Consumer Misperceptions, Uncertain Fundamentals, and the Business Cycle  Online Appendix Patrick Hürtgen

*

December 30, 2013

1 Model appendix The core of the model builds on conventional medium-scale New Keynesian DSGE models such as Smets and Wouters (2007), Justiniano et al. (2010) and Fernández-Villaverde (2010). 1.1 Households

The economy is inhabited by a continuum of households indexed by h ∈ [0, 1]. Preferences are additively separable in consumption and labor supply

U (Ct (h), Nt (h)) = log (Ct (h) − hc Ct−1 (h)) − ϑ

Nt (h)1+φ , 1+φ

(1)

where Ct (h) denotes household h's consumption and Nt (h) the amount of hours worked. The parameter hc denotes habit persistence in consumption, ϑ determines labor supply in steady state and φ is the inverse Frisch elasticity of labor supply. Each household h supplies a dierent type of labor Nt (h) and has some monopoly power in the labor market, posting the nominal wage

Wt (h) at which it is willing to supply specialized labor services to rms that demand them (see Erceg et al., 2000). Households have access to a complete set of state-contingent Arrow-Debreu securities to fully insure against idiosyncratic income risk that derives from the limited ability to adjust wages in each period.1 Let Dt+1 (h) denote the payo in period t + 1 of the portfolio

* 1

University of Bonn. Contact details : Kaiserplatz 7-9, 53113 Bonn, Germany, e-mail: [email protected]. With complete markets, consumption and the marginal utility of consumption are equalized across households and states at all times in equilibrium (given identical endowments).

1

of state-contingent securities held by household h at the end of period t and let Qt,t+1 denote the stochastic discount factor. The budget constraint of household h is given by

Pt Ct (h) + Pt It (h) + Et {Qt,t+1 Dt+1 (h)} − Dt (h) ) ( 1 k = Wt (h)Nt (h) + Rt ut (h) − IS Pt a(ut (h)) Kt−1 (h) + Tt (h) + Υt (h) , µt

(2) (3)

where It (h) is investment, Kt (h) is the capital stock, Tt (h) are lump-sum payments and Υt (h) are the prots of rms. The utilization rate of capital, ut (h), transforms physical capital into eective capital rented to rms at real rate rtk . The cost of physical capital utilization is a quadratic function a(ut (h)) = δ1 (ut (h)−1)+ δ22 (ut (h)−1)2 where in steady state, u = 1, a(1) = 0 with curvature

a′′ (1) a′ (1)

=

δ2 δ1 .

Households own and invest in capital facing investment adjustment

costs as in Christiano et al. (2005). The law of motion for capital is:

Kt (h) = (1 − δ)Kt−1 (h) +

µIS t

( ( )) It 1−S It (h) . It−1

(4)

Investment adjustment costs S(·) are specied as in Christiano et al. (2005): S(It /It−1 ) = ( )2 It κ I − µ , which are introduced to dampen the volatility of investment over the business 2 It−1 cycle. The parameter κ ≥ 0 measures the curvature of investment adjustment costs and µI is the long-run growth rate of investment. In steady state it holds that S = S ′ = 0 and S ′′ > 0. The exogenous investment-specic technological shock µIS t measures the variation in the eciency at which the nal good can be transformed into physical capital and follows an AR(1) process:

) ) ( ( = ρis log µIS log µIS t−1 + ϵis,t t

2 ϵis,t ∼ N (0, σis ).

(5)

A representative household h maximizes the expected discounted lifetime utility with respect to Ct (h), ut (h), Kt (h), It (h), Wt (h), Nt (h) and Dt+1 (h) subject to the budget constraint (2) and a standard no-Ponzi scheme condition. Households have identical rst-order conditions as consumers have access to complete nancial markets where they insure their idiosyncratic income risk. 1.2 Optimal wage setting

Dierentiated labor services are bundled to a homogeneous labor good Nt according to a Dixit-Stiglitz aggregator

[∫

1

Nt =

Nt (h)

ηw,t −1 ηw,t

] η ηw,t−1 w,t

dh

(6)

,

0

where ηw,t denotes the intratemporal elasticity of substitution across dierent varieties of labor types. The time-varying gross markup µw,t =

( log

µw,t µw

)

( = ρw log

µw,t−1 µw

ηw,t ηw,t −1

follows an exogenous ARMA(1,1) process

) + ϵw,t − ρwlag ϵw,t−1 ,

2

2 ϵw,t ∼ N (0, σw ),

(7)

where ϵw,t is a wage markup shock. The optimal bundling of dierentiated labor services based on cost minimization yields the labor demand schedule:

( Nt (h) =

Wt (h) Wt

)−ηw,t

(8)

Nt .

The aggregate wage index Wt is a composite of all labor type specic wage rates:

[∫

1 1−ηw,t

Wt =

Wt (h)

] 1−η1 w,t . dh

(9)

0

The fraction (1 − θw ) of households can adjust their posted nominal wage. Wage ination and infrequent wage adjustments induce relative wage distortions that facilitate an inecient allocation of labor. Each period optimizing households choose their wage Wt⋆ (h) = Wt (h) for their labor type in order to maximize the expected discounted lifetime utility subject to the labor demand schedule. The wage of the remaining fraction of households θw is indexed to past ination. A household that is not allowed to change wages for τ periods has a normalized wage ∏ Πχw of τs=1 Πt+s−1 Wt (h), where the indexation parameter is χw ∈ [0, 1]. The relevant terms of the t+s optimization problem are:

] τ w ∏ Πχt+s−1 Wt (h) Nt+τ (h)1+φ + λt+τ Nt+τ (h) max Et (βθw ) −ϑ 1+φ Πt+s Pt+τ Wt⋆ (h) s=1 τ =0 ( τ )−ηw t+τ w ∏ Πχt+s−1 Wt (h) Nt+τ (h) = Nt+τ . Πt+s Wt ∞ ∑

[

τ

s.t.

(10) (11)

s=1

Given the assumption of complete markets (assuming identical initial conditions) and separable utility in labor (see Erceg et al., 2000), I consider a symmetric equilibrium where Ct (h) =

Ct , λt (h) = λt , ut (h) = ut , Kt (h) = Kt , It (h) = It and Wt⋆ (h) = Wt⋆ . 1.3 Final good producers

Perfectly competitive nal good producers bundle intermediate goods Yt (i) to a nal good Yt following a Dixit-Stiglitz aggregation technology:

[∫

1

Yt =

Yt (i)

ηp,t −1 ηp,t

] η ηp,t−1 p,t

di

(12)

.

0

The time-varying intratemporal elasticity of substitution across dierent varieties of consumption goods is denoted ηp,t . The time-varying gross price markup µp,t = ARMA(1,1) stochastic process

( log

µp,t µp

)

( = ρp log

µp,t−1 µp

ηp,t ηp,t −1

follows an exogenous

) + ϵp,t − ρplag ϵp,t−1 ,

3

ϵp,t ∼ N (0, σp2 ) ,

(13)

where ϵp,t is a price markup shock. Prot maximization yields the input demand schedule for intermediate goods:

( Yt (i) =

Pt (i) Pt

)−ηp,t

(14)

Ytd .

The minimum costs of a bundle of intermediate goods that provides one unit of composite good amounts to the aggregate price index:

[∫

] 1−η1

1

Pt =

Pt (i)

1−ηp,t

p,t

(15)

.

di

0 1.4 Intermediate goods producers

A continuum of monopolistically competitive rms is indexed by i ∈ [0, 1], where each rm produces a dierentiated good using the same technology

Yt (i) = At Ktα (i)Nt (i)1−α − ϕztprof its ,

(16)

where the xed cost of production is ϕ. The aggregate level of technology is At = Xt Zt , where

Xt and Zt denote the permanent and the temporary component, respectively. The growth rate of the permanent component follows an AR(1) process, which implies that the level Xt builds up gradually over time. The stochastic processes are:

Xt Xt−1

( =

Xt−1 Xt−2

)ρx exp(ϵx,t ) ,

ρz Zt = Zt−1 exp(ϵz,t ) ,

ϵx,t ∼ N (0, σx2 )

(17)

ϵz,t ∼ N (0, σz2 ) .

(18)

In addition, consumers observe a noisy signal about the permanent component

St = Xt exp(ϵs,t ) ,

ϵs,t ∼ N (0, σs2 ) ,

(19)

where σs measures the signal precision and ϵs,t is referred to as noise shock. Firms set prices in a staggered fashion ` a la Calvo (1983), i.e. rms can re-optimize prices with probability (1 − θp ) each period and, therefore, take into account that they may not be able to adjust prices in the next period. Prices of those rms that cannot change prices are indexed to past ination for which the degree of indexation is governed by χp ∈ [0, 1]. Firms set prices

P ⋆ = P (i) to maximize expected prots subject to the demand schedule (14) (

) P (i) χp t max Et θpτ Qt,t+τ Yt+τ (i) Πt+s−1 − M Ct+τ (i) Pt⋆ Pt+τ τ =0 s=1 )−ηpt ( τ ∏ χp Pt (i) d Yt+τ (i) = Πt+s−1 Yt+τ , Pt+τ ∞ ∑

τ ∏

s.t.

(20) (21)

s=1

where Qt,t+s is the households' stochastic discount factor as dened before and M Ct (i) is rm

i's real marginal cost.

4

1.5 Monetary policy and aggregation

The central bank operates a Taylor rule where the nominal interest rate Rt responds to changes in ination and output growth as well as the lagged interest rate

Rt R

( =

  d γdy 1−γR yt )γR ( )γπ d Rt−1  Πt  yt−1   2 exp(ϵm,t ) , ϵm,t ∼ N (0, σm ).   y   R Π µ

(22)

Aggregate demand in the economy is:

( )−1 Ytd = Ct + It + µIS a(ut )Kt−1 . t

(23)

1.6 Detrended model equilibrium conditions

ˆt = λ ˆt = λ qtK µA t+1 = − 1 =

( ) ( )−1 1 −1 ˆ ˆ ˆ Ct − hc Ct−1 A − βhc Et Cˆt+1 µA − h C c t t+1 µt ˆ t+1 Rt λ βEt A µt+1 πt+1 ) ˆ t+1 ( λ K k βEt (1 − δ) qt+1 + rt+1 ut+1 ˆt λ ) ˆ t+1 1 ( λ δ2 2 βEt δ1 (ut+1 − 1) + (ut+1 − 1) ˆ t µIS 2 λ t+1   ( )2 ( ) ˆ ˆ ˆ It A It A It A  1 − κ qtK µIS µ − µI µt − µI µt −κ t ˆ ˆ 2 Iˆt−1 t It−1 It−1

ˆ K λt+1 1 + βEt qt+1 µIS t+1 κ ˆ t µA λ t+1

(

Iˆt+1 A µt+1 − µI Iˆt

)(

Iˆt A µt Iˆt−1

(24) (25) (26)

(27)

)2

rtk =

(28)

gt1

1 (δ1 + δ2 (ut − 1)) µIS t ( χp )−ηp,t+1 1 ˆ t mct yˆd + βθp Et Πt = λ gt+1 t Πt+1 ( χp )1−ηp,t+1 ∗ Πt 2 Πt ∗ d ˆ g = λt Πt yˆt + βθp Et Πt+1 Π∗t+1 t+1

(29)

ηp,t gt1 = (ηp,t − 1) gt2 η − 1 ( ˆ ∗ )1−ηw,t ˆ ( ˆ )ηw,t d ft = Wt λt Wt lt η )ηw,t+1 −1 ( χw )1−ηw,t+1 ( ˆ ∗ Wt+1 A Πt + βθw Et µ ft+1 ˆ ∗ t+1 Πt+1 W

(31)

gt2

t

5

(30)

(32)

(

ft

ˆt W = ϑ ˆ∗ W

)ηw,t (1+φ)

t

( + βθw Et ut kˆt−1 ltd

=

mct = 1 = 1 = Rt R

=

yˆtd

(33)

)ηw,t+1 (1+φ) )−ηw,t+1 (1+φ) ( ˆ ∗ Wt+1 A µ ft+1 ˆ ∗ t+1 W t

α w ˆt A µ 1 − α rtk t ( )1−α ( )α ( )α 1 1 w ˆt1−α rtk 1−α α ( χ )1−ηp,t p Πt−1 θp + (1 − θp ) (Π∗t )1−ηp,t Πt ( χw )1−ηw,t ( ) Πt−1 w ˆt−1 1 1−ηw,t 1−ηw,t θw + (1 − θw ) (Π∗w t ) Πt w ˆt µA t   d γY 1−γR y ˆ A t ( )γR ( )γΠ µ d t Rt−1  Πt  yˆt−1   exp (εm,t )     R Π µy (

yˆtd =

Πχt w Πt+1

( )1+φ ltd

µA t

)−α (

ut kˆt−1

)α ( ) 1−α ltd −ϕ

υtp ) ( ( )−1 ( IS )−1 δ2 2 ˆ ˆ = cˆt + It + µt δ1 (ut − 1) + (ut − 1) kt−1 µA t 2

lt = υtw ltd ( χ )−ηp,t p Πt−1 p p υt = θp υt−1 + (1 − θp ) (Π∗t )−ηp,t Πt ( χw )−ηw,t ( ) Πt−1 w ˆt−1 1 −ηw,t w −ηw,t w υt = θw υt−1 + (1 − θw ) (Π∗w t ) Πt w ˆt µA t  )2  ( ˆ 1 It A 1 − κ 0 = kˆt − (1 − δ) kˆt−1 A − µIS µ − µI  Iˆt t ˆ 2 It−1 t µt Exogenous processes are specied in the main text and in Sections 1.1-1.5.

6

(34) (35) (36) (37)

(38)

(39) (40) (41) (42) (43) (44)

2 Data appendix Table 1: Data sources Label

Frequ.

Description

Source

GDP GDPQ GCND GCS NRI RI P16 E16 TFP LBCPU FYFF E5Y

Q Q Q Q Q Q Q Q Q Q M Q

Gross domestic product Real gross domestic product Pers. cons. expendit. on nondurable goods Personal consumption expendit. on services Real nonresidential investment Real residential investment Civilian non-institutional pop. over 16 Civilian employment (S.A.) Real-time utilization adj. TFP Hourly non-farm business compensation Eective federal funds rate Business conditions expected during the next ve years

BEA (Table 1.1.5, Line 1) BEA (Table 1.1.6, Line 1) BEA (Table 1.1.5, Line 4) BEA (Table 1.1.5, Line 5) BEA (Table 1.1.6, Line 8) BEA (Table 1.1.6, Line 11) BLS (LNU00000000Q) BLS (LNS12000000) Fernald (2010) BLS (PRS85006103) St. Louis FRED Michigan Consumer Sentiment Survey Table 16

Table 2: Data construction Label

Description

Construction

GDPDEF C Y I W FFR TFP

GDP deator Real per-capita consumption Real per-capita output Real per-capita investment Real wages Eective federal funds rate Real-time utilization adj. TFP

GDPQ/GDP (GCND+GCS)/P16/GDPDEF GDPQ/P16 (NRI+RI)/P16 LBCPU/GDPDEF Quarterly average of FYFF TFP

Notes: The data set constructed with U.S. data is transformed to match the model equivalents specied in the observation equation (45).

The observation equation Yt relates the observed data dened in Table 2 to the respective counterparts in the model, i.e.

      Yt =     

 cˆt − cˆt−1 + a ˆt−1      ˆit − ˆit−1 + a ˆt−1  ∆ log (It )      w r r ˆt−1 +a ˆt−1  ∆ log (Wt )    ˆt − w  =    a ˆt − a ˆt−1 ∆ log (At )       πt ∆ log (GDPDEFt )    rt FFRt ∆ log (Ct )





where ∆ denotes the temporal dierence operator.

7

(45)

3 Consumers' Kalman lter Dene the matrices:



ρx −ρx −1 ρx

  1 C=  0  1 [ D=

0

0

0

ρz

0

1

1 0 1 0



σx2 0

0

0

0

   0   , Σ1 =  0   0 0   0 0

]

1 0 0 0



[ , Σ2 =

0

0

0

 0   0   0

0 σz2 0



0

]

0 σs2

.

( ) The process for ξt = x ˆt , x ˆt−1 , zt , µ ˆA t−1 is described compactly as (46)

ξt = Cξt−1 + Rϵt , and the observation equation for consumers is

( A )′ yt = µ ˆt , sˆt = Dξt + Sϵt ,

(47)

where yt is the vector of observables, ϵt contains all structural shocks, E [Rϵt ϵ′t R′ ] = Σ1 and

E [Sϵt ϵ′t S ′ ] = Σ2 . Let P = Vart−1 [ξt ]. The value of P is found by solving the following equation:

[ ] )−1 ( P = C P − P D′ DP D′ + Σ2 DP C ′ + Σ1 .

(48)

According to the updating equation of a linear projection (see Hamilton (1994), equation 13.2.15) the evolution of the unobserved state is:

ξt|t = ξt|t−1 + P D(DP D′ + Σ2 )−1 (yt − Dξt|t−1 ) ′

−1

= (I − BD)ξt|t−1 + P D(DP D + Σ2 )

(49)

yt

(50)

= Aξt−1|t−1 + BDCξt−1 + B(DR + S)ϵt .

(51)

The last step uses ξt|t−1 = Cξt−1|t−1 , B = P D(DP D′ +Σ2 )−1 and A = (I − BD) C . Equation (10) in the main text follows the notation based on matrices A and B .

8

4 Model solution The solution to the full information log-linearized model can be obtained using standard methods, e.g. Klein (2000). The vector of control variables is X1,t and the vector of state variables is denoted by X2,t . The full information model solution is given in recursive form by the policy and transition function respectively

[ where X2,t = x ˆt x ˆt−1 zt

X1,t = ΠX2,t−1 ,

(52)

˜ t, X2,t = M X2,t−1 + Rϵ

(53)

]′ Xobs , t

ϵt = [ϵx,t ϵz,t ϵs,t ϵm,t ϵp,t ϵw,t ϵis,t ]′ , and

Xobs t is the vector

of observed states. Introducing imperfect information necessitates an adjustment of solution methods as proposed in Baxter et al. (2011). Agents cannot directly observe the components of [ ]′ productivity, i.e. x ˆt and zt . Dene the vector of unobserved states as ξt = x ˆt x ˆt−1 zt µ ˆA t−1 ,

which is a subset of all state variables X2,t .2 Agents form contemporaneous estimates about the

states, i.e. ξt|t , stemming from solving the Kalman ltering problem (Section 3 in the Online Appendix contains a detailed derivation). The following system describes the evolution of the actual states and the beliefs of the agents

[

ξt ξt|t

]

[ =

N11

0

N21 N22

][

ξt−1

]

ξt−1|t−1

[ +

R B(DR + S)

] ϵt ,

where N11 = C . Solving the consumers' Kalman ltering problem yields a recursive solution for the contemporaneous beliefs (see equation (51)), i.e.

ξt|t = Aξt−1|t−1 + Byt = Aξt−1|t−1 + BDCξt−1 + B(DR + S)ϵt ,

(54)

such that N21 = BC and N22 = A. The matrices A, B, C and D were already introduced in the ltering problem (see Section 3 in the Online Appendix). Given the contemporaneous estimates about the unobserved states ξt−1|t−1 and the linearity of the model, certainty equivalence3 applies (see Baxter et al., 2011) and hence

X1,t = ΠX2,t−1|t−1 , [ where X2,t−1|t−1 = ξt−1|t−1

2 3

4

xobs i,t−1

]′

and

(55)

4 xobs i,t−1 represents all observable lagged state variables.

Note that µˆA ˆt . t−1 is perfectly observed, but it is required to pin down x Certainty equivalence implies that even though consumers know that they imperfectly observe the fundamentals of the economy, their decisions are as if they knew the true value of the unobserved state variable (i.e. under full information). For example, Pearlman et al. (1986), Pearlman (1992), Svensson and Woodford (2004) and Lorenzoni (2009) also use certainty equivalence in a linear model with partial information.

9

5 Additional tables Table 3: Parameters xed prior to estimation Parameter

Value

Description/Target

β ϑ u ϕ µA δ ηp ηw

0.99 8 1 0.045 1 0.025 9 9

Stochastic discount factor Steady state labor hours: 0.35 Steady state capacity utilization Zero prots in steady state Growth rate of productivity Annual depreciation rate of 10% Steady state price markup of 12.5% Steady state wage markup of 12.5%

Table 4: Prior and posterior distribution of complete information model

Prior

Posterior

Parameter

Description

Distr.

Mean

Std

Mean

5%

95%

hc κ α φ δ2

Habit persistence Investment adj. costs Capital share Inverse Frisch elasticity Capital utilization costs

B N B G N

0.6 6 0.25 2 0.15

0.1 2 0.1 0.75 0.05

0.82 9.07 0.10 1.19 0.14

0.79 6.68 0.05 0.38 0.05

0.86 11.26 0.15 1.95 0.22

θp θw χp χw γdy γπ γR

Price stickiness Wage stickiness Price indexation Wage indexation Taylor rule: output growth Taylor rule: ination Interest rate smoothing

B B B B N N B

0.66 0.66 0.7 0.7 0.1 1.4 0.7

0.1 0.1 0.15 0.15 0.1 0.125 0.1

0.90 0.67 0.81 0.59 0.22 1.18 0.80

0.86 0.55 0.67 0.38 0.09 1.07 0.77

0.93 0.79 0.96 0.83 0.35 1.30 0.83

ρis ρx ρz ρp ρw ρplag ρwlag

Inv.-specic TFP Autocorr. perm. TFP Autocorr. temp. TFP Price markup Wage markup Lagged price markup Lagged wage markup

B B B B B B B

0.8 0.8 0.8 0.7 0.7 0.5 0.5

0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.66 0.96 0.93 0.56 0.91 0.49 0.57

0.56 0.95 0.90 0.38 0.86 0.39 0.47

0.77 0.98 0.96 0.74 0.96 0.60 0.66

100σx 100σz 100σm 100σis 100σp 100σw

Perm. TFP Temp. TFP Monetary policy Investment-spec. TFP Price markup Wage markup

IG IG IG IG IG IG

0.5 1 1 1 1 1

5 5 5 5 5 5

0.17 0.73 0.26 9.50 3.22 14.56

0.13 0.66 0.24 6.32 2.13 4.90

0.23 0.80 0.28 12.64 4.22 28.32

Notes: The complete information model M2 is estimated with a precise signal, i.e. σs = 0. B is beta distribution, G is gamma distribution, IG is inverse gamma distribution, N is normal distribution.

10

Table 5: Prior and posterior distribution of incomplete information model with identical productivity parameters

Prior

Posterior

Parameter

Description

Distr.

Mean

Std

Mean

5%

95%

hc κ α φ δ2

Habit persistence Investment adj. costs Capital share Inverse Frisch elasticity Capital utilization costs

B N B G N

0.6 6 0.25 2 0.15

0.1 2 0.1 0.75 0.05

0.59 7.02 0.14 1.45 0.08

0.54 4.58 0.06 0.58 0.00

0.64 9.44 0.24 2.21 0.18

θp θw χp χw γdy γπ γR

Price stickiness Wage stickiness Price indexation Wage indexation Taylor rule: output growth Taylor rule: ination Interest rate smoothing

B B B B N N B

0.66 0.66 0.7 0.7 0.1 1.4 0.7

0.1 0.1 0.15 0.15 0.1 0.125 0.1

0.91 0.78 0.88 0.47 0.08 1.001 0.67

0.89 0.67 0.80 0.23 -0.01 1.000 0.62

0.94 0.87 0.97 0.74 0.18 1.005 0.71

ρis ρx = ρz ρp ρw ρplag ρwlag

Inv.-specic TFP Autocorr. TFP Price markup Wage markup Lagged price markup Lagged wage markup

B B B B B B

0.8 0.8 0.7 0.7 0.5 0.5

0.1 0.1 0.1 0.1 0.1 0.1

0.70 0.95 0.41 0.90 0.46 0.61

0.58 0.93 0.28 0.85 0.36 0.52

0.80 0.97 0.53 0.94 0.56 0.71

100σs 100σa 100σm 100σis 100σp 100σw

Noise shock Neutral TFP Monetary policy Investment-spec. TFP Price markup Wage markup

IG IG IG IG IG IG

1 1 1 1 1 1

5 5 5 5 5 5

0.51 1.03 0.3 6.89 3.92 32.3

0.25 0.93 0.28 3.97 2.88 9.93

0.77 1.13 0.33 9.5 4.86 53.69

Notes: The incomplete information model M3 is estimated based on ρz = ρx . B is beta distribution, G is gamma distribution, IG is inverse gamma distribution, N is normal distribution.

11

Table 6: Prior and posterior distribution of incomplete information model 1982:1 to 2007:4

Prior

Posterior

Parameter

Description

Distr.

Mean

Std

Mean

5%

95%

hc κ α φ δ2

habit persistence investment adj. costs labor share Inverse Frisch elasticity capital utilization costs

B N B G N

0.6 6 0.25 2 0.15

0.1 2 0.1 0.75 0.05

0.60 6.00 0.25 2.00 0.15

0.74 7.74 0.10 1.18 0.12

0.69 5.39 0.03 0.37 0.03

θp θw χp χw γdy γπ γR

price stickiness wage stickiness price indexation wage indexation Taylor rule: output growth Taylor rule: ination interest rate smoothing

B B B B N N B

0.66 0.66 0.7 0.7 0.1 1.4 0.7

0.1 0.1 0.15 0.15 0.1 0.125 0.1

0.91 0.53 0.45 0.59 0.26 1.45 0.81

0.88 0.40 0.25 0.33 0.12 1.28 0.78

0.94 0.65 0.64 0.86 0.39 1.61 0.84

ρis ρx ρz ρp ρw ρplag ρwlag

inv.-specic TFP autocorr. perm. TFP autocorr. temp. TFP price markup wage markup lagged price markup lagged wage markup

B B B B B B B

0.8 0.8 0.8 0.7 0.7 0.5 0.5

0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.78 0.97 0.95 0.69 0.90 0.46 0.49

0.66 0.96 0.93 0.56 0.84 0.34 0.38

0.90 0.99 0.98 0.82 0.96 0.58 0.60

100σs 100σx 100σz 100σm 100σis 100σp 100σw

noise shock perm. TFP temp. TFP monetary policy investment-spec. TFP price markup wage markup

IG IG IG IG IG IG IG

1 0.5 1 1 1 1 1

5 5 5 5 5 5 5

0.38 0.14 0.71 0.18 4.33 2.56 8.13

0.22 0.10 0.62 0.16 2.59 1.67 3.39

0.54 0.17 0.78 0.2 6.05 3.36 13.52

Notes: The incomplete information model M4 is estimated for the sub-sample 1982:1 to 2007:4. B is beta distribution, G is gamma distribution, IG is inverse gamma distribution, N is normal distribution.

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Table 7: Forecast error variance decomposition 1982:1 to 2007:4 Temp. Perm. Invest.-spec. Monetary Price Wage Noise TFP TFP TFP policy markup markup Quarters 1 4 12 Quarters 1 4 12 Quarters 1 4 12 Quarters 1 4 12

Consumption growth 6.5 39.1 51.7

18.4 12.0 9.0

0.1 0.1 0.2

8.7 5.6 4.9

10.8 6.3 5.3

34.4 21.0 16.2

21.2 15.7 12.7

7.0 5.6 5.4

7.7 8.0 8.4

0.0 0.1 0.1

12.2 8.1 7.1

28.1 20.2 16.3

14.7 12.4 10.4

21.4 16.7 13.2

66.1 51.7 41.9

6.0 6.1 5.2

Investment growth 0.0 0.0 0.5

1.0 2.1 2.9

81.0 81.0 79.6

3.2 3.1 3.1

Output growth 4.5 30.7 42.7

12.7 10.2 8.1

19.4 12.1 9.8

8.5 6.4 5.7

Wage growth 1.5 19.3 34.5

2.7 2.9 2.4

0.3 0.8 0.7

2.0 2.5 2.1

Notes: Forecast error variance decomposition with parameter values at their posterior mean of model M4 which is based on the estimation of the sub-sample 1982:1 to 2007:4.

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6 Additional Figures Figure 1: Impulse responses to a monetary policy shock Productivity and beliefs 1 0.05

0.5

Consumption

Investment

Output

0.2

0

0

−0.05

−0.2

0.1 0

−0.1

−0.4

−0.1

−0.15

−0.6

−0.2

−0.2

−0.8

−0.25

−1

0

−0.5

−1

0

10

20

−0.3

0

10

Wage

20

−1.2

−0.3

0

Labor

0.05

0.1

0

0

10

20

Inflation

−0.4

0

0.2 0.15

−0.02

−0.1

−0.2

−0.03

0.1

−0.04

0.05

−0.05

0

−0.2

0

10

20

−0.4

0

10

20

−0.06

20

0.25

−0.01 −0.1

−0.3

10

Nominal interest rate 0.3

0.01

−0.05

−0.15

0

0

10

20

−0.05

0

10

20

Notes: Impulse responses to a one standard deviation monetary policy shock with parameters at their posterior mean value. The dashed lines are 90 percent condence bands. All variables are measured in percentage deviations from steady state (x-axis). A time unit is a quarter (y-axis).

Figure 2: Sensitivity to signal precision Sensitivity to signal precision 0.35

Impact consumption volatility due to noise shock (in %)

0.3

0.25

0.2

0.15

0.1

0.05

0 0

2

4 6 Precision of the signal σs (in %)

8

10

Notes: Fraction of impact consumption volatility attributed to noise shocks with parameters at posterior mean and varying the standard deviation of the noise shock between zero and ten percent.

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References Baxter, B., Graham, L., Wright, S., March 2011. Invertible and non-invertible information sets in linear rational expectations models. Journal of Economic Dynamics and Control 35 (3), 295311. Calvo, G. A., September 1983. Staggered prices in a utility-maximizing framework. Journal of Monetary Economics 12 (3), 383398. Christiano, L. J., Eichenbaum, M., Evans, C. L., February 2005. Nominal Rigidities and the Dynamic Eects of a Shock to Monetary Policy. Journal of Political Economy 113 (1), 145. Erceg, C. J., Henderson, D. W., Levin, A. T., October 2000. Optimal monetary policy with staggered wage and price contracts. Journal of Monetary Economics 46 (2), 281313. Fernald, J., 2010. A Quarterly, Utilization-Adjusted Series on Total Factor Productivity. Tech. rep., FRBSF Working Paper. Fernández-Villaverde, J., March 2010. The econometrics of DSGE models. SERIEs 1 (1), 349. Hamilton, J. D., 1994. Time Series Analysis. Princeton, N.J. : Princeton University Press. Justiniano, A., Primiceri, G. E., Tambalotti, A., March 2010. Investment shocks and business cycles. Journal of Monetary Economics 57 (2), 132145. Klein, P., September 2000. Using the generalized Schur form to solve a multivariate linear rational expectations model. Journal of Economic Dynamics and Control 24 (10), 14051423. Lorenzoni, G., December 2009. A Theory of Demand Shocks. American Economic Review 99 (5), 22502284. Pearlman, J., Currie, D., Levine, P., April 1986. Rational expectations models with partial information. Economic Modelling 3 (2), 90105. Pearlman, J. G., April 1992. Reputational and nonreputational policies under partial information. Journal of Economic Dynamics and Control 16 (2), 339357. Smets, F., Wouters, R., June 2007. Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach. American Economic Review 97 (3), 586606. Svensson, L. E. O., Woodford, M., January 2004. Indicator variables for optimal policy under asymmetric information. Journal of Economic Dynamics and Control 28 (4), 661690.

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