Banque de France's Workshop on Granularity: Basile Grassi's slides, June 2016

Large Firm Dynamics and the
Business Cycle
Vasco M. Carvalho (Cambridge/CREi & CEPR)
Basile Grassi(Oxford & Nufﬁeld College)
Banque de France, 24 June 2016
The Granularity of Macroeconomic Fluctuations: where do we stand?

Large Firm Dynamics and the Business Cycle
1965 1971 1976 1982 1987 1993 1998 2004 2009
%points
-6
-5
-4
-3
-2
-1
0
1
2
3
4
Large Firms Corr: 0.564 (1e-05)
GDP
Large Firms
Observation: Large ﬁrms comove with the business cycle More

1965 1971 1976 1982 1987 1993 1998 2004 2009
%points
-6
-5
-4
-3
-2
-1
0
1
2
3
4
GDP
Large Firms
Option A: Analyze sensitivity of large ﬁrms to the business cycle

1965 1971 1976 1982 1987 1993 1998 2004 2009
%points
-6
-5
-4
-3
-2
-1
0
1
2
3
4
GDP
Large Firms
Option B: Analyze sensitivity of cycle to large ﬁrm dynamics

Employment
100
101
102
103
104
Pr(FirmSize>x)
10-4
10-3
10-2
10-1
100
Why?
GM’s 1970 Sales ≃2.5% of US
GDP
Walmart’s 2014 US sales
≃1.9% of US GDP
"The release of iPhone 5
could potentially add
between 1/4 to 1/2%-point to
fourth quarter annualized
GDP growth" [JP Morgan]
Largest 0.02% of ﬁrms ≃20%
of total employment

Employment
100
101
102
103
104
Pr(FirmSize>x)
10-4
10-3
10-2
10-1
100
Granular hypothesis (Gabaix
2011): whenever ﬁrm-size
distribution
has a "fat tail", σGDP ∝ σ
N1−1/ζ
has a "thin tail", σGDP ∝ σ√
N
N = 5 ∗ 106
, σ ≃ 0.2, ζ ≃ 1.2 ⇒
σGDP ≃ 1.7%vs σGDP ≃ 0.005%
These are Central Limit Theorems

Employment
100
101
102
103
104
Pr(FirmSize>x)
10-4
10-3
10-2
10-1
100
Economic questions:
Why is the firm
size-distribution Pareto?
What induces it to fluctuate
over time?
If a large firm declines why
don’t its competitors expand?
Is this consistent with other
facts on firm growth &
churning?
How much does this matter
quantitatively?

Introduction
This paper seeks to evaluate the impact of large firm dynamics on
aggregate dynamics.
1 Building on standard firm dynamics model, we develop a
quantitative theory of aggregate fluctuations arising from
firm-level shocks alone.

Introduction
aggregate dynamics.
2 We derive an analytical characterization of the law of motion of
the ﬁrm size distribution and show that the resulting aggregate
output is endogenously:
i Persistent
ii Volatile
iii and Exhibits time-varying second moment

Introduction
aggregate dynamics.
2 We derive an analytical characterization of the law of motion of
the firm size distribution and show that the resulting aggregate
output is endogenously:
i Persistent
ii Volatile
iii and Exhibits time-varying second moment
3 Quantitatively large firm dynamics account for 1/4 of aggregate
fluctuations

Related literature
Micro-origins of aggregate ﬂuctuations literature:
o Carvalho (2010), Gabaix (2011), Acemoglu et al (2012), di Giovanni
and Levchenko (2012) Carvalho and Gabaix (2013) building on
older literature by Jovanovic (1987), Bak et al (1993), Scheinkman
and Woodford (1994) and Horvath (1998)
o di Giovanni, Levchenko and Mejean (2012) for some empirical
evidence
Firm dynamics literature:
o Hopenhayn (1992), Hopenhayn and Rogerson (1993), Khan and
Thomas (2003, 2008), Clementi and Palazzo (2010), Moscarini and
Postel Vinay (2012), Chari, Christiano and Kehoe (2013), Bloom et
al (2014)
Random Growth:
o Gabaix (1999), Luttmer (2007,2012)

Roadmap
Setup
o Hopenhayn (1992) with a discrete number of ﬁrms
Theoretical results
Quantitative results

Model: Overview
Finite number, Nt , of heterogeneous incumbent ﬁrms:
differ in their productivity level dicrete on a grid
Φ = {ϕ, ϕ2, . . . , ϕS}
Perfect competition and decreasing return to scale.
Subject to a ﬁxed operating cost ⇒ Exit.
Finite (but large) number of potential entrants subject to an
entry cost.
Elastic Labor Supply with respect to the wage.

Incumbent’s Problem
For an aggregate state µ and idiosyncratic pdty level ϕs, the value of
an incumbent is V (µ, ϕs)
V (µ, ϕs
) = π∗
(µ, ϕs
) +max



0, β
µ′∈Λ
∑
ϕs′
∈Φ
V (µ′
, ϕ′
)F(ϕs′
|ϕs
)Γ(dµ′
|µ)




V (µ, ϕs
) = π∗
(µ, ϕs
) +max



0, β
µ′∈Λ
∑
ϕs′
∈Φ
V (µ′
, ϕ′
)F(ϕs′
|ϕs
)Γ(dµ′
|µ)



Instantaneous proﬁt: π∗(µ, ϕs) = Maxn ϕsnα − w(µ)n − cf

V (µ, ϕs
) = π∗
(µ, ϕs
) +max



0, β
µ′∈Λ
∑
ϕs′
∈Φ
V (µ′
, ϕ′
)F(ϕs′
|ϕs
)Γ(dµ′
|µ)



Firm level productivity markovian process: F(ϕs′
|ϕs) on the
grid Φ

V (µ, ϕs
) = π∗
(µ, ϕs
) +max



0, β
µ′∈Λ
∑
ϕs′
∈Φ
V (µ′
, ϕ′
)F(ϕs′
|ϕs
)Γ(dµ′
|µ)



Firm level productivity markovian process: F(ϕs′
|ϕs) on the
grid Φ
Law of motion of aggregate state: Γ(µ′|µ) is here endogenous

Incumbent and Entrants Problem
Threshold rule: for ϕs ≥ ϕs∗(µ) ﬁrm continues and for
ϕs ≤ ϕs∗(µ)−1 ﬁrm decides to exit next period
10 20 30 40
0.001
0.002
0.003
10 20 30 40
0.001
0.002
0.003
Incumbent Distribution Entrant Distribution

Aggregation and Market Clearing
Aggregate output: Yt = ∑
Nt
i=1 yt,i = At (Ld
t )α where
At =
Nt
∑
i=1
(ϕsi,t )
1
1−α
1−α
=
S
∑
s=1
(ϕs
)
1
1−α µs,t
1−α

Nt
i=1 yt,i = At (Ld
t )α where
At =
Nt
∑
i=1
(ϕsi,t )
1
1−α
1−α
=
S
∑
s=1
(ϕs
)
1
1−α µs,t
1−α
Labor supply Ls
t (w) = Mwγ implies
wt = α
1
1−α
At
1
1−α
M
1−α
γ(1−α)+1

Nt
i=1 yt,i = At (Ld
t )α where
At =
Nt
∑
i=1
(ϕsi,t )
1
1−α
1−α
=
S
∑
s=1
(ϕs
)
1
1−α µs,t
1−α
Labor supply Ls
wt = α
1
1−α
At
1
1−α
M
1−α
γ(1−α)+1
⇒ Behaves as a one factor model with aggregate TFP At

Nt
i=1 yt,i = At (Ld
t )α where
At =
Nt
∑
i=1
(ϕsi,t )
1
1−α
1−α
=
S
∑
s=1
(ϕs
)
1
1−α µs,t
1−α
Labor supply Ls
wt = α
1
1−α
At
1
1−α
M
1−α
γ(1−α)+1
⇒ Behaves as a one factor model with aggregate TFP At
The distribution µt pins down At ⇒ Aggregate state is µt

Roadmap
Setup
Theoretical results
o General characterization of LOM of aggregate state
o Particular case of random growth
. Stationary distribution
. Close form solution of the stationary case
. Aggregate dynamics
. Persistence and Volatility

Productivity Distribution Dynamics: an example

Productivity Distribution Dynamics
Theorem 1
In the continuum case µt+1 (Hopenhayn 1992)
µt+1 = (P∗
t )′
µt + (P∗
t )′
MG+εt+1
The distribution at t is the sum of
The evolution of incumbents
And the contribution of entry/exit
⇒ The distribution µt+1 is a deterministic object.
This law of motion converges to a stationary distribution µ, the
steady-state of our model.

Productivity Distribution Dynamics
Theorem 1
In the general case µt+1
µt+1 = (P∗
t )′
µt + (P∗
t )′
MG + εt+1
The distribution at t is the sum of
The evolution of incumbents
And the contribution of entry/exit
A random vector εt+1 variance-covariance matrix Σ(µt ) More
⇒ The distribution µt+1 is a stochastic object.

Assumption: Gibrat’s Law
Assumption: Firm-level productivity evolves as a Markov Chain
such that for ﬁrm i at date t with productivity level si,t ∈ [1 . . . S]
si,tsi,t − 1 si,t + 1
a
b = 1 − a − c
c

Assumption: Gibrat’s Law
Assumption: Firm-level productivity evolves as a Markov Chain
such that for ﬁrm i at date t with productivity level si,t ∈ [1 . . . S]
si,tsi,t − 1 si,t + 1
a
b = 1 − a − c
c
This Markovian process satisfy Gibrat’s law:
ϕsi,t+1
ϕsi,t
= ρe + σ2
e ǫt+1
with Et [ǫt+1] = 0 and Var[ǫt+1] = 1. More

With an ∞ Number of ﬁrms
Corollary
If the potential entrants’ productivity distribution is Pareto i.e
Gs = Ke (ϕs
)−δe
then
as N → ∞, the stationary productivity distribution
ˆµs = K1
ϕs
ϕs∗
−δ
+ K2
ϕs
ϕs∗
−δe
for s ≥ s∗
where δ = log(a/c)
log(ϕ)
.

Corollary
Gs = Ke (ϕs
)−δe
then
ˆµs = K1
ϕs
ϕs∗
−δ
+ K2
ϕs
ϕs∗
−δe
for s ≥ s∗
where δ = log(a/c)
log(ϕ)
.
This a mixture of Paretos:
i The distribution of entrants
ii A Pareto determined by the Gibrat’s law: δ =
log(a/c)
log(ϕ)

Corollary
Gs = Ke (ϕs
)−δe
then
ˆµs = K1
ϕs
ϕs∗
−δ
+ K2
ϕs
ϕs∗
−δe
for s ≥ s∗
where δ = log(a/c)
log(ϕ)
.
This a mixture of Paretos:
i The distribution of entrants
ii A Pareto determined by the Gibrat’s law: δ =
log(a/c)
log(ϕ)
This is the stationary distribution in the continuum case.

Incumbents’ Value Function: Stationary Equilibrium
case S → ∞
The exit thresholds is s∗ = ⌈s∗⌉ where
s∗
= (1 − α)
log
cf
1−β
1−r2
1−r2/ϕ1/(1−α)
1−βρ
ρ(1−α)
(α)
−α
1−α
log ϕ
+ α
log w
log ϕ
The value function of incumbents is
V (s) =
−cf
1 − β
1 − βr
[s−s∗+1]+
2 +
1 − α
1 − βρ
α
w
α
1−α
ϕ
1
1−α
s

1 − βρ
r2
ϕ
1
1−α
[s−s∗+1]+ 

with the function of deep parameters r2 < 1 < ϕ1/(1−α)
Proof S < ∞

case S → ∞
The value function is decreasing in the wage w.
and in the operating cost cf
The value function can be described as the present dicounted
value of intantaneous proﬁt adjust by the exit risk.
The further an incumbent is from the exit threshold s∗ (large s)
the closer is its value to the present discounted value of
instantaneous proﬁt.
The thresholds s∗ is convex combinaison of the log w and a
constant function of parameters.
Proof S < ∞

Aggregate Dynamics: A Characterization
Theorem 2
Deﬁne Tt = A
1/(1−α)
t = ∑S
s=1(ϕs)
1
1−α µs,t
i.e a non-linear transformation of aggregate productivity
(the aggregate state in the model)
More

Theorem 2
Deﬁne Tt = A
1/(1−α)
t = ∑S
s=1(ϕs)
1
1−α µs,t
Tt ∝ to the average ﬁrm size
More

Theorem 2
Define Tt = A
1/(1−α)
t = ∑S
s=1(ϕs)
1
1−α µs,t
Same logic, define Dt := ∑S
s=st −1 (ϕs)
1
1−α
2
µs,t ∝ the dispersion
of firm size
More

Theorem 2
Deﬁne Tt = A
1/(1−α)
t = ∑S
s=1(ϕs)
1
1−α µs,t
s=st −1 (ϕs)
1
1−α
2
of ﬁrm size
The dynamics of average size is
Tt+1 = ρTt + ρEt (ϕ) + OT
t + σt εt+1
persistent
contribution of entry/exit
stochastic term
More

Theorem 2
Deﬁne Tt = A
1/(1−α)
t = ∑S
s=1(ϕs)
1
1−α µs,t
s=st −1 (ϕs)
1
1−α
2
of ﬁrm size
The dynamics of average size is
Tt+1 = ρTt + ρEt (ϕ) + OT
t + σt εt+1
persistent
contribution of entry/exit
stochastic term
The time-varying volatility is
σ2
t = ̺Dt + ̺Et (ϕ2
) + Oσ
t
is determined by the second moment of incumbents
and the same object for entry/exit
More

Aggregate Persistence and Volatility
Proposition 1:
The persistence of the aggregate output ρ is increasing in
ﬁrm-level persistence, in the fatness of the stationary
distribution. If that distribution is Zipf, then ρ = 1. More
Proposition 2:
i The rate of decay of volatility: role of productivity process,
decreasing returns, tail of entrant and is much slower than than
1/N (predicted by CLT). More
ii Aggregate volatility is an increasing function of ﬁrms dispersion:
Dt := ∑S
s=st −1 (ϕs)
1
1−α
2
µs,t . More

Roadmap
Setup
Theoretical results
o Stationary Steady State Calibration
o Business Cycle Statistics
o 1st Prediction: Large ﬁrms drive the cycle
o 2nd Prediction: Cross-Sectional Dispersion & Aggregate Volatility

Calibration I
Firm productivity follows the Gibrat’s law.
Entrants’ signal is distributed according to a Pareto.
Set the value of deep parameters: α, β, γ and ce = 0.
Use the remaining parameters to match the following firm level
targets:
Statistic Model Data References
Entry Rate 0.109 0.109 BDS firm data
Idiosyncratic Vol. σe 0.08 0.1 − 0.2 Castro et al. (forthcoming)
Tail index of Firm size dist. 1.097 1.097 BDS firm data
Tail index of Entrant Firm size dist. 1.570 1.570 BDS firm data
Share of Employment of the largest firm 0.2% 1% Share of Wall-Mart
Number of firms 4.5 × 106
4.5 × 106
BDS firm data
Parameters σ2
e

The Firm Size Distribution: Model vs Data
10-2
100
102
104
106
10-5
10-4
10-3
10-2
10-1
100
Incumbent Distribution against Data
—– 10-2
100
102
104
106
10-8
10-6
10-4
10-2
100
Entrant Distribution against Data

Business Cycle Statistics
Simulating a path of 10,000 periods yields:
Model Data
σ(x) σ(x)
σ(y)
ρ(x, y) σ(x) σ(x)
σ(y)
ρ(x, y)
Output 0.47 1.0 1.0 1.83 1.0 1.0
Hours 0.31 0.66 1.0 1.78 0.98 0.90
Agg. Productivity 0.21 0.46 1.0 1.04 0.57 0.66
The model accounts for 0.47/1.83= 26% of output volatility.
Numerical Method Mechanism

1st
Prediction: Large Firms Drive the Cycle
Employment
10 0
10 1
10 2
10 3
10 4
Pr(firms>x)
10 -6
10 -5
10 -4
10 -3
10 -2
10 -1
10 0
Variation of CCDF (simulation)
– Employment
10 0
10 1
10 2
10 3
10 4
Pr(firms>x) 10 -5
10 -4
10 -3
10 -2
10 -1
10 0
Variation of CCDF (BDS-Compustat)

1st
Prediction: Large Firms Drive the Cycle
Sample Firms with more than 10k 15k 20k
Model Correlation in level −0.64
(0.000)
−0.57
(0.000)
−0.48
(0.000)
Correlation in growth rate −0.41
(0.000)
−0.42
(0.000)
−0.44
(0.000)
Data Correlation in (HP ﬁltered) level −0.34
(0.008)
−0.51
(0.000)
−0.46
(0.000)
(0.011)
−0.43
(0.001)
−0.38
(0.004)
Robustness Check

2nd
Prediction: Cross-Sectional Dispersion Drive
Aggregate Volatility
Correlation of Dispersion and Aggregate Volatility:
Sample Aggregate Volatility Dispersion of Dispersion of
Real Sales Employment
Model Aggregate Volatility 0.9968
(0.000)
0.9983
(0.000)
Data Aggr. Vol. in TFP growth 0.3461
(0.016)
0.2690
(0.065)
Aggr. Vol. in GDP growth 0.2966
(0.041)
0.1782
(0.226)
Robustness Check

Conclusion
We build a quantitative firm dynamcis model in which the origin of
aggregate dynamics is cast in the large firms dynamics only:
We show -analytically- that aggregate output is:
i Persistent
ii Volatile
iii Exhibit time-varying second moments
We explore quantitatively and in the data the role of the firm
size distribution in shapping aggregate fluctuations
Future work: Introduce frictions for small firms, pricing
behavior

Distributional Dynamics and the Business Cycle
The firm size distribution is a “sufficient statistic” for
understanding aggregate fluctuations.
If we observe the firm size distr. over time: µt
If we use our calibrated model as an aggregating device:
At =
S
∑
s=1
(ϕs
)
1
1−α µs,t
1−α
Yt = At (Ld
t )α
Vart Yt+1 =
̺
T2
S
∑
s=st −1
(ϕs
)
1
1−α
2
µs,t
What would be the implied history of agg. fluctuations and
volatility based on this data alone?

Distributional Dynamics and the Business Cycle
Year
76 78 80 82 84 86 88 90 92 94 96 98 00 02 04 06 08 10
ppdev
-5
-4
-3
-2
-1
0
1
2
3
Corr:0.288 (0.0878)
Aggregate Productivity
Year
76 78 80 82 84 86 88 90 92 94 96 98 00 02 04 06 08 10
ppdev
-5
-4
-3
-2
-1
0
1
2
3
Corr:0.513 (0.001)
Aggregate Output
Year
76 78 80 82 84 86 88 90 92 94 96 98 00 02 04 06 08 10
ppdev
-5
-4
-3
-2
-1
0
1
2
3
Corr:0.369 (0.026)
Aggregate Volatility

Large Firm Dynamics over the Cycle
Sales growth of large vs. small over the business cycle following
Gertler and Gilchrist (1994) and Chari, Christiano and Kehoe
(2013)
Use Quartely Financial Reports data 1987-2013; reports sales by
asset size
Find (time-varying) asset size cutoff such that:
o Small firms: those accounting for 30% of sales in manufacturing
over two consecutive quarters
o Large firms: converse
Correlate with HP-deviations of aggregate manufacturing
output
Results unchanged if simply use “large firms” as > 1 Billion USD
assets
Back

Compustat percentile
¡ ¡¢ ¡£ 56 ¡¤ ¥ ¥¢ ¥£ 66 ¥¤ § §¢ §£ 76 §¤ ¤ ¤¢ ¤£ ¤¥ ¤¤ ¨ ¨¢ ¨£ 96 ¨¤ ¢ £ ¥ ¤
-15
-10
-5
0
5
10
15
20
top 1% Corr:0.373 (0.0035)
top 5% Corr:0.357 (0.0054)
top 10% Corr:0.402 (0.0015)
Back

More on the random vector εt+1
The random vector εt+1 has mean zero and a variance-covariance
matrix:
Σ(µt ) =
S
∑
s=s∗(µt )
(MGs + µs,t ).Ws
where P∗
t is the transition matrix P with the ﬁrst (s∗(µt ) − 1) rows
replaced by zeros. Ws = diag(Ps,.) − P
′
s,.Ps,. where Ps,. denotes the
sth-row of the transition matrix P.
Back

More on the Gibrat’s Law (Córdoba 2008)
Firm-level productivity evolves as a Markov Chain on the state
space Φ = {ϕs}s=1..S with transition matrix
P =







a + b c 0 · · · · · · 0 0
a b c · · · · · · 0 0
· · · · · · · · · · · · · · · · · · · · ·
0 0 0 · · · a b c
0 0 0 · · · 0 a b + c







Under this assumption, ﬁrm’s productivity follows Gibrat’s law:
E
ϕ
si,t+1 − ϕ
si,t
ϕ
si,t
|ϕ
si,t = a(ϕ−1 − 1) + c(ϕ − 1) = ρe − 1 and Var
ϕ
si,t+1 − ϕ
si,t
ϕ
si,t
|ϕ
si,t = σ2
e
The stationary distribution of this Markovian Process is
K (ϕs)−δ
(i.e Pareto) with δ = log(a/c)
log ϕ
.
Back

The value function of incumbents is
V (s) =
−cf
1 − β



1 −
βr
[s−s∗+1]+
1
1 −
r1
r2
S−s∗+1
−
βr
[s−s∗+1]+
2
1 −
r2
r1
S−s∗+1



 . . .
. . . +
1 − α
1 − βρ
α
w
α
1−α ϕ
1
1−α
s











1 − β



r1
ϕ
1
1−α



[s−s∗+1]+
1 −
r1
r2
S−s∗+1



ρ +
1 − ρ − a(1 − ϕ
−1
1−α )
1 − β + aβ



ϕ
1
1−α
r2



S−s∗+1



 . . .
. . . − β



r2
ϕ
1
1−α



[s−s∗+1]+
1 −
r2
r1
S−s∗+1



ρ +
1 − ρ − a(1 − ϕ
−1
1−α )
1 − β + aβ



ϕ
1
1−α
r1



S−s∗+1















with
r1 =
1/β − b + (1/β − b)2 − 4ca
2c
> ϕ
1
1−α > r2 =
1/β − b − (1/β − b)2 − 4ca
2c
and s∗ = ⌈s∗⌉ such that s∗ solve
cf
1 − β
rS−s∗ +1
2 (r1 − 1) + rS−s∗ +1
1 (1 − r2 ) =
1 − α
1 − ρβ
α
w
α
1−α
ρ ϕ
1
1−α
s∗
rS−s∗ +1
2
r1
ϕ
1
1−α
− 1 + rS−s∗ +1
1 1 −
r2
ϕ
1
1−α
+
1 − ρ − a(1 − ϕ
−1
1−α )
1 − β + aβ
1 − α
1 − ρβ
α
w
α
1−α
ϕ
1
1−α
S
(r1 − r2 )
Back

Sketch of the Proof
The Bellman equation to solve is
V (s) = (1− α)
α
w
α
1−α
ϕ
1
1−α
s
−cf + β Max {0, aV (s − 1) + bV (s) + cV (s + 1)}
For s > s∗, we can rewrite this equation as
aV (s − 1) + b −
1
β
V (s) + cV (s + 1) = −
(1 − α)
β
α
w
α
1−α
ϕ
1
1−α
s
+ cf
which is a second order linear difference equation in V (s) ⇒
solutions are in a 2-dimensional vector space generate by rs
1
and rs
2 roots of a + b − 1
β X + cX2.
Used the boundary conditions at s∗ and S to solve for the
(unique) solution
Back

More On Aggregate Dynamics: A Complete
Characterization
The equations governing the evolution of the ﬁrst moment (∝
aggr. productivity) are Tt+1 = ρTt + ρEt (ϕ) + OT
t + σt εt+1 and
σ2
t = ̺Dt + ̺Et (ϕ2) + Oσ
t .
The persistence of the aggregate state is ρ = aϕ
−1
1−α + b + cϕ
1
1−α and
̺ = aϕ
−2
1−α + b + cϕ
2
1−α − ρ2
The terms Et (ϕ) and Et (ϕ2) are the respective contribution of
net entry to respectively aggregate productivity and aggregate
volatility: Et (x) = M ∑S
s=st
Gs (xs)
1
1−α − xst −1
1
1−α
µst −1,t
The terms OT
t and Oσ
t are correction terms arising from having
imposed bounds on the state-space.
Back

Aggregate Persistence
Proposition 1
If δ ≥ 1
1−α then the persistence of the aggregate output, ρ:
i) is increasing in ﬁrm-level persistence:
∂ρ
∂b
≥ 0
Back

Proposition 1
If δ ≥ 1
∂ρ
∂b
≥ 0
ii) is increasing in the fatness of the stationary productivity
distribution:
∂ρ
∂δ ≤ 0
Back

Proposition 1
If δ ≥ 1
∂ρ
∂b
≥ 0
ii) is increasing in the fatness of the stationary productivity
distribution:
∂ρ
∂δ ≤ 0
iii) if the productivity distribution is Zipf (δ = 1
1−α ), aggregate state
dynamics contain a unit root: ρ = 1
Back

Level of Aggregate Volatility
Proposition 2 i)
If 1 < δ(1 − α) < 2 and 1 < δe(1 − α) < 2, the unconditional
expectation of aggregate variance satisﬁes:
E
σ2
t
T2
∼
N→∞
̺D1
N
2− 2
δ(1−α)
+
̺D2
N
1+ δe
δ − 2
δ(1−α)
where D1 and D2 are functions of model parameters but
independent of N and M.
Back

Proposition 2 i)
E
σ2
t
T2
∼
N→∞
̺D1
N
2− 2
δ(1−α)
+
̺D2
N
1+ δe
δ − 2
δ(1−α)
The rate of decay is much slower than 1/N (predicted by CLT)
Back

Proposition 2 i)
E
σ2
t
T2
∼
N→∞
̺D1
N
2− 2
δ(1−α)
+
̺D2
N
1+ δe
δ − 2
δ(1−α)
Role of productivity process, decreasing returns, tail of entrant
Back

Proposition 2 i)
E
σ2
t
T2
∼
N→∞
̺D1
N
2− 2
δ(1−α)
+
̺D2
N
1+ δe
δ − 2
δ(1−α)
Role of productivity process, decreasing returns, tail of entrant
Counterpart of Gabaix (2011) with endogenous δ, ﬁrm choice,
entry/exit
Back

(Time-varying) Aggregate Volatility
Proposition 2 ii)
The dynamics of conditional aggregate volatility depend on the
dispersion of ﬁrm size:
∂Vart Yt+1
∂Dt
=
̺
T2
≥ 0
where Dt := ∑S
s=st −1 (ϕs)
1
1−α
2
µs,t is the second moment of the
ﬁrm size distribution.
Intuition:
Back

Proposition 2 ii)
∂Vart Yt+1
∂Dt
=
̺
T2
≥ 0
where Dt := ∑S
s=st −1 (ϕs)
1
1−α
2
Intuition:
When the dispersion is high, small firms are really small and
large firms are really large. Large firms matter a lot in the
aggregate.
Back

Proposition 2 ii)
∂Vart Yt+1
∂Dt
=
̺
T2
≥ 0
where Dt := ∑S
s=st −1 (ϕs)
1
1−α
2
Intuition:
When the dispersion is high, small firms are really small and
large firms are really large. Large firms matter a lot in the
aggregate.
Shocks to these large firms will generate large aggregate effects.
Back

Calibration
Parameters Value Description
a 0.6129 Pr. of moving down
c 0.3870 Pr. of moving up
S 36 Number of productivity levels
ϕ 1.0874 Step in pdty bins
Φ {ϕs}s=1..S Productivity grid
γ 2 Labor Elasticity
α 0.8 Production function
cf 1.0 Operating cost
ce 0 Entry cost
β 0.95 Discount rate
M 4.8581 ∗ 107 Number of potential entrants
G {MKe(ϕs)−δe }s=1..S Entrant’s distr. of the signal
Ke 0.9313 Tail parameter of the distr. G
δe(1 − α) 1.570 Scale parameter of the distr. G
Back

Calibration of σ2
e
Comin and Phillipon (2006) and Davis et al (2007) report sales
growth volatility estimates for publicly listed firms between 10%
and 20%.
Gabaix (2011) find standard deviations of 12%, and 14% for,
respectively, growth rates of the sales per employee, of sales,
and of employees among the top 100 firms.
Davis et al (2007) report even higher values for employment
volatility at privately held firms, based on the Longitudinal
Database of Businesses.
Foster et al (2008) and Castro et al (forthcoming) report an
average value for annual productivity (TFPR) volatility of about
20%.
Back

Numerical Method
We are using the fact that the law of motion of the aggregate
state Tt is solved for close form.
Here firms form expectations assuming that Et (ϕ), Et (ϕ2) and
Dt is fixed at its steady-state level.
Limited rationality assumption: agent pay attention to only the
first moment (Gabaix).
Usual assumption to solve heterogeneous agents model with
aggregate risk as in Krusell and Smith (1998).
But very different because the law of motion of Tt is a known
function of the deep parameters (no need simulation step).
Back

Inspecting the Mechanism: Shock on the Largest
Firm
5 10 15 20 25 30
ppDev.
-0.035
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
Aggregate Output: Y
5 10 15 20 25 30
ppDev.
-0.025
-0.02
-0.015
-0.01
-0.005
0
Aggregate Hours: L
5 10 15 20 25 30
ppDev.
-0.018
-0.016
-0.014
-0.012
-0.01
-0.008
-0.006
-0.004
-0.002
0
Aggregate Productivity: A
Back

Robustness Check: Tails over the Cycle
Sample Firms with more than 1k 5k 10k 15k 20k
Model Correlation in level −0.50
(0.000)
−0.71
(0.000)
−0.64
(0.000)
−0.57
(0.000)
−0.48
(0.000)
(0.000)
−0.35
(0.000)
−0.41
(0.000)
−0.42
(0.000)
−0.44
(0.000)
Data Correlation in (HP ﬁltered) level −0.36
(0.005)
−0.17
(0.20)
−0.34
(0.008)
−0.51
(0.000)
−0.46
(0.000)
(0.030)
−0.21
(0.114)
−0.33
(0.011)
−0.43
(0.001)
−0.38
(0.004)
Back

Robustness Check: Dispersion and Volatility
(1) (2) (3)
IQR of Real Sales STD of Pdy (Durables) IQR of real sales
(Compustat) (Kehrig 2015) (Bloom et al. 2014)
Aggregate Volatility in TFP growth 0.2532
(0.0825)
0.3636
(0.0269)
0.3583
(0.030)
Aggregate Volatility in GDP growth 0.1911
(0.1932)
0.2923
(0.079)
0.3504
(0.034)
NOTE: In column (1) the Inter Quartile Range (IQR) of real sales is computed using
Compustat data from 1960 to 2008 for manufacturing ﬁrms. Nominal values are
deﬂated using the NBER-CES Manufacturing Industry Database 4-digits price index.
In column (2) we take the establishment-level median standard deviation of
productivity (levels) from Kherig (2015) who, in turn, computes it from Census data.
In column (3) we take the establishment-level IQR of sales growth from Bloom at al.
(2014).
Back

Banque de France's Workshop on Granularity: Basile Grassi's slides, June 2016

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