A Framework for Analyzing the Impact of Business Cycles on Endogenous Growth

1. Introduction Model Results Conclusions
A Framework for Analyzing the Impact
of Business Cycles on Endogenous Growth
Marcin Bielecki
University of Warsaw
Faculty of Economic Sciences
PhD Students’ Seminar
March 16, 2015

2. Introduction Model Results Conclusions
Motivation
Business cycles literature typically employs a variant of the
so-called neoclassical growth model, where the trend growth is
assumed to be exogenous and constant over time
Endogenous growth literature typically abstracts from the
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 seek links between the two

3. Introduction Model Results Conclusions
Literature review
The seminal paper of Aghion and Howitt (1992) rekindled
interest in Schumpeterian-type endogenous growth theory
Klette and Kortum (JPE 2004) develop a very rich and
powerful model of product innovation performed by
heterogeneous firms
Acemoglu et al. (NBER 2013) provide further refinements by
including firms’ heterogeneity with respect to their innovative
capacity
Bilbiie et al. (JPE 2012) use the closed economy Melitz model
to relate endogenous firm entry decisions to the business cycle

4. Introduction Model Results Conclusions
Households
Standard CRRA utility function
Ut =
∞
τ=0
βτ C1−θ
t+τ − 1
1 − θ
(1)
No storage technology – the entire output is consumed
Ct = Yt (2)
Constant inelastic labor supply with a fraction s of workers
supplying skilled labor Ls and a fraction 1 − s supplying
unskilled labor Lu
Lt = L (3)
Ls
t = sL (4)
Lu
t = (1 − s) L (5)

5. Introduction Model Results Conclusions
Final Goods Producer
A perfectly competitive representative firm producing final
goods output from mass Mt of intermediate goods yit,
where σ denotes the elasticity of substitution between varieties
Yt =
Mt
0
y
σ−1
σ
it di
σ
σ−1
(6)
There exists an associated price index P of the final good,
where pit denotes the price of i-th variety
Pt =
Mt
0
p1−σ
it di
1
1−σ
(7)
The resulting demand function for an intermediate good
yit = YtPσ
t p−σ
it (8)

6. Introduction Model Results Conclusions
Intermediate Goods Producers
There exists a mass Mt ∈ (0, 1) of active intermediate goods
producing establishments
Each period an establishment hires f units of skilled labor and
gains access to the following production technology
yit = ztqit it (9)
where zt is an aggregate productivity shock,
qit is an establishment-specific quality parameter,
and it denotes units of employed unskilled labor
An establishment’s (nominal) marginal cost is proportional to
the (nominal) unskilled wage wt and inversely proportional to
zt and qit, and the optimal pricing rule is in fact a constant
mark-up applied to the marginal cost
pit =
σ
σ − 1
mark-up
wt
ztqit
marginal cost
(10)

7. Introduction Model Results Conclusions
Intermediate Goods Producers
An establishment’s operating profit can be expressed as follows
πo
it =
(σ − 1)σ−1
σσ
YtPσ
t zσ−1
t w1−σ
t qσ−1
it − ws
t f (11)
where ws
t is the skilled labor wage
Establishments can improve their goods’ quality by investing
in R&D activities. The resulting improvements are best
thought of as process (rather than product) innovations
R&D costs are expressed as follows
cR&D (ws
t , xit) = ws
t · x (qit, Qt, αit) (12)
where xit denotes the demand for R&D capable labor, which
depends on the establishment’s quality, the aggregate quality
index of all intermediates Qt and the success probability αit

8. Introduction Model Results Conclusions
Success probability function
R&D intensity x
0
1
Successprobabilityα

9. Introduction Model Results Conclusions
R&D Decision
The success probability αit is a function of R&D productivity
parameter a and the adjusted R&D intensity
xit
(qit/Qt)σ−1
(similar approach to Ericson and Pakes (RES 1995))
α (xit, qit, Qt) =
a xit
(qit/Qt)σ−1
1 + a xit
(qit/Qt)σ−1
(13)
The idea behind the adjustment is that if an establishment is
significantly more productive than the others, it has harder
time generating new ideas, whereas the ones that are less
productive can imitate the successful establishments
At this point I introduce a new variable φit ≡ (qit/Qt)σ−1
, so
that
α (xit, φit) =
axit/φit
1 + axit/φit
(14)

10. Introduction Model Results Conclusions
R&D Decision
The success probability function can be inverted to yield the
demand function for R&D capable labor
α (xit, φit) =
axit/φit
1 + axit/φit
(15)
x (φit, αit) =
1
a
αit
1 − αit
φit (16)
The operating profit of an establishment can be rewritten
using the relative productivity variable φit as follows
πo
it =
PtYt
σMt
φit − ws
t f (17)
The total profit equation is a linear function in φit
πit =
PtYt
σMt
φit − ws
t f
operating profit
−ws
t
1
a
αit
1 − αit
φit (18)

11. Introduction Model Results Conclusions
Value Function
The profit function
πit =
PtYt
σMt
−
ws
t
a
αit
1 − αit
φit − ws
t f (19)
The value function
Vt (φit) = max
αit∈[0,1]
πit (φit)
Pt
+ max {0, Et [Λt,t+1Vt+1 (φi,t+1|φit, αit)]}
(20)
where Λt,t+1 = β Yt+1
Yt
−θ
(1 − δ) is the stochastic discount factor
with δ denoting the incumbent exit probability and
φi,t+1 =



ιφit
ηt
with probability αit
φit
ηt
with probability 1 − αit
(21)
where ι is the incremental innovation step size and ηt is the rate of
growth of the aggregate quality index

12. Introduction Model Results Conclusions
Value Function
I drop the subscript i since an establishment’s solution
depends only on its relative quality variable φ. Also, I use
the notation to denote the t + 1 period’s variables
V (φ, Y, M, ωs
) = max
α∈[0,1]
Y
1
σM
−
ωs
a
α
1 − α
φ − ωs
f (22)
+ max 0, E β (Y /Y )
−θ
(1 − δ) V φ , Y , M , (ωs
) |φ, Y, M, ωs
, α
where ωs
≡ ws
/Y

13. Introduction Model Results Conclusions
Balanced Growth Path
Along the BGP the value function is linear (γ ≡ Y /Y )
V (φ) = max
α∈[0,1]
Y
1
σM
−
ωs
a
α
1 − α
φ − ωs
f (23)
+ max



0, E

βγ1−θ
(1 − δ)
ϑ
V φ |φ, α






14. Introduction Model Results Conclusions
R&D Intensity (Partial Equilibrium)
α∗
=
a
σMωs − 1−ϑ
ϑ
η
ι−η
1 + a
σMωs
(24)
R&D intensity α∗ is the larger:
the lower is the number of active establishments M
the higher is the innovative step size ι
the closer to 1 is the discounting factor ϑ
the lower is the high skilled wage relative to output ωs
the higher is the unit R&D productivity a
the lower is the aggregate quality index growth rate η
(good for numerical stability)

15. Introduction Model Results Conclusions
Entry and General Equilibrium
A prospective entrant solves the following problem (the R&D
cost function is the same as for the φ = 1 incumbent)
VE = max
αE∈[0,1]
αEβγ1−θ
EV − ws
fE +
1
aE
αE
1 − αE
(25)
Free Entry Condition ensures that VE = 0
A successful entrant has a (1 − δexo) M chance of replacing
an incumbent and 1 − (1 − δexo) M chance of starting a new
product line, with δexo being the ‘pure’ exogenous exit
probability. The resulting incumbent survival probability is
(1 − δ) = (1 − δexo)
1 − M
1 − (1 − δexo) M
(26)
Constant Mass of Firms
Labor Market Clears

16. Introduction Model Results Conclusions
General Equilibrium – Numerical Solution Procedure
1 Compute αE
2 Guess M and ωs
3 Compute δ
4 Jointly determine α and η in a loop
5 Update M and ωs and iterate steps 2-5 until convergence

17. Introduction Model Results Conclusions
Entry Costs and Fixed Costs vs Growth
Entry cost fE
0.0
0.5
1.0
1.5
2.0
Fixed
costf1.0
1.2
1.4
1.6
1.8
2.0
2.2
Growthrateγ+1.002
0.001
0.002
0.003
0.004
0.005

18. Introduction Model Results Conclusions
Entry Costs and Fixed Costs vs Active Establishments
Entry cost fE
0.0
0.5
1.0
1.5
2.0
Fixed
costf
1.0
1.2
1.4
1.6
1.8
2.0
2.2
MassofactiveproductlinesM
0.04
0.05
0.06
0.07
0.08
0.09
0.10

19. Introduction Model Results Conclusions
Entry Costs and Fixed Costs vs Welfare
Entry cost fE
0.0
0.5
1.0
1.5
2.0
Fixed
costf1.0
1.2
1.4
1.6
1.8
2.0
2.2
UtilityU
−0.612
−0.610
−0.608
−0.606
−0.604
−0.602
−0.600

20. Introduction Model Results Conclusions
Growth and Business Cycles (Work In Progress)
V (φ, Y, M, ωs
) = max
α∈[0,1]
Y
1
σM
−
ωs
a
α
1 − α
φ − ωs
f
+ max 0, E β (Y /Y )
−θ
(1 − δ) V φ , Y , M , (ωs
) |φ, Y, M, ωs
, α
Y = M
1
σ−1 ZQL
ln Z = ρ ln Z + εZ
Along the business cycle, a change in Y can be caused by change in
M, Q or Z. Thus, if changes in Z influence M or Q, then large
variation in Y may not require large variation in Z if the
amplification mechanism is strong enough.

21. Introduction Model Results Conclusions
Preview of Future Results
Impulse response functions for output under the RBC model (red)
and this model (blue)
0 5 10 15 20 25 30 35 40
-0.5
0
0.5
1

22. Introduction Model Results Conclusions
Preview of Future Results
The persistence and amplitude of exogenous shocks has an
impact on the average growth rate of an economy
Entry costs have a decisive role in the behavior of entry along
the business cycle
Fixed costs impact mainly Balanced Growth Path behavior
Hypothesis: given the nature of the exogenous shocks, there
is a welfare-optimal combination of f and fE

23. Introduction Model Results Conclusions
Thank you for your attention