Introduction Model Results Conclusions

A Framework for Analyzing the Impactof Business Cycles on Endogenous Growth

Marcin Bielecki

University of WarsawFaculty of Economic Sciences

PhD Students’ Seminar

March 16, 2015

Introduction Model Results Conclusions

Motivation

Business cycles literature typically employs a variant of theso-called neoclassical growth model, where the trend growth isassumed to be exogenous and constant over time

Endogenous growth literature typically abstracts from theshort-term fluctuations and focuses on the balanced growthpath results or the transition dynamics

My research program aims to fill the gap in the literature byemploying a single framework to analyze both business cycleand growth phenomena and seek links between the two

Introduction Model Results Conclusions

Literature review

The seminal paper of Aghion and Howitt (1992) rekindledinterest in Schumpeterian-type endogenous growth theory

Klette and Kortum (JPE 2004) develop a very rich andpowerful model of product innovation performed byheterogeneous firms

Acemoglu et al. (NBER 2013) provide further refinements byincluding firms’ heterogeneity with respect to their innovativecapacity

Bilbiie et al. (JPE 2012) use the closed economy Melitz modelto relate endogenous firm entry decisions to the business cycle

Introduction Model Results Conclusions

Households

Standard CRRA utility function

Ut =

∞∑τ=0

βτC1−θt+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 workerssupplying skilled labor Ls and a fraction 1− s supplyingunskilled labor Lu

Lt = L (3)

Lst = sL (4)

Lut = (1− s)L (5)

Introduction Model Results Conclusions

Final Goods Producer

A perfectly competitive representative firm producing finalgoods output from mass Mt of intermediate goods yit,where σ denotes the elasticity of substitution between varieties

Yt =

[∫ Mt

0yσ−1σ

it di

] σσ−1

(6)

There exists an associated price index P of the final good,where pit denotes the price of i-th variety

Pt =

[∫ Mt

0p1−σit di

] 11−σ

(7)

The resulting demand function for an intermediate good

yit = YtPσt p−σit (8)

Introduction Model Results Conclusions

Intermediate Goods Producers

There exists a mass Mt ∈ (0, 1) of active intermediate goodsproducing establishmentsEach period an establishment hires f units of skilled labor andgains 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 tothe (nominal) unskilled wage wt and inversely proportional tozt and qit, and the optimal pricing rule is in fact a constantmark-up applied to the marginal cost

pit =σ

σ − 1︸ ︷︷ ︸mark-up

wtztqit︸︷︷︸

marginal cost

(10)

Introduction Model Results Conclusions

Intermediate Goods Producers

An establishment’s operating profit can be expressed as follows

πoit =(σ − 1)σ−1

σσYtP

σt z

σ−1t w1−σ

t qσ−1it − wst f (11)

where wst is the skilled labor wage

Establishments can improve their goods’ quality by investingin R&D activities. The resulting improvements are bestthought of as process (rather than product) innovations

R&D costs are expressed as follows

cR&D (wst , xit) = wst · x (qit, Qt, αit) (12)

where xit denotes the demand for R&D capable labor, whichdepends on the establishment’s quality, the aggregate qualityindex of all intermediates Qt and the success probability αit

Introduction Model Results Conclusions

Success probability function

R&D intensity x0

1

Succ

ess

prob

abili

tyα

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 intensityxit

(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 issignificantly more productive than the others, it has hardertime generating new ideas, whereas the ones that are lessproductive 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)

Introduction Model Results Conclusions

R&D Decision

The success probability function can be inverted to yield thedemand 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 rewrittenusing the relative productivity variable φit as follows

πoit =PtYtσMt

φit − wst f (17)

The total profit equation is a linear function in φit

πit =PtYtσMt

φit − wst f︸ ︷︷ ︸operating profit

−wst1

a

(αit

1− αit

)φit (18)

Introduction Model Results Conclusions

Value Function

The profit function

πit =

[PtYtσMt

− wsta

(αit

1− αit

)]φit − wst f (19)

The value function

Vt (φit) = maxαit∈[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 =

ιφitηt

with probability αit

φitηt

with probability 1− αit(21)

where ι is the incremental innovation step size and ηt is the rate ofgrowth of the aggregate quality index

Introduction Model Results Conclusions

Value Function

I drop the subscript i since an establishment’s solutiondepends only on its relative quality variable φ. Also, I usethe ′ notation to denote the t+ 1 period’s variables

V (φ, Y,M, ωs) = maxα∈[0,1]

Y

{[1

σM− ωs

a

(α

1− α

)]φ− ωsf

}(22)

+ max{

0,E[β (Y ′/Y )

−θ(1− δ)V

(φ′, Y ′,M ′, (ωs)

′ |φ, Y,M, ωs, α)]}

where ωs ≡ ws/Y

Introduction Model Results Conclusions

Balanced Growth Path

Along the BGP the value function is linear (γ ≡ Y ′/Y )

V (φ) = maxα∈[0,1]

Y

{[1

σM− ωs

a

(α

1− α

)]φ− ωsf

}(23)

+ max

0,E

βγ1−θ (1− δ)︸ ︷︷ ︸ϑ

V(φ′|φ, α

)

Introduction Model Results Conclusions

R&D Intensity (Partial Equilibrium)

α∗ =

aσMωs − 1−ϑ

ϑηι−η

1 + aσMωs

(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)

Introduction Model Results Conclusions

Entry and General Equilibrium

A prospective entrant solves the following problem (the R&Dcost function is the same as for the φ = 1 incumbent)

VE = maxαE∈[0,1]

{αEβγ

1−θEV − 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 replacingan incumbent and 1− (1− δexo)M chance of starting a newproduct line, with δexo being the ‘pure’ exogenous exitprobability. The resulting incumbent survival probability is

(1− δ) = (1− δexo)[

1−M1− (1− δexo)M

](26)

Constant Mass of Firms

Labor Market Clears

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

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 co

stf

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Gro

wth

rateγ

+1.0

02

0.001

0.002

0.003

0.004

0.005

Introduction Model Results Conclusions

Entry Costs and Fixed Costs vs Active Establishments

Entry cost fE0.0

0.5

1.0

1.5

2.0

Fixed co

stf

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Mas

sof

acti

vepr

oduc

tlin

esM

0.04

0.05

0.06

0.07

0.08

0.09

0.10

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 co

stf

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Uti

lityU

−0.612

−0.610

−0.608

−0.606

−0.604

−0.602

−0.600

Introduction Model Results Conclusions

Growth and Business Cycles (Work In Progress)

V (φ, Y,M, ωs) = maxα∈[0,1]

Y

{[1

σM− ωs

a

(α

1− α

)]φ− ωsf

}+ max

{0,E

[β (Y ′/Y )

−θ(1− δ)V

(φ′, Y ′,M ′, (ωs)

′ |φ, Y,M, ωs, α)]}

Y = M1

σ−1ZQL

lnZ ′ = ρ lnZ + εZ

Along the business cycle, a change in Y can be caused by change inM , Q or Z. Thus, if changes in Z influence M or Q, then largevariation in Y may not require large variation in Z if theamplification mechanism is strong enough.

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

Introduction Model Results Conclusions

Preview of Future Results

The persistence and amplitude of exogenous shocks has animpact on the average growth rate of an economy

Entry costs have a decisive role in the behavior of entry alongthe business cycle

Fixed costs impact mainly Balanced Growth Path behavior

Hypothesis: given the nature of the exogenous shocks, thereis a welfare-optimal combination of f and fE

Introduction Model Results Conclusions

Thank you for your attention