A Baseline DSGE Model

A Baseline DSGE Model

Jes´

us Fern´

andez-Villaverde

University of Pennsylvania

Pablo A. Guerr´

on-Quintana

Boston College

December 22, 2020

1

Introduction

In this online appendix, we present the canonical DSGE model that we estimate in Section 6 of “Estimating DSGE Models: Recent Advances and Future Challenges” in more detail. The basic structure of the economy is as follows. We have a representative household that consumes, saves, holds money, and supplies labor. A final good firm produces output using a continuum of intermediate good producers. These intermediate goods, in turn, are produced by monopolistic competitors by renting capital and labor from the household. The interme-diate good producers are subject to nominal price rigidities `a la Calvo. The representative household is the owner of all of these firms. The model is closed with a government that sets up monetary and fiscal policy. Since there are trends in the data, we introduce two unit roots, one in the level of neutral technology and one in the investment-specific technology, that induce stochastic long-run growth.

1.1

Representative household

The representative household has a lifetime utility function on consumption, ct, real money

balances, mt/pt (where pt is the price level), and hours worked, lt:

E0 ∞ X t=0 βtdt  log (ct− hct−1) + υ log  mt pt  − ϕtψ l_t1+ϑ 1 + ϑ  , (1)

where β is the discount factor, h controls habit persistence, and ϑ is the inverse of the Frisch labor supply elasticity. We pick a utility function (log utility in consumption) whose marginal relation of substitution between consumption and leisure is linear in consumption to ensure the presence of a balanced growth path (BGP) with constant hours.

The intertemporal preference shock dt follows a law of motion

log dt = ρdlog dt−1+ σdεd,t,

where εd,t∼ N (0, 1). The labor supply shock ϕt follows a law of motion

log ϕt= ρϕlog ϕt−1+ σϕεϕ,t,

where εϕ,t ∼ N (0, 1). We should understand both shocks as a stand-in for more complex

mechanisms, such as financial frictions, demographic shifts, or changes in risk attitudes. The household’s budget constraint is:

ct+ xt+ mt pt + bt+1 pt + Z qt+1,tat+1dωt+1,t = wtlt+ rtut− µ−1t Φ [ut]
kt−1+ mt−1 pt + Rt−1 bt pt + at+ Tt+ zt.

In terms of uses –the left-hand side of equation (1.1)– and beyond consumption, the household can save in physical capital, by investing xt in new capital, holding real money

balances, purchasing government debt, bt, and trading in Arrow securities. More concretely,

at+1 is the amount of those securities that pays one unit of consumption in event ωt+1,t

purchased at time t at (real) price qt+1,t. This price will be such that, in equilibrium, the net

supply of the Arrow securities would be zero and, from the next equation on, we drop the terms associated with these securities.

In terms of resources –the right-hand side of equation (1.1)– the household gets income by renting its labor supply at the real wage wt and its capital at real rental price rt. The

household chooses the utilization rate of capital, ut> 0, given the depreciation cost µ−1t Φ [ut],

where µt is an investment-specific technological shock that we will introduce below. We

interpret ut= 1 as the “normal” utilization rate. The household also has access to its money

balances, the government debt (with a nominal gross interest rate of Rt), the Arrow security

that pays in the actually realized event, the lump-sum transfers (of taxes if negative) from the government, Tt, and the profits of the economy’s firms, zt.

 h

S0[Λx] = 0, and S00[·] > 0, where Λx is the growth rate of investment along the BGP.

The investment-specific technological shock follows a unit-root process µt= µt−1exp (Λµ+ zµ,t) where zµ,t= σµεµ,t and εµ,t ∼ N (0, 1).

The value of µt is also the inverse of the relative price of new capital in consumption terms.

The Lagrangian function associated with the household’s problem is:

E0 ∞ X t=0 βt        dt n log (ct− hct−1) + υ log  mt pt  − ϕtψ l1+ϑ_t 1+ϑ o −λt ( ct+ xt+ m_ptt +_pbt_t −wtlt− rtut− µ−1t Φ [ut]
kt−1− m_pt−1_t − Rt−1bt−1_pt − Tt− zt ) −Qt n kt− (1 − δ) kt−1− µt  1 − Sh xt xt−1 i xt o        where the household maximizes over ct, lt, bt, ut, ut, and xt, λt is the Lagrangian multiplier

associated with the budget constraint and Qt the Lagrangian multiplier associated with

in-stalled capital. Also, define the (marginal) Tobin’s Q as qt = Q_λtt, (loosely the value of installed

capital in terms of its replacement cost)

The first-order conditions with respect to ct, lt, bt, ut, and xt are:

dt(ct− hct−1) −1 − hβEtdt+1(ct+1− hct) −1 = λt λt= βEt  λt+1 Rt Πt+1  rt = µ−1t Φ 0_[u t] qt= βEt  λt+1 λt (1 − δ) qt+1+ rt+1ut+1− µ−1t+1Φ [ut+1]
 1 = qtµt  1 − S  xt xt−1  − S0  xt xt−1  xt xt−1  + βEtqt+1µt+1 λt+1 λt S0 xt+1 xt   xt+1 xt 2 . The last equation is important. If S [·] = 0 (i.e., there are no adjustment costs), we get qt = _µ1

t, i.e., the marginal Tobin’s Q is equal to the replacement cost of capital (the relative

price of capital). Furthermore, if µt= 1, as in the standard neoclassical growth model, qt= 1.

1.2

The final good producer

There is one final good is produced using intermediate goods with the production function: y_td= Z 1 0 y ε−1 ε it di ε−1ε , (2)

where ε is the elasticity of substitution.

Final good producers are perfectly competitive and maximize profits subject to the pro-duction function (2), taking as given all intermediate goods prices pti and the final good price

pt. As a consequence their maximization problem is:

max yit ptytd− Z 1 0 pityitdi.

The input demand functions associated with this problem are, for all i, yit =

 pit pt −ε yd t, where yd

t is the aggregate demand. If we combine this input demand functions with the zero

profit condition ptytd= R1 0 pityitdi, we get: pt= Z 1 0 p1−ε_it di 1−ε1 .

1.3

Intermediate good producers

Each intermediate good producer i has a technology yit = Atkit−1α ldit

1−α

, where kit−1

is the capital rented by the firm, ld

it is the labor input rented, and At is the technology

level that evolves as a unit-root process At = At−1exp (ΛA+ zA,t), where zA,t = σAεA,t and

εA,t∼ N (0, 1). Also, define zt = A 1 1−α t µ α 1−α

t , a variable that encodes the joint effect of both technology

shocks along the BGP. Simple algebra tells us that zt = zt−1exp (Λz+ zz,t) where zz,t = zA,t+αzµ,t

1−α and Λz =

ΛA+αΛµ

1−α .

Intermediate good producers solve a two-stage problem. In the first stage, taken the input prices wt and rt as given, firms rent ldit and kit−1 in perfectly competitive factor markets to

minimize real cost:

min

ld it,kit−1

wtlitd + rtkit−1

subject to their supply curve yit= Atkαit−1 ldit

1−α . The first-order conditions for this problem are:

wt = % (1 − α) Atkαit−1 l d it
−α rt = %αAtkα−1it−1 l d it
1−α , where % is the Lagrangian multiplier.

Given that the firm has constant returns to scale, we can find the real marginal cost mct

by setting the level of labor and capital equal to the requirements of producing one unit of good Atkit−1α litd
1−α = 1 or: Atkit−1α l d it
1−α = At  α 1 − α wt rt l_itd α ld_it
1−α = At  α 1 − α wt rt α l_itd = 1 that implies that:

ld_it=  α 1−α wt rt −α At . Then: mct =  1 1 − α 1−α_{ 1} α α w1−α_t r_tα At .

This marginal cost does not depend on i: all firms receive the same technology shocks and all firms rent inputs at the same price.

In the second stage, intermediate good producers choose the price that maximizes dis-counted real profits subject to Calvo pricing. In each period, a fraction 1 − θp of firms can

change their prices. All other firms can only index their prices by past inflation. Indexation is controlled by the parameter χ ∈ [0, 1], where χ = 0 is no indexation and χ = 1 is total indexation.

The problem of the firms is then: max pit E t ∞ X τ =0 (βθp)τ λt+τ λt ( _τ Y s=1 Πχ_t+s−1 pit pt+τ − mct+τ ! yit+τ ) subject to yit+τ = τ Y s=1 Πχ_t+s−1 pit pt+τ !−ε yd_t+τ,

where the marginal value of a dollar to the household, is treated as exogenous by the firm. Since we have complete markets in securities and the utility function is separable in consump-tion, λt+τ/λt is the correct valuation of future profits.

Substituting the demand curve in the objective function and the previous expression, we get: max pit E t ∞ X τ =0 (βθp)τ λt+τ λt      τ Y s=1 Πχ_t+s−1 Πt+s pit pt !1−ε − τ Y s=1 Πχ_t+s−1 Πt+s pit pt !−ε mct+τ  yd_t+τ   

whose solution p∗_it implies the first-order condition: Et ∞ X τ =0 (βθp) τ λt+τ     (1 − ε) τ Y s=1 Πχ_t+s−1 Πt+s !1−ε p∗_it pt + ε τ Y s=1 Πχ_t+s−1 Πt+s !−ε mct+τ  y_t+τd    = 0,

where we have dropped irrelevant constants and used the fact that we are in a symmetric equilibrium.

This expression nests the usual result in the fully flexible prices case θp = 0:

p∗_it = ε

ε − 1ptmct+τ,

i.e., the price is equal to a mark-up over the nominal marginal cost.

Since we only consider a symmetric equilibrium across firms, p∗_it = p∗_t and:

Et ∞ X τ =0 (βθp) τ λt+τ     (1 − ε) τ Y s=1 Πχ_t+s−1 Πt+s !1−ε p∗_t pt + ε τ Y s=1 Πχ_t+s−1 Πt+s !−ε mct+τ  y_t+τd    = 0.

To express the previous first-order condition recursively, we define: g_t1 _{= E}t ∞ X τ =0 (βθp) τ λt+τ τ Y s=1 Πχ_t+s−1 Πt+s !−ε mct+τyt+τd and g2_{t = E}t ∞ X τ =0 (βθp)τλt+τ τ Y s=1 Πχ_t+s−1 Πt+s !1−ε p∗_t pt yd_t+τ and then the first-order condition is εg_t1 = (ε − 1)g_t2. We need (βθp)

τ

λt+τ to go to zero

suffi-ciently fast in relation to the rate of inflation for g_t1 and g_t2 to be well-defined and stationary. Then, we can write the g’s recursively as:

g_t1 = λtmctytd+ βθpEt  Πχ_t Πt+1 −ε g1_t+1 and g_t2 = λtΠ∗tytd+ βθpEt  Πχ_t Πt+1 1−ε Π∗_t Π∗ t+1  g2_t+1 where Π∗_t = p∗t pt.

or, dividing by p1−ε_t , 1 = θp  Πχ_t−1 Πt 1−ε + (1 − θp) Π∗1−εt .

1.4

The government problem

The government determines monetary and fiscal policy. We assume there is no interac-tion between monetary and fiscal policy, that is, the results of open market operainterac-tions are distributed in a lump-sum fashion to the household.

In terms of monetary policy, the government sets the nominal interest rate following a Taylor rule: Rt R =  Rt−1 R γR    Πt Π γΠ   yd t yd t−1 Λyd   γy  1−γR exp (mt)

through open market operations. The variable Π is the target level of inflation (equal to inflation in the BGP), R is the BGP nominal gross return of capital (equal to the BGP real gross returns of capital plus Π), and Λyd is the BGP growth rate of y_td. The term m_t= σ_mε_mt

is a random shock to monetary policy, where εmt∼ N (0, 1).

Government consumption is given by gt=egtzt, whereegtfollows an autoregressive process: log_egt= (1 − ρg) log g + ρglogegt−1+ σgεg,t where εg,t∼ N (0, 1)

Since the economy is growing, the level of real government consumption is multiplied by the term zt to keep it as a stationary share of output. Government consumption is financed by

lump sum taxes that ensure that the deficit is zero.

1.5

Aggregation

First, we derive an expression for aggregate demand:

yd_t = ct+ gt+ xt+ µ−1t Φ [ut] kt−1.

With this value, the demand for each intermediate good producer is: yit = ct+ gt+ xt+ µ−1t Φ [ut] kt−1

 pit pt

and using the production function is: Atkαit−1 litd
1−α = ct+ gt+ xt+ µ−1t Φ [ut] kt−1
 pit pt −ε . Since all the firms have the same optimal capital-labor ratio:

kit−1 ld it = α 1 − α wt rt

and by market clearing

Z 1 0 ld_itdi = ld_t and Z 1 0 kit−1di = utkt−1,

it must be the case that:

kit−1 ld it = utkt−1 ld t . Then: Atkαit−1 l d it
1−α = At  kit−1 ld it α l_itd
= At  utkt−1 ld t α l_itd Integrating out Z 1 0 At  utkt−1 ld t α ld_itdi = At  utkt−1 ld t αZ 1 0 ld_itdi = At(utkt−1)α ldt
1−α and we have At(utkt−1) α ld_t
1−α− φzt = ct+ gt+ xt+ µ−1t Φ [ut] kt−1
Z 1 0  pit pt −ε di. Define vp_t =R₀1  pit pt −ε

di. By the properties of the index under Calvo’s pricing

vp_t = θp  Πχ_t−1 Πt −ε v_t−1p + (1 − θp) Π∗−εt , we get: ct+ gt+ xt+ µ−1t Φ [ut] kt−1 = At(utkt−1) α ld_t
1−α v_tp .

2

Equilibrium

A definition of equilibrium in this economy is standard and the symmetric equilibrium policy functions are determined by the following equations:

• The first-order conditions of the household: dt(ct− hct−1) −1 − hβEtdt+1(ct+1− hct) −1 = λt φtψ(lst)θ = wtλt λt = βEt  λt+1 Rt Πt+1  rt= µ−1t Φ 0 [ut] qt = βEt  λt+1 λt (1 − δ) qt+1+ rt+1ut+1− µ−1t+1Φ [ut+1]
 1 = qtµt  1 − S  xt xt−1  − S0  xt xt−1  xt xt−1  + βEtqt+1µt+1 λt+1 λt S0 xt+1 xt   xt+1 xt 2

• The firms that can change prices set them to satisfy: g_t1 = λtmctytd+ βθpEt  Πχ_t Πt+1 −ε g1_t+1 g_t2 = λtΠ∗ty d t + βθpEt  Πχ_t Πt+1 1−ε Π∗_t Π∗_t+1  g2_t+1 εg_t1 = (ε − 1)g2_t

where they rent inputs to satisfy their static minimization problem: utkt−1 ld t = α 1 − α wt rt mct=  1 1 − α 1−α_{ 1} α α w_t1−αrα_t At

The price level evolves as 1 = θp

_Πχ t−1

Πt

1−ε

+ (1 − θp) Π∗1−εt .

• The government follows: Rt R =  Rt−1 R γR    Πt Π γΠ   ydt yd t−1 Λyd   γy  1−γR exp (mt)

• Markets clear: y_td = At(utkt−1) α ld t
1−α v_tp y_td = ct+ gt+ xt+ µ−1t Φ [ut] kt−1 where v_tp = θp  Πχ_t−1 Πt −ε v_t−1p + (1 − θp) Π∗−εt and ls t = ldt and kt− (1 − δ) kt−1− µt  1 − Sh xt xt−1 i xt= 0.

3

Stationary equilibrium

Since we have growth in this model induced by technological change, most of the vari-ables are growing in average. Thus, before solving the model, we need to make all varivari-ables stationary.

3.1

Manipulating the equilibrium conditions

First, we work on the first-order conditions of the household: dt  ct zt − hct−1 zt−1 zt−1 zt −1 − hβEtdt+1  ct+1 zt+1 zt+1 zt − hct zt −1 = λtzt φtψ(lt)θ = wt zt λtzt λtzt= βEt  λt+1zt+1 zt zt+1 Rt Πt+1  µtrt= Φ0[ut] qtµt = βEt  λt+1zt+1 λtzt zt zt+1 µt µt+1 [(1 − δ) qt+1µt+1+ µt+1rt+1ut+1− Φ (ut+1)]  1 = qtµt 1 − S " _xt zt xt−1 zt−1 zt zt−1 # − S0 " _xt zt xt−1 zt−1 zt zt−1 # _xt zt xt−1 zt−1 zt zt−1 ! +βEtqt+1µt+1 λt+1 λt S0 "xt+1 zt+1 xt zt zt+1 zt # xt+1 zt+1 xt zt zt+1 zt !2 .

The firms that can change prices set them to satisfy: g1_t = λtztmct yd_t zt + βθpEt  Πχ_t Πt+1 −ε g_t+11 g2_t = λtztΠ∗t yd_t zt + βθpEt  Πχ_t Πt+1 1−ε Π∗_t Π∗_t+1  g_t+12 εg1_t = (ε − 1)g_t2,

where they rent inputs to satisfy their static minimization problem: ut ld t kt−1 zt−1µt−1 = α 1 − α wt zt 1 rtµt zt zt−1 µt µt−1 mct=  1 1 − α 1−α_{ 1} α α  wt zt 1−α (rtµt)α z 1−α t µα t At . The price level evolves as:

1 = θp

 Πχ_t−1 Πt

1−ε

+ (1 − θp) Π∗1−εt .

The government follows:

Rt R =  R_t−1 R γR        Π_t Π γΠ       yd_t zt yd_t−1 zt−1 zt zt−1 Λ_yd       γy_      1−γR exp (mt)

log_egt = (1 − ρg) log g + ρglogegt−1+ σgεg,t. Markets clear: yd t zt = µα_t−1z_t−1α At zt  ut_z kt−1 t−1µt−1 α ld_t
1−α v_tp but since µα_t−1z_t−1α = zt−1 At−1, we have yd t zt = zt−1 At−1 At zt  ut_z kt−1 t−1µt−1 α ld_t
1−α v_tp yd t zt = ct zt +xt zt +_egt+ zt−1 zt µt−1 µt Φ [ut] kt−1 zt−1µt−1

where vp_t = θp _Πχ t−1 Πt −ε v_t−1p + (1 − θp) Π∗−εt and lt = ltd kt ztµt ztµt zt−1µt−1 − (1 − δ) kt−1 zt−1µt−1 − µt µt−1 zt zt−1 1 − S " _x t ztzt xt−1 zt−1zt−1 #! xt zt = 0.

3.2

Change of variables

We now redefine that variables to obtain a system on stationary variables that we can easily manipulate. Hence, we define_ect = _zc_tt, eλt= λtzt, ret = rtµt, qet= qtµt, xet =

xt zt, wet = wt zt, e w_t∗ = wt∗ zt, ekt= kt ztµt, and ey d t = yd t zt.

Then, the set of equilibrium conditions are: • The first-order conditions of the household:

dt  ect− hect−1 zt−1 zt −1 − hβEtdt+1  e ct+1 zt+1 zt − h_ect −1 = eλt φtψ(lt)θ =wetλet e λt= βEt{eλt+1 zt zt+1 Rt Πt+1 } e rt= Φ0[ut] e qt = βEt ( e λt+1 e λt zt zt+1 µt µt+1 ((1 − δ)q_et+1+ert+1ut+1− Φ (ut+1)) ) 1 = _eqt  1 − S  e xt e xt−1 zt zt−1  − S0  e xt e xt−1 zt zt−1  e xt e xt−1 zt zt−1  +βEtqet+1 e λt+1 e λt zt zt+1 S0  e xt+1 e xt zt+1 zt   e xt+1 e xt zt+1 zt 2

• The firms that can change prices set them to satisfy: g_t1 = eλtmctye d t + βθpEt  Πχ_t Πt+1 −ε g1_t+1 g_t2 = eλtΠ∗tye d t + βθpEt  Πχ_t Πt+1 1−ε Π∗_t Π∗_t+1  g2_t+1 εg_t1 = (ε − 1)g2_t

where they rent inputs to satisfy their static minimization problem: utek_t−1 ld t = α 1 − α e wt e rt zt zt−1 µt µt−1 mct=  1 1 − α 1−α_{ 1} α α (w_et)1−αer α t

• The price level evolves 1 = θp

_Πχ t−1

Πt

1−ε

+ (1 − θp) Π∗1−εt .

• The government follows: Rt R =  Rt−1 R γR    Πt Π γΠ   e ytd e yd t−1 zt zt−1 Λyd   γy  1−γR exp (mt)

log_egt = (1 − ρg) log g + ρglogegt−1+ σgεg,t. • Markets clear: e yd_t = _ect+ext+eg + zt−1 zt µt−1 µt Φ [ut] ekt−1 e yd_t = At At−1 zt−1 zt  utekt−1 α ld_t
1−α v_tp where v_tp = θp  Πχ_t−1 Πt −ε v_t−1p + (1 − θp) Π∗−εt and lt= ldt and ekt_zt−1zt _µµ_t−1t − (1 − δ) ekt−1−_zt−1zt _µµ_t−1t  1 − Sh ext e xt−1 zt zt−1 i e xt= 0.

4

The steady state

Now, we will find the deterministic steady-state of the model. First, let _ez = exp (Λz),

e

µ = exp (Λµ), and eA = exp (ΛA). Also, given the definition of ec, xet, wet, we

∗

t, and ye

d

t, we have

Then, in steady state, the first-order conditions of the household can be written as: 1 e c − h e zec − hβ 1 e c_ez − h_ec = eλ ψ(l)θ =we_eλ 1 = β1 e z R Π e r = Φ0[1] e q = β 1 e zµ_e((1 − δ)q +e eru − Φ [1]) 1 =q (1 − S [_e z] − S_e 0[_ez]_ez) + βqe e zS 0_[ e z]z_e2, the first-order conditions of the firm as:

g1 = eλmcy_ed+ βθp  Πχ Π −ε g1 g2 = eλΠ∗_eyd+ βθp  Πχ Π 1−ε g2 εg1 = (ε − 1)g2 uek ld = α 1 − α e w e rezµe mc =  1 1 − α 1−α_{ 1} α α e w1−αr_eα the law of motion for prices as:

1 = θp

 Πχ

Π 1−ε

+ (1 − θp) Π∗1−ε

and the market clearing conditions as: e c +x +_{e e}g =y_ed vp_eyd= A z(uek) α (ld)1−α l = ld vp = θp  Πχ Π −ε vp+ (1 − θp) Π∗−ε

To find the steady-state, we need to choose functional forms for Φ [·] and S [·]. For Φ [u] we pick: Φ [u] = Φ1(u − 1) +Φ₂2(u − 1)2. Since, in the steady state, u = 1, thener = Φ

0_{[1] = γ} 1

and Φ [1] = 0. The investment adjustment cost function is Sh xt

xt−1

i

= κ₂( xt

xt−1 − Λx)

2_{. Then,}

along the BGP, S [z] = S0[Λx] = 0. Using this two expressions, we can rearrange the system

of equations that determine the steady state as: (1 − hβ/_ez) 1 1 − h e z 1 e c = eλ ψ(l)θ =we_eλ R = Πez β e r = γ1 e r =  1 − β e zµ_e(1 − δ)  / β e z_eµ 1 − βθpΠ(1−χ)ε
g1 = eλmcye d 1 − βθpΠ−(1−χ)(1−ε)
g2 = eλΠ∗ye d εg1 = (ε − 1)g2 e k ld = α 1 − α e w e rzeµe mc =  1 1 − α 1−α_{ 1} α α e w1−αr_eα 1 − θpΠ−(1−χ)(1−ε) 1 − θp = Π∗1−ε e c +x +_{e e}g =y_ed vp_eyd= Ae e z(ek) α_(ld₎1−α l = ld 1 − θpΠ(1−χ)ε 1 − θp vp = Π∗−ε e k = zeµe e zµ − (1 − δ)_e x.e First, notice that there is some restrictions on γ1

˜ r = 1 − β e zµe (1 − δ) β e zµe = γ1

and that the nominal interest rate is:

R = Πez β

The relationship between inflation and optimal relative prices is: Π∗ = 1 − θpΠ

−(1−ε)(1−χ)

1 − θp

1−ε1

. The expression for the dispersion of prices is given by:

vp = 1 − θp 1 − θpΠ(1−χ)ε

Π∗−ε.

We can calibrate ψ such that in steady state l = 1 (the utility function we selected does not have natural units for labor, so l = 1 is just a normalization). Labor demand must equal labor supply, ld_{= 1.}

Once we have ld_{, we can solve for wages, capital, investment, output, and consumption.}

Wages are given by:

1 − βθpΠ(1−χ)ε
(1 − βθpΠ−(1−χ)(1−ε)) ε − 1 ε Π ∗ = mc mc =  1 1 − α 1−α_{ 1} α α e w1−αr_eα That allows us to find ek = _1−αα we

e rzeµ.e We also have: e yd = e A e z(ek) α_(ld₎1−α vp .

But, since in steady-state ek = zeeµ e

zeµ−(1−δ)e

x, it is the case that:

e c + zeµ − (1 − δ)e e zµ_e ek +eg =ey d₌ e A e z(ek) α vp . Finally: e c = e A e zek α_{− φ} vp − e zµ − (1 − δ)_e e zµ_e ek −eg.

5

Log-linear approximations

For each variable vart, we define vardt = log vart− log var, where var is the steady-state value for the variable vart. Then, we can write vart= var expdvart.

We start by log-linearizing the marginal utility of consumption: dt  e ct− hect−1 zt−1 zt −1 − hβEtdt+1  e ct+1 zt+1 zt − h_ect −1 = eλt. (3)

It is helpful to define the auxiliary variable auxt= dt



ect− hect−1

zt−1

zt

−1

. Then, we have that: auxt− hβEt 1 e zt+1 auxt+1 = eλt wherez_et+1 = zt+1 zt .

Then, (3) can be written as: aux expauxdt−

hβ z auxEtexp d auxt+1−bzet+1 = eλe b e λt

which can be log-linearized as: aux  d auxt− hβ z Et(auxdt+1− bzet+1)  = eλbeλ_t. (4) Using the following two steady state relationship that aux(1 −hβ

e z ) = eλ and that Etbzet+1 = 0, we can write: d auxt− hβ z Etauxdt+1=  1 −hβ z  b e λt.

Next, we log-linearize the auxiliary variable: auxeauxdt = d expdbt

 e c expb_ect₋h e zec exp b_ect−1−bzet −1 . The log-linear approximation is:

auxaux_dt = d  e c − h e zec −1 b dt− d  e c − h e zec −2 e cb_ect +d  e c − h e zec −2 h_ec e z  b e ct−1− bezt  .

Making use of the fact that aux = d _ec − h e zec

−1

, we find:

auxaux_dt= aux db_t−  ec − h e zec −1 e cb_ect+  e c − h e zec −1 h_ec e z  b ect−1− bezt  ! , that simplifies to:

auxaux_dt= aux db_t−  1 − h e z −1 b e ct+  1 −h e z −1 h e z  b ect−1− bezt  ! d auxt= bdt−  1 − h e z −1 b e ct− hb_ect−1 e z + hbz_et e z ! . (5)

Putting the (4) and (5) together:  1 −hβ z  b e λt = b dt−  1 −h e z −1 b ect− hb_ect−1 e z + hb_ezt e z ! − hβ z Et ( b dt+1−  1 − h e z −1 b e ct+1− hb_ect e z + hb_ezt+1 e z !) . After some algebra, we arrive to the final expression:

(1 − hβ z )beλt = bdt− hβ z Etdbt+1− 1 + h2/z2β 1 −h e z
bect+ h e z 1 −h e z
bect−1+ βh/z 1 − h e z
Etbect+1− h e z 1 −h e z
bezt. Now, we log-linearize the Euler equation:

e λt = βEt{eλt+1 zt zt+1 Rt Πt+1 }. To do so, we write the expression as

e λeb e λ = βEt{eλe b e λt+1 1 e zezz,t ReRbt ΠeΠbt+1 } By using the fact that

R = Πez β we simplify to: ebeλ = Et{e b e λt+1 1 e b Rt }.

Now, it is easy to show that: b e

λt= Et{beλ_t+1+ bR_t− bΠ_t+1}.. (6) Next, we log-linearize the static consumption/labor optimal condition φtψ(lt)θ =weteλt:

b

φt+ θblt= bwet+ beλt. (7) Let us now consider _ert = Φ0[ut]. First, we write ree

b e

rt _{= Φ}0u expubt, where the log-linear

approximation is:

e

rbr_et= Φ00[u] ubut. Since _er = Φ0[u], then:

b e rt= Φ00[u] u Φ0_[u] ubt, or b e rt= Φ2 Φ1 ˆ ut. (8)

The next equation relates the shadow price of capital to the return on investment:

e qt = βEt ( e λt+1 e λt 1 e zt+1 1 e µt+1 ((1 − δ)q_et+1+ert+1ut+1− Φ [ut+1]) )

whereµ_et+1= µ_µt+1_t . We can we write this expression as

e qebq_et = β e z_eµEtexp ∆beλt+1−bezt+1−bµet+1 ( (1 − δ)qe_eb e qt+1+ e ru expb e rt+1+ubt+1−Φu expubt+1  ) . Log-linearization delivers: e qbq_et = β e zµ_eEt  4beλ_t+1− b_ez_t+1− bµ_et+1  ((1 − δ)q +_{e e}ru − Φ [u]) + β e zµ_e  Et(1 − δ)qbeqet+1+eru  b e rt+1+but+1  − Φ [u] uu_bt+1  .

Making use of the following steady-state relationships: u = 1, Φ [u] = 0, _er = Φ0[u], q = 1_e and 1 = β e zµee r + β e zµe(1 − δ), and Et  −bz_et+1− beµt+1 

= 0, the previous expression simplifies to:

b e q_t = ((1 − δ) +_er) β e z_eµEt∆beλt+1+ β (1 − δ) e z_eµ Etbqet+1+ β e zµ_eeruEtbert+1,

that implies: b e q_{t= E}t∆beλ_t+1+ β (1 − δ) e zµ_e Etbeqt+1+  1 − β(1 − δ) e zµ_e  Etb e rt+1. (9)

The next equation to log-linearize is: 1 = _eqt  1 − S  e xt e xt−1 zt zt−1  − S0  e xt e xt−1 zt zt−1  e xt e xt−1 zt zt−1  +βEteqt+1 e λt+1 e λt zt zt+1 S0  e xt+1 e xt zt+1 zt   e xt+1 e xt zt+1 zt 2

which can be rearranged as:

1 = q exp_e bq_et  1 − Sh_ez exp∆bxet+bzet i − S0hz exp_e ∆bext+bezt i e z exp∆bext+bezt  + +βq˜ e zEtexp b e qt−bzet+1+4beλt+1S 00h e z exp∆bext+1+bzet+1 i e z2exp2(∆bext+1+bzet+1) .

Taking the log-linear approximation (and using the fact thatq = 1) we get:_e 0 = b_eq_t− S00[_ez]z_e2∆bx_et+ bzet  + β zS 00 [z]_{e e}z3_Et  ∆b_ext+1+ bezt+1  . Reorganizing: κ_ez2∆bx_et+ bezt  = b_eq_t+ βκz_e2_Et∆bxet+1, (10) where κ comes from the adjustment cost function.

Let us log-linearize the law of motion for g1

t and gt2. First consider

g_t1 = eλtmctye d t + βθpEt  Πχ_t Πt+1 −ε g1_t+1 that can be rewritten as:

g1expbg 1 t = eλmc e ydexpb e λt+mcct+bye d t _+βθ pg1Πε(1−χ)Etexpε( bΠt+1−χ bΠt)+bg 1 t+1.

If we log-linearize that last expression, we get: g1_bg1_t = eλmcy_ed  b e λt+mcct+ bye d t  + βθpg1Πε(1−χ)Et  ε( bΠt+1− χbΠt) +bg 1 t+1  . Since, in steady state 1 − βθpΠε(1−χ) =

e λmcey

d

Let us now consider: g2_t = eλtΠ∗tye d t + βθpEt  Πχ_t Πt+1 1−ε Π∗_t Π∗_t+1  g_t+12 , that can rewritten as:

g2expbg 2 t _{= eλΠ}∗ e ydexpb_eλ t+ bΠ∗t+bye d t _+βθ pΠ−(1−ε)(1−χ)g2Etexp−(1−ε)( b Πt+1−χ bΠt)−(Πb_t+1∗ − bΠ∗_t)+_bg_t+12 . If we log-linearize that last expression, we get:

g2_bg_t2 = eλΠ∗y_ed  b e λt+ bΠ∗t + bye d t  +βθpΠ−(1−ε)(1−χ)g2Et  −(1 − ε)Πb_t+1− χbΠ_t  −Πb∗_t+1− bΠ∗_t  +_bg_t+12 . It can be shown that:

1 − βθpΠ−(1−ε)(1−χ) = e λΠ∗_eyd g2 , therefore: b g_t2 = 1 − βθpΠ−(1−ε)(1−χ)
 b e λt+ bΠ∗t + bye d t  +βθpΠ−(1−ε)(1−χ)Et  −(1 − ε)Πb_t+1− χbΠ_t  −Πb∗_t+1− bΠ∗_t  +_bg2_t+1. (12) Note that is easy to show that εg1

t = (ε − 1)gt2 log-linearizes to:

b

g_t1 =_bg2_t. (13)

The relationship between the capital-labor ratio and the real wage-real interest rate utek_t−1 ld t = α 1 − α e wt e rt zt zt−1 µt µt−1 . becomes: b ut+ bek_t−1− bld_t = b e wt− bret+ bezt+ bµet. (14) Let us log-linearize the marginal cost

mct=  1 1 − α 1−α_{ 1} α α (w_et)1−αer α t

to get

c

mct = (1 − α)bwet+ αbert. (15) Now we concentrate on the aggregate price law of motion:

1 = θp

 Πχ_t−1 Πt

1−ε

+ (1 − θp) Π∗1−εt

that can be rewritten as:

1 = θpΠ−(1−ε)(1−χ)exp−(1−ε)( bΠt−χ bΠt−1)+ (1 − θp) (Π∗)(1−ε)exp(1−ε) bΠ

∗ t

that log-linearizes to:

θpΠ−(1−ε)(1−χ)

(1 − θp) (Π∗)(1−ε)

( bΠt− χbΠt−1) = bΠ∗t. (16)

The Taylor rule log-linearizes to: b Rt = γRRb_t−1+ (1 − γ_R)  γΠΠb_t+ γ_y(4b e yd_t + b_ezt)  +m_bt. (17)

(the government consumption rule is already linear). The market clearing conditions:

lt = ltd, e ct+ext+egt+ Φ [ut] ekt−1 e µtzet =_ey_td, and vp_ty_etd= Aet e zt  utek_t−1 α l_td
1−α− φ can be written as:

l expblt _{= l}d_expbl_td , e c expb_ect₊ e x expbx_et₊ e g expb_eg_t + Φh_eu expbu_et i e k expb_ek t−1 e zµ exp_e b e µt+bezt =_eydexpby_e d t_, and vp_eydexpbv p t+bye d t ₌ Ae e z  uekαeld 1−α exp b_e At−bzet+α  b ut+bekt−1  +(1−α)eld t −φ respectively, where eAt+1= At+1 At .

˜ cb_ect+ ˜xbext+ ˜gbegt+ γ1ek e zµ_eubt = ˜y d b e yd_t, (19) and (˜ydvp)_bv_tp+ by_ed_t= Ae e z  uek α e ld 1−α b e At− bezt+ α  b ut+ bek_t−1  + (1 − α) eld_t  . (20) Let us consider vp_t = θp _Πχ t−1 Πt −ε

v_t−1p + (1 − θp) Π∗−εt , that can be written as

vpexpbv p t = θ pΠε(1−χ)vpexpε( bΠt−χ bΠt−1)+bv p t−1+ (1 − θ p) Π∗−εexpε bΠ ∗ t . Using 1 − θpΠε(1−χ)
vp = (1 − θp) Π∗−ε we get: b vp_t = θpΠε(1−χ)  ε( bΠt− χbΠt−1) +bv p t−1  − 1 − θpΠε(1−χ)
εΠb ∗ t. (21)

Finally, let us log-linearize the law of motion of capital: ˜ ktzeteµt = (1 − δ) ˜kt−1+µetzet  1 − S  ˜ xt ˜ xt−1e zt  ˜ xt.

If we rearrange terms, we get: ˜ kz_eµ exp_e b_k˜ t+bzet+beµt = (1 − δ) ˜k exp b ˜ kt−1₊ e zµ exp_e b e zt+bµet  1 − Shzeb e zt+4bext i ˜ x expb e xt_. Log-linearizing: ˜ kz_eµ_e  b˜ kt+ bzet+ bµet  = (1 − δ) ˜kb˜kt−1+ezeµ˜x(bzet+ bµet+ bxet). Using ˜k = zeeµ e zeµ−(1−δ) ˜

x, we can rearrange the previous expression to get: b˜ kt+ bezt+ bµet = (1 − δ) e z_eµ b ˜ kt−1+ e zµ − (1 − δ)_e e zµ_e (bzet+ bµet+ bext) or b˜ kt= (1 − δ) e zµ_e b˜ kt−1+ e zµ − (1 − δ)_e e zµ_e bext− 1 − δ e z_eµ  b e zt+ bµet  . (22)

6

System of linear stochastic difference equations

We now collect the whole system of linear stochastic difference equations. Equation 1 The first equation is:

θpΠ−(1−ε)(1−χ)

(1 − θp) (Π∗)(1−ε)

( bΠt− χbΠt−1) = bΠ∗t

In order to make notation for compact, define a2 = θpΠ

−(1−ε)(1−χ)

(1−θp)(Π∗)(1−ε). Substituting for a2:

a2Πb_t− a₂χ bΠ_t−1− bΠ∗_t = 0. (23) Equation 2 The second equation is

b e

rt = φuuˆt,

where φu = Φ2/Φ1. Then,

− b_ert+ φuuˆt= 0. (24)

Equation 3 The third equation is

b g_t1 =_bg_t2 Rearranging:

b

g1_t −_bg2_t = 0. (25) Equation 4 The fourth equation is:

b ut+ bek_t−1− bl_td= bw_et− b_er_t+ bz_et+ bµ_et Rearranging: b ut+ bert+ bekt−1− bl d t − bwet− bezt− bµet. (26) Equation 5 The fifth equation is:

c

Equation 6 The sixth equation is: b Rt= γRRb_t−1+ (1 − γ_R)  γΠΠb_t+ γ_y(4b e yd_t + bz_et)  +m_bt Rearranging: − bRt+ γRRbt−1+ (1 − γR)γΠΠbt+ (1 − γR)γyb e zt+ (1 − γR)γyb e yd_t− (1 − γR)γyb e yd_t−1+m_bt= 0. (28)

Equation 7 The seventh equation is: ˜ cb_ect+ ˜xbxet+ ˜gbegt+ γ1ek e zµ_ebut= ˜y d b e yd_t or, rearranging: ˜ cb_ect+ ˜xbext+ ˜gbegt+ γ1ek e zµ_eubt− ˜y d b e yd_t = 0. (29)

Equation 8 The eighth equation is: (˜ydvp)_bv_tp+ b_eyd_t= Ae e z  uekαeld 1−α b e At− bzet+ α  b ut+ bek_t−1  + (1 − α) eld_t  .

Define the parameter produc = Ae e z  uek α eld 1−α . This is defined as produc = Ae e z n

exp [log(u)] exphlog(ek)io

αn

exphlog(eld)io

1−α

.

In terms of our code, we have that ˜yd_{= explog ˜y}d
_{, v}p _{= exp [log (v}p_{)]. Substituting:}

(˜ydvp)_bv_tp+ b_eyd_t= produc  b e At− bzet+ α  b ut+ bek_t−1  + (1 − α) eld_t  . Rearranging:

(˜ydvp)_bv_tp+ (˜ydvp)b_eyd_t − (produc)beAt+ (produc)bzet− (α)(produc)but

−(α)(produc)bek_t−1− (1 − α) (produc)el_td= 0. (30) Equation 9 The ninth equation is:

b v_tp = θpΠε(1−χ)  ε( bΠt− χbΠt−1) +bv p t−1  − 1 − θpΠε(1−χ)
εΠb∗_t

Define a3 = βθpΠε(1−χ). Then, θpΠε(1−χ)= a_β3. This type of parameter definition will become

clearer when we analyze the price setting equations. Then, substituting:

b v_tp = a3 β ε bΠt− χε a3 βΠbt−1+ a3 β bv p t−1−  1 −a3 β  ε bΠ∗_t And rearranging: a3ε β Πbt− a3εχ β Πbt−1+ a3 βbv p t−1−  1 − a3 β  ε bΠ∗_t −_bvp_t = 0. (31) Equation 10 The tenth equation is:

b˜ kt= (1 − δ) e zµ_e b˜ kt−1+ e zµ − (1 − δ)_e e zµ_e bext− 1 − δ e z_eµ  b e zt+ bµet  , which we rearrange to:

(1 − δ) e zµ_e b ˜ kt−1+  1 − (1 − δ) e zµ_e  b e xt− 1 − δ e zµ_e  b e zt+ beµt  − bk˜t= 0. (32)

Equation 11 The eleventh equation is:

b e zt= b e At+ αbeµt 1 − α which we rearrange to:

1 1 − α b e At+ α 1 − αbµet− bezt= 0. (33) Equation 12 The twelfth equation is:

b

φt+ θblt = bwet+ beλt which we rearrange to:

b

φt+ θblt− bwet− beλt = 0. Equation 13 The thirteenth equation is:

(1 −hβ z )beλt= bdt− hβ z Etdbt+1− 1 + h2_/z2_β 1 − h e z
bect+ h e z 1 − h e z
bect−1+ βh/z 1 −h e z
Etbect+1− h e z 1 − h e z
bezt

Rearranging: b dt− hβ e z Etdbt+1− 1 + h2/_ez2β 1 − b e z
bect+ b e z 1 − h e z
bect−1+ βh/˜z 1 − b e z
Etbect+1− b e z 1 − b e z
bezt−  1 − hβ e z  b e λt= 0. (34) Equation 14 The fourteenth equation is:

b e

λt= Et{beλ_t+1+ bR_t− bΠ_t+1} which we rearrange to:

Et{beλ_t+1− beλ_t+ bR_t− bΠ_t+1} = 0. (35) Equation 15 The fifteenth equation is:

b e q_{t= E}t∆beλ_t+1+ β (1 − δ) e zµ_e Etbqet+1+  1 −β(1 − δ) e zµ_e  Etb e rt+1

which we rearrange to: Etbeλ_t+1− beλ_t+ β (1 − δ) e zµ_e Etbqet+1+  1 − β(1 − δ) e z_eµ  Etb e rt+1− bqet= 0. (36) Equation 16 The sixteenth equation is:

κz_e2∆b_ext+ bzet 

= bq_et+ βκz_e2_Et∆bxet+1 which we rearrange by undoing the first-difference operator:

b e q_t+ βκz_e2_Etb e xt+1− (1 + β)κez 2 b e xt+ κze 2 b e xt−1− κez 2 b e zt= 0. (37)

Equation 17 The seventeenth first equation is

b g_t1 = 1 − βθpΠε(1−χ)
 b e λt+mcct+ bey d t  + βθpΠ ε(1−χ) t Et  ε( bΠt+1− χbΠt) +bg 1 t+1 

As before, define a3 = βθpΠε(1−χ). Then:

(1 − a3) beλ_t+ (1 − a₃) c mct+ (1 − a3) bye d t + εa3EtΠb_t+1− χεa₃Πb_t+ a₃E_t b g1_t+1−_bg_t1 = 0. (38)

Equation 18 The eighteenth equation is b g_t2 = 1 − βθpΠ−(1−ε)(1−χ)
 b e λt+ bΠ∗t + bye d t  +βθpΠ−(1−ε)(1−χ)Et  −(1 − ε)Πb_t+1− χbΠ_t  −Πb∗_t+1− bΠ∗_t  +_bg2_t+1 Define: a7 = βθpΠ−(1−ε)(1−χ). Substituting: (1 − a7) beλt+ bΠ∗t+ (1 − a7) bye d t+ (ε − 1)a7EtΠbt+1− χ(ε − 1)a7Πbt− a7EtΠb∗_t+1+ a7Etbg 2 t+1−bg 2 t = 0.

Shocks The preference shocks has the following structure: b

dt = ρddbt−1+ εd,t

b

ϕt = ρϕϕbt−1+ εϕ,t. The government expenditure process is

b e

g_t= ρgb e

g_t−1+ σgεg,t.

The following shocks are i.i.d.:

b e µ_t = zµ,t b e At = zA,t mt = σmεm,t.

6.1

Solving the model

Now, let statet=  b Πt,bg 1 t,bg 2 t, bekt, bRt, bey d t, bect,bv p t, bqet, bext, beλt, bzet 0 , nstatet =  b e rt,but, bΠ ∗ t,mcct, blt, bwet 0 , exot=  zµ,t, bdt,ϕbt, zA,t, mt 0 , and εt= (εµ,t, εd,t, εϕ,t, εA,t, εm,t, εg,t) 0 .

Then, we write the system defined above as:

0 = AA ∗ statet+ BB ∗ statet−1+ CC ∗ nstatet+ DD ∗ exot,

0 = Et

F F ∗ statet+1+ GG ∗ statet+ HH ∗ statet−1

+J J ∗ nstatet+1+ KK ∗ nstatet+ LL ∗ exot+1+ M M ∗ exot

! , and

exot+1= N N ∗ exot+ Σ1/2∗ εt+1 with Etεt+1= 0.

The matrices in this notation are:

AA =                           a2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 (1 − γR)γΠ 0 0 0 −1 (1 − γR)γy 0 0 0 0 0 (1 − γR)γy 0 0 0 0 0 −˜yd _{˜c 0 0 x˜ 0 0} 0 0 0 0 0 y˜d_vp _{0 y˜}d_vp _{0 0 0 produc} a3ε/β 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 −1 0 0 0 0 0 1 − (1−δ)_{z ˜˜µ} 0 −(1−δ)_{z ˜˜µ} 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 −1 0                           BB =                           −χa2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γR −(1 − γR)γy 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −α(produc) 0 0 0 0 0 0 0 0 −a3εχ β 0 0 0 0 0 0 a3 β 0 0 0 0 0 0 0 1−δ_{˜z ˜µ} 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0                          

CC =                           0 0 −1 0 0 0 −1 φu 0 0 0 0 0 0 0 0 0 0 1 1 0 0 −1 −1 α 0 0 −1 0 (1 − α) 0 0 0 0 0 0 0 γ1˜k ˜ z ˜µ 0 0 0 0 0 −α(produc) 0 0 −(1 − α)(produc) 0 0 0 −(1 − a3 β)ε 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 θ −1                           DD =                        0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 g 0 0 0 −produc 0 0 0 0 0 0 0 0 −1−δ ˜ z ˜µ 0 0 0 0 0 α 1−α 0 0 1 1−α 0 0 0 0 1 0 0 0                        F F =            0 0 0 0 0 0 βh/˜z 1−h_z˜ 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 β(1−δ)_{˜z ˜µ} 0 1 0 0 0 0 0 0 0 0 0 0 βκ˜z2 0 0 a3ε a3 0 0 0 0 0 0 0 0 0 0 a7(ε − 1) 0 a7 0 0 0 0 0 0 0 0           

GG =            0 0 0 0 0 0 GG1,7 0 0 0 −(1 − hβ/˜z) GG1,12 0 0 0 0 1 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 −1 0 −1 0 0 0 0 0 0 0 0 0 1 GG4,11 0 −κ˜z2 −a3εχ −1 0 0 0 1 − a3 0 0 0 0 1 − a3 0 −a7(ε − 1)χ 0 −1 0 0 1 − a7 0 0 0 0 1 − a7 0            where GG1,7 = − 1 + βh2_/ e z2 1 − h_z˜ GG1,12 = − h ˜ z(1 − h_z˜) GG4,11 = −(1 + β)κ˜z2 HH =            0 0 0 0 0 0 h ˜ z(1−h_z˜) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 κ˜z2 _{0 0} 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0            J J =            0 0 0 0 0 0 0 0 0 0 0 0 1 − β(1−δ)_{z ˜˜µ} 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −a7 0 0 0 0            KK =            0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 − a3 0 0 0 1 0 0 0 0           

LL =            0 −βh/˜z 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0            M M =            0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0            N N =            0 0 0 0 0 0 0 ρd 0 0 0 0 0 0 ρϕ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ρg            Σ =            σ_µ2 0 0 0 0 0 0 σ2_d 0 0 0 0 0 0 σ_ϕ2 0 0 0 0 0 0 σ_A2 0 0 0 0 0 0 σ2 m 0 0 0 0 0 0 σ2 g           

7

State space representation

A standard solution method gives us for the previous system of stochastic difference equa-tions gives us:

statet = P P ∗ statet−1+ QQ ∗ exot

and

Our observables are inflation, the federal funds rate, and the first differences of real wages, output, the relative price of investment, and labor; or in our notation:

Yt= log Πt, log Rt, 4 log wt, 4 log yt, log lt, 4 log µ−1t

0

Therefore, to write the likelihood function, we need to write the model in the following state space form:

St = F ∗ St−1+ G ∗ εt

obst = H0+ H1∗ St

where St= (statet−2, exot, exot−1).

Hence, F =    P P 0 QQ 0 N N 0 0 I 0    and G =    0 I 0   ∗ Σ 1/2_.

Building the H matrix requires more work. Note that: b

Πt = P P (1, :) ∗ statet−1+ QQ (1, :) ∗ exot=

P P (1, :) ∗ P P ∗ statet−2+ QQ (1, :) ∗ exot+ P P (1, :) ∗ QQ ∗ exot−1,

b

Rt = P P (5, :) ∗ statet−1+ QQ (5, :) ∗ exot=

P P (5, :) ∗ P P ∗ statet−2+ QQ (5, :) ∗ exot+ P P (5, :) ∗ QQ ∗ exot−1,

b e

wt− bwet−1 = RR (6, :) ∗ statet−1+ SS (6, :) ∗ exot− RR (6, :) ∗ statet−2− SS (2, :) ∗ exot−1= RR (6, :) ∗ P P ∗ statet−2+ RR (6, :) ∗ QQ ∗ exot−1+

SS (6, :) ∗ exot− RR (6, :) ∗ statet−2− SS (2, :) ∗ exot−1=

(RR (6, :) ∗ P P − RR (6, :)) ∗ statet−2+ SS (6, :) ∗ exot+

b e

y_t− by_et−1 = P P (6, :) ∗ statet−1+ QQ (6, :) ∗ exot− P P (6, :) ∗ statet−2− QQ (6, :) ∗ exot−1 =

(P P (6, :) ∗ P P − P P (6, :)) ∗ statet−2+ QQ (6, :) ∗ exot+

(P P (6, :) ∗ QQ − QQ (6, :)) ∗ exot−1

b

lt = RR (5, :) ∗ statet−1+ SS (5, :) ∗ exot =

RR (5, :) ∗ P P ∗ statet−2+ SS (5, :) ∗ exot+ RR (5, :) ∗ QQ ∗ exot−1

b e

µ_t= (1 0 0 0 0)exot.

Also, remember that bΠt = log Πt− log Πss, bRt= log Rt− log Rss, bwet− bwet−1 = 4 logwet = 4 log wt− 4 log zt, byet− beyt−1 = 4 logeyt = 4 log yt− 4 log zt., blt = log lt − log lss = log lt, b e µ_t= log µt µt−1 − Λµ, 4 log µ −1 t = − log µt µt−1, ⇒ 4 log µ −1 t = −bµet− Λµ. Since 4 log zt= αΛµ+ΛA 1−α + αzµ,t+zA,t 1−α , we have: obst= H0+ H1∗ St. where H0 =            log Πss log Rss αΛµ+ΛA 1−α αΛµ+ΛA 1−α 0 −Λµ            and H1 =            P P (1, :) P P (1, :) QQ (1, :) P P (1, :) QQ (1, :) P P (5, :) P P (5, :) QQ (5, :) P P (5, :) QQ (5, :) RR (6, :) (RR (6, :) − I) SS (6, :) SS (6, :) (RR (6, :) − I) P P (6, :) (P P (6, :) − I) QQ (6, :) QQ (6, :) (P P (6, :) − I) RR (5, :) RR (5, :) SS (5, :) RR (5, :) SS (5, :) 0 (−1 0 0 0 0 0) 0            .