The Competition E¤ect in Business Cycles Appendix A: Model Derivations Vivien Lewis Ghent University and Goethe University Frankfurt, IMFS
Arnoud Stevens Ghent University
November 26, 2012
Contents 1 Introduction
A2
2 Non-linear Model 2.1 Final Goods Producer . . . . . . . . 2.2 Labor Bundler . . . . . . . . . . . . 2.3 Intermediate Goods Producers . . . 2.3.1 Cost Minimization . . . . . . 2.3.2 Price Setting . . . . . . . . . 2.4 Firm Entry . . . . . . . . . . . . . . 2.5 Households . . . . . . . . . . . . . . 2.5.1 Risk-free Bonds . . . . . . . 2.5.2 Equity . . . . . . . . . . . . 2.5.3 Investment in Startup . . . . 2.5.4 Firm Accumulation . . . . . 2.5.5 Intensive Margin Investment 2.5.6 Wage Setting . . . . . . . . . 2.6 Budgetary Government . . . . . . . 2.7 Aggregation and Market Clearing . 2.7.1 Labor Market Clearing . . . 2.7.2 Capital Market Clearing . . 2.7.3 Aggregate Production . . . . 2.7.4 National Income Account . . 2.7.5 Total Investments . . . . . . 2.8 Monetary Policy . . . . . . . . . . . 2.9 Identities . . . . . . . . . . . . . . . 2.10 Shock Processes . . . . . . . . . . .
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A2 A2 A4 A5 A6 A6 A10 A11 A13 A13 A13 A14 A14 A15 A17 A17 A17 A17 A18 A18 A19 A19 A19 A20
3 Model Summary and Linearization 3.1 Model Summary . . . . . . . . . . 3.2 Linearized Model Summary . . . . 3.2.1 Steady State Shares . . . . 3.2.2 Shock Normalization . . . .
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A20 A20 A24 A28 A32
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A1
4 Details Steady State A32 4.1 Recursive steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A34 5 Details Bayesian estimation procedure DSGE models
1
A37
Introduction
The model combines the entry mechanism and the translog preference structure proposed by Bilbiie et al. (2012) with a set of real and nominal frictions as in Christiano et al. (2005) and Smets and Wouters (2007).
2
Non-linear Model
2.1
Final Goods Producer
There is a mass Nt of …rms, each producing one di¤erentiated intermediate good, indexed by f 2 (0; Nt ). the
…rms’intermediate goods ytf are bundled into a …nal good YtC by a …nal goods …rm. Building on Feenstra (2003), which assume that preferences take the translog form, the elasticity of demand for an individual good is allowed to vary with the number of competing goods. We start by postulating the optimal expenditure function Pt YtC , where Pt is the aggregate price index. As in Feenstra (2003), page 82, equation (5), the log price index function is given by ln Pt =
0
+
1 2
e N
P t
Nt Nt 1 X ln pft + + e Nt N t N f =1
where f; j = 1; : : : ; Nt and bf j =
P t
2
Nt X Nt X
bf j ln pft ln pjt ,
(1)
f =1 j=1
Nt 1 Nt ; 1 Nt ;
f =j . f 6= j
e > Nt is the (constant) number of all conceivable goods. Nt is the (time-varying) number of available goods and N The parameter
scales the demand elasticity. Variable
P t
represents a shock to this scaling parameter and
captures— as we discuss below— exogenous shifts in the desired price markup. We have normalized original Feenstra (2003)-equation. Note that we have Nt squared prices ln pft
2
and Nt (Nt
0
= 0 in the
1) cross products
of prices ln pft ln pjt . Shephard’s lemma states that the demand for a particular good f , for a given price pft , equals the derivative of the expenditure function with respect to the price of the relevant good, ytf = YtC
@Pt @pft
.
Hence the expenditure share on an individual good f is sft
pft ytf @Pt pft = . Pt YtC @pft Pt
A2
(2)
The right hand side of (2) is the elasticity of the price index to the price of product f . Thus, the spending share is approximately computed as the derivative of the log price index function with respect to the log price of good f , sft
P t
@ ln Pt
1 = = f Nt @ ln pt
(Nt Nt
1)
ln pft
Nt X
P t
+
Nt
ln pjt .
(3)
j=1;j6=f
NB: we have two of each cross-product, since ln pft ln pjt = ln pjt ln pft , hence the 2 in the denominator disappears. The expenditure share on product f can be written as @ ln Pt @
where ln pt =
ln pft
PNt
ln pjt j=1 Nt
=
1 Nt
=
1 Nt
=
1 + Nt
P t
ln pft +
P t
Nt
P t
Nt X
P t
ln pft +
P t
ln pft +
Nt
Nt
ln pjt ,
j=1;j6=f
ln pjt ,
j=1
ln pft
ln pt
Nt X
P t
,
(4) P t
is the mean log price. Therefore,
measures the (exogenous) elasticity of the
expenditure share with respect to the price of product f ; i.e., if pft increases by 1%, the expenditure share decreases by
P t
percentage points.
Denote the price elasticity of demand by @ytf pft
"ft or approximately "ft =
@ ln ytf . @ ln pft
@pft ytf
,
Taking logs of the expenditure share (2), we have ln sft = ln pft + ln ytf
ln Pt YtC .
Di¤erentiating with respect to ln pft , we …nd that the derivative of the log expenditure share with respect to the log product price equals 1
"ft ,
@ ln sft @
ln pft
=1+
@ ln ytf @
ln pft
=1
"ft .
Therefore, using the approximation @ ln sft = @sft =sft and di¤erentiating (3) with respect to ln pft , we can derive the price elasticity of demand as "ft For large Nt , the term
Nt 1 Nt
=1
@
@sft
1
ln pft
sft
P t
=1+
(Nt Nt sft
1)
.
(5)
approaches 1. Then, the demand elasticity (5) simpli…es to "ft = 1 + A3
P t f st
.
(6)
Because the share sft is positive, we need to impose
P t
> 0 to ensure that the price elasticity exceeds unity.
Di¤erentiating the demand elasticity (6) with respect to the log price of good f , we get @"ft @ ln pft
=
@sft
P t
= h i2 f f @ ln p t st
P t
st (f )
!2
> 0.
Thus, the demand elasticity is increasing in the price.
Symmetric Equilibrium Conditions Under symmetry of goods prices pft = pjt = pt , the price index (1) becomes ln Pt =
e N
1 2
P t
which simpli…es to
Nt + ln pt + e Nt N
1 ln Pt = 2
e N
P t
P t
Nt 1 1 Nt (ln pt )2 + Nt (Nt Nt Nt
2
Nt + ln pt e Nt N
,
Pt = exp
1 2
Thus, the product price relative to the price index is pt = exp t (Nt ) = Pt
1 2
e N
P t
Nt e Nt N
e N
P t
!
Nt e Nt N
1) (ln pt )2 ; !
pt .
.
(7)
Under price symmetry, we have ln pt = ln pft , which simpli…es the expression for the expenditure share (4), @ ln Pt 1 = @ ln pt Nt
st
(8)
The demand for a single variety is then found by rearranging the de…nition of the expenditure share (2) and using (8) to substitute out st , yt = Furthermore, using st =
1 Nt
YtC . t Nt
in (6), the demand elasticity becomes P t
"t = 1 +
2.2
(9)
Nt .
(10)
Labor Bundler
~ t and sell it to the intermediate goods Labor bundlers buy the di¤erentiated labor types Lt , aggregate them to L producers under perfectly competitive conditions. A bundler maximizes his pro…ts, ~t Wt L
Z1
Wt Lt d ,
0
A4
(11)
subject to the following constant elasticity of substitution (CES) aggregation function: 0
~t = @ L
Z1
Lt
11+
1 1+ W t
W t
d A
0
,
(12)
where Wt is the aggregate nominal wage, Wt is the wage of labor type , and
W t
is a stochastic parameter which
determines the time-varying markup in the labor market. The …rst order condition yields the demand curve for labor type , Wt Wt
Lt =
1+ W t W t
~ t. L
(13)
The aggregate wage index is given by 0
Wt = @
Z1
(Wt )
1 W t
0
The following derivative will be useful later: #Lt #Wt
1+
=
W t
1 Wt
W t
Lt . Wt
W t
1+
=
W t
1
W t
d A
.
1+ W t W t
Wt Wt
(14)
1
~ t, L (15)
Symmetric Equilibrium Conditions In a symmetric equilibrium, the wage index (14) implies Wt = Wt .
(16)
Consequently, the demand for a single labor type (13) simpli…es to ~ t. Lt = L
2.3
(17)
Intermediate Goods Producers
s . The production function is CobbEach …rm produces a single variety f using labor ~lC;t and capital services kC;t
Douglas with
denoting the capital share, ytf =
Total factor productivity
Z t
Z t
s;f kC;t
~lf C;t
1
.
(18)
is assumed to follow an exogenous process. Real pro…ts in period t are given by dft =
pft f y Pt t
f wt ~lC;t
s;f rtk kC;t
pacft ,
(19)
where wt denotes the real wage rate, rtk represents the rental rate of capital services, and the variable pact measures price adjustment costs at the …rm level. A5
2.3.1
Cost Minimization
f s;f The …rm chooses labor ~lC;t and capital services kC;t as to minimize costs, subject to the production function (18).
The Lagrangian of the cost minimization problem reads as pft f y Pt t
f s;f LfCM;t = wt ~lC;t + rtk kC;t + pacft
+ with
CM;t
ytf
CM;t
s;f kC;t
Z t
(20)
~lf C;t
1
,
denoting the Lagrange multiplier on the output constraint. The …rst order conditions are @LfCM;t ~f @L
= wt
CM;t (1
@LfCM;t ~f @L
= rtk
CM;t (
Z t
)
s;f kC;t
~lf C;t
= 0,
C;t
)
Z t
1
s;f kC;t
~lf C;t
1
= 0.
C;t
Rearrange to get the input prices, wt =
CM;t (1
rtk =
CM;t (
)
Z t
s;f kC;t
Z t
)
~lf C;t 1
s;f kC;t
1
~lf C;t
,
(21)
.
(22)
Combining (21) and (22) shows that the capital-labor ratio is identical across intermediate goods producers and equal to the aggregate capital-labor ratio, f wt ~lC;t
=
s;f rtk kC;t
~ C;t wt L (1 ) = k s , ( ) rt KC;t
(23)
~ C;t = PNt ~lf and Kts = PNt k s;f . The Lagrange multiplier on the output constraint where L f =0 C;t f =0 C;t
CM;t
is the real
marginal cost at the optimum mct . Combining (21) and (23), we can, therefore, derive mct =
1
rtk
wt
Z t
1
.
1
(24)
Note that the real marginal cost is symmetric across …rms. 2.3.2
Price Setting
Firms set prices pt (f ) to maximize: LfP S;t = dft + vtf .
(25)
where dft and vtf respectively denote …rm pro…ts and value in period t. The …rst order condition for price setting reads as:
@LfP S;t @pft
=
@dft @pft A6
+
@vtf @pft
= 0.
(26)
Pro…ts Given real marginal costs (24), real pro…ts (19) in period t can be rewritten to obtain pft f y Pt t
dft =
mct ytf
pacft .
(27)
The derivative of …rm pro…ts with respect to the product price is ytf (1 Pt
"t ) + mct
ytf
@pacft
pt
@pft
" f t
.
Using the demand elasticity (10), this becomes @dft @pft
ytf Pt
=
P t
Nt + mct
ytf
P t
1+
pft
@pact (f ) . @pt (f )
Nt
(28)
Price Adjustment Costs Firm-level price adjustment costs are proportional to real …rm revenues, pacft where
p
p
pft
2
[
p (1 +
!2
1 ) + (1
p;t
p )]
1
pft f y , Pt t
(29)
0. Price adjustment costs are higher, the more the change in the …rm’s price pft =pft
weighted average by
=
pft
p
p (1
+
p;t 1 ) + (1
p ),
and the higher is the parameter
= 0. We introduce indexation as in Ireland (2007). When
p
p.
1
diverges from the
Perfectly ‡exible prices are given
is equal to zero, there is no indexation to past
in‡ation and we have the case of Rotemberg (1982) with a purely forward-looking New Keynesian Phillips Curve. Instead, when pft =pft
1
p
> 0, the price adjustment cost is a function of the di¤erence between the …rm’s price change
and the weighted average of past in‡ation,
p;t 1 ,
(with weight
p)
and steady state in‡ation, which is
equal to 0. For simplicity, we assume that entrants, too, pay this price adjustment cost. Bilbiie et al. (2007) show that the impulse responses to shocks change negligibly under the alternative assumption that entrants can change their price costlessly. Iterating pacft one period, we see that pacft+1 also depends on pft , pacft+1
=
pft+1
p
[
pft
2
p (1
+
p;t )
+ (1
!2
p )]
pft+1 f y . Pt+1 t+1
We compute the derivative of pacft with respect to the product price pft , @pacft @pft
=
pft
pft
p f pt 1
+
p
2
pft pft pft
[
p (1 +
p;t
1 ) + (1
1
[
p (1
+
p;t 1 )
1
A7
+ (1
!
p )]
!2
p )]
ytf Pt
1 Pt
ytf
+
pft
@ytf @pft
!
.
@ytf , @pft
Using the price elasticity of demand from (10) to replace @pacft
pft
=
@pft
pft
p f pt 1
pft
[
2
+
p;t 1 )
!
+ (1
p )]
1
pft pft 1
p
p (1
this becomes
[
p (1 +
p;t
!2
1 ) + (1
p )]
ytf Pt
ytf Pt
P t
Nt .
(30)
We also di¤erentiate pacft+1 with respect to pft , @pacft+1 @pft Substituting
@pacft @pft
!2
pft+1
=
pft
pft+1
p
[
pft
p (1
+
p;t )
+ (1
!
p )]
f yt+1 . Pt+1
(31)
in (28) using (30), we can derive the change in …rm pro…ts with respect to a change in the
product price, @dft @pft
2
41
=
pft pft 1
p
2
pft p f pt 1
pft pft
[
[
p (1
p (1
+
+
p;t 1 )
p;t 1 )
!2 3 f P 5 yt p )] t Nt Pt ! ytf yf + mct tf 1 + p )] Pt pt
+ (1
+ (1
1
P t
Nt .
(32)
Firm Value Firm value at time t, in real terms, equals the expected present discount value of the pro…t stream from t + 1 into the in…nite future,
1 X
vtf = Et
f t;s ds ,
(33)
s=t+1
where
t;s
= [ (1
s t
N )]
UC;s =UC;t is the marginal utility value in period s to the representative household of
an additional unit of pro…ts in period t. The only term in the in…nite sum in (33) that depends on pft is dft+1 . Real pro…ts in t + 1 depend on the …rm price in t through pacft+1 . We compute the derivative of …rm value with respect to the product price, @vtf @pft Combining with (31) to replace @vtf @pft
=
p Et
@Et =
@pact+1 (f ) @pt (f ) ,
8 < :
t;t+1
n
f t;t+1 dt+1
@pft
o
@Et =
n
f t;t+1 pact+1
@pft
o
.
(34)
we have pft+1 pft
!2
pft+1 pft
A8
[
p (1
+
p;t )
+ (1
!
p )]
9 f = yt+1 . Pt+1 ;
(35)
@dft @pft
Price Setting Condition Using (32) and (35) to replace
@vtf , @pft
and
respectively, the …rst order condition
(26) becomes P t
0 = mct 1 +
p f pt 1
p Et
pft 2
8 < :
pft [
p (1
0 = mct
pft
1+
P t
p
2 pft+1
t;t+1
pft
t (f )
=
pft
1
Et
8 < :
pft
+ (1
[
p (1
+
[
p )]
p
pft+1 pft
ytf Pt
!2 3
+ (1
p;t )
p (1
p )]
!
+ (1
+
p )]
p;t 1 )
f
5 yt Pt
9 f = yt+1 . Pt+1 ; !2 3
+ (1
P t
"
pft+1 pft
Nt 1
P t
1+ p
[
p (1
+
p;t )
2
[
p (1
f t,
(36)
!9 = . p )] ;
+ (1
Nt 2
pft pft 1
p
p )]
Rearrange (36) to get the pricing equation, pft = mct Pt
5
p )]
!
+ (1
!2
+
[
pft 1
p;t 1 )
p;t 1 )
p (1
pft
2
Pt Pt+1
+
[
pft
1
f yt+1 t;t+1 f yt
p (1
pft+1
Nt 41
where pft
!2
2
P t
Nt
pft
p;t 1 )
pft pft 1
Divide both sides by ytf =Pt to get Pt
+
!
1
Nt 41
P t
+
Nt
pft
pft
ytf
+
p;t 1 )
+ (1
p )]
#
. +
p
f t
Using the (symmetric) price elasticity of demand (10), this simpli…es to pft = mct Pt ("t
"
1) 1
"t p
2
pft pft 1
2
[
p (1
+
p;t 1 )
More simply, we can write the relative product price as a markup f t
=
f t
+ (1
p )]
#
. +
p
f t
over real marginal cost mct ,
f t mct .
(37)
The markup is given by f t
"t
= ("t
1) 1
p
2
f p;t
A9
2 p p;t 1
. +
p
f t
(38)
where
f p;t
= pft =pft
1
1 is …rm-level in‡ation, f t
=
1 + fp;t (
= pt =pt
p;t f p;t
1 is aggregate in‡ation and
1
p p;t 1
f yt+1 Pt 1+ t;t+1 f yt Pt+1
Et
Note that if prices are perfectly ‡exible, i.e.,
f p;t
f p;t+1
)
p p;t
.
! 0, the markup reads as
p
"t
d;f t
2
("t
1)
P t
1+
=
P t
Nt
Nt
,
(39)
which we refer to as the desired price markup. Using (39), the actual markup (38) simpli…es to f t
d;f t
= 1
2
f p;t
p
2
p p;t 1
. +
p
(40)
f t
Symmetric Equilibrium Conditions In a symmetric equilibrium, real pro…ts, the relative product price and markup read, respectively, as dt = t t
t yt
= =
mct yt
pact ,
(41)
t mct ,
h
(42) d t p
1
2
(
p;t
p p;t
with, d t
2.4
=
P t
1+ P t
Nt
Nt
i 2 + ) 1
, p
(43)
t
.
(44)
Firm Entry
s , Setting up a …rm requires a composite of labor ~lE;t and capital services kE;t E t
The variable
E t
=
Z t
s kE;t
~lE;t
1
.
captures exogenous entry costs per …rm, measured in terms of a composite of labour and capital
services. Thus the technology for setting up all entrants is NE;t
E t
=
Z t
~ E;t L
s KE;t
1
,
(45)
~ E;t = NE;t ~lE;t and K s = NE;t k s . The entrants’cost minimization problem is to choose L ~ E;t and K s where L E;t E;t E;t ~ E;t + rtk K s , subject to the production function (45). The Lagrangian of the cost minimization as to minimize wt L E;t problem reads as ~ E;t + rk K s + LE;t = wt L t E;t
E;t
NE;t A10
E t
Z t
s KE;t
~ E;t L
1
,
(46)
with
E;t
denoting the Lagrange multiplier on the technology constraint. We see from this that the marginal cost
of setting up a …rm is
LE;t NE;t
=
E E;t t
E E;t t
in real terms or Pt
in nominal terms. The …rst order conditions for
this problem are @LE;t ~ E;t @L
= wt
E;t (1
@LE;t s @KE;t
= rtk
E;t (
Z t
) Z t
)
~ E;t L
s KE;t 1
s KE;t
~ E;t L
1
= 0, = 0,
which we can rearrange to get wt =
E;t (1
)
rtk =
E;t ( )
NE;t E t , ~ LE;t
(47)
NE;t E t . s KE;t
(48)
Combining these two equations shows that the entrants’wage bill over their rental bill is constant, ~ E;t wt L 1 = k s rt KE;t
.
(49)
Considering perfect cross-sectoral factor mobility, factor prices are equalized across sectors. Therefore, we have from (23) and (49) that the capital-labor ratio is the same across sectors, S S KC;t KE;t = . ~ C;t ~ E;t L L
(50)
Finally, by combining (49) and (47), we can derive that the Lagrange multiplier
E;t
equals the real marginal cost
of intermediate goods producers, mct , i.e., E;t
2.5
=
1
rtk
1
wt
Z t
= mct .
1
(51)
Households
The economy is made up by a continuum of di¤erentiated households, indexed by lifetime utility, E0
1 X
2 (0; 1), which seek to maximize
t TI t Ut .
(52)
t=0
TI t
is a disturbance that can be interpreted as a ‘time-impatience shock ’ to the subjective discount factor
2 (0; 1). Period utility is given by Ut =
1 1
c
(Ct
hCt
1 1)
A11
c
1+
l
(Lt )1+
l
.
(53)
First, households derive utility from consumption Ct , where the curvature parameter
c
> 0 and the external
habit coe¢ cient h 2 (0; 1) govern the inter-temporal elasticity of substitution. Second, utility depends negatively on hours worked Lt , with
l
> 0 denoting the reciprocal of the Frisch elasticity of labor.
> 0 is the weight on
labor disutility in household welfare. The marginal utilities of consumption and labor are UC;t = [Ct UL;t =
hCt
1]
c
,
(54)
(Lt ) l .
(55)
Maximization of (52) is subjected to the demand function (13) for household ’th di¤erentiated labor type and the households’period-by-period budget constraint, which in real terms reads as
>
Rt 1 k B + (1 N ) (dt + vt ) Et 1 + FN;t 1 ( ) NE;t 1 + wt Lt + rt ut Kt Pt t 1 Bt + mct E t NE;t + vt Et + Ct + It + Tt + W ACt + a (ut ) Kt . Pt
(56)
There are three assets: risk-free nominal bonds Bt , capital Kt and equity Et Capital services, Kts; , are related to the physical stock of capital through Kts; = ut Kt ,
(57)
where ut is the capital utilization rate set by the household. The income side includes labor income Pt wt Lt , rental income on capital holdings Pt rtk ut Kt , the return on bonds Rt on equity. Previously …nanced startups NE;t
1,
1 Bt 1 ,
and the returns on startup investment and
survive with probability FN;t
1(
) described below. Successful
startups and shares purchased in the previous period pay a dividend Pt dt and are worth (1 of period t. We assume that a proportion
N
N ) Pt vt
of incumbents and new entrants are hit by an exit shock, as in Bilbiie
et al. (2012). The timing assumption is as follows. At the beginning of period t, a fraction %t are successfully established. Of those, a constant fraction
N
Pt dt . Likewise incumbents exit with a constant probability
N.
(1
N ) Pt vt
N ) Pt dt .
at the end
The value of the shares and entrants is (1
1
of entrants NE;t
1
exit, the remaining ones produce and earn pro…ts The dividend on equity holdings Et
1
is therefore
at the end of period t. The expenditure side
includes bond purchases.Bt , consumption Pt Ct , investment Pt It , taxes Pt Tt , wage adjustment costs Pt W ACt and the costs of adjusting the utilization rate. The latter cost is denoted by the increasing convex function a (ut ). As in Christiano et al. (2005), we impose that in steady state,u = 1, a (ut ) = 0 and
a
=
households purchase equity Et (h) in incumbent …rms at nominal price Pt vt and spend Pt mct
a00 (u) a0 (u) . E t
In addition,
NE;t on startup
costs. Since we are concerned with a symmetric optimization problem, we drop the h-index for the remainder of this section so as to simplify notation. A12
2.5.1
Risk-free Bonds
The household chooses bonds Bt to maximize utility (52) subject to the budget constraint (56). Let
BC;t
denote
the Lagrange multiplier on the household’s budget constraint. Then the …rst order condition w.r.t. consumption is @L = @Ct
TI t UC;t
+
@
BC;t [: : :
@Ct
Ct ]
,
such that in a symmetric equilibrium the Lagrange multiplier is BC;t
TI t UC;t .
=
(58)
The Euler equation for bond holdings reads TI t UC;t
2.5.2
= Et
(
Rt 1+
TI t+1 UC;t+1 C p;t+1
)
.
(59)
Equity
Maximizing utility w.r.t. equity Et , subject to the budget constraint, we get BC;t vt
+ Et
Substituting out the Lagrange multiplier vt = 2.5.3
(1
BC;t+1 (1
N ) (dt+1
+ vt+1 ) = 0.
BC;t ,
this becomes ( ) TI U t+1 C;t+1 (dt+1 + vt+1 ) . N ) Et TIU C;t t
(60)
Investment in Startup
Households invest in startups (new …rms). Following Beaudry et al. (2011), we assume that not all …rm startups are successful. Only a fraction FN;t becomes operational one period later. The success probability of startups is speci…ed as FN; t (NE;t ; NE;t
1)
=1
SN
NE;t NE;t 1
,
where SN ( ) is the hazard rate, i.e., the failure rate of entrants. The failure rate is an increasing function of the change in entry. Mata and Portugal (1994) document that failures of new …rms is positively related to entry rates. The failure rate could also be viewed as a ‡ow adjustment cost to extensive margin investment similar to the physical capital (or intensive margin) investment adjustment costs in Christiano et al. (2005). SN ( ) is an 0 (1) = 0 and S 00 (1) > 0. adjustment cost function, which has the following steady state properties: SN (1) = SN N
Maximizing utility w.r.t. entry NE;t , subject to the budget constraint at the symmetric equilibrium, we get E BC;t mct t
+ Et
BC;t+1 (1
+
N ) (dt+1
BC;t+2 (1
+ vt+1 ) [FN 1;t ( ) NE;t + FN;t ( )] N ) (dt+2 + vt+2 ) FN 2;t+1 ( ) NE;t+1
A13
= 0.
Rearranging and substituting out the Lagrange multiplier BC;t , we have 8 9 TI U t+1 C;t+1 < = (1 N ) (dt+1 + vt+1 ) [FN 1;t ( ) NE;t + FN;t ( )] TIU C;t t mct E = E . TI U TI t t C;t+1 t+2 UC;t+2 : + t+1 ; (1 N ) (dt+2 + vt+2 ) FN;2t+1 ( ) NE;t+1 TIU TI U C;t
t
C;t+1
t+1
Using the …rm value equation (60), we …nally get mct E t 2.5.4
= vt [FN 1;t ( ) NE;t + FN;t ( )] + Et
(
TI U t+1 C;t+1 vt+1 FN 2;t+1 ( TIU C;t t
) NE;t+1
)
.
(61)
Firm Accumulation
The law of motion for the number of …rms is given by Nt+1 = (1 2.5.5
N ) [Nt
+ FN t (NE;t ; NE;t
1 ) NE;t ] .
(62)
Intensive Margin Investment
The capital stock Kt evolves according to the law of motion, Kt+1 = (1
K ) Kt
I t FK
+
(It ; It
1) .
(63)
Physical capital investment is determined by the function FK ( ), de…ned as FK (It ; It
1)
= 1
It
SK
It
It . 1 I t
SK ( ) is an adjustment cost function with the same properties as SN ( ).
is a shock to the domestic investment-
speci…c technology process. The household owns the capital stock Kt and …nances (intensive margin) physical capital investment It . The household budget constraint can be written more compactly as XtK > It
rtk ut Kt + a (ut ) Kt .
The residual term, XtK
=
Rt 1 Bt 1 + (1 N ) (dt + vt ) [Et 1 + Ft 1 ( ) NE;t Pt Bt mct E vt Et Ct Tt W ACt ; t NE;t Pt
1]
+ wt Lt
can be treated as exogenous for the investment decision. We have the following optimization problem: max E0
It ;Kt+1
h
1 X t=0
XtK
+
BC;t
+
BC;t qt
(1
t TI t
fU (Ct ; Lt )
It + rtk ut Kt K ) Kt
+
A14
a (ut ) Kt
I t FK
(It ; It
i
1)
Kt+1
;
where
BC;t
and
BC;t qt
are the Lagrange multipliers on the household budget constraint (56) and the capital
accumulation (63), respectively. Variable qt denotes the household’s shadow price of physical capital in real terms. The …rst order condition for investment It reads as BC;t
=
I BC;t qt t FK1;t
Substituting out the Lagrange multipliers
I BC;t+1 qt+1 t+1 FK2;t+1
+ Et
.
(64)
in the investment equation (64) and rearranging, we get ( ) TI U C;t+1 t+1 1 = qt It FK1;t + Et qt+1 It+1 FK2;t+1 . TIU C;t t BC;t
(65)
The …rst order condition for capital holdings Kt+1 is Et
n
BC;t+1
h
i a (ut+1 ) + (1
k rt+1 ut+1
K)
BC;t+1 qt+1
o
Substituting out the Lagrange multiplier BC;t in (??) using (58), we have ( h TI U t+1 C;t+1 k qt = E t rt+1 ut+1 a (ut+1 ) + (1 TIU C;t t The …rst order condition for ut is
rtk Kt
BC;t
BC;t qt
K ) qt+1
i
)
= 0.
.
(67)
a0 (ut ) Kt = 0, which simpli…es to rtk = a0 (ut ) .
2.5.6
(66)
(68)
Wage Setting
We introduce Rotemberg (1982)-type adjustment costs into the wage setting equation. Note that we need to re-introduce the household index
here. Quadratic wage adjustment costs, in real terms, are
where
w
2
Wt Wt 1
w
W ACt =
2
[
(1 +
w
p;t 1 )
+ (1
w )]
Wt , Pt
(69)
> 0 measures the degree of wage rigidity. Iterating W ACt (h) one period, we see that W ACt+1 also
depends on Wt ,
2
Wt+1 Wt
w
W ACt+1 =
2
[
w
(1 +
p;t )
[
(1 +
w )]
Wt+1 . Pt+1
+ (1
w )]
+ (1
Di¤erentiating W ACt and W ACt+1 w.r.t. Wt , we …nd @W ACt @Wt
=
w
+ @W ACt+1 = @Wt
w
Wt Wt 1 w
2
Wt Wt 1
Wt Wt 1
Wt+1 Wt
2
w
p;t 1 )
2
[
w (1 +
Wt+1 Wt A15
[
w
p;t
1 ) + (1
(1 +
p;t )
+ (1
w )]
1 Pt
(70)
1 , Pt
w )]
1 Pt+1
.
(71)
Using the household budget constraint (56), the Lagrangian for the wage setting problem is given by maxLW S;t = Et Wt
1 X
s
TI t+s Ut+s
+
Wt+s Lt+s (h) Pt+s
::: +
BC;t+s
s=0
W ACt+s (h)
.
The …rst order condition with respect to the wage rate reads as @LW S;t
@Lt 1 @Lt Wt + BC;t Lt + @Wt Pt @Wt Pt @W ACt+1 @W ACt (h) Et BC;t+1 BC;t @Wt (h) @Wt
TI t UL;t
=
@Wt
= 0. Using (15), (70) and (71), to replace 0 =
Wt Wt 1
w
+ Et
(
Lt + Wt Wt [ Wt 1
W t
BC;t w
2
@W ACt @Wt
,
[
w;t (h)
w
W t
Lt wt
1 Pt
(1 +
+
w
p;t 1 )
Wt+1 Wt
Wt Wt 1
=
+ (1
w )]
TIU t C;t ,
W t
UL;t UC;t
w;t
+1
w
(1 +
p;t )
+ (1
BC;t
w )]
with
1 Pt+1
TIU t C;t ,
)
.
and multiplying by
(72) w p;t 1
w;t
TI U t+1 C;t+1 TIU t C;t
Note that if wages are perfectly ‡exible, i.e.,
1 Pt
Lt
2
Et
[
w )]
1 Pt
we can write this as
w p;t 1
w;t
(
+ (1
1, using (58) to replace
w
2
Lt
W t
2
W t w
w
we have
1
p;t 1 )
Wt+1 Wt
1+
=
BC;t
(1 +
the nominal wage Pt wt = Wt and dividing by 1
@W ACt+1 , @Wt
2
Wt Wt 1
BC;t+1 w
De…ning the wage in‡ation rate
and
W t
1+
TI t UL;t
BC;t
@Lt @Wt
wt
wt w;t+1
+1
w;t+1
w p;t
wt+1
)
! 0, the wage setting condition reads as # " UL;t W wt = 1 + t . UC;t w
A16
.
Symmetric Equilibrium Conditions In a symmetric equilibrium, the wage setting condition reads as 1 W t
UL;t Lt UC;t w ( w;t + 1) ( w;t
(73)
W t
w
2 +
2.6
W t
1+
Lt wt =
w
(
2 w p;t 1 ) wt
w;t
Et
(
w p;t 1 ) wt
TI U t+1 C;t+1 TIU C;t t
(
w;t+1
+ 1) (
w;t+1
w p;t ) wt+1
)
.
Budgetary Government
The government collects lump-sum taxes Pt Tt from household as well as issues one-period bonds Bt to …nance its exogenously given consumption,
G. t
The government budget constraint is given by Pt Tt + Bt = Pt
2.7 2.7.1
G t
+ Rt
1 Bt 1 .
(74)
Aggregation and Market Clearing Labor Market Clearing
Nt ~ t equals labor used in the production of intermediate goods L ~ C;t = P ~lf , plus labor used Total labor demand L C;t f =1
~ E;t , i.e., in the production of new …rms L
~t = L ~ C;t + L ~ E;t . L
(75)
~ t equals the supply of the labor bundler, which in a symmetric equilibrium matches Aggregated labor demand L Z1 aggregate labor supply Lt = Lt d of all households, i.e., 0
~ t. Lt = L 2.7.2
(76)
Capital Market Clearing
s = Similarly, total capital services Kts equals capital services used in the production of intermediate goods KC;t Nt P s;f s , i.e., kC;t , plus capital services used in the production of new …rms KE;t f =1
s s Kts = KC;t + KE;t .
(77)
Kts = ut Kt .
(78)
Aggregate capital services are given by
A17
2.7.3
Aggregate Production
The aggregate production of goods Ytagg reads as follows: Ytagg = where Ytagg =
Nt P
f =1
Z t
~ C;t L
s KC;t
1
,
Nt Nt ~ C;t = P ~lf and K s = P k s;f . In a symmetric equilibrium this simpli…es to ytf , L C;t C;t C;t f =1
f =1
Ytagg = Nt yt =
Z t
s KC;t
~ C;t L
1
,
(79)
~ C;t = Nt ~lC;t and K s = Nt k s . where L C;t C;t 2.7.4
National Income Account
Integrating the budget constraints of all domestic households
2 (0; 1), yields:
R 1 Rt 1 k B + (1 N ) (dt + vt ) Et 1 + FN;t 1 ( ) NE;t 1 + wt Lt + rt ut Kt d 0 Pt t 1 R 1 Bt E = 0 P + mct t NE;t + vt Et + Ct + It + Tt + W ACt + a (ut ) Kt d . t
By using Ct = Ct , It = It , Kt
1
= Kt
1,
(80)
Lt = Lt , ut = ut , Tt = Tt , Bt = Bt , Et = Et , wt = wt , NE;t = NE;t ,
W ACt = W ACt , the fact that total equity equals the number of …rms Et = Nt , and the …rm accumulation equation (62), we obtain dt Nt + wt Lt + rtk ut Kt Bt Rt 1 = Bt 1 + Tt + mct Pt Pt
(81) E t NE;t
+ Ct + It + W ACt + a (ut ) Kt .
The aggregated real pro…ts of all intermediate goods producers, in a symmetric equilibrium, are given by dt Nt =
t N t yt
~ C;t wt L
s rtk KC;t
P ACt ,
(82)
where P ACt = Nt pact are the economy-wide price adjustment costs. Combining the aggregated budget constraint of households (81) with the government budget constraint (74), the aggregate pro…ts of intermediate goods producers (82), the demand function for intermediate goods (9), the capital market clearing condition (78), and the labor market clearing conditions (75) and (76), yields s ~ E;t + rtk KE;t YtC + wt L
= Ct + It +
G t
+ mct
(83)
E t NE;t
+ W ACt + P ACt + a (ut ) Kt .
A18
~ E;t + rk K s = mct Finally, pro…ts in the …rm entry sector are zero, implying wt L t E;t YtC = Ct + It +
G t
EN E;t , t
and
+ W ACt + P ACt + a (ut ) Kt .
(84)
Using the income approach, GDP Yt is de…ned and measured as the sum of all factor payments, dt Nt + wt Lt + rtk ut Kt .
Yt
Use the government budget constraint (74).to substitute out
(85)
Rt 1 Pt Bt 1 + Tt
Bt Pt
in equation (81) and subsequently
combine the resulting equation with the goods market clearing condition (84), to obtain Yt 2.7.5
dt Nt + wt Lt + rtk ut Kt = YtC
P ACt + mct
E t NE;t .
(86)
Total Investments
As in Bilbiie et al. (2012), we introduce the additional variable, total investment, as the sum of intensive and extensive margin investment, i.e., T It = It + mct
2.8
E t NE;t .
(87)
Monetary Policy
The model is closed with the monetary authority following a simple empirical Taylor-type rule to set the nominal interest rate Rt , targeting product price in‡ation and the level as well as changes in the output gap, and is subject to exogenous monetary policy shocks Rt = R 1
R
(Rt
R, t
1)
R
y (1
Yt Y
R)
1+ 1+
Yt =Y Yt 1 =Y
p;t p
dy
R t ,
(88)
where the outputgap is de…ned as the di¤erence between the actual and steady state level of output.
2.9
Identities
The relative price
t
= pt =Pt , product price in‡ation 1 +
p;t
=
pt pt 1 ,
and welfare-based in‡ation 1 +
C p;t
=
Pt Pt 1
can be linked through t
=
t 1
1+ 1+
Further, we can link welfare-based in‡ation, wage in‡ation 1+
C p;t
= (1 +
A19
p;t . C p;t
(89)
w;t ,
and the real wage wt through
w;t )
wt 1 . wt
(90)
2.10
Shock Processes
Finally, we need to de…ne the various shock processes. Except for the exogenous spending and markup shocks, all disturbances follow an AR(1) process in logarithmic terms. Following Smets and Wouters (2007), exogenous spending is also a¤ected by the productivity shock and disturbances in market power are assumed to follow an ARMA(1,1) process; i.e., Z t
=
Z
TI t
=
TI
ln
I t
=
I
ln
I t 1
+ "I;t ,
ln
G t
=
G ln
G t 1
+ "G;t +
ln
R t
=
R ln
R t 1
+ "R;t ,
ln
P t
=
P
ln
P t 1
+ "P;t
W t
=
W
E t
=
E
ln ln
ln ln
3
ln
Z t 1 TI t 1
ln
(91)
+ "T I;t ,
W t 1
ln ln
+ "Z;t ,
E t 1
(92) (93) GZ "Z;t ,
(94) (95)
P "P;t 1 ,
+ "W;t
(96)
W "W;t 1 ,
(97)
+ "E;t .
(98)
Model Summary and Linearization
3.1
Model Summary
~ E;t , L ~ C;t , mct , wt , rtk , dt , NE;t , The equilibrium of the model of time t is de…ned by 39 endogenous variables: Lt , L s , K s , K s, K , q , u , Nt ,vt , Yt , YtC , yt , Ct , It , T It , KE;t t t t t C;t R, t
P, t
W, t
and
E, t
d, t
t,
t,
Rt , UL;t , UC;t ,
C , p;t
p;t ,
w;t ,
Z, t
TI, t
I, t
G, t
that satisfy the optimality conditions of all agents, the market clearing conditions and the
shock processes; which are given by: Firms (1) Price index (7): t
1 2
= exp
(2) Firm output (9): yt =
e N
P t
YtC . t Nt
Nt e Nt N
!
(3) Real pro…ts (41): dt =
t yt
mct yt
A20
pact .
.
(4) Price setting (42): t
=
t mct .
(5) Price markup (43): t
(6) Natural price markup (44):
=h
d t p
1
2
(
d t
=
2 p p;t 1 )
p;t
P t
1+ P t
Nt
Nt
i
. +
p
t
.
(7) Real marginal costs (24): mct =
rtk
1
1
wt
Z t
.
1
(8) Capital-labor ratio in the goods producing sector (23): s KC;t wt ( ) = k . ~ C;t rt (1 ) L
(9) Aggregate (intensive margin) production function (79): N t yt =
Z t
1
~ C;t L
s KC;t
(10) Aggregate (extensive margin) production function (45): NE;t
E t
=
Z t
s KE;t
1
~ E;t L
.
(11) Capital-labor ratio in the …rm entry sector (49) s KE;t wt ( ) = k . ~ E;t rt (1 ) L
Households (12) Marginal utility of labor (55): (Lt ) l .
UL;t = (13) Marginal utility of consumption (54): UC;t = [Ct
hCt
1]
c
.
(14) Euler equation for bonds (59): TI t UC;t
= Et
(
Rt 1+
A21
TI t+1 UC;t+1 C p;t+1
)
.
(15) Firm value (60): vt =
(1
N ) Et
(
TI U t+1 C;t+1 TIU C;t t
)
(dt+1 + vt+1 ) .
(16) Firm entry condition (61): mct
E t
= vt [FN 1;t ( ) NE;t + FN;t ( )] + Et
(17) Shadow price of physical capital (67): ( h TI U t+1 C;t+1 k qt = E t rt+1 ut+1 TIU C;t t
(
TI U t+1 C;t+1 vt+1 FN 2;t+1 ( TIU C;t t
a (ut+1 ) + (1
K ) qt+1
) NE;t+1
i
)
)
.
(18) Physical capital investment condition (65): 1=
qt It FK1;t
+ Et
(
TI U t+1 C;t+1 qt+1 It+1 FK2;t+1 TIU C;t t
)
.
(19) Capital utilization rate condition (68): rtk = a0 (ut ) . (20) Capital services (78): Kts = ut Kt . (21) Wage setting (73): 1 W t
W t
UL;t Lt UC;t w ( w;t + 1) ( w;t 1+
Lt wt =
W t
w
2 +
w
(
w;t
Et
(
w p;t 1 ) wt
2 w p;t 1 ) wt TI U t+1 C;t+1 TIU C;t t
(
w;t+1
+ 1) (
Market Clearing (22) Labour market clearing (75) and (76): ~ C;t + L ~ E;t . Lt = L (23) Capital market clearing condition (77) s s Kts = KC;t + KE;t .
A22
w;t+1
w p;t ) wt+1
)
.
.
(24) & (25) National income accounts (84) and (86): YtC
G t
= Ct + It +
Yt = YtC
+ W ACt + P ACt + a (ut ) Kt ,
P ACt + mct
E t NE;t .
(26) Total investments (87): E t NE;t .
T It = It + mct Laws of Motion (27) Law of motion for the number of …rms (62): Nt+1 = (1
N ) [Nt
+ FN t (NE;t ; NE;t
1 ) NE;t ] .
(28) Law of motion for the physical capital stock (63): Kt+1 = (1
K ) Kt
I t FK
+
(It ; It
1) .
Identities (29) Link product price in‡ation and welfare-based in‡ation (89): t
=
t 1
1+ 1+
p;t . C p;t
(30) Link wage in‡ation and the real wage rate (90): C p;t
1+
= (1 +
w;t )
wt 1 . wt
Monetary Policy (31) Equation (88) Rt = R
1
R
(Rt
1)
R
y (1
Yt Ytn
R)
1+ 1+
p;t
(1
R)
p
Shock Processes (32) Process for the disturbance to total factor productivity (91): ln
Z t
=
Z
ln
Z t 1
+ "Z;t .
(33) Process for the disturbance to general preferences (92): ln
TI t
=
TI
ln
A23
TI t 1
+ "T I;t .
Yt =Ytn Yt 1 =Ytn 1
dy
R t .
(34) Process for the disturbance to investment-speci…c technology (93): ln
I t
=
I t 1
ln
I
+ "I;t .
(35) Process for the disturbance to government spending (94): G t
ln
=
G ln
G t 1
+ "G;t +
GZ "Z;t .
(36) Process for the disturbance to monetary policy (95): R t
ln
=
R t 1
R ln
+ "R;t .
(37) Process for the disturbance to price markup (96): P t
ln
=
P
ln
P t 1
+ "P;t
P "P;t 1 .
+ "W;t
W "W;t 1 .
(38) Process for the disturbance to wage markup (97): ln
W t
=
W
ln
W t 1
(39) Process for the disturbance to …rm entry costs (98): ln
3.2
E t
=
E
E t 1
ln
+ "E;t .
Linearized Model Summary
Linearizing the economy’s equilibrium conditions summarized in section 3.1, the model reads as: Firms (1) Linearizing the price index (7): ^t =
^t + N
N ~ N
1
^Pt ,
(99)
1 . 2 N
where (2) Linearizing …rm output (9): y^t = Y^tC
^t N
^t .
(100)
(3) Linearizing real pro…ts (41), noting that pac = 0 and using the price setting condition (42): d^t = ^t + y^t + ("
A24
1) ^ t .
(101)
(4) Linearizing the price setting condition (42): ^t = ^ t + mc c t.
(102)
(5) The linearized price markup (43) can be rewritten in terms of in‡ation: ^ p;t =
p p;t 1
+
(1
N ) Et ^ p;t+1
(1
N)
"
1
p p;t p
h
^t
i ^ dt .
(103)
(6) Linearizing the desired price markup (44): ^t N
^ dt =
where
^Pt ,
(104)
1 . 1+ N
=
(7) Linearizing marginal costs (24): mc c t = r^tk + (1
^Z t .
)w ^t
(105)
(8) Linearizing the goods producing sector’s capital-labor ratio (23): b ^s = w ~ C;t . r^tk + K ^t + L C;t
(106)
(9) Linearizing the aggregate (intensive margin) production function (79): b ~ C;t . )L
^t + y^t = ^Z + K ^ s + (1 N t C;t
(107)
(10) Linearizing the aggregate (extensive margin) production function (45): Z ^ ^s ^E t + NE;t = ^t + KE;t + (1
(11) Linearizing the …rm entry sector’s capital-labor ratio (49):
b ~ E;t . )L
b ^s = w ~ E;t . r^tk + K ^t + L E;t
(108)
(109)
Households (12) Linearizing marginal utility of labor (55): ^L;t = U
^
l Lt .
(110)
(13) Linearizing marginal utility of consumption (54): ^C;t = U
c
1
h A25
C^t
hC^t
1
.
(111)
(14) Linearizing the Euler equation for bonds (59): ^C;t = R ^t ^Tt I + U
TI ^ Et ^ C p;t+1 + Et ^t+1 + Et UC;t+1 .
(112)
(15) Linearizing …rm value (60), using the Euler equation for bonds (59) and the steady state relationship d v
=
1
(1 (1
N) N)
: ^t R
v^t =
Et ^ C p;t+1 + [1
^
(1
N )] Et dt+1
+ [ (1
^t+1 : N )] Et v E
(16) Linearizing the …rm entry condition (61), using the steady state relationship mc
(113)
= v and the steady state
properties of FN : FN 1 (NE ; NE ) = FN 2 (NE ; NE ) = 0 and FN (NE ; NE ) = 1, we get after rearranging terms: 1 v^t mc c t + ^E t (1 + ) 'N 00 where 'N = SN (1) .
^E;t = N
+
1 ^ NE;t 1+
1
+
^E;t+1 , Et N
1+
(114)
(17) Linearizing the shadow price of physical capital (67), using the Euler equation for bonds (59) and the steady rk + (1
state relationships q =
^t R
q^t =
K) q
, q = 1, u = 1 and a (u) = 0, we get:
Et ^ C p;t+1 + [1
k ^t+1 K )] Et r
(1
+ [ (1
^t+1 . K )] Et q
(115)
(18) Linearizing the physical capital investment condition (65), using the steady state relationship q = 1 and the steady state properties of FK1;t :FK1 (I; I) = 1 and FK2 (I; I) = 0, we get after rearranging terms: 1
1 ^ It (1 + ) 'K 1+ 00 where 'K = SK (1) .
I^t =
(^ qt ) +
1
+
1+
Et I^t+1 +
1 (1 + ) 'K
^It ,
(116)
(19) Linearizing the Capital utilization rate condition (68), using the steady state conditions u = 1 and a (u) = 0: r^tk =
^t , au
(117)
where
a
=
a00 (u) a0 (u)
.
(20) Linearizing the expression for capital services (78): ^ ts = u ^ t. K ^t + K
(118)
(21) The linearized wage setting equation (73) can be rewritten in terms of wage and price in‡ation, using the de…nition for the long run elasticity of substitution between labour types ^ w;t =
(
1)
w w
w ^t
^L;t U
^C;t U
1 w
A26
^W t
+
w
w ^ p;t 1
=
1+
W
W
:
+ ^ w;t+1
w ^ p;t .
(119)
Market Clearing (22) Linearizing the labor market clearing conditions (75) and (76): ~ b ~ b ^ t = Lc L ~ C;t + LE L ~ E;t . L L L
(120)
(23) Linearizing the capital market clearing condition (77) Kts =
KE ^ s KC ^ s KC;t + K . K K E;t
(121)
(24) & (25) Linearizing the national income accounts (84) and (86), using the steady state relationships rk = a0 (u) and mc
E
= v: Y^tC
C ^ I ^ G G rk K C + I + ^ + C u ^t , t t YC YC YC t Y i Y C ^ C vNE h ^ Yt + mc c t + ^E t + NE;t . Y Y
=
Y^t =
(26) Linearizing total investments (87), using the steady state relationship wZ i I ^ vNE h ^E;t . TcI t = It + mc c t + ^E + N t TI TI
(122) (123) E
= v: (124)
Laws of Motion
(27) Linearizing the law of motion for the number of …rms (62): ^t+1 = (1 N
^
^ +
N NE;t .
N ) Nt
(125)
(28) Linearizing the law of motion for the physical capital stock (63): ^ t+1 = (1 K
^ +
^+
K ) Kt
K It
I K ^t .
(126)
Identities (29) Linearizing the link between product price in‡ation and welfare-based in‡ation (89): ^t
^t
1
^C p;t .
= ^ p;t
(127)
(30) Linearizing the link between wage in‡ation and the real wage rate (90): ^ w;t = ^ C ^t p;t + w
w ^t
1.
(128)
Monetary Policy (31) Linearizing the monetary policy rule (88): ^t = R
^
R Rt 1
+
dy
h
+ (1 Y^t
R)
Y^tn
h
A27
^ p;t + Y^t
1
y
Y^tn 1
Y^t i
Y^tn + ^R t .
i
(129)
Shock Processes (32) Linearized process for the disturbance to total factor productivity (91): ^Z t =
Z Z ^t 1
+ d"Z;t .
(130)
(33) Linearized process for the disturbance to general preferences (92): ^Tt I =
TI T I ^t 1
+ d"T I;t .
(131)
(34) Linearized process for the disturbance to investment-speci…c technology (93): ^It =
I I ^t 1
+ d"I;t .
(132)
(35) Linearized process for the disturbance to government spending (94): ^G t =
G
tG t 1 + d"G;t +
GZ d"Z;t .
(133)
(36) Linearized process for the disturbance to monetary policy (95): ^R t =
R R ^t 1
+ d"R;t .
(134)
(37) Linearized process for the disturbance to price markup (96): ^Pt =
P P ^t 1
+ d"P;t
P d"P;t 1 .
(135)
(38) Linearized process for the disturbance to wage markup (97): ^W t =
W W ^t 1
+ d"W;t
W d"W;t 1 .
(136)
(39) Linearized process for the disturbance to …rm entry costs (98): ^E t = 3.2.1
E E ^t 1
+ d"E;t .
(137)
Steady State Shares
Preliminaries The shadow price of physical capital (67) and the physical capital investment condition (65) together imply the following steady state condition for the rental rate of capital: rk = (1
(1
K ))
1
.
(138)
The …rm value condition (60) entails that pro…ts in steady state can be expressed as a function of the steady state …rm value, d=
1
(1 (1 A28
N) N)
v.
(139)
By combing the price setting condition (42) with real pro…ts (41), steady state real pro…ts can also be written as d=
y . "
(140)
Using the …rm entry condition (61), the value of …rms in steady state reads as E
v = mc
.
(141)
The steady state share of …rm entry in the total stock of …rms follows immediately from the law of motion for the number of …rms (62), NE = N (1
N N)
.
(142)
Analogously, use the law of motion for the physical capital stock to derive the steady state share of intensive margin investment in the total stock of physical capital, I = K
K.
(143)
Equations (23) and (49) imply that the steady state ratio of the wage bill to the rental bill is equal across sectors and given by wLc wLE 1 = k = k r Kc r KE
.
(144)
Finally, using (9) the steady state share of the number of …rms in …nal output of manufactured goods reads as N 1 = . c Y y SS share
(145)
vNE YC
The share of extensive margin investment in …nal output of manufactured goods is vNE N NE =v c . Yc Y N Replace
N Y c using
(145), and
NE N
using (142), to get vNE 1 =v c Y y (1
Combining (139) and (140), we obtain
v y
=
(1 1 " 1 (1
N) N)
N N)
.
. Using this result, the share of extensive margin
investment in …nal output of manufactured goods …nally reads as vNE 1 = Yc "1
A29
N
(1
N)
.
(146)
SS shares
Yc Y
and
vNE Y
Noting that Y = Y c + vNE , we can derive the ratio of …nal manufactured goods production to GDP, Yc " [1 (1 1 N )] = = vNE Y " [1 (1 )] 1+ Yc N +
.
(147)
N
Then, the share of extensive margin investment in total income can be derived as vNE vNE Y c = = Y Yc Y " [1 SS shares
rk Kc Kc Kc Y c , KE , K
and
N
(1
N )]
+
.
(148)
N
KE K
Aggregate real pro…ts at steady state read as ~c wL
N d = yN
r k Kc .
Using the steady state conditions (140) and (145) to substitute out d and yN , respectively, and rearranging terms, we obtain Yc =
" "
~ c + r k Kc . wL
1
(149)
Then, the share of rental income in …nal manufactured goods production can be written as r k KC = Yc
1 ~c wL rk K
"
1 "
+1
.
Substitute in the steady state ratio of the wage bill to the rental bill (144), to get r k KC = Yc
"
1 "
.
(150)
The ratio of capital used in the production of manufactured goods to capital used in startup activities can be written as Kc r k KC Y c Y Y c = . KE Y c Y Y c r k KE Noting that Y = Y c + vNE , v = mc
E
and mcNE
E
= wLE + rk KE , this can be simpli…ed to
Kc r k KC Y c = KE Yc Y Yc
wLE +1 . r k KE
Substituting in the steady state ratio of the wage bill to the rental bill (144) as well as (150) and (147), we …nally get Kc = KE
"
1
" [1
(1
"
N )]
.
(151)
N
Given that K = Kc + KE , the share of capital used in the production of manufactured goods reads as Kc = K
Kc KE Kc KE
A30
+1
,
(152)
whereas the share of capital used in startup activities equals KE = K SS shares
Kc KE
1 . +1
(153)
rk K Yc ,
The steady state ratio of total rental income to …nal manufactured goods production can easily be obtained by combining equations (150) and (152), i.e., rk K r k KC = c Y Yc SS share
Kc K
1
(154)
I Yc
The share of intensive margin investment in …nal manufactured goods production equals I I 1 rk K = . Yc K rk Y c Combine this result with equations (138) and (143), to get: I = Yc 1
rk K , Yc K)
K
(1
which can easily be calculated once we know the steady state ratio SS shares
~c L ~c L ~E , L L
and
rk K Yc ,
(155) given in (154).
~E L L
By equation (144), the ratio of workers employed in the production of manufactured goods to workers employed in startup activities equals the ratio of capital used in both sectors, Kc Lc = , LE KE
(156)
where the latter has been de…ned in equation (151). ~c + L ~ E ; the share of workers employed in the production of manufactured goods reads as Given that L = L ~c L = L
~c L ~E L ~c L ~E L
,
(157)
+1
whereas the share of workers employed in startup activities equals ~E L = L
1 ~c L ~E L
A31
+1
.
(158)
SS shares
I TI
and
vNE TI
Noting that T I = I + vNE , the share of intensive margin investment in total investment can be derived as I 1 1 = . = c vN Y E TI 1 + IE 1 + I vN Yc Using (155) and (146) to substitute out
Yc I
and
vNE Yc ,
(159)
respectively, we can easily assess this share.
Finally, given (159), the share of extensive margin investment in total investment reads as vNE =1 TI 3.2.2
I . TI
(160)
Shock Normalization
Prior to estimation we normalize several shocks, such that they all enter the estimation with a unit coe¢ cient. Denoting the scaled shocks by the superscript ‘e’, we apply the following normalizations to respectively the price
and wage markup shocks, the time impatience shock, the disturbance to the investment-speci…c technology, and, …nally, the exogenous government spending process: P b ~
W b ~
TI b ~
I b ~
4
Details Steady State
"
=
1
^Pt ,
p
=
(
1) 1
w w
=
G b ~ =
^W t ,
w
I = Et ^Tt+1
(161)
^Tt I = ( 1
TI
(162) 1) ^Tt I ,
^It ,
(1 + ) 'K G G ^ . YC t
(163) (164) (165)
In the deterministic steady state, wage and price in‡ation are equal to zero, i.e., w
=
p
=
C p
= 0.
(166)
The capital utilization rate u is assumed to be equal to 1 in steady state, u = 1.
(167)
For the remaining steady state values, …rst note that the optimality conditions and market clearing conditions in the deterministic steady state simplify to:
A32
Firms e N 1N eN 2 ( )N
= exp YC . N y . " mc.
y = d = =
rk
Ny = =
KEs
=
~E L
(171)
(173) 1
.
1
w( ) . (1 )
(174) (175)
Z
(KCs )
~C L
Z
(KEs )
~E L
rk
(172)
w
Z
=
E
(169) (170)
rk
1
mc =
NE
(168)
. " 1+ N = . N " 1
=
KCs ~C L
.
d
= d
!
1 1
.
(176)
.
(177)
w( ) . (1 )
(178)
(L) l .
(179)
Households UL = UC Rt
= [(1 1 = .
d = mc
E
rk
h) C]
c
.
(181)
1
(1 (1
= v. 1 (1 =
N) N)
v.
(182) (183)
(1
K )) .
q = 1.
(184) (185)
rk = a0 (1) .
(186)
K s = K. w =
(180)
(187) 1+
A33
W
UL . UC
(188)
Market Clearing ~C + L ~E . L = L
(189)
K s = KCs + KEs .
(190)
YC
= C +I +
G
Y
= Y C + mc
E
E
T I = I + mc
.
(191) NE .
(192)
NE .
(193)
Law of Motion N
(1
=
N)
NE;t .
(194)
N KK
4.1
= It .
(195)
Recursive steady state
We already know the steady state of
w,
p,
C, p
u, ,
d,
R, q and rk from equations (166), (167), (172), (173),
(181), (185) and (184), respectively. To …nd the remaining variables in steady state, we …rst derive a system of six equations in , , w, C, Y C and N , which we solve numerically, given that we normalize L and given that we know the steady state price elasticity of demand ". Equation 1 In the deterministic steady state, the variety e¤ect (168) reads as = exp Using equation ,
1 2 N
1 2 N
1
can be expressed as a function of ", i.e.,
N e N
1 2 N
=
. 1 2(" 1) .
(196) The steady state ratio of the number
of available goods to the number of all conceivable goods is calibrated to be high at
N e N
= 0:95.
Equation 2 Combining (188) with equations (179) and (180), yields the steady state wage rate, w = 1+
W
[(1
(L) l h) C]
c
.
(197)
Equation 3 We can derive two expressions for …rm pro…ts. First, substituting out …rm output y from the pro…t equation (170) by using (169), yields d=
1YC . " N
A34
(198)
Second, combining …rm value (182) with the free entry equation (183) to eliminate v, yields 1
d= Given that
=
" " 1,
(1 (1
N)
E
mc
N)
.
(199)
the price setting equation (171) implies mc =
"
1 "
.
(200)
Together (198), (199) and (200) can be combined to eliminate pro…ts d and marginal costs mc, "
1
E
=
"
1YC . N) " N
(1 (1
1
N)
(201)
Equation 4 Plugging (200) into the marginal cost equation (174), we obtain "
1 "
(w)1
rk
1
=
Z
(1
.
)1
(202)
Equation 5 First, rewrite the aggregate production function (176) as KCs ~C L
Z
Ny =
~C . L
~ C using (175), Substitute out the capital-labor ratio KCs =L w rk
Z
Ny =
1
~C . L
~C , Use the labor market clearing condition (189) to replace L Ny =
w rk
Z
1
~E . L
L
(203)
Second, rewrite the aggregate …rm production function (177) as E
NE
=
Z
KEs ~E L
~E . L
Replace the capital-labor ratio using (178), and the number of entrants using (194), to get N
E
N
1
N
Ny =
Z
=
w rk
Z
1
~E . L
(204)
Combining (203) and (204), yields:
Finally, multiply both sides by
1
w rk
L
N
1
N
E
.
N
and use Y C = yN , to obtain YC =
Z
1
w rk A35
L
N
1
N N
E
.
(205)
Equation 6 From (191) the relationship between consumption and …nal manufactured goods production reads as C =YC Given the steady state shares I=Y C and
G =Y C ,
I
G
.
this expression can be rewritten to obtain
C =YC
G
I YC
1
.
YC
(206)
Summary 6-equation system Finally, we have a six-equation system: (196), (197), (201), (202), (205) and (206), for six unknowns: , w, L, C, Y C and N ; i.e., 1
= exp
1) (L) l 1+ W [(1 h) C] C (1 N) 1 Y , 1 (1 N) " N
w = "
1
E
"
1
" " YC
1
=
1
=
2 ("
(w)1
rk
Z
)1 w rk
(1 Z
=
1
C = YC
1
I YC
N e N c
,
(207)
,
(208) (209)
,
(210) N
L
1
N
E
,
(211)
N
G
YC
.
(212)
As is customary, we set L = 1. Then, our six equations jointly determine , w, C, Y C , N and state values for
Z,
E,
W,
rk ,
I YC
and
G
YC
, given steady
.
Remaining steady state values Finding C by solving the above system gives us UC through (180). Similarly, the solution to
yields UL through
(179) Once we have derived N , we can …nd NE from (194). Real marginal costs mc are given by (174). Knowing mc, we can solve for v through (183). Pro…ts d are derived plugging v into (182). The above system’s solutions to N ,
~ E from (204), then the labor market clearing and Y C can be combined to …nd y using (169). Compute L
~ C and L ~ E , we have K s and K s from the respective capital-labor ratios (175) condition (189) yields LC . Given L C E and (178). Total capital services K s are given by (190) and equal the total capital stock K through (187). Knowing K, we also know I through (195). Finally, GDP and total investments are given by (192) and (193), respectively.
A36
5
Details Bayesian estimation procedure DSGE models
First, we log-linearize the model around the steady state. The solution to the log-linearized version of the model can be expressed as follows: y ^t = A (z) y ^t
1
+ B (z)
where y ^t represents the vector of endogenous variables and
t
t,
(213)
denotes the vector of exogenous white noise shocks.
A (z) and B (z) are the transition and impact matrices, respectively, which are non-linear functions of the structural parameters z. Next, the model is brought to the data by linking the observable variables Yt to the model variables y ^t . This can be represented by the measurement equation, Yt = C^ yt + where
t
t,
(214)
characterizes the independent and identical distributed measurement errors. Together, equations (213)
and (214) form the state space representation of the model. The white noise shocks
t
are assumed to be Gaussian distributed. Consequently, the Kalman Filter can be
employed to compute the likelihood function (denoted by L
=YT , where YT = fY1 ; Y2 ; :::; YT g) from the
state space representation. Applying the Bayesian theorem, we can combine this likelihood function with the joint prior distribution p ( ) in order to obtain the joint posterior distribution p p
=YT = Z
=YT , which reads as L
=YT p ( )
.
(215)
fL ( =YT ) p ( )g d
Because it is most unlikely to obtain a closed-form solution for the posterior distribution, we need to approximate this distribution numerically. One way to conduct the numerical integration is to apply the Markov Chain Monte Carlo sampling method. In this paper we implement this method by employing the Random Walk Metropolis Hastings algorithm.
References [1] Beaudry, P., Collard, F. and Portier, F., 2011, “Gold Rush Fever in Business Cycles”, Journal of Monetary Economics, 58, 2, pp. 84-97. [2] Bilbiie, F.O., Ghironi, F. and Melitz, M., 2007, “Monetary Policy and Business Cycles with Endogenous Entry and Product Variety”, NBER Macroeconomics Annual, 22, pp. 299-353. A37
[3] Bilbiie, F.O., Ghironi, F. and Melitz, M., 2012, “Endogenous Entry, Product Variety and Business Cycles”, Journal of Political Economy, 120, 2, pp. 304-345. [4] Christiano, L.J., Eichenbaum, M. and Evans, C.L., 2005, “Nominal Rigidities and the Dynamic E¤ects of a Shock to Monetary Policy”, Journal of Political Economy,113, 1, pp. 1-45. [5] Feenstra, R.C., 2003, “A Homothetic Utility Function for Monopolistic Competition Models, without Constant Price Elasticity”, Economics Letters, 78, 1, pp. 79-86. [6] Ireland, P.N., 2007, “Changes in the Federal Reserve’s In‡ation Target: Causes and Consequences”, Journal of Money, Credit and Banking, 39, 8, pp. 1851-1882. [7] Mata, J. and Portugal, P., 1994, “Life Duration of New Firms”, Journal of Industrial Economics, 42, 3, pp. 227-245. [8] Rotemberg, J.J., 1982, “Monopolistic Price Adjustment and Aggregate Output”, Review of Economic Studies, 49, 4, pp. 517-31. [9] Smets, F. and Wouters, R., 2007, “Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach”, American Economic Review, 97, 3, pp. 586-606. [10] Lewis, V. and Poilly, C., 2012, “Firm Entry, Markups and the Monetary Transmission Mechanism”, Journal of Monetary Economics, 59, 7, November.
A38
