This document summarizes Marc Bielecki's research on modeling the impact of business cycle fluctuations on endogenous growth rates. The research aims to develop a single framework to analyze both business cycle and growth phenomena. Key contributions include a microfounded aggregate R&D intensity function and modeling of innovating, heterogeneous firms hit by aggregate and idiosyncratic shocks. Empirical evidence shows entry, expansions and contractions are procyclical. The model can replicate these features and shows that temporary productivity shocks can have permanent effects by shifting the long-run growth rate.

1. The impact of business cycle fluctuations
on aggregate endogenous growth rates
Marcin Bielecki
University of Warsaw and Narodowy Bank Polski
June 30, 2016

2. Motivation
Modern business cycle theory starts with the view that
growth and fluctuations are not distinct phenomena
to be studied with separate data and different analytical tools.
– Cooley and Prescott (1995) Economic Growth and Business Cycles

3. Motivation
Business cycles literature typically employs a variant
of the neoclassical growth model, where the trend growth
is assumed to be exogenous and constant over time
Endogenous growth literature typically features
no short-term fluctuations and focuses
on the balanced growth path results or the transition dynamics
My research program aims to fill the gap in the literature
by employing a single framework to analyze both business cycle
and growth phenomena and examine links between the two

4. Literature review
Seminal paper of Aghion and Howitt (ECTA 1992)
rekindles interest in Schumpeterian-type endogenous growth theory
Klette and Kortum (JPE 2004) develop a model
of product innovation performed by heterogeneous firms
Bilbiie et al. (JPE 2012) use the closed economy Melitz model
to relate endogenous firm entry decisions to the business cycle
Acemoglu and Cao (JET 2015) develop a model
with innovation performed both by incumbents and entrants
Comin and Gertler (AER 2006, NBER 2016 et al.)
use procyclical R&D intensity function
Gourio et al. (2016) finds long-lasting
negative effects of low entry on macroeconomic variables

5. Contributions
This paper: microfounded aggregate R&D intensity function
Economic contribution: evaluate the effects of temporary shock
on the endogenous growth rate and long-run BGP level
Technical contribution: modeling of innovating, heterogenous firms
hit by both aggregate and idiosyncratic shocks

6. Cyclical properties of establishment dynamics
Data from US BLS Business Employment Dynamics
Sample: 1992q3-2015q1
-15
-10
-5
0
5
10
15
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
Recession Expansions Contractions
-15
-10
-5
0
5
10
15

7. Cyclical properties of establishment dynamics
Data from US BLS Business Employment Dynamics
Sample: 1992q3-2015q1
2013
2014
-15
-10
-5
0
5
10
15
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
Recession Entries (Births) Exits (Deaths)

8. Results preview and intuition
0 20 40 60 80 100
Quarter
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Productivityshock(%)
0 20 40 60 80 100
Quarter
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
Qualitylevel(%)
Transitory shock shifts level of BGP path permanently
Despite lack of frictions, 3.5% of shock becomes permanent
Intuition: positive transitory shock temporarily encourages
both entry and incumbent R&D
Even after activity returns to normal, invented ideas remain
This creates a lasting shift in BGP level path

9. Setup overview
Closed economy version of Melitz and Redding (HIE 2014)
Discrete time
No capital, two types of labor
Skilled labor: “managers” and R&D, unskilled: production
Skilled with mass s and unskilled with mass 1 − s
Monopolistic competition, heterogeneity w.r.t. quality level
Establishments invest in R&D to raise their quality level
Both horizontal and vertical innovation
Endogenous exit, entry and R&D intensity

10. Households
Maximize utility function w.r.t. consumption ct and hours worked t:
U0 = Et
∞
t=0
βt c1−θ
t
1 − θ
− ψt
1+κ
t
1 + κ
(1)
where ψt increases over time to ensure constant BGP hours
subject to budget constraint where sht are firm share holdings:
ct + psh
t sht = wt t + sht−1 psh
t + πt + τt (2)
All households share the same stochastic discount factor Λt,t+1:
1 = Etβ
ct+1
ct
−θ
Λt,t+1
psh
t+1 + πt+1
psh
t
(3)

11. Final goods producers
Purchase intermediates yt (i) from Mt ∈ (0, 1) active establishments
Aggregate them into final goods Yt with elasticity of substitution σ:
Yt =
ˆ Mt
0
yt (i)
σ−1
σ
di
σ
σ−1
(4)
Aggregate price index Pt results from intermediates’ prices pt (i):
Pt =
ˆ Mt
0
pt (i)
1−σ
di
1
1−σ
(5)
Profit maximization yields conditional demand for i-th intermediate:
yt (i) = YtPσ
t pt (i)
−σ
(6)

12. Intermediate goods producers – static problem
Differ with respect to their quality level qt (i)
Hire f units of skilled labor to access production function
linear in stochastic productivity shock Zt and unskilled labor u
t (i):
yt (i) = Ztqt (i) u
t (i) (7)
Optimal pricing strategy:
pt (i) =
σ
σ − 1
markup
W u
t
Ztqt (i)
marginal cost
(8)
where W u
t is nominal unskilled wage
Real operating profit is linear in qt (i)
σ−1
:
πo
t (i) =
Yt
σ
σ − 1
σ
Zt
wu
t
σ−1
qt (i)
σ−1
− ws
t f (9)
where wu
t and ws
t are real wages of unskilled and skilled, respectively

13. Aggregation and relative quality φ
Aggregate quality index Qt, Melitz (ECTA 2003):
Qt =
1
Mt
ˆ Mt
0
qt (i)
σ−1
di
1
σ−1
(10)
Define establishment’s relative quality level φt (i):
φt (i) = (qt (i) /Qt)
σ−1
(11)
Real operating profit is linear in φt (i):
πo
t (i) =
Yt
σMt
φt (i) − ws
t f (12)
Establishment differences summarized by φ – drop subscript i

14. Intermediate goods producers – dynamic problem
Incumbents invest in R&D to improve their future quality:
φt+1 =
ι · φt/ηt with probability α
φt/ηt with probability 1 − α
(13)
where ι is innovative step size and ηt is aggregate growth rate
Success probability function α, Ericson and Pakes (RES 1995):
αt ( x
t , φt) =
a x
t /φt
1 + a x
t /φt
(14)
where a is skilled R&D labor ( x
t ) effectiveness parameter
Total profit is linear in φt:
πt (φt, αt) =
Yt
σMt
−
ws
t
a
αt
1 − αt
φt − ws
t f (15)

15. Recursive representation
Establishment chooses αt that maximizes the value function:
vt (φt) = max
αt ∈[0,1]
πt (φt, αt) +
Et [Λt,t+1 (1 − δt) vt+1 (φt+1|φt, αt)]
(16)
where Λt,t+1 is SDF and δt is endogenous exit shock probability
For high enough φ value function is linear in φ
– incumbents choose the same level of α – Gibrat’s law
Piecewise linear approximation makes problem tractable
System reduces to functions of two state variables:
exogenous shock Zt and endogenous establishment mass Mt

16. Recursive representation
Establishment chooses αt that maximizes the value function:
vt (φt) = max
αt ∈[0,1]
πt (φt, αt) +
max {0, Et [Λt,t+1 (1 − δt) vt+1 (φt+1|φt, αt)]}
(17)
where Λt,t+1 is SDF and δt is endogenous exit shock probability
For high enough φ value function is linear in φ
– incumbents choose the same level of α – Gibrat’s law
Piecewise linear approximation makes problem tractable
System reduces to functions of two state variables:
exogenous shock Zt and endogenous establishment mass Mt

17. Piecewise linear approximation along the φ dimension
Point where piecewise linear functions overlap is φ∗Policyfunctionα
true
approx.
φ∗
Relative quality level φ
Valuefunctionv

18. Incumbents’ policy function
0.95 1.00 1.05
Productivity shock Z
0.90
0.95
1.00
1.05
1.10
EstablishmentmassM
Success probability α (%)
47
48
49
50
51
52
53

19. Incumbents’ policy function
Productivity shock Z
0.90
0.95
1.00
1.05
1.10 Establishm
ent m
ass M
0.90
0.95
1.00
1.05
1.10
Successprobabilityα(%)
47
48
49
50
51
52
53

20. Entrants
Entry success function mirrors incumbents’ R&D function
Prospective entrants choose αe
t maximizing their value function:
ve
t = max
αe
t ∈[0,1]
−ws
t f e
+ 1
ae
αe
t
1−αe
t
+αe
t Et Λt,t+1vt+1 φe
t+1
(18)
Free entry condition allows for negative value of entry:
ve
t ≤ 0 (19)

21. Entrants’ policy function
0.95 1.00 1.05
Productivity shock Z
0.90
0.95
1.00
1.05
1.10
EstablishmentmassM
Entry rate Me
/M (%)
0
2
4
6
8
10
12
14
16

22. Entrants’ policy function
Establishment mass M
0.90
0.95
1.00
1.05
1.10 Productivity shock
Z
0.90
0.95
1.00
1.05
1.10
EntryrateMe
/M(%)
−2
0
2
4
6
8
10
12
14
16

23. Aggregate quality dynamics
Aggregate quality index improves at the following rate:
ηt = [1 + αt (ι − 1)]
(1 − δ) Mt + σ
σ−1 Me
t
Mt+1
(20)
Faster growth when higher αt and Me
t , ceteris paribus
Along the BGP, entry accounts for around 1/3
of quality improvements (see Acemoglu and Cao (JET 2015))

24. Calibration
Parameters chosen to match BGP stylized facts
Par. Description Value Justification
s Share of skilled workers 0.1 Ballpark estimate1
β Discount factor 0.99 Standard (quarterly)
θ Inverse of IES 2 Standard
κ Inverse of Frisch 4 Standard (macro)
σ Elasticity of substitution 4 Average markup ≈ 1.332
ι Innovative step size 1.015 Annual TFP growth ≈ 2%
a Incumbent R&D eff. 10 Expansions ≈ contractions
ae
Entrant R&D eff. 10 a = ae
f Incumbent labor req. 1 R&D employment ≈ 1%
f e
Entrant labor req. 1 Profits in income ≈ 5%
δexo
Exog. exit shock prob. 0.02 Exit rate ≈ 3% (quarterly)
1Acemoglu et al. (NBER 2013)
2Christopoulou and Vermeulen (EE 2012)

25. Cross-correlations with output
Data moments based on 1992q3-2015q1 sample (92 periods)
Model moments based on 10000 simulated periods
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Output, Establishments (+i)
Data
Model
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Output, Net entry (+i)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Output, Contractions (+i)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Output, Expansions (+i)
Good qualitative fit to the data
Excellent match for expansions
Slightly worse for contractions and net entry

26. Autocorrelations
Data moments based on 1992q3-2015q1 sample (92 periods)
Model moments based on 10000 simulated periods
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Data
Model
Good qualitative fit to the data
Excellent match for expansions
Slightly worse for contractions and net entry

27. Impulse response functions
Shock hysteresis on BGP: 3.5% of shock remains
0 20 40 60 80 100
Quarter
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Productivityshock(%)
0 20 40 60 80 100
Quarter
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Output(%)
0 20 40 60 80 100
Quarter
−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Netentry(%)
0 20 40 60 80 100
Quarter
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Establishments(%)
0 20 40 60 80 100
Quarter
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Expansions(%)
0 20 40 60 80 100
Quarter
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
Qualitylevel(%)
Quantitatively volatility smaller than in the data

28. Conclusions
Able to replicate qualitative business cycle features
of establishment dynamics
Lack of salient frictions generates smaller volatility than in the data
Found long-run effects of short-run fluctuations:
3.5% of shock is “permanent”
Frictions expected to be amplifiers
– obtained lower bound estimate on hysteresis
Implications for welfare costs of business cycles