Inference. Jesús Fernández-Villaverde University of Pennsylvania

Transcription

1 Inference Jesús Fernández-Villaverde University of Pennsylvania 1

2 A Model with Sticky Price and Sticky Wage Household j [0, 1] maximizes utility function: X E 0 β t t=0 G t ³ C j t 1 1 σ 1 1 σ ³ N j t 1+γ 1+γ + η 1 ξ Mj t P t 1 ξ 0 < β < 1 is the discount factor, σ > 0 the elasticity of intertemporal substitution, ξ > 1 the elasticity of money holdings, and γ > 0the inverse of the elasticity of labor supply with respect to real wages. 2

3 Subject the budget constrain given by: P t Ct j + Mj t Mj t 1 + X Q t (h t+τ )Dt j (h t+τ)+ B j h t+τ = W j t Nj t + Πj t + T j t + Dj t + Bj t, t+1 R t where Π j t are firms profits, T t j are nominal transfers, Dt j (h t+τ) denotes holdings of contingent bonds, B j t+1 denotes holdings of an uncontingent bonds, and Wt j is the hourly nominal wage. 3

4 Technology Intermediate Goods producer i [0, 1] use the following production function: Y i t = A t K δ sr " Z 1 0 ³ φ 1 ij N t φ dj # φ φ 1 1 δ A t is a technology factor, which is common to the whole economy. Nt ij is the amount of hours of type j labor used by intermediate good producer i. φ > 1 is the elasticity of substitution between different types of labor, and 0 > δ > 1 is the capital share of output. The production function is concave in the labor aggregate, and we assume that capital is fixedintheshortrunatalevelk sr. 4

5 Final good: Y t = " Z 1 0 ³ ε Y i t 1 ε t t di # ε t ε t 1 ε t > 1 the elasticity of substitution between intermediate goods, Λ t = ε t / (ε t 1) price markup. There is a shock to the elasticity of substitution, ε t. 5

6 Final Goods Price Setting Final good producers are competitive and maximize profits. The input demand functions associated with this problem are Y i t = " P i t P t # εt Y t i, The zero profit condition the price of the final good P t = " Z 1 0 P i t 1 ε t di # 1 1 ε t 6

7 Intermediate Goods Producers Problem Operate in a monopolistic competition environment. They maximize profits taking as given all prices and wages but their own price. The profit maximization problem of the intermediate good producers is n divided o into two stages: In the first stage, given all wages, firms choose ij N t to obtain the optimal labor mix. Hence, the demand of j [0,1] producer i for type of labor j is N ij t = W j t W t φ " Y i t A t # 1 1 δ j, 7

8 Where the aggregate wage W t canbeexpressedas W t = " Z 1 0 ³ W j t 1 φ dj # 1 1 φ. 8

9 In the second stage, they set prices. They can reset their price only when they receive a stochastic signal to do so. This signal is received with probability 1 θ p. If they can change the price, they choose the price that maximizes: subject to E t X τ=0 θ τ pq t+τ t Y i t+τ = P i t Y i t+τ W t+τ " P i t P t+τ Ã Y i! 1 t+τ A t+τ # εt+τ Y t+τ i, τ 1 δ 9

10 The solution is: E t X τ=0 θ τ pq t+τ t P t i, P t+τ Λ t MC i t Yt i =0, Theevolutionoftheaggregatepricelevelis: P t = h θ p (P t 1 ) 1 ε t +(1 θ p )(P t ) 1 ε t i 1 1 ε t 10

11 Consumers problem Intertemporal susbstitution Equation G t C t 1 σ = βe t {G t+1 C 1 σ t+1 R t P t P t+1 } Demand for money, at a given interest rate, always satisfied. 11

12 Wage Setting Problem Consumers operate in a monopolistic competition environment. They maximize utility given all wages, but their own. They reset wages if signal to do so. They receive the signal with probability (1 θ w ). As before, the signal is independent across intermediate good producers and past history of signals. If they can change their wage, they choose the wage, Wt j,, that maximizes: X E t (βθ w ) τ τ=0 G t+τ C 1 σ t+τ W j, t P t+τ ϑ ³ N j t+τ γ N j t+τ =0 12

13 subject to N j t+τ = W t j, W t+τ φ Z 1 0 Ã Y i! 1 1 δ t+τ di A t+τ j, τ The evolution of the aggregate wage level is: W t = h θ w W 1 φ t 1 +(1 θ w)(w t ) 1 φi 1 1 φ. 13

14 Fiscal and Monetary Policy On the fiscal side, the government cannot run deficits or surpluses, so itsbudgetconstraintis Z 1 0 T (h t,j)dj = M(h t ) M(h t 1 ), On the Monetary side, as suggested by Taylor (1993), we assume that the monetary authority conducts monetary policy using the nominal interest rate, through a Taylor rule. µ r t rt 1 r = r ρr µ Ã! πt γπ γy yt e ms t π y 14

15 Dynamics a t +(1 δ)n t y t =0 mc t (w t p t )+y t n t =0 1 σ c t + γn t g t mrs t =0 ρ r r t 1 +(1 ρ r ) h γ π π t + γ y y t i + mst r t =0 y t + c t =0 w t p t (w t 1 p t 1 w t + π t )=0 15

16 E t [ σr t + σπ t+1 σg t+1 + σg t c t + c t+1 ]=0 E t [κ p mc t + κ p µ t π t + βπ t+1 ]=0 E t [κ w mrs t κ w (w t p t ) w t + β w t+1 ]=0 16

17 where a t = ρ a a t 1 + ε at µ t = ε µt ms t = ε mst g t = ρ g g t 1 + ε gt and κ p =(1 δ)(1 θ p β)(1 θ p )/(θ p (1 + δ( ε 1))) κ w =(1 θ w )(1 βθ w )/ [θ w (1 + φγ)] 17

18 Solve the Model (Uhlig Algorithm) Derive the system: 0=As t + Bs t 1 + Ce t + Dz t 0=E t [Fs t+1 + Gs t + Hs t 1 + Je t+1 + Ke t + Lz t+1 + Mz t ] z t+1 = Nz t + ε t+1 ε t+1 N (0, Σ) s t =(w t p t,r t, π t, w t,y t ) 0 is the endogenous state, e t =(n t,mc t,mrs t,c t ) are endogenous variables, and z t =(a t,ms t,µ t,g t ) 0 is the exogenous state. 18

19 N = ρ a ρ g Solution s t = PPs t 1 + QQz t e t = RRs t 1 + SSz t 19

20 Writing the Solution in State Space Form Transition equation Ã! à st PP QQ N = 0 N z t à st z t!!ã st 1 = F à st 1 z t 1 z t 1!! + + Gε t à Q I! ε t = Measurement equation à I 0 s t = 0 0!à st z t! = H à st z t! Evaluate the Likelihood function using the Kalman Filter. 20

21 Apply Metropolis-Hastings Algorigthm to get a draw from the posterior. Compute moments and marginal likelihood. 21

22 Prior Distribution Mean/std 1 1 θ p gamma(2, 1) (1.42) 1 1 θ w gamma(3, 1) (1.71) γ π normal(1.5, 0.25) 1.5 (0.25) γ y normal(0.125, 0.125) (0.125) ρ r uniform[0, 1) 0.5 (0.28) σ 1 gamma(2, 1.25) 2.5 (1.76) γ normal(1, 0.5) 1.0 (0.5) ρ a uniform[0, 1) 0.5 (0.28) ρ g uniform[0, 1) 0.5 (0.28) σ a (%) uniform[0, 1) 50.0 (28.0) σ m (%) uniform[0, 1) (28.0) σ λ (%) uniform[0, 1) 50.0 (28.0) σ g (%) uniform[0, 1) 50.0 (28.0)

23 Algorithm (gencoeffsehl.m) Step 0 Read data (usadefl1d.txt) Step 1 Intial value for θ 0, N and set j =1. Step 2 Evaluate f(y T θ 0 )andπ(θ 0 )andmakesuref(y T θ 0 ), π(θ 0 ) > 0 (a) Given θ 0 evaluate prior π(θ 0 )(priorehl.m) (b) Given θ 0 : Uligh algorithm to solve the model (modelehl.m and solve2.m) (c) Kalman Filter to evaluate f(y T θ 0 ) (likeliehl.m) 23

24 Step 3 θ j = θ j 1 + ε N (0, Σ ε )andu from Uniform[0, 1] (a) Given θ j evaluate prior π(θ j )(priorehl.m) (b) Given θ j : Uligh algorithm to solve the model (modelehl.m and solve2.m) (c) Kalman Filter to evaluate f(y T θ j ) (likeliehl.m) Step 4 If u α ³ θ j 1, θ j θ j 1 otherwise. =min f(y T θ j )π ³θ j f(y T θ j 1 )π(θ j 1 ), 1 then θ j = θ j, θ j = Step 5 If j N then j à j + 1 and got to 3. 24

25 Prior Mean (Std) 1 1 θ p gamma(2, 1) (1.42) 1 1 θ w gamma(3, 1) (1.71) γ π normal(1.5, 0.25) 1.5 (0.25) γ y normal(0.125, 0.125) (0.125) ρ r uniform[0, 1) 0.5 (0.28) σ 1 gamma(2, 1.25) 2.5 (1.76) γ normal(1, 0.5) 1.0 (0.5) ρ a uniform[0, 1) 0.5 (0.28) ρ g uniform[0, 1) 0.5 (0.28) σ a (%) uniform[0, 1) 50.0 (28.0) σ m (%) uniform[0, 1) (28.0) σ λ (%) uniform[0, 1) 50.0 (28.0) σ g (%) uniform[0, 1) 50.0 (28.0) Mean of (Std) 4.37 (0.35) 2.72 (0.27) 1.08 (0.09) 0.26 (0.06) 0.74 (0.02) 8.33 (2.50) 1.74 (0.29) 0.74 (0.05) 0.82 (0.03) 3.88 (1.09) 0.33 (0.02) (5.32) (3.28) Posterior