.99 κ=1 ϕ=1 φ = 6 (implies a steady-state mark-up of 20%) γ = 0.75 aπ = 1.5, ay = 0.5 (Taylor, 1993) ρ = 0.95, σ = 0.01 κ and φ do not appear in the log-linearized version of the model Marcin Kolasa (WSE) NK model 24 / 36 Technology shock GDP Real marginal cost 0.8 0 0.7 -0.1 0.6 0.5 -0.2 0.4 -0.3 0.3 0.2 -0.4 0.1 -0.5 0 0 10 20 30 40 50 60 70 80 90 0 100 10 20 Inflation 30 40 50 60 70 80 90 100 80 90 100 Nominal interest rate 0 0 -0.1 -0.1 -0.2 -0.2 -0.3 -0.3 -0.4 -0.4 -0.5 -0.5 -0.6 -0.6 -0.7 -0.7 -0.8 -0.8 0 10 20 30 40 50 60 70 80 90 100 70 80 90 100 0 10 20 30 40 50 60 70 Productivity 1.2 1 0.8 0.6 0.4 0.2 0 0 10 20 30 40 50 60 Notes: grey line - flexible prices (γ → 0); black line - sticky prices (our baseline model); time unit - quarters; all variables expressed as percentage (inflation and interest rate - percentage point) deviations from their steady-state values Marcin Kolasa (WSE) NK model 25 / 36 Monetary shock I Monetary shock: an additive stochastic component RRt in the interest rate rule, so that (14) becomes: Rt = R + aπ (πt − π̄) + ay (ln Yt − ln Y ) + RRt RRt follows a first-order autoregressive process: RRt = ρR RRt−1 + εR,t where 0 ≤ ρR < 1 Marcin Kolasa (WSE) NK model 26 / 36 Monetary shock II GDP Real marginal cost 0 0 -0.1 -0.25 -0.5 -0.2 -0.75 -0.3 -1 -0.4 -1.25 -1.5 -0.5 0 10 20 30 40 50 60 70 80 90 0 100 10 20 Inflation 30 40 50 60 70 80 90 100 70 80 90 100 70 80 90 100 Nominal interest rate 0 0 -0.25 -0.25 -0.5 -0.5 -0.75 -0.75 -1 -1 -1.25 -1.5 -1.25 -1.75 -1.5 -2 -1.75 0 10 20 30 40 50 60 70 80 90 100 0 10 20 Monetary shock 30 40 50 60 Real interest rate 1.2 0.1 1 0.08 0.8 0.06 0.6 0.04 0.4 0.02 0.2 0 0 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 Notes: ρR = 0.90; time unit - quarters; all variables expressed as percentage (inflation and interest rate - percentage point) deviations from their steady-state values Marcin Kolasa (WSE) NK model 27 / 36 Government spending shock I Government spending Gt fully financed by lump sum taxes Vt levied on households, so that Gt = Vt holds every period Modifications to the model: Households’ budget constraint (1) becomes: Wt Lt + Divt + Rt−1 Bt = Pt Ct + Bt+1 + Vt Goods market clearing condition (15) becomes: Yt = Ct + Gt It is easy to verify that first-order conditions of households’ maximixation problem (4)-(6) remain unchanged We assume that government spending is stochastic and follows a first-order autoregressive process: ln Gt = (1 − ρG )G + ρG ln Gt−1 + εG ,t where: 0 ≤ ρG < 1 G is the steady state level of government spending Marcin Kolasa (WSE) NK model 28 / 36 Government spending shock II GDP Real marginal cost 0.16 0 0.14 -0.01 0.12 -0.02 0.1 -0.03 0.08 -0.04 0.06 -0.05 0.04 -0.06 0.02 -0.07 0 0 10 20 30 40 50 60 70 80 90 0 100 10 20 Inflation 30 40 50 60 70 80 90 100 70 80 90 100 Nominal interest rate 0 0 -0.02 -0.02 -0.04 -0.04 -0.06 -0.06 -0.08 -0.08 -0.1 -0.1 -0.12 -0.14 -0.12 0 10 20 30 40 50 60 70 80 90 100 0 10 20 Government spending 30 40 50 60 Consumption 1.2 0 1 -0.02 0.8 -0.04 0.6 -0.06 0.4 -0.08 0.2 0 -0.1 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 Notes: ρG = 0.95; government spending share in output is 20%; time unit - quarters; all variables expressed as percentage (inflation and interest rate - percentage point) deviations from their steady-state values Marcin Kolasa (WSE) NK model 29 / 36 Shock decomposition in a standard medium-sized closed-economy model (Smets and Wouters, 2007) Marcin Kolasa (WSE) NK model 30 / 36 Shock decomposition in a standard medium-sized closed-economy model (Smets and Wouters, 2007) Marcin Kolasa (WSE) NK model 31 / 36 Shock decomposition in a standard medium-sized open-economy model (Christoffel et al., 2008) Marcin Kolasa (WSE) NK model 32 / 36 Shock decomposition in a model with financial frictions (Christiano et al., 2010) Marcin Kolasa (WSE) NK model 33 / 36 The role of expectations Equation (18) implies that current inflation is affected by inflation expectations Modern monetary policy: management of expectations Woodford: For not only do expectations about policy matter, (...) but very little else matters Marcin Kolasa (WSE) NK model 34 / 36 Optimal monetary policy Equation (16) implies that price dispersion (i.e. ∆t > 1) is costly Price dispersion can be eliminated if the central bank chooses to stabilize inflation at zero (i.e. sets the inflation target to zero and responds very aggresively to any deviations from the target) Hence, a policy strictly stabilizing inflation can replicate the flexible price equilibrium However, monetary policy may face a trade-off between stabilizing inflation and keeping output at a desired (not constant, in general) level This trade-off vanishes if: steady state output is efficient (i.e. distortions related to monopolistic competition are eliminated, e.g. by proper subsidies to firms) there are no cost-push shocks (i.e. shocks to the Phillips curve) In this case perfect price stability is optimal Marcin Kolasa (WSE) NK model 35 / 36 New Keynesian model - summary Very simple dynamic stochastic general equilibrium model (DSGE) with monopolistic competition and sticky prices Monopolistic power of firms =⇒ decentralized allocations are not Pareto optimal (production not at an efficient level) Price stickiness restores the role of monetary policy: Monetary policy has real effects (affects output, consumption, real wages) The case for price stabilization: price stability eliminates price distortion Pursuing strict price stabilization is optimal if steady state distortions (due to monopolistic competition) are eliminated (e.g. by production subsidies) The workhorse model in central banks Marcin Kolasa (WSE) NK model 36 / 36

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