Dynamic Stochastic General Equilibrium Models

Transcription

1 Dynamic Stochastic General Equilibrium Models Dr. Andrea Beccarini M.Sc. Willi Mutschler Summer 2014 A. Beccarini () Advanced Macroeconomics DSGE Summer / 33

2 The log-linearization procedure: One starts from the Taylor-series approximation of the rst order of the function f (x t, y t ) around the non-stochastic steady-state values x and y: f (x t, y t ) t f (x, y) + f 0 x (x, y)(x t x) + f 0 y (x, y)(y t y) Having de ned x t xe xˆ t it follows xˆ t = log(x t ) log(x) x t x x substituting it in the Taylor-series approximation yields: f (x t, y t ) t f (x, y) + xfx 0 (x, y) xˆ t + yfy 0 (x, y) yˆ t t follows for example: x t y t = xye xˆ t + yˆ t xy(1 + xˆ t + yˆ t ) (x t y t ) θ t (x y) θ + θ(x y) θ 1 (xxˆ t yyˆ t ) nally recall xˆ t yˆ t t 0 A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

3 The linearized model: Equation of consumption The linearization of the Euler equation of consumption is the following: ˆ C t = h 1 + h Ĉt h E tĉt h ˆε B t E t ˆε B t+1 (1 + h)σ c 1 h Rˆ t E t π t+1 ˆ + (1 + h)σ c A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

4 By the external habit, present consumption depends on both the expected consumption of the next period and on consumption of the last period. For h = 0 one obtains the usual Euler equation. The elasticity of consumption with respect to interest rates depends both on the intertemporal elasticity of substitution σ c and on the degree of the habit h. A higher h reduces the e ect on consumption of a variation of interest rate holding σ c. A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

5 From the maximization problem of households, combine i λ t = ε B t (Ct τ H t ) σ c R with E t hβλ t t+1 P t+1 = λ t P t E t hβε B t+1 (C t+1 hc t ) σ c R t P t+1 i t E t 2 4 E t βε B (C hc ) σ c R P + βεb (C hc ) σ c R P ˆεB t+1+ σ c βε B (C hc ) σ c 1 (CĈ t+1 hccˆ t ) R P + +βε B (C hc ) σ c R ˆ P R t βε B (C hc ) σ c R P ˆP t = βε B (C hc ) σ c R P (1 + ˆε B t+1 σ c (C hc ) 1 (CĈ t+1 hccˆ t ) + Rˆ t ˆP t+1 ) A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

6 ε B t (C t hc t 1 ) σ c Pt 1 t ε B (C hc ) σ c P 1 + ε B (C hc ) σ c P 1ˆε B t + σ c ε B (C hc ) σ c 1 P 1 (CĈ t hc ˆ C t 1 ) ε B (C hc ) σ c P 1 ˆP t = ε B (C hc ) σ c P 1 (1 + ˆε B t σ c (C hc ) 1 (CĈ t hc ˆ C t 1 ) ˆP t ) and considering that in the steady-state it holds that βr h ε B (C i hc ) σ c = ε B (C hc ) σ c P P, β = 1 R Putting both sides together yields: R t + E t ˆε B σ c t+1 1 h E t (Ĉ t+1 ) hcˆ t = εt ˆ σ c 1 h ˆ C t hc t ˆ 1 + Et ˆ Π t+1 from which one obtains the explicit expression for ˆ C t A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

7 The linearized model: Equation of investment The linearization of the investment equation yields ˆ t = β ˆ t 1 + β 1 + β E ˆ t t S 00 (1)(1 + β) ˆQ t + βe t ˆε t+i 1 + β ˆε t The dependency of the optimal investment of past values introduce the dynamic in this equation. A positive shock of the adjustment costs function reduces investment and hence it can be considered a negative investment shock. A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

8 Recall the optimal condition for investment: 2 L K t = E t 4 = 0 β t λ t β t λ t Q t 1 + S ε t t t 1 + t S 0 ε t t t 1 β t+1 λ t+1 Q t+1 t+1 S 0 ε t+1 t+1 t ε t+1 t+1 2 t ε t t A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

9 ε, Q t S0 t t t 1 ε t t t 1 βe t ε = Q t 1 S t t t Q t+1 λ t+1 λ t S0 ε t+1 t+1 ε t+1 t+1 t t+1 t t = (1) n the steady-state it 2holds: ε Q S 0 ε β 6 ε 4 Q λ λ S 0 {z } {z } =0 =0 Q = 1, ε = Q ε S C A, {z } =0 since in the steady-state ε = 1 and S(1) = S 0 (1) = 0. A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

10 The approximation of the single terms yields ε S 0 t t ε t t t 1 t S 0 (1) ε 1 + S 00 (1) ε + S 00 (1) ε S 00 (1) ε ε ε + S 0 (1) ε + S 0 (1) ε 2 = S 00 (1) hît Î t 1 + ˆε t + S 0 (1) ( t ) + ( t 1 ) i ε t ε + A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

11 Note that since S 0 (1) = 0 one does not need the approximation of Q t, Q t+1, λ t and λ. βe t Q t+1 λ t+1 Q t λ t " β Qλ Qλ S 0 (1) ε β Qλ Qλ + β Qλ Qλ S 0 ε t+1 t+1 t + β Qλ Qλ S 00 (1)! ε t+1 2 t+1 S 00 (1) ε ε ε t ε 2 2 # S 00 (1) ε S 0 (1) S 0 (1) ε 2 (E t t+1 ) ( t ) + 2S 0 (1) ε = βs 00 (1) 3 E t ε t+1 ε = E t Î t+1 Î t + E t ˆε t+1 A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

12 since Q = 1 : 1 Q t 1 Q 1 Q 2 (Q t Q) = 1 ˆQ t ε S t t t 1 S(1) + S 0 (1) (ε t ε) + ε ( t ) ε 2 ( t 1 ) = 0 (2) A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

13 and hence it follows S 00 (1) hît Î t 1 + ˆε t i h i βs 00 (1) E t Î t+1 Î t + E t ˆε t ˆQ t, ˆ t = β ˆ t 1 + β 1 + β E ˆ t t S 00 (1)(1 + β) ˆQ t + βe t ˆε t+i 1 + β ˆε t (3) A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

14 The linearized model: The Q-equation The linearization of the Q-equation yields: ˆQ t = ( ˆR t E t ˆπ t+1 ) + 1 τ 1 τ + r k E t ˆQ r t+1 + k 1 τ + r k E t ˆr t+1 k + η Q t whereby because of the steady-state condition it holds, that β = 1/(1 τ + r k ). An equity premium shock is added in order to take into account the e ect of changes of the external re nancing premium over the value of the capital stock: η Q t which is n.i.d.. A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

15 Starting from Q t = E t β λ t+1 λ t Qt+1 (1 τ) + z t+1 r k t+1 Ψ(z t+1 ) n the steady-state: Q = 1, z = 1 and ψ(1) = 0. t follows: Q = β λ λ h i Q(1 τ) + zr k ψ(z) (4) 1 = β(1 τ + r k ) (5) The linearization of the left side yields Q t 1 + ˆQ t. A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

16 The right-hand side becomes: E t β λ t+1 λ t h i Q t+1 (1 τ) + z t+1 rt+1 k Ψ(z t+1 ) β(1 τ + r k )+ + β 1 h i Q(1 τ) + zr k ψ(z) (E t λ t+1 λ) + λ {z } =β 1 β λ i hq(1 λ 2 τ) + zr k ψ(z) (λ t λ) + {z } =β 1 + β λ λ (1 τ)(e tq t+1 Q) + β λ λ z(e trt+1 k r)+ β λ r k ψ 0 (z) (z t+1 z) = λ = 1 + E t ˆλ t+1 ˆλ t + β(1 τ)e t ˆQ t+1 + βe t r k ˆr t+1 whereby one has exploited that r k = ψ 0 (z). A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

17 From the Euler equation it follows: E t λ t+1 λ t = ( ˆR t ˆπ t+1 ). Putting these conditions together yields: ˆQ t = ( ˆR t ˆπ t+1 ) + +β(1 τ)e t ˆQ t+1 + βe t r k ˆr t+1 whereby β = 1/(1 τ + r k ). A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

18 The linearized model: Equation of capital accumulation The linearization of the equation of the capital accumulation yields: ˆK t = (1 τ) ˆK t 1 + τî t 1 Starting from K t = K t 1 (1 τ) + t t S ε t t t 1. n the steady-state it holds K = (1 τ)k + S(1), = τk. {z} =0 The approximation of the left side yields K t = K + (K t K ) K + ˆK t K. A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

19 For the right side, one obtains : K t 1 (1 τ) + t t S ε t t t 1 (1 τ)k + + (1 τ)(k t 1 K ) + ( t ) and considering the steady-state relationships: ˆK t K = (1 τ)k ˆK t 1 + Î t = (1 τ)k ˆK t 1 + Î t τk (6) ˆK t = (1 τ) ˆK t 1 + τî t 1 A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

20 The linearized model: The Phillips curve Through the partial indexation it follows that the linearized equation depends not only on the present marginal costs and on the expectations of the future in ation but also on the past in ation rate. ˆπ t = β E t ˆπ t+1 + γ p ˆπ t βγ p 1 + βγ p + (1 βξ p )(1 ξ p ) αˆr t k + (1 α)ŵ t ˆε a t + η p t (1 + βγ p )ξ p {z } mc ˆ t For γ p = 0 the usual forward-looking Phillips curve A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

21 Starting from E t β i ξ i λ t+i p y j ept (Pt+i 1 /P t 1 ) γ p t+i i=0 λ t P t P t+i /P t (1 + λ p,t+i ) MC t+i P t+i = 0 denote ϕ t = log( ep t ), p t = log(p t ), mc t = log(mc t /P t ) and µ = log(1 + λ p ), then it follows : E t β i ξ i λ h t+i p y j t+i e ϕ t p t+i +γ p (p t+i 1 p t 1 ) i=0 λ t i e µ+mc t+i = 0 n the steady-state it holds ϕ t = p t = p. t follows: e 0 e µ+mc = 0, mc = µ A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

22 The linearization yields (having noted that the internal bracket is null in the steady-state). E t i=0 β i ξ i λ p λ y e 0 (ϕ t p) + e 0 ( 1)(p t+i p) +e 0 γ p (p t+i 1 p) e 0 γ p (p t 1 p) e 0 (mc t+i mc) = 0 (7) E t 2 3 β i ξ i p ϕ t = E t β i ξ i 4 p {z mc } +mc t+i + p t+i i=0 i=0 =µ γ p (p t+i 1 p t 1 ) 5 (8) 1 1 βξ p ϕ t = 1 1 βξ p µ + E t 1 β i ξ i p mct+i + p t+i γ p (p t+i p t 1 ) i=0 (9) A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

23 1 ϕ t = µ + (1 βξ p )E t β i ξ i p mct+i + p t+i γ p (p t+i p t 1 ) i=0 ˆϕ t = ϕ t p = (1 βξ p )E t β i ξ i p mct+i ˆ + ˆp t+i γ p (p t+i 1 p t 1 ) i=0 (10) ˆϕ t βξ p E t ˆϕ t+1 = (1 βξ p )( mc ˆ t + ˆp t ) (11) A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

24 The linearization of the equation of motion of the price level yields: (P t ) λp,t 1 Pt 1 = ξ p P t 2 γp Pt 1 1 λp,t h i 1 = ξ p Π γ p t 1 Π t 1 1/λp,t + (1 ξp ) + 1 ξ p ept 1 λp,t ept P t 1/λp,t 0 = 1 λ p,t ξ p Π γ p Π 1 1 λp,t 1 λ p,t (1 ξ p ) P P 1 λp,t 1 γp π γ p 1 (Π t 1 Π) Π 2 (Π t Π) 1 1 P (( ep t P) (P t P)) (12) ˆϕ t ˆp t = ξ p 1 ξ p ˆπ t γ p ˆπ t 1 A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

25 The substitution in eq. (10) yields the new-keynsian Phillips curve ˆπ t = β 1 + βγ p E t ˆπ t+1 + γ p 1 + βγ p ˆπ t 1 + (1 βξ p )(1 ξ p ) αˆr t k + (1 α)ŵ t ˆε a t + η p t (1 + βγ p )ξ p {z } mc ˆ t A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

26 The linearized model: The wage equation Similarly one obtains the linearized equation of motion for the real wage: ŵ t = β 1 + β E tŵ t βŵt 1 + β 1 + β E 1 + βγ t ˆπ w t β ˆπ t + γ w 1 + β ˆπ t 1 (1 βξ w )(1 ξ w ) (1 + β) 1 + (1+λ w )σ l λ w σ c ŵ t σ l ˆL t 1 h Ĉt hĉ t 1 ξ w ˆε L t ηˆ w t (13) The real wage is a function of expected and past real wages and the expected, current and past in ation rate if γ w > 0. There is a negative e ect of the deviation of the actual real wage from the wage that would prevail in a exible labour market. A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

27 and hence: ˆL t = ˆr k t ŵ t + ẑ t + ˆK t 1 A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33 The linearized model: The labour demand curve The linearized demand of labour is: W Starting from: t L t = 1 α rt k z t K t 1 α. n the steady-state it holds WL ˆL t = ŵ t + (1 + ψ0 (1) ψ 00 (1) )ˆr k t + ˆK t 1 r k zk = 1 α α, whereby z = 1. substracting from the logarithmic transformation of (??) the logarithmic steady-state: log(l t ) log(l) = log(r k t ) log(r k ) (log(w t ) +log(p t ) (log(w ) log(p t )))+ + log(z t ) log(z) + log(k t 1 log(k )) (14)

28 r k t = Ψ 0 (z t ) =) r k + r k ˆr k t = Ψ 0 (z) + Ψ 00 (z)z t ẑ t Ψ 0 (z t ) r k = Ψ 0 (z), z = 1 =) ˆr t k Ψ 0 (z) = Ψ 00 (1)ẑ t ẑ t = Ψ0 (1) Ψ 00 (1) ˆr t k t follows: ˆL t = ŵ t Ψ0 (1) ˆr Ψ 00 (1) t k + ˆK t 1 This works like the derivation of the equation of in ation. Given a capital stock, the demand of labour negatively depends on real wages. A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

29 The market equilibrium: The good market is in equilibrium if the production is equal to the public expense G t, the household demand of consumption and investment plus related costs. Y t = C t + G t + t + ψ(z t )K t 1 The capital market is in equilibrium if the demand (of rms of the intermediate sector) of capital goods equals the supply of households. The labour market is in equilibrium if the demand of labour of rms equals the supply of households at wages set by households. A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

30 The good market equilibrium in steady-state deviations is: Ŷ t = (1 τk y g y )Ĉ t + τk y Î t + g y ε G t = = φˆε a t + φα ˆK t 1 + φα ψ0 (1) ψ 00 (1) ˆr k t + φ(1 α)ˆl t where k y and g y denote the steady-state proportion of capital and public spending on output. 1 + φ is the proportion of x costs of the production. The shock of the public spending ε G t follows a AR(1) process. A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

31 The monetary policy: The monetary policy follows a generalized Taylor-Rule: ˆR t = ρ ˆR t 1 + (1 ρ) π t + r π ( ˆπ t 1 π t ) + r Y Ŷ t + +r π ( ˆπ t ˆπ t 1 ) + r Y Ŷ t Ŷ t 1 + η R t (15) A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

32 The Taylor rule is obtained by minimizing the central bank loss function subject to the Phillips curve and the S curve. The central bank loss function is generally set to be equal to a weighted sum of the squared di erence between the actual and the target in ation and the squared output gap. The output gap Ŷ t is the di erence between the actual and the potential output. The potential output is the production that theoretically the economy performs under perfect exibility of prices and wages. A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33

33 The central bank reacts to deviations of both the lagged in ation with respect to the target in ation (normalized to 0) and of the lagged output gap. The monetary policy is subject to two di erent kinds of shock: a shock of the target in ation process, π t = ρ π π t 1 + ηt π, and shock of the interest rate: η R t. The central smooths the interest rates by setting the present rate as a weighted average of the past interest rate and the optimal rate as a function of the above quantities. A. Beccarini (CQE) Advanced Macroeconomics DSGE Summer / 33