The new Kenesian model

Transcription

1 The new Kenesian model Michaª Brzoza-Brzezina Warsaw School of Economics 1 / 4

2 Flexible vs. sticky prices Central assumption in the (neo)classical economics: Prices (of goods and factor services) are fully exible An increase in money supply immediately increases prices 1:1 Classical dichotomy: money is neutral and monetary policy has no real eects 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 eects Also, additional propagation channels for other shocks Consequences for models: money and nominal variables important 2 / 4

3 Sticky prices: empirical evidence Price duration: US: average time between price changes is 2-4 quarters (Blinder et al., 1998; Bils and Klenow, 24; Klenow and Kryvstov, 25) Euro area: average time between price changes is 4-5 quarters (Rumler and Vilmunen, 25; Altissimo et al., 26) The higher ination, 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 3 / 4

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 Long-term relationships with customers 'Good' causes of price stickiness: in a stable economic environment agents trust in price stability 4 / 4

5 Plan of the Presentation 1 Introduction 2 Model 3 Log-linearization 4 Numerical simulations 5 Implications for monetary policy 6 Summary 5 / 4

6 New Keynesian model - basic features General equilibrium model Two stages of production, at one of them rms 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 eects 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 Simplications: No capital accumulation - labour is the only factor No trend productivity growth, only stationary stochastic shocks (hence, no need to normalize trending real variables) 6 / 4

7 Household's problem - objective Households maximize lifetime utility: max U = E t= β t [ c1 σ t 1+ϕ t where n t is the work eort and c t is consumption. Subject to: 1 σ n 1 + ϕ ] (1) W t n t + Div t + R t 1B t = P t c t + B t+1 (2) where P t is the price level of consumption and W t is the nominal wage rate and the transversality condition: E lim t ( B t P t t s=1 ) 1 (3) R s 7 / 4

8 FOCs Lagrangean: L t = E t i= 1+ϕ β t+i [[ c1 σ t+i 1 σ n t+i 1 + ϕ ] + + λ t+i (W t+i n t+i + R t+i 1B t+i + Div t+i P t+i c t+i B t+1+i)] P t+i First order conditions are: c t : c σ t = λ t (4) B t+1 P t : λ t + βe t λ t+1r t P t P t+1 = (5) n t : n ϕ t + λ t W t P t = (6) 8 / 4

9 Household's equilibrium conditions FOCs yield two equilibrium conditions. Intertemporal - choice between consumption today and tomorrow: c σ t = E t βc σ P t t+1 R t P t+1 Intratemporal - choice between consumption and leisure (labor supply): (7) n ϕ t c σ t = W t P t w t (8) 9 / 4

10 Production The production process is assumed to take place in two stages Intermediate goods rms produce dierentiated goods under monopolistic competition Here price stickiness is introduced - some rms will not be able to adjust their prices. This is the reason we assume products are dierentiated. Under perfect competition everybody has the same price (equal to marginal cost). Final good rms produce a homogeneous good from intermediate inputs 1 / 4

11 Final-goods rms I Final-goods rms produce according to the CES production function (Dixit-Stiglitz aggregator): where: y t = 1 y t (i) ε 1 ε di ε ε 1 y t (i) is output produced by intermediate-goods rm i ε > 1 is elasticity of substitution between individual intermediate goods When ε, y t is a sum of intermediate products (perfect competition) (9) 11 / 4

12 Final-goods rms II Maximization problem of nal-goods rms: max y t (i) P ty t 1 P t (ι)y t (ι)di subject to constraint (9) Final-goods rms are competitive, so they maximize their prots by chosing the inputs y t (i), taking prices as given FOC: ( ) Pt (ι) ε y t (i) = y t (1) Equation (1) gives the demand for intermediate good i P t 12 / 4

13 Intermediate-goods rms I Linear production function in labour only: y t (ι) = a t n t (ι). Productivity a t follows a rst-order autoregressive process: where: ρ < 1 and ε iid(, σ 2 ) a t = a ρ t 1 eε t (11) Labour inputs rented from households, technology available for free Prices are set according to the Calvo (1983) mechanism 13 / 4

14 Marginal cost The (real) cost function is q t = w t n t (ι) Real marginal cost is: mc t (ι) δq t(ι) δy t (ι) = w t a t (12) Note that the RHS is independent of ι. Hence, so is the LHS. The marginal cost is the same for every rm. 14 / 4

15 Optimal price setting This is somwhat more complicated. Assume that every period only a fraction 1 θ of rms are allowed to change their prices. The rm maximizes expected lifetime prots: maxπ t = E t i=(βθ) i λ t+i subject to the demand functions of households: ( P ) t (ι) ε y t+i(ι) = y t+i ( P ) t (ι) mc t+i y t+i(ι) P t+i P t+i where P t is the price set by rms that are allowed to reoptimize in period t. Note that rms are owned by households. Hence, their prots are discounted with β and vauled according to the household's marginal utility of consumption λ t+i. 15 / 4

16 Price as mark-up After some transformation (details in the supplement) we arrive at: E t i=(βθ) i λ t+i ( P t P t+i ε ) ( P ) ε ε 1 mc t t+i y t+i = (13) P t+i Note that all rms set the same new price Note that under exible prices (θ = ) and monopolistic competition the price chosen in period t is set as a mark-up over nominal marginal cost: ε P t = MP t mc t where M ε 1 is the gross markup. Hence, under monopolistic competition the price is set as a mark-up over marginal cost. 16 / 4

17 Price level First-order condition (13) is the same for each rm allowed to reset its price Therefore, all rms allowed to reoptimize at time t choose the same price The aggregate price level P t P t = 1 is then: P t (i) 1 ε di = ( (1 θ)p 1 ε t 1 1 ε where the rst equality follows from (1) = + θpt 1 1 ε ) 1 1 ε (14) 17 / 4

18 Monetary policy To complete the model we have to decide upon monetary policy The standard assumption is that it follows a Taylor rule This is motivated by empirical observations of central bank behavior R t R = ( Rt 1 R ) ( ρ ( πt ) ( ) ) φy 1 ρ φπ yt e ε i,t (15) π y 18 / 4

19 General equilibrium Market clearing conditions: Final output equals consumption: y t = c t (16) Labour supply equals labour demand: 1 1 y t (i) 1 ( ) Pt (i) ε yt y t n t = n t (i)di = di = di = t a t P t a t a t where: t = 1 ( ) P t (i) ε di 1 measures price dispersion P t with the law of motion: t = (1 θ) ( Pt P t ) ε + θ ( Pt 1 P t (17) ) ε t 1 (18) Note that price dispersion generates ineciency y t = a t n t t 19 / 4

20 Equilibrium conditions We have a set of 11 equations (4), (5), (6), (11), (12), (13), (14), (15), (16), (17) and (18) in 11 variables c t, n t, λ t, y t, mc t, P t, P t, t, w t, R t, a t. 2 / 4

21 Plan of the Presentation 1 Introduction 2 Model 3 Log-linearization 4 Numerical simulations 5 Implications for monetary policy 6 Summary 21 / 4

22 Household's equilibrium conditions Euler equation: ĉ t = E t ĉ t+1 1 σ ( ˆRt E t ˆπ t+1) (19) Labor supply: ϕˆn t + σĉ t = ŵ t (2) 22 / 4

23 Phillips curve Some derivations bring us to the (log-linearized) new Keynesian Phillips curve ˆπ t = (1 βθ)(1 θ) θ mc ˆ t + βe t ˆπ t+1 (21) Ination depends on marginal cost and expected ination. The latter because rms have to be forward looking. They do not know when they will be able to reset prices. 23 / 4

24 Marginal cost Marginal cost: mc t = w t a t Log-linear approximation: mc(1 + mc ˆ t ) = w (1 + wˆ a t â t ) Steady state: mc= w a Divide to get: mc ˆ t = wˆ t â t 24 / 4

25 Monetary policy Taylor rule becomes: ˆR t = ρ ˆRt 1 + (1 ρ)(φ π ˆπ t + φ y ŷ t ) + ε R,t 25 / 4

26 Price dispersion and production function Assuming zero steady state ination ˆ t = Then production function becomes ŷ t = â t + ˆn t 26 / 4

27 The NK model (large version) We now have a complete model: Euler equation ĉ t = E t ĉ t+1 1 σ ( ˆRt E t π t+1) Consumption-leisure choice: ϕˆn t + σĉ t = ŵ t Marginal cost: mc ˆ t = wˆ t â t Productivity: â t = ρ a â t 1 + ε a,t Production function: ŷ t = â t + ˆn t Phillips curve: ˆπ t = (1 βθ)(1 θ) θ mc ˆ t + βe t ˆπ t+1 Taylor rule: ˆRt = ρ ˆRt 1 + (1 ρ)(φ π ˆπ t + φ y ŷ t ) + ε R,t Market clearing: ĉ t = ŷ t 27 / 4

28 Large version - comments The large version is a good starting point for applied (say at central banks) DSGE models Of course it still lacks many aspects of reality Several additional elements are added to make them match the data better On the other hand, for educational reasons the NK model is often reduced to three equations 28 / 4

29 The NK model (compact version) Derivation in Gali (28), ch. 3 The three-equation model: ˆπ t = (1 βθ)(1 θ) (σ + ϕ)ˆx t + βe t ˆπ t+1 + ε c,t θ ˆx t = E t ˆx t+1 1 σ ( ) ˆRt E t π t+1 + εa,t ˆR t = ρ ˆRt 1 + (1 ρ)(φ π ˆπ t + φ y ˆx t ) + ε R,t 29 / 4

30 Calibration Standard calibration: θ = 2 β =.99 ϕ = 1 ε = 6 (implies a steady-state mark-up of 2%) θ =.75 φ π = 1.5, φ y =.5 (Taylor, 1993) ρ =.95, σ =.1 3 / 4

31 Plan of the Presentation 1 Introduction 2 Model 3 Log-linearization 4 Numerical simulations 5 Implications for monetary policy 6 Summary 31 / 4

32 Technology shock GDP Inflation Real marginal cost Nominal interest rate Productivity / 4

33 Monetary shock I Introduce monetary shock ε R,t Taylor rule - unexpected deviation from 33 / 4

34 Monetary shock II GDP Real marginal cost Inflation Nominal interest rate Monetary shock Real interest rate / 4

35 Government spending shock I Government spending G t fully nanced by lump sum taxes V t levied on households, so that G t = V t holds every period Modications to the model: Households' budget constraint (2) becomes: W t L t + Div t + R t 1B t = P t C t + B t+1 + V t Goods market clearing condition (16) becomes: Y t = C t + G t It is easy to verify that rst-order conditions of households' maximixation problem (4)-(6) remain unchanged We assume that government spending is stochastic and follows a rst-order autoregressive process: where: ln G t = (1 ρ G )G + ρ G ln G t 1 + ε G,t ρ G < 1 G is the steady state level of government spending 35 / 4

36 Government spending shock II GDP Inflation Real marginal cost Nominal interest rate Government spending Consumption / 4

37 Plan of the Presentation 1 Introduction 2 Model 3 Log-linearization 4 Numerical simulations 5 Implications for monetary policy 6 Summary 37 / 4

38 The role of expectations Equation (21) implies that current ination is aected by ination expectations Modern monetary policy: management of expectations Woodford: For not only do expectations about policy matter, (...) but very little else matters 38 / 4

39 Optimal monetary policy Equation (17) implies that price dispersion (i.e. t > 1) is costly Price dispersion can be eliminated if the central bank chooses to stabilize ination at zero (i.e. sets the ination target to zero and responds very aggresively to any deviations from the target) Hence, a policy strictly stabilizing ination can replicate the exible price equilibrium However, monetary policy may face a trade-o between stabilizing ination and keeping output at a desired (not constant, in general) level This trade-o vanishes if: steady state output is ecient (i.e. distortions related to monopolistic competition are eliminated, e.g. by proper subsidies to rms) there are no cost-push shocks (i.e. shocks to the Phillips curve) 39 / 4

40 New Keynesian model - summary Very simple dynamic stochastic general equilibrium model (DSGE) with monopolistic competition and sticky prices Monopolistic power of rms = decentralized allocations are not Pareto optimal (production not at an ecient level) Price stickiness restores the role of monetary policy: Monetary policy has real eects (aects 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 4 / 4