New Keynesian model. Marcin Kolasa. Warsaw School of Economics Department of Quantitative Economics. Marcin Kolasa (WSE) NK model 1 / 36


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1 New Keynesian model Marcin Kolasa Warsaw School of Economics Department of Quantitative Economics Marcin Kolasa (WSE) NK model 1 / 36
2 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
3 Sticky prices: empirical evidence Price duration: US: average time between price changes is 24 quarters (Blinder et al., 1998; Bils and Klenow, 24; Klenow and Kryvstov, 25) Euro area: average time between price changes is 45 quarters (Rumler and Vilmunen, 25; Altissimo et al., 26) The higher inflation, the more frequently price changes occur Crossindustry 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
4 Why are prices sticky? Lucas (1972): imperfect information Extensions: rational inattention (Sims, 23; Mackowiak and Wiederholt, 29), sticky information (Mankiw and Reis, 27) Costs of changing prices (explicit or implicit): Menu costs Explicit contracts which are costly to renegotiate Longterm 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
5 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
6 Households I Rent labour (the only production factor) to firms Own firms and so get their profits Div t Hold nominal bonds B t, paying a nominal and riskfree (i.e. determined in the previous period) interest rate R t 1 (expressed in gross terms) Make optimal consumptionsavings (by adjusting bond holdings) and workleasure decisions Intertemporal budget constraint: W t L t + Div t + R t 1 B t = P t C t + B t+1 (1) where P t is the price level of consumption and W t is the nominal wage rate Marcin Kolasa (WSE) NK model 6 / 36
7 Households II Households maximize their expected lifetime utility : U = E β t u(c t, L t ) (2) t= with instantaneous utility function u(c t, L t ) given by: u(c t, L t ) = C t 1 θ 1 θ κ L1+ϕ t 1 + ϕ The optimization is subject to the constraints: Budget constraint (1) Transversality condition: ( ) B t t 1 E lim (3) t P t R s=1 s Marcin Kolasa (WSE) NK model 7 / 36
8 Households optimization Lagrange function: ( LL = E β t Ct 1 θ [ 1 θ κ L1+ϕ t 1 + ϕ + Wt L λ t + Div t P t C t t +R t 1 B t B t+1 t= ] ) First order conditions: LL C t = = C θ t = λ t P t (4) LL L t = = κl ϕ t = λ t W t (5) LL } = = λ t = βe t { λ t+1 R t B t+1 (6) Marcin Kolasa (WSE) NK model 8 / 36
9 Firms Two stages of production: Finalgoods firms produce output by combining intermediate goods Intermediategoods firms produce using labour as the only input Contrary to the RBC model, finalgoods production nontrivial 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
10 Finalgoods firms I Finalgoods firms produce according to the CES production function (DixitStiglitz aggregator): Y t = 1 Y t (i) φ 1 φ di φ φ 1 (7) where: The continuum of intermediategoods firms (indexed by i) is normalized to 1 Y t (i) is output produced by intermediategoods firm i φ > 1 is elasticity of substitution between individual intermediate goods Note: When φ, Y t is a simple sum of intermediate products (like in the RBC model, where all producers are perfectly competitive) Marcin Kolasa (WSE) NK model 1 / 36
11 Finalgoods firms II Maximization problem of finalgoods firms: 1 max P ty t P t (i)y t (i)di Y t,y t(i) subject to constraint (7) Finalgoods firms are competitive, so they maximize their profits by chosing the inputs Y t (i), taking all prices (P t (i) and P t ) as given First order conditions: ( ) Pt (i) φ Y t (i) = Y t (8) Equation (8) defines the demand for intermediate input i P t Marcin Kolasa (WSE) NK model 11 / 36
12 Intermediategoods firms I Linear production function in labour only: Y t (i) = X t L t (i) (9) Productivity X t is common to all firms and follows the firstorder autoregressive process: where: ρ < 1 and ε iid(, σ 2 ) ln X t = ρ ln X t 1 + ε t (1) 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
13 Price setting with flexible prices I Maximization problem of intermediategoods firm i: max {P t(i)y t (i) W t L t (i)} P t(i),y t(i),l t(i) subject to the demand function (8) and the production function (9) Maximization problem rewritten using the demand and production function constraints: { ( P t (i) W ) ( ) } t Pt (i) φ Y t max P t(i) X t P t Each intermediategoods firms takes the economywide wage rate W t and output Y t as given Marcin Kolasa (WSE) NK model 13 / 36
14 Price setting with flexible prices II Firstorder condition: P t (i) = φ W t (11) φ 1 X t Note that Wt X t is marginal cost So, imperfectly competitive intermediategoods firms set their prices as a (constant) markup over marginal costs, where the markup equals to φ φ 1 Note that since neither markups nor marginal costs are firmspecific, all intermediategoods firms choose the same prices Marcin Kolasa (WSE) NK model 14 / 36
15 Price setting with sticky prices I Calvo scheme: Each period a constant proportion 1 γ ( < γ < 1) of randomly selected intermediategoods firms is allowed to reset their prices The remaining intermediategoods 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 intermediategoods firm i is still P t (i) equals γ s The expected time of a price remaining fixed equals (1 γ) 1 Marcin Kolasa (WSE) NK model 15 / 36
16 Price setting with sticky prices II Maximization problem of intermediategoods firm i: { E t λ t+s β s γ (P s t (i) W ) ( ) t+s Pt (i) φ Y t+s} max P t(i) s= X t+s P t+s 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 Firstorder condition: ( E t λ t+s β s γ s P t (i) s= φ φ 1 W t+s X t+s ) ( ) Pt (i) φ Y t+s = (12) P t+s Marcin Kolasa (WSE) NK model 16 / 36
17 Price setting with sticky prices III Firstorder 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 P t The aggregate price level P t is then: P t = = 1 P t (i) 1 φ di ( 1 φ (1 γ) P t where the first equality follows from (8) 1 1 φ = ) + γp 1 φ 1 1 φ t 1 (13) Marcin Kolasa (WSE) NK model 17 / 36
18 Monetary policy Prices are sticky, so monetary policy has real effects Monetary authorities set the shortterm (one period) nominal interest rate according tho the Taylorlike feedback rule (see Taylor, 1993): R t = R + a π (π t π) + a y (ln Y t ln Y ) (14) where: R = π β is steady state nominal (gross) interest rate π = Pt P t 1 is (gross) inflation and π is the target (steady state) inflation Y is steadystate output a π > 1, a y 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 a π ) Marcin Kolasa (WSE) NK model 18 / 36
19 General equilibrium Market clearing conditions: Output produced by firms must be equal to households total spending (consumption): Y t = C t (15) Labour supplied by households must be equal to labour demanded by firms: L t = 1 1 where: t = L t (i)di = ( Pt(i) P t 1 Y t (i) 1 di = X t ( Pt (i) P t ) φ Y t X t di = t Y t X t (16) ) φ di 1 is a measure of price dispersion ( t = 1 i : P t (i) = P t ) Proceeding similarly to (13) one can show that: ( ) φ ( ) φ Pt Pt 1 t = (1 γ) + γ t 1 (17) P t P t Marcin Kolasa (WSE) NK model 19 / 36
20 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: C t, L t, W t, λ t, Y t, L t, P t, P t and R t. This system can actually be reduced to just 5 endogenous variables: Y t, P t, P t, t and R t The only exogenous driving force in the model is stochastic productivity X t defined by (1) In principle, the New Keynesian model usually includes also other shocks Marcin Kolasa (WSE) NK model 2 / 36
21 Loglinear approximation of the model Due to nonlinearities and presence of expectations, the model does not have a closedform solution Standard technique: loglinear approximation of the model equations around the (nonstochastic) steadystate Nonstochastic steadystate: no productivity shocks and all variables in the model constant Marcin Kolasa (WSE) NK model 21 / 36
22 New Keynesian Phillips curve Loglinearized (12) and (13) imply the following New Keynesian Phillips curve: π t = βe t π t+1 + (1 γ)(1 βγ) mc t (18) γ where: W mc t is log devation of real marginal cost (i.e. t X tp t ) from its steadystate value we assumed for simplicity that the target (and so steadystate) inflation rate is zero Marcin Kolasa (WSE) NK model 22 / 36
23 New Keynesian IS curve Loglinearized (4), (6) and (15) imply the following New Keynesian IS curve: y t = E t y t+1 1 θ (R t R E t π t+1 ) (19) where: y t = ln Y t ln Y is log devation of output from its steadystate value we assumed for simplicity that the target (and so steadystate) inflation rate is zero Marcin Kolasa (WSE) NK model 23 / 36
24 Calibration Standard calibration: θ = 2 (more reasonable than 1 if there is no capital) β =.99 κ = 1 ϕ = 1 φ = 6 (implies a steadystate markup of 2%) γ =.75 a π = 1.5, a y =.5 (Taylor, 1993) ρ =.95, σ =.1 κ and φ do not appear in the loglinearized version of the model Marcin Kolasa (WSE) NK model 24 / 36
25 Technology shock GDP Inflation Real marginal cost Nominal interest rate Productivity Notes: grey line  flexible prices (γ ); black line  sticky prices (our baseline model); time unit  quarters; all variables expressed as percentage (inflation and interest rate  percentage point) deviations from their steadystate values Marcin Kolasa (WSE) NK model 25 / 36
26 Monetary shock I Monetary shock: an additive stochastic component RR t in the interest rate rule, so that (14) becomes: R t = R + a π (π t π) + a y (ln Y t ln Y ) + RR t RR t follows a firstorder autoregressive process: where ρ R < 1 RR t = ρ R RR t 1 + ε R,t Marcin Kolasa (WSE) NK model 26 / 36
27 Monetary shock II GDP Real marginal cost Inflation Nominal interest rate Monetary shock Real interest rate Notes: ρ R =.9; time unit  quarters; all variables expressed as percentage (inflation and interest rate  percentage point) deviations from their steadystate values Marcin Kolasa (WSE) NK model 27 / 36
28 Government spending shock I Government spending G t fully financed by lump sum taxes V t levied on households, so that G t = V t holds every period Modifications to the model: Households budget constraint (1) becomes: W t L t + Div t + R t 1 B t = P t C t + B t+1 + V t Goods market clearing condition (15) becomes: Y t = C t + G t It is easy to verify that firstorder conditions of households maximixation problem (4)(6) remain unchanged We assume that government spending is stochastic and follows a firstorder autoregressive process: ln G t = (1 ρ G )G + ρ G ln G t 1 + ε G,t where: ρ G < 1 G is the steady state level of government spending Marcin Kolasa (WSE) NK model 28 / 36
29 Government spending shock II GDP Real marginal cost Inflation Nominal interest rate Government spending Consumption Notes: ρ G =.95; government spending share in output is 2%; time unit  quarters; all variables expressed as percentage (inflation and interest rate  percentage point) deviations from their steadystate values Marcin Kolasa (WSE) NK model 29 / 36
30 Shock decomposition in a standard mediumsized closedeconomy model (Smets and Wouters, 27) Marcin Kolasa (WSE) NK model 3 / 36
31 Shock decomposition in a standard mediumsized closedeconomy model (Smets and Wouters, 27) Marcin Kolasa (WSE) NK model 31 / 36
32 Shock decomposition in a standard mediumsized openeconomy model (Christoffel et al., 28) Marcin Kolasa (WSE) NK model 32 / 36
33 Shock decomposition in a model with financial frictions (Christiano et al., 21) Marcin Kolasa (WSE) NK model 33 / 36
34 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
35 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 tradeoff between stabilizing inflation and keeping output at a desired (not constant, in general) level This tradeoff 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 costpush shocks (i.e. shocks to the Phillips curve) In this case perfect price stability is optimal Marcin Kolasa (WSE) NK model 35 / 36
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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