Who Quits Next?
Firm Growth in Growing Economies
Julieta Caunedo
Emircan Yurdagul∗
Cornell University
Universidad Carlos III
June 2015
Abstract
This paper provides a theory linking characteristics of the firm dynamics to the
nature of aggregate growth in an economy. We analyze firms’ life-cycle productivity,
age-employment profiles, and firm selection across countries. Using a large cross-country
dataset, we document (i) more frequent labor productivity growth for firms operating
in fast growing economies, (ii) younger firms in faster growing economies, (iii) lack
of systematic relationship between the tail of the employment size distribution and
growth, and (iv) steeper age-employment profiles in slow growing economies. We build
a tractable general equilibrium model that displays endogenous long run growth compatible with a stationary size distribution and the documented empirical facts. We
explore the role of i) intensive and ii) extensive margin firms’ investment for economic
growth. We show that disparities in the extensive margin of firm growth, the number
of firms successfully innovating in a period of time, account for about a third of crosscountry growth disparity. The extensive margin is tightly related to the age distribution
of firms observed in an economy. The intensive margin of firm growth is closely related
to rank reversals in firm productivity growth but not necessarily aggregate economic
growth.
Keywords: Productivity, Firm Selection, Economic Growth.
JEL codes: E23, O4.
∗
Contact: julieta.caunedo@cornell.edu or yurdagul@wustl.edu
1
1
Introduction
Entrepreneurs, in braving uncertainty, face the risk of being replaced if their ventures are
not sufficiently successful given the performance of other firms in the market. Selection
and return uncertainty are key in determining the characteristics of firms operating in
an economy and aggregate economic growth. Economies where i) firms invest little
or ii) few firms invest in productivity, may grow slower not only because the levels
of investment are low, but also because the process of firm selection in the market
slows down. This paper explores the role of i) intensive and ii) extensive margin firms’
investment for economic growth. It provides a theory that links aggregate growth to
firm dynamic characteristics.
In particular, we argue that patterns of firm dynamic, such as characteristics of the
age distribution of firms and cross sectional employment age profiles, hold information
as of the nature of economic growth. Using a cross country dataset we document the
relationship between growth and features of the industry dynamic. We build a stylized
general equilibrium model of industry dynamic with endogenous firm investment and
selection (entry and exit) consistent with those patterns. We show that disparities in the
extensive margin of firm growth, the number of firms successfully innovating in a period
of time, account for about a third of cross-country growth disparity. The extensive
margin is tightly related to the age distribution of firms observed in an economy. The
intensive margin of firm growth is closely related to rank reversals in firm productivity
growth but not necessarily aggregate economic growth. The intensive margin of firm
growth relates to employment-age profiles across firms, i.e. average employment per
firm age.
Firm investment behavior is key in understanding the nature of economic growth.
When investing, firms face two different layers of uncertainty: a) on whether a given
outcome is realized (i.e. introduction of a new product to the market); but also, conditional on success, b) on the return to any given investment (i.e. profits associated to the
2
new product in the market). Often times, ventures that entail high uncertainty in both
dimensions, such as those associated to disruptive technologies, are the ones pushing
growth the most1 . We argue that disparities in the probability with which firms turn
their investments into actual labor productivity growth (probability of success hereafter), and uncertainty on returns if successful, can explain observed differences in firm
employment growth, age distribution, selection and ultimately, aggregate growth.
We start by describing cross-country patterns of aggregate growth and industry dynamics2 : (i) establishments’ labor productivity growth (productivity growth hereafter),
(ii) firm size and employment distribution, and iii) firm size and age in the cross-section.
First, the frequency of firm level productivity growth is higher in countries with a higher
growth rate of GDP per capita. Second, firms are younger in faster growing economies.
Third, the share of firms that are large, employment-wise, does not vary systematically with aggregate economy growth3 . Finally, there is a non-monotonic relationship
between the steepness of the age-employment profiles and aggregate growth rates. In
particular, we show that the cross sectional employment age profile in slow growing
economies can, on average, be steeper than that in faster growing economies.
The first fact is to be expected if aggregate growth is connected to the activity of
the firms operating in the market. However, it is not tautological. It is possible for
a larger share of firms to be growing at relatively low rates, which would induce slow
rather than fast aggregate growth. The second empirical regularity is explained by
more severe selection in faster growing economies (higher firm turn around via entry
and exit). The last two are less understood and important in describing the nature of
aggregate growth.
The presence of large firms (employment-wise) in a frictionless economy, is related to
their relative productivity versus the average in the economy4 . If there are economies
1
See Christensen (1997) for a detailed description disruptive technologies.
We use the 2006-2011 Enterprise Survey data by the World Bank.
3
Formally, the estimated tail of the employment distribution (linear in log-log space) does not vary
systematically with growth.
4
In an economy with frictions such as Restuccia and Rogerson (2008), the size of the firm can or
2
3
of scale, and large firms are relatively more productive, they could function as an
engine for growth. Our empirical evidence and previous literature suggest otherwise.
The literature finds that small young firms account for a large portion of productivity
gains in an economy (Haltinwanger, et.al., 2010, Eslava & Haltiwanger, 2012). Such
finding is consistent with the lack of correlation that we find in our data. In terms of
mechanisms that could generate the lack of correlation, in this paper we point that it is
necessary to distinguish between those that generate generate rank reversals in the firm
productivity space (and hence employment) from those that do not, but may improve
average productivity.
Steep age-employment profiles in slow growth economies are at odds with a pure selection theory. Such theory predicts positive correlation between selection and the speed
of growth both at the firm level and in the aggregate. Two issues arise. First, the data
analysis is cross sectional, so these profiles cannot be interpreted as firm employment
growth paths necessarily (longitudinal data). Instead, they reflect the fact that in slow
growing economies, relatively large older firms coexist with small young ones. Second,
aggregate growth in the economy is related to the pace of average productivity improvements, whereas firm employment growth, reflects firm productivity improvements
relative to the mean. There are two types of slow growing economies: very unproductive poor ones, and very productive rich ones. In the former, steep employment age
profiles are consistent with weak selection, which allows small unproductive firms to
survive, and coexist with large slow growth firms. In the latter, this fact is related
to strong selection, young small firms entering and exiting the market but sometimes
experiencing high productivity growth relative to the mean. If those successful firms
are relatively few, average output growth is low.
In order to understand the relationship between the probability of success, return
uncertainty and the highlighted empirical facts, we build a tractable model with encannot be correlated with it’s productivity. Employment would be determine by a combination of
productivity and distortions.
4
dogenous firm selection. In the model, firms are allowed to invest to grow. In our
benchmark, whether investment returns are realized or not is uncertain. Towards the
end of the paper, we allow further uncertainty in returns, consistent with productivity
rank reversals. In the model, firms can be innovative and attempt to improve productivity, or alternatively, they can operate with a constant productivity technology. At the
beginning of each period, innovative firms decide whether or not to invest in technology
improvements and how much to produce in the current period. If they invest, they
pay a cost that is a non-linear function of their investment and get returns one period
ahead with some probability. The probability of success is the same for all the firms in
the market, but the realizations of course need not be. Innovative firms can also decide
to liquidate, at which point their technology is taken over by a non-innovative firm.
Non-innovative firms decide whether to stay in the market, produce and pay operating
costs, or exit.
The study of a general equilibrium economy is critical for our purposes for two
reasons. First, in an economy with higher probability of positive returns to investment,
firms’ productivity growth would be faster; which implies that relatively unsuccessful
firms exit quicker (strong selection). Hence, growth in this economy will be higher
than the level predicted by a simple partial equilibrium model where selection is not
accounted for. Second, suppose we compare two economies, one with high probability
of success and one with low probability of success. As long as returns to investment are
common across firms operating in an economy (possibly proportional to productivity),
the allocation of employment and sales among firms may not differ between the low
and how probability of success economies. This helps our model replicate the absence
of a systematic relationship between growth and the allocation of employment across
firms.
In our paper, the key fundamental difference across economies is the probability
of realizing returns to investment. Uncertainty of such realization relates to political
instability, changes in tax regimes, changes in terms of trade. Such uncertainty has a
5
direct impact on the life-cycle of firms through the frequency of growth episodes. The
probability of success also alters the expected return per dollar invested, and hence the
size of those investments in equilibrium. Together, these two channels will generate firm
productivity growth profiles that differ across economies. We identify the probability of
success in the data via law of large numbers. We argue that at the top of the employment
distribution, the probability of success is proxy by the number of firms who successfully
increase their productivity in a year. Preliminary computations show that variations in
the probability of success have quantitative impact on both the rate of investment at
the firm level and aggregate growth of the economy. Moreover, it determines the share
of firms investing in productivity growth, as well as the equilibrium measure of firms
operating in the market changes.
The existence of a balance growth path poses tight restrictions on equilibrium productivity growth of firms operating in the market.Without return uncertainty conditional of investment success, there is no rank reversal in equilibrium, which in turn
implies no employment growth. Intuitively, firms’ productivity grows at a constant
rate along the balance growth path. If the firm is successful, it grows at everybody’s
rate. Hence, its employment (which depends on its relative productivity vis a vis the
average) does not change. If instead the firm is unsuccessful, employment shrinks, as
it is now relatively less productive than before. When we augment our benchmark
economy to allow return uncertainty (and hence for rank reversals), we show that the
probability of success jointly with firm uncertainty are relevant to determine the relationship between age-employment profiles and firms’ age distribution with aggregate
growth in the economy. Furthermore, return uncertainty and success probability jointly
determine the pace of selection in the market and the decoupling between average firm
growth and aggregate growth. However, once the model is able to replicate disparities
in employment-age profiles observed in the economy, it’s predictive power for differences
in aggregate growth is very similar to the one obtained by varying the probability of
success alone.
6
Related literature. This paper provides a theory that links characteristics of the
industry dynamics to aggregate growth, consistently with the patterns documented in
the data. Allocation of factors across firms has been shown to be key in understanding aggregate productivity differences across countries, and through them, income per
capita (Hsieh and Klenow (2009), Restuccia and Rogerson (2008)). Until recently,
most of the literature that studies the link between the micro structure of the economy
and productivity focused on static allocations. However, static allocations can reflect
firm level distortions that can have lasting effects on incentives to innovation, and firm
growth (as in Hsieh and Klenow (2012), Da-Rocha, Tavares, and Restuccia (2014) and
Cole, Greenwood, and Sanchez (2012)) and ultimately, aggregate growth (as in Akcigit,
Alp, and Peters (2014), Peters (2011)).
Peters (2011) links static misallocation (through markup variation) with innovation
incentives and growth. Akcigit, Alp, and Peters (2014) provides a theory of firm dynamics in developing countries based on contracting frictions that prevent managerial
delegation in weak institutional environments, i.e. poor economies. Both Akcigit, Alp,
and Peters (2014) and Peters (2011) follow the Klette and Kortum (2004) tradition in
that firms innovate by deciding the frequency of upgrades. Distinctively, we fix the
frequency of innovation episodes, but we allow firms to choose the intensity in the improvement. This is important because in an environment where firms only decide the
frequency of upgrades, steep employment age profiles are always coupled with faster
growing economies. As mentioned before, this fact is at odds with the data. We are
able to accommodate slow growing economies and steep employment age profiles by
decoupling success rates from return uncertainty.
Hsieh and Klenow (2012) analyze employment age profiles in India, Mexico and the
US to argue that about 25% of the differences in aggregate TFP can be accounted by
disparities in firm employment growth. When analyzing a general equilibrium model
with endogenous productivity growth (as in Atkeson and Burstein (2010)), they adjust returns to innovation in each country by the revenue distortions uncovered from
7
the Mexican and Indian data. Because employment profiles are flatter in these countries, implied tax distortions are higher for firms with higher productivity. Higher tax
distortions for more productive firms, induce the model to predict flatter employment
profiles. In this paper we take an alternative path. We argue that it is not the actual
return that is lower for more productive larger firms, but the expected return, once
adjusted by uncertainty. Whereas the same technologies might available in developed
and developing countries, their expected returns are different. Furthermore, our result
holds even assuming that uncertainty on returns is the same irrespective of firm size.
Slower growth as the firm ages is not generated by productivity dependent penalties
on innovation, but through a general equilibrium effect via endogenous selection, i.e.
entry and exit.
Another piece of research that argues that while the same technologies might be
available across countries, features of the environment in which firms operate can generate disparities in the technological ladders that firms adopt, is Cole, Greenwood,
and Sanchez (2012). The authors assert that poorer financial markets can explain the
disparities in total factor productivity and employment size distribution. Contracting problems are undoubtedly a relevant mechanism that explains flatter productivity
profiles. Akcigit, Alp, and Peters (2014), analyze contracting frictions that affect the
ability of entrepreneurs to delegate managerial activities. Such friction can explain
why it is optimal in certain economies for firms to remain small through time. Improvements in the contractual environment improve the net benefit of delegation and
induces firm growth. In our paper, we vary the probability of obtaining returns to any
given investment, conditional on the investment taken place. The presence of contracting frictions is definitely an important mechanism that would affect what we call the
probability of success. There are other sources that our framework can also accommodate. For example, the existence of ”social capital” as documented in Knack and
Keefer (1997), and trust relationships, documented in Bloom and Sadun (2012); or the
degree of technology experimentation, which has been shown key to predict innovation
8
(Thomke (2003)). By keeping this reduced form approach, we gain model tractability
and intuition as of the link between these alternative sources of uncertainty, the industry dynamics and growth. Certainly, the mechanism underlying firm uncertainty need
to be further understood.
There is also a growing literature that studies firm growth in general equilibrium and
its impact on aggregate allocation. Atkeson and Burstein (2010) study an open economy
with endogenous growth. They focus on the impact of trade barriers on the equilibrium
entry and exit rates and incentives to innovate. In their economy, firms choose the
probability of realizing an improvement in productivity of a given size. Distinctively,
our model takes the probability of success as a fundamental of the economy, and lets
firms choose their path of productivity improvement. This is important because it
allows us to fully characterize the equilibrium firm size distribution, which is key in
validating our results.
Our paper is also related to the literature studying growth as the outcome of technological investment of heterogeneous firms operating in the market. Luttmer (2010)
provides conditions such that the thick tail in the firm size distribution widely documented in the data can arise as the outcome of firms’ growth under uncertainty. In
our paper, we assume an initial distribution of productivity that has such a tail and
hence we are silent about the process that originates it. This is also a feature of Perla
and Tonetti (2014) that study diffusion of technology and growth in an economy with
heterogeneous firms. Two features differentiate our model from theirs. First, when
firms decide to invest in searching new technologies, the technology that is assigned to
them is purely exogenous. In our model, firms decide on the optimal level of technology that they would like to run. Second, in Perla and Tonetti (2014) growth is lead
by the firms at the bottom of the size distribution, who find it profitable to search
better technologies. Bigger and more productive firms do not innovate. In our model,
growth is lead by successful and fast growing firms. Small firms in the market invest
in technology and only if successful, they turn large. If unsuccessful, they may opt out
9
of innovation. The smallest firms in the market survive for a finite number of periods,
and then endogenously exit.
The rest of the paper is organized as follows. Section 2 gathers empirical evidence
across countries. Section 3 presents a stylized model that is consistent with those facts.
Section 4 gives the analytical results from the model. Section 5 gives a quantitative
assessment of the model. Section 6 introduces an extension of our benchmark model by
allowing for additional idiosyncratic shocks affecting the size of firm growth. Section 7
concludes.
2
Evidence
We use the standardized dataset for 2006-2011 of the World Bank Enterprise Surveys
(ES). We use data for 92 countries and for each country we pick the most recent survey
available. For aggregate statistics such as income we use the Penn World Table 8.0.
When we split countries into growth quartiles, we consider the average growth rate
of GDP per capita (cgdpo/pop) since 2000. We omit countries with negative average
growth rate in that interval.
Since we argue that firms’ probability of success has implications on income levels,
we first analyze sales and employment information in the ES dataset. In particular,
our objective is to show how the frequency of firms with positive growth in labor
productivity relates with output growth across countries. The fraction of firms that
have positive productivity growth does not correspond to the probability of success
in attempts to increase productivity, a variable that we do not observe in the data.
However, we expect such fraction to be increasing in the probability of success. High
probability of success would increase the frequency of episodes with productivity growth
for a given set of firms operating in the market. Selection may induce firms to exit more
often, but if anything those would be relatively unsuccessful firms. Hence, we expect
the fraction of firms with productivity growth to be increasing in the probability of
10
.1
Figure 1: Fraction of firms with productivity growth and GDP per capita growth
KAZ
MNG
ZWE
TJK
OMN
CHN
GEO
GABMDA
GDP per capita growth
.05
0
AGO
UKR
VNM BLR
ZMB
ROMARM UZB
EST
PER MOZ
ECU
RWA
TZA LVA BGR
ETH
MNE
PAN BTN
SUR
SVK LTU
ALB
TUR
CHL
SRB
POL
DOM
MRT
BIH
GHA
LKA
IDNUGAHRV
ZAF BWA
THA
DMA
HUN
BOL
MWI
COL
GMB
SLE
PRY
LCA
BLZ
CZE
BGD
BDI
BFA
ZARMEX
NAM
PAK
HND
BRA
GRD
SVN
URY
GNB
NPL
MLI
ATG
TGO
GTM
LSO
CRI
BEN PHL
NER
JAM
MAR
CAF CMR
BRB
KEN
SEN MDG MUS
FJI
TCD
LBR
SWZ
-.05
GIN
SLV
Correlation=0.3175
20
40
60
% with productivity growth
80
100
success.
To compute growth in labor productivity, we compute the ratio of sales per worker
in the last year, and three years before the survey. Figure 1 shows this relationship
between the fraction of firms with positive sales-per-worker growth within the two years
and average annual GDP growth as described before. The correlation coefficient is 0.32
in spite of the large variation across countries. This initial set of evidence suggests that
there is a link between probability of success and income growth.
Second, we focus on the firm age across countries. Figure 2 plots the average firm
age against the growth rate of GDP per capita in our sample. It shows that faster
growing countries tend to have younger firms (with correlation -0.11).
Next, we turn to the relationship between the distribution of employment among
firms and income growth. For this, we construct the Pareto tails for each country
following Axtell (2001). We are interested in the tail of the distribution rather than
other summary statistics for two reasons. First, the largest firms in each country
contribute disproportionately more to value added, and explain most of the changes
11
Figure 2: Average firm age and GDP per capita growth
30
ZWE
LBN
ARG
BOL
CHL
GTM
URY
HND
LKA
GRD
BRA IRL
SUR
20
JAM
CRI
SLV
PAKPRY
DEU
DOM
PRTESP
MEX
GRC
SRB
BIH
PHL SVN BLZ
POL
COL TURPAN
PER
CMR ATG ZAR
ECU
SLE
OMN
HRV
IND
MYS
BWA
THA
ZMB
ZAF
GHA
MUS
MOZTCD
IDN
MWI
BLR
DMA
BRB
BGD
HUN JOR BTN
NER
LCA
UKR TJK
LSO
GAB
BGR UZB
CZE
SVK EST
ROM
BFA
LVA
UGA LTU
CAF MLI
MNE
KEN
CHN
MDA
TZA
SEN
BEN NAM
GNB
IRQ
AGO
NPL
TGO
SWZ
MRT ETH ARMVNMGEO
GMB
RWA
BDI
GIN
LBR
ALB
MDG
10
Mean age
FJI
AZE
MNG
NGA
KAZ
KHM
-.05
0
.05
Correlation=-0.1257
.1
.15
GDP per capita growth
in aggregate output (as in Carvalho and Gabaix (2013)). Second, the survey considers
formal firms for most countries. In the presence of informal firms, the nature of this data
could bias the analysis of the relationship between the employment size distribution
and GDP growth. Our results are robust to analyzing firms that are larger than 5
employees.5
Our estimates for the tails of the distribution go from 1 to 4. A higher parameter
indicates a thinner tail, i.e. there are relatively few large establishments employmentwise. Figure 4 shows that the thickness of the tail in the employment size distribution
lacks systematic relationship with GDP growth (with correlation -0.06). In addition, it
5
In order to get the Pareto tail indices, we first form 15 employment categories, where category i
corresponds to 3i−1 integers. (The first category includes firms with 1 employee, the second category
includes firms with 2 to 4 employees, so on.) For each country, we drop the categories that have less
frequency that the one with higher order. Then we regress the logarithm of the number of firms on
the logarithm of the median point of each category. The coefficient of the latter is the tail index
for that economy. Pareto distribution captures the allocation of employment in these tails very well.
Specifically, the average R-squared statistic in the regressions that we run to get the tail indices is
0.977.
12
.15
Figure 3: Tail indices and GDP per capita growth
NGA
GDP per capita growth
.05
.1
AZE
KAZ
LBN
ZWE
MNG
TJK
OMN
GEO
0
BLR
CHN
AGO
UKR
MDA
GABVNM
TCDUZB
ZMB
ROM
ARM
EST
MOZ PER
ECU
BGR
RWA
ETH
MNE
PANTZA
SUR SVK BTN LVA
LTU
ALB
TUR
CHL
SRB
POL
JOR
BIH DOM
GHA MRT
BWA
UGA
IDN LKA
MYS
ZAFHUN
HRV
THA
DMA
BOL
MWI
COL
GMB
IRL
SLE
PRY
ARG
LCA
BLZ
CZE
BGD
BDI
BFA
ZAR
MEX
NAM
PAK
DEU BRA HND SVN
GRCESP
GRD
URY
LBR
GNB NPL
PRT
MLI
ATG
TGO
GTM
LSO
BEN CRI
PHL
NER
JAM
CMR
MAR
CAF
BRB
KEN
SEN
MUS
MDG
FJI
SWZ
KHM
IND
-.05
GIN
SLV
1
1.5
2
Correlation=-0.0763
2.5
Tail index
also lacks systematic relationship with the fraction of firms with positive productivity
growth (with correlation -0.08).
Finally, we analyze firm employment growth and its relationship to aggregate output
growth. We construct age-employment profiles by using the cross sectional dimension of
our dataset. In particular, we compute the predicted employment by fitting a quadratic
polynomial of age on employment for each particular growth quartile. In order to make
profiles comparable across growth groups we normalize to 1 the predicted employment
at age 5. Figure 5 shows that faster growing economies do not necessarily have steeper
employment profiles. In fact, the figure illustrates that the slowest quartile have the
steepest employment profile with age, and the fastest quartile has the flattest.
We close this section by highlighting the four empirical facts we document here.
First, there is a positive correlation between frequency of productivity growth at the
firm-level and the growth rate of income per capita. Second, the latter growth rate is
not correlated with the tail of employment size distribution. Third, there is a negative
correlation between average firm age and the growth rate of income per capita. Finally,
13
100
Figure 4: Tail indices and fraction of firms with productivity growth
% with productivity growth
40
60
80
ZWE
AGO
BLR
BGR
GEO
MNE
MAR
MUS
HRV
LBR
MLI
ZMB
SRB
AZE
UZB
GHA EST
MWI
SLE
SVNUKR
TJK
LTU
LVA
VNM
MNG
KAZ
ARM
CZE
MEX
LKA
BIH
URY
TZA
NAM
MDA
ROM
SVK
BRA BWA
SWZ JAM
KEN
CHL ZARGAB
POL
MOZ
ZAFHUN BOL
BRBCOL
TUR
NGA
THA
ECU
PAK
GMB
CMR
BDI
GTM
PRY
BTN OMN
LCA
GIN
PER
GNB
LSO
SLV
CAF
PHL
MDG
PAN
UGA
BEN
TCD
IDN
BFA
FJI BGD SEN
NPL
DMACHN GRD
TGO
HND
DOM
ALB
BLZ
MRT
NER
ETH
CRI
RWA
ATG
20
SUR
Correlation=-0.0904
1
1.5
2
2.5
Tail index
employment age profiles are not necessarily steeper in faster growing economies. If
anything, faster growing economies tend to have flatter employment profiles.
14
.8
Average employment (normalized, age 5=1)
1.2
1.6
1
1.4
1.8
Figure 5: Age-employment profiles by growth quartile
0
10
20
30
40
50
60
70
Age
0%-1.7%
3
1.7%-2.8
2.9%-4.8%
5%+
Model
In the economy, there is an infinitely-lived representative consumer that derives utility
from consumption of a final good y.
There is a continuum of heterogeneous firms. Each firm operates a technology that
uses labor l to produce a homogeneous good y. The technology displays decreasing
returns in labor, and has a Hicks neutral productivity shifter z ζ . Hence, after paying
workers their wage rate w the firm bears positive profits that are paid as dividends to
the representative consumer in the economy.
There are two alternative type of firms. The non-innovative ones, whose productivity
level z is constant in time; and the innovative ones, whose productivity can change via
investment. The technology for improvements in productivity is such that whenever a
firm with productivity z undertakes an investment φ in period t, the productivity in
period t + 1 is z 0 = φz with probability q and z otherwise. The probability of success
15
in investment returns is the same for all firms operating in the economy. Investment
is costly, as characterized by C(φ, z, w) = c φ
τ zη
θ
.
w 1−θ
There is an inelastic supply of labor equal to 1. A firm has an overhead cost of labor
of fj , j = {N, I} that depends on whether or not it belongs to the innovative group.
At any point in time a non-innovative firm can decide to exit the market at no
cost. An innovative firm has also an option of liquidation. If the firm is liquidated and
turn into a non-innovative project it receives a scrap value equal to the expected value
of the non-innovative firm with the same productivity z. If the innovative firm exits
it gets a scrap value of zero, so liquidation and transformation into a non-innovative
project is preferable as long as the non-innovative projects are valuable. Finally, firms
are liquidated exogenously at rate δ after production and investment takes place.
Notice that the mirror process of that of liquidation of innovative projects is that
of entry into the non-innovative sector. For a new firm to be created into the noninnovative sector a price P (z, w) is paid. If there is free entry in the market, this price
equals the expected value of the firm, which is the scrap value received by the liquidated
innovative firm. Fundamentally, the model dictates that less productive firms will tend
to imitate incumbents in the innovative sector (as they inherit productivity z).
Finally, we allow endogenous entry to the innovative sector. Any entrant starting a
new firm pays the entry cost is xs w, and draws a productivity from a Pareto distribution
with threshold zs > 0. We assume that zs is smaller than the productivity level of the
least productive firm in the economy, and grows at the same rate with the average
productivity in the economy.
We now characterize the optimization of each of the agents in the economy. We can
solve the problem of the firms in two parts. First we solve for the allocation of labor
given the distribution of productivities in the market. Then we solve for the dynamic
decisions of the firm, which include technology investment.
16
3.1
Firm’s problem (Static)
The static problem of a firm is:
Π(z, w) = max z ζ lθ − wl
l
where θ is the share of labor in production.
Then the optimality gives
l=
For notational convenience, define η ≡
θz ζ
w
1
1−θ
ζ
.
1−θ
The total labor supply that can be used for productive purposes is equal to 1 − M f ,
if there are M firms operating in the market (M = MI + MN where MI is the measure
of innovative firms and MN that of non-innovative firms). We can solve for the cost of
labor
θ
w=
(1 − (fN MN + fI MI ))1−θ
Z
η
Z
I
z dv (z) +
η
N
1−θ
z dv (z)
where v I (v N ) is the equilibrium distribution of productivities for innovative (nonR
innovative) projects. By definition, Mj = dv j (z) for j = I, N .
Using the equilibrium cost of labor we can characterize profits, labor demand and
output for an arbitrary firm with productivity z:
zη
θ
Π(z, w) = (1 − θ)θ 1−θ
θ
w 1−θ
1
l(z, w) = θ 1−θ
y(z, w) = θ
zη
1
w 1−θ
θ
zη
1−θ
θ
w 1−θ
17
3.2
3.2.1
Firm’s Problem (Dynamic)
Non-Innovative Firm
Let VN (z, w) be the value of a firm operating in the non-innovative sector, when the
cost of labor is w and its productivity is z. The value of a firm operating in sector N
satisfies:
θ
VN (z, w) = (1 − θ)θ 1−θ
zη
w
θ
1−θ
− fN w +
1−δ
max{0, VN (z, γw)}
R
Hence, the value of the firm is negative if and only if the the flow value in that period
is negative. i.e.
zη
θ
VN (z, w) ≤ 0 ⇔ (1 − θ)θ 1−θ
θ
w 1−θ
− fN w ≤ 0.
Notice that the only decision for this firm is when to exit the market, conditional
on survival.
3.2.2
Innovative firms
An innovative firm has the same static revenues as a non-innovative firm, but is different
in two important aspects. First it can engage in risky investment in productivity which
results in successful innovation only with probability q. Second, at any period it can
sell its technology to an entrant into the non–innovative sector. We can write the value
of an innovative firm as:
VI (z, w) = max(1 − θ)θ
θ
1−θ
φ≥1
+
zη
θ
w 1−θ
−c
φτ z η
θ
w 1−θ
− fI w
1−δ
[q max{V (φz, w0 ), P (z, w0 )} + (1 − q) max{V (z, w0 ), P (z, w0 ))}]
R
Here we assume that if a firm sells the technology in a period, this nullifies any
improvement in the technology that was made on the productivity in the last period.
18
This is analogous to assume that the liquidation of the firm takes place before the firm
knows whether their technology investment where successful or not.
If the firm would like to liquidate, at the beginning of every period it meets one
potential entrant that will use its technology z to operate a non–innovative firm. We
assume that the latter makes a take-it-or-leave-it offer P (z, w) to the former to buy the
entire technology.
As long as the technology’s worth as an N -firm is more than its worth as an I-firm,
it is optimal for the entrant to offer minimum price that leaves the I-firm indifferent
between selling or not. If the technology is more valuable as an I-firm, the entrant
is indifferent between offering any amount less than VI (z, w) because any price in this
range will imply that the transaction will not go through; and any price above that
gives negative payoff to the N -firm. Without loss of generality, we assume that in this
case the offer reads the price equal to the value of the technology as an N -firm:
P (z, w) = min{VI (z, w), VN (z, w)}
and the innovative firm accepts if and only if
P (z, w) ≥ VI (z, w)
In other words the transaction occurs when its surplus is positive. With this pricing
scheme, we have max{V (φz, w0 ), P (z, w0 )} = V (φz, w0 ) and max{V (z, w0 ), P (z, w0 )} =
V (z, w0 ). We assume that there is an infinite mass of potential entrant non-innovative
firms. Hence, an innovative firm makes its technology investment decisions as if it were
operating an infinitely lived firm.
θ
VI (z, w) = max(1 − θ)θ 1−θ
φ≥1
zη
w
θ
1−θ
−c
φτ z η
w
θ
1−θ
− fI w +
1−δ
[qV (φz, w0 ) + (1 − q)V (z, w0 )]
R
When we describe the solution of the model, we will show that any innovative firm
19
is bound to sell its technology to an entrant non-innovative firm in finite number of
periods.
4
Solution of the model
For tractability reasons, we focus our attention on the balanced growth path.
Definition 1 A balanced growth path (BGP) in this economy is a sequence of aggregate
output, a measure of aggregate productivity and wages that grow at a constant rate.
Additionally, we require the existence of an invariant distribution of firm productivities
except possibly for a time trend.
To characterize the BGP we guess there exist one. Under this assumption we solve
for the optimal policies of the firms and show the existence of an invariant distribution.
Once we solve for the equilibrium distribution and allocation of firms across innovative
and non-innovative projects, we compute the equilibrium growth rate for aggregate
output and wages. Both are constant which confirms the existence of the BGP.
Aggregate output in the economy can be characterized by
Y (w) =
w
θ
hence along a BGP output and wages have to grow at the same rate. i.e.
[Yt , wt ] = [Y, w]γ t
for some γ ≥ 1. Using the growth rate for wages we can go back to the problem of the
firms.
20
4.1
4.1.1
Solution of firms’ problems
Non-Innovative Firms
Define the function for the number of periods left in the market as:
θ
1
t
T (z, w) ≡ max{t : (1 − θ)θ 1−θ z η ≥ fN w 1−θ γ 1−θ }.
Here notice that we need to subtract 1 period since the first term gives the first period
that the firm will not operate in the market. We will simply denote this as T .
Proposition 1 Along the BGP, the value of a firm at the beginning of a period can be
characterized by
VN (z, w) = BN T
zη
θ
w 1−θ
− DN T w.
where for any T ≥ 1:
BN T = (1 − θ)θ
θ
1−θ
T −1 X
1−δ
t=0
DN T
R
γ
θ
− 1−θ
1−
t
= (1 − θ)θ
θ
1−θ
1−
1−δ
T
θ
Rγ 1−θ
1−δ
θ
Rγ 1−θ
T
t
T −1 X
1 − 1−δ
γ
1−δ
R
=f
γ = fN
1−δ
R
1
−
γ
R
t=0
and for T = 0 both coefficients are 0.
If the functions BN T and DN T are as characterized above the non-innovative firm
exits the market in T periods, and VN (z, w) satisfies the Bellman equation which assures
optimality.
21
4.1.2
Innovative firms
We guess that the value of an innovative firm is
VI (z, w) = BI
zη
− DI w
θ
w 1−θ
Replacing it in the Euler equation yields
BI
+
zη
θ
w 1−θ
zη
θ
− DI w = max(1 − θ)θ 1−θ
φ≥1
θ
−c
w 1−θ
φτ z η
θ
w 1−θ
− fI w
1−δ
φη z η
zη
0
[q(BI
−
D
w
)
+
(1
−
q)(B
− DI w0 )]
I
I
θ
θ
0
0
R
w 1−θ
w 1−θ
Then, solving the FOC with respect to φ yields
"
z 0 = φz =
1−δ
qBI η
R
1
# τ −η
θ
z
cτ γ 1−θ
Substituting back
"
BI η = η(1 − θ)θ
θ
1−θ
− c(η − τ )
1−δ
qBI η
R
cτ γ
θ
1−θ
τ
# τ −η
+
1−δ
(1
R
− q)BI η
θ
γ 1−θ
gives the first equation relating φ and γ through BI . Meanwhile, the other component
of the value function, DI reads:
DI =
fI
.
1 − 1−δ
γ
R
Here, an important reminder is that we assumed interiority in this solution. In
particular, we assumed that the firm with optimal investment rule as defined above,
and successful innovation, remains in the market next period. We will come back to
this later, and show that a firm that innovates successfully remains in the market.
22
4.1.3
Sale of the innovative firm to an entrant non-innovative firm
Proposition 2 For every innovative firm, there exists a finite time that the technology
will be sold to an entrant non-innovative firm, if the firm is not hit with an exogenous
exit shock until then.
As shown in the proof in the Appendix, for this mechanism to be optimal for the
firms we need (i) the surplus in the transaction to increase in unsuccessful innovations.
Hence, unlucky firms eventually would like to liquidate. We also need that (ii) for a
sufficiently large z relative to w the surplus is negative (a very productive firm does not
find profitable to liquidate), and (iii) for a sufficiently small z relative to w the surplus
is positive. If this is true there is a finite point in time such that an innovative firm is
traded. Hence, the population of both types of firms is non-degenerate.
4.2
Distribution dynamics
Define the threshold productivity such a non innovative firm exits in exactly in t periods,
Z(t, w) as
1
t
fN w 1−θ γ 1−θ
Z(t, w) =
! η1
θ
(1 − θ)θ 1−θ
The lower bound in productivity for firms operating in the market is z̃, which solves:
θ
1
(1 − θ)θ 1−θ z̃ η = fN w 1−θ
In fact, z̃ = Z(0, w). Under the BGP with growth rate γ, this implies that
z̃ 0 = z̃µ
23
1
where µ ≡ γ η(1−θ) . Also, the productivity threshold for selling for an innovative firm ẑ,
equals
ẑ = Z(t̂, w)
which also grows at rate µ.
Proposition 3 If the initial distribution of productivities in the market is Pareto with
shape parameter λ, and entrants in the innovative sector draw their productivity from
the incumbent distribution
1. the growth rate of the threshold levels, µ, is the same as the investment rate, φ.
2. the equilibrium distribution of productivities across innovative firms is also Pareto
with shape parameter λ
This result implies that
1
φ = γ η(1−θ) ,
which is the second equation relating γ and φ. It also shows that there is an invariant
distribution of firmsin the market. It remains to be shown that the relative population
of innovative and non-innovative projects in the market is constant along the BGP. Let
α be the proportion of firms in the market in the innovative sector, i.e. α ≡
MI
.
M
Proposition 4 The share of innovative firms in the market is constant along the BGP
and solves
(1 − α) = α(1 − q)[1 − µ−λ ]
1−δ
[1 − (1 − δ)t̂ ]
δ
Finally, we give the implications of the free-entry condition in our model. Remember
that the assumptions regarding the entry of innovative firms are (i) new entrants are
assumed to draw their productivity from a Pareto distribution with threshold zs < ẑ,
(ii) they pay the entry cost before their productivity draw is realized, and (iii) the
24
threshold zs grow at the same rate as the average productivity. Hence, the free-entry
condition reads:
(1 − Fs (ẑ))(E(VI (z, w)|z > ẑ) + Fs (ẑ)0 = BI
E (z η |z < ẑ) (1 − Fs (ẑ))
θ
w 1−θ
− DI w = xs w
where Fs is the cumulative distribution function of new entrants. Using the properties
of the Pareto distribution, we have:
λ zg λ ẑ η
− DI = xs
BI
1
λ − η ẑ
w 1−θ
Notice that:
ẑ η
w
Since we obtain
z̃ η
1
1
1−θ
=
z̃ η
w
1
1−θ
(1)
µ−ηt̂ .
from the exit condition of the marginal non-innovative firm,
w 1−θ
and t̂ from the exit condition of the marginal innovative firm; equation (1) gives us the
threshold productivity level ẑ given the distribution parameter of the new entrants, zs .
5
Quantitative exploration
This section is split in two parts. First, we show the quantitative properties of our
model, particularly showing how the variation in the probability of success shape aggregate implications. Then, we present our calibration exercise and document how this
variation alone can account for cross-country differences in the growth rates and other
relevant macroeconomic outcomes.
5.1
Role of the probability of success in aggregate outcomes
In showing the quantitative properties of the model, we set the parameters other than q
to their benchmark calibrated values that we explain in the calibration. Then we vary
25
Figure 6: Probability of success and aggregate growth
0.12
0.1
GDP growth
0.08
0.06
0.04
0.02
0
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
q
q to show the sensitivity of various aggregate moments to the probability of success.
First of all, Figure 6 shows that there is a strong relationship between the probability
of success and the implied aggregate growth rate of the economy.
Higher probability of success also makes survival more difficult, especially for the
non-innovative firms that are simply stuck with their current productivity level and
observing the wage rate in the economy increase as the income level grows. This can
be seen in Figure 7, which illustrates shortening life-time of non-innovative firms in
economies with higher levels of probability of success.
Higher probability of success affects the allocation of firms into (non-)innovative
activity not only through a shorter life-span for non-innovative firms, but also because
it changes the likelihood that the innovative firms lag behind their same-sector counterparts. The former channel increases the relative measure of innovative firms in the
economy. However, a higher growth rate might make being an innovative firm a less
26
Figure 7: Probability of success and average survival
80
70
Life−time of N−firms
60
50
40
30
20
10
0
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
q
appealing option, relative to selling the technology to a non-innovative firm, because of
the additional costs of investment in innovation. Hence it is possible to have a U-shaped
relationship between the probability of success and the measure of innovative firms, as
Figure 8 shows.
5.2
Calibration
Our calibration strategy is to set as many parameters as possible following the conventional values in the literature, while we replicate the average features of our sample for
key moments. We assume that the cost of innovative activity is a quadratic function of
the attempted jump in productivity. In setting the weights of productivity and labor,
we assume that the production function exhibits limited span-of-control in a specific
manner:
ζ η
y = z l = zl
27
θ
ζ
ζ
.
Figure 8: Probability of success and fraction of innovative firms
0.86
0.84
0.82
Fraction of I−firms
0.8
0.78
0.76
0.74
0.72
0.7
0.68
0.66
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
q
We set the span-of-control parameter equal to 0.85, a value that is standard in the
literature.6 With our approach to the production function, this gives ζ =0.85. Using
the standard labor share of 0.66, we set θ equal to 0.66×0.85=0.561. We set the risk-free
interest rate, r, equal to 0.07, which is the average for our sample countries in the World
Development Indicators dataset provided by the World Bank. For the tail parameter
of the Pareto distribution, we use the average of our estimates for each country that we
use in Section 2.
The main novelty of our calibration strategy is in determining the probability of
success for each country in the sample. In our benchmark model, all the innovative
firms try to innovate and they succeed with probability q. In case of not succeeding,
small innovative firms that are already at the edge of exiting might endogenously leave
the market; however, larger firms remain in the regardless of their success in innovating.
6
See, for instance, Midrigan and Xu (2010). Other studies using similar values include Buera,
Kaboski, and Shin (2011) with a value of 0.79 and Cagetti and De Nardi (2006) with 0.88.
28
Hence, according to our model, at the top of the size distribution the fraction of firms
that grow in productivity should be equal to q.7 Accordingly, we look at the top 10
percent of the size distribution for each country, and take the frequency cases with of
productivity growth to be equal to 1 − (1 − q)1/2 . This gives us a q for each country in
the sample.
Our remaining parameters are the level parameter for the investment cost, c, and the
exogenous exit rate, δ. Since our calibration exercise is aimed at matching the average
profile among our sample-economies, we calibrate these two parameters to match the
average growth and average firm age in our sample. Calibrated model parameters are
summarized in Table ??.
Parameter
Value
Basis
Investment cost steepness, τ
2
Quadratic
Weight of productivity, ζ
0.85
Span of control
Weight of labor, θ
0.56
0.66× Span of control
Interest rate, R − 1
0.07
WDI
Overhead costs, (fI , fN )
(0, 0.25)
Pareto tail, λ
1.96
Average in the sample
Investment cost (level), c
0.17
Average growth 3.7%
Exogenous exit rate, δ
0.015
Average firm age 15.3
Once we calibrate our model parameters to match the average growth and firm
age in our sample, our objective is to show how the model accounts for the crosscountry variation in these moments.8 We start by showing the model implications of
the differences in the probability of success on the aggregate growth. In line with the
earlier discussion of how the variation in q relates to the aggregate growth, Figure
9 shows that the model implies a monotone relationship between the probability of
7
Notice that exogenous exit shocks might also hit firms. However, since these shocks are independent
of the size and the innovation status of the firm, they do not matter for this equality.
8
Here, we focus attention on the countries with estimated q larger than 0.2, so that we avoid
negative implied growth rates. This costs us 10 countries in the sample.
29
GDP growth (model)
Figure 9: Probability of success and aggregate growth, model
0.1
0.05
0
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
q
success and the growth rate.
The main question, however, is how well these model-implied growth rates square
with the ones observed in the data. This is depicted in Figure 10. Model explains 32
percent of the variation in the growth rates through the variation in q.9
Next, we turn our attention to the variation in the firm age across countries. First,
Figure 11 shows that model implies that firms generally get younger as countries’ growth
rates are higher. This is qualitatively in line with empirical evidence we document
regarding the same relationship in our sample. To see how the model fits the observed
variation in average firm age, we plot the model implied average age against the data
counterparts in Figure 12. Specifically, the model explains about 10 percent of the
variation in average firm age across countries.
Even though the benchmark model does a decent job in explaining the cross-country
9
Specifically, 0.32 is the slope of the linear fit in Figure 10.
30
Figure 10: Aggregate growth, model fit
45−degree line
linear fit
NGA
GDP growth (data)
AZE
0.1
KAZ
MNG
ZWE
TJK
CHN
GAB
AGO
MDA UKR
BLR
VNM
TCD
UZB ZMB
ROM
ARM
SLV
PER
MOZ
0.05
ECU
BGR
TZA
EST
MNE
LVA
PAN
BTNSVK
LTU
TUR
CHLPOL
SRB
BIH
GEO
BWA
UGA
LKA
IDNHRV
ZAF
THA
HUN
BOL
MWI
COL
FJI BFA BGD
SLE
PRY
LCA
BLZ
CZE
BDI MEX 491
384 PAKSVN
GHA NAM
HND
URY BRA
LBR
NPL
MLI
TGO
GTM
LSO
CRI
BEN
PHL
JAM
0
ETH
0
0.05
0.1
0.15
GDP growth (model)
variation in growth rates of output per capita, a major shortcoming of this simple framework is that firms can never increase their ranking in the productivity ladder, hence are
never able to grow in size, employment-wise. We document this feature of the model in
Figure 13, which shows the benchmark model’s quadratic fit of employment as a function of age. The lack of employment growth at the firm level is an undesirable property,
which is a simple artifact of the increase in productivity of successful firms being equal
to the growth rate of the average productivity. Absent idiosyncratic movements among
successful firms, none of them can climb up the ladder.
In the next section, we extend our benchmark economy by introducing idiosyncratic
productivity shocks, and show how it can help the model generate the age-employment
profiles observed in the data.
31
Figure 11: Average firm age and aggregate growth, model
19
18.5
Average age (model)
18
17.5
17
16.5
16
15.5
15
14.5
14
0
0.02
0.04
0.06
0.08
0.1
0.12
GDP growth (model)
Figure 12: Average firm age, model fit
ZWE
30
45−degree line
linear fit
BOL
25
CHL
Mean age (data)
FJI
GTM
URY
LKA
BGD
HND
BRA
20
PAK
PRY
JAM
CRI
SLV
MEX
SRB
BIH
SVN
15
10
5
14
PHL
POL
COL
PER
SLE
PAN
TUR
491
HRV
BWA
ZMB
ZAF THA
GHA
IDN
MWI TCDBTNMOZ
BLR
HUN LSO
UKR TJKLCA
AZE
GAB
BGR
UZB
CZE
SVK
EST
ROM
BFA
LTU
LVA
UGA
MLI
MNG MNE
CHN
MDA
TZA
NAM
AGO VNMARM GEO
NPL ETH
TGO
NGA
KAZ
384
BDI
LBR
15
16
17
Mean age (model)
32
18
19
20
Figure 13: Age-employment patterns, model
Labor−Age profile no shocks, normalized quadratic fit
1.05
model
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0
5
10
15
20
33
25
30
35
40
6
Extension: Idiosyncratic investment returns
In this section, we add further uncertainty in productivity increments. In particular,
we assume that if the firm successfully innovates, the actual jump in its productivity
depends on an i.i.d. shock in addition to the investment that it makes:

 φεz, with probability q;
0
z =
 z,
with probability 1 − q.
where ε v U (ε, ε).
Our strategy for calibrating (ε, ε) is to minimize the distance between the average
age-employment profile in the data and in the model. Figure 14 shows this fit with the
calibrated (ε, ε), which are equal to (0.955, 1.100). We do not change the parameter
values used in the benchmark, including the calibrated parameters c and δ.
With this modification of the benchmark model, the model is ready for the comparison of the variation in the age-employment patterns with the data.10 For illustration
purposes, we group countries into four quartiles of income growth; the practice that we
followed for the data counterpart in Section 2. Figure 15 shows the age-employment
profiles in these four groups. In particular, it documents that the model also generates
non-monotonicity of the steepness of these profiles as in the data. In particular, the
steepest profile is exhibited by the slowest group, and the flattest one is that of the
fastest, which is also the case in the data.
10
Most of the analytical results in the benchmark follow with this extension, including the linear
policy rule for the investment, φ. We will include the modifications in the results in the appendix of
later drafts.
34
Figure 14: Age-employment patterns, model with idiosyncratic shocks
Employment−age profiles, normalized quadratic fit
1.4
1.35
1.3
1.25
1.2
1.15
1.1
1.05
1
0.95
0.9
0
5
10
15
20
25
30
35
40
Figure 15: Age-employment profiles by growth quartile, model with idiosyncratic shocks
Employment−age profiles, normalized quadratic fit
3
2.5
1.1%
3.1%
4.2%
6.7%
2
1.5
1
0.5
0
0
5
10
15
20
35
25
30
35
40
7
Final remarks
We build a stylized model of firm dynamics with growth where the aggregate growth
of the economy is linked to the probability of success that firms face in their returns to
investment in firm growth.
In particular, our model highlights two channels through which such probability
affects firm-level as well as aggregate growth. First, lower probability of success has
an impact on firm growth directly by decreasing the frequency of successful growth
episodes in a firm’s life-cycle, and indirectly by lowering incentives to invest in growth.
This partial equilibrium mechanism, increases the firm-level and aggregate growth in
economies with higher probability of success.
What is also relevant, and helpful to fit important features in the data, is the general
equilibrium channel. Economies with higher probability of success in turning investments into actual growth will have higher growth, making survival harder. Average
productivity will grow faster, and the distribution of productivity will shift upwards,
possibly preserving its shape. In addition to accounting for an important part of the
cross-country variation in growth rates these features enable our model replicate two important features of our cross-country data: younger firms in faster growing economies,
and the lack of systematic relationship between the tail of the size distribution and
growth.
Firm level uncertainty is built into our benchmark model only through the probability of realizing positive returns to investment. In order to see the link between
such probability of success and the firm level patterns of employment growth, we extend our benchmark economy allowing for return uncertainty. We are able to generate
age-employment patterns comparable to those in the data.
36
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Journal of Political Economy, 114(5), 835–870.
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American Economic Review, 103(5), 1697–1727.
Christensen, C. (ed.) (1997): The Innovators Dilemma. Harvard Business Review
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St. Louis 2012-040, Federal Reserve Bank of St. Louis.
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and Aggregate Productivity with Endogenous Establishment-Level Productivity,”
Working Papers tecipa-523, University of Toronto, Department of Economics.
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Papers 18133, National Bureau of Economic Research, Inc.
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38
8
Appendix A
Proposition 2 (Proof ).
We first argue that BI > BN t̂ .
θ
BI = max(1 − θ)θ 1−θ − cφτ +
φ≥1
1−δ
1
1
[qBI φη θ + (1 − q)BI θ ]
R
γ 1−θ
γ 1−θ
θ
≥ (1 − θ)θ 1−θ − c +
1−δ
1
BI θ
R
γ 1−θ
Meanwhile,
θ
BN t̂ < (1 − θ)θ 1−θ +
1−δ 1
θ BN t̂
R γ 1−θ
Hence for c small enough,
θ
(1 − θ)θ 1−θ − c
BI ≥
> BN t̂ .
1
1 − 1−δ
θ
R
γ 1−θ
Under the condition that fN − fI is not too large, we also know that DI > DN T since
DI − DN T
t
t
T −1 ∞ X
X
1−δ
1−δ
= fI
γ − fN
γ .
R
R
0
0
Next, we use the last findings to show that the surplus is increasing unsuccessful innovations, in other words it decreases with T (z, w). Define the surplus function:
s(z, w) ≡ −VI (z, w) + VN (z, w) = −(BI − BN T )
zη
θ
w 1−θ
+ (DI − DN T )w
Then in one unsuccessful innovation the surplus becomes:
s(z, wγ) = −(BI − BN T −1 )
zη
θ
(wγ) 1−θ
39
+ (DI − DN T −1 )wγ
= −(BI −(BN T −(1−θ)θ
θ
1−θ
θ
1 − δ − 1−θ
γ
R
s(z, wγ) − s(z, w) = −(BI − BN T )
−
zη
θ
(1 − θ)θ
θ
1−θ
(wγ) 1−θ
−
1−δ
R
zη
))
θ
(wγ) 1−θ
zη
w
zη
w
T −1
))wγ
(γ − 1−θ − 1) + (DI − DN T )w(γ − 1)
T −1
+ fN
1−δ
γ
R
T −1
wγ
θ
θ
1−θ
T −1
+(DI −(DN T −f
1−δ
γ
R
θ
θ
1−θ
θ
1 − δ − 1−θ
γ
R
s(z, wγ) − s(z, w) = −(BI − BN T )
T −1
(γ − 1−θ − 1) + (DI − DN T )w(γ − 1)
zη
θ
[(1 − θ)θ 1−θ
w
θ
1−θ
γ
θ
T
1−θ
− fN wγ T ]
We know γ > 1, BI > BN T and DI > DN T . Hence, the first line in the equation below
is positive. Moreover, by definition of T ,
zη
θ
(1 − θ)θ 1−θ
w
θ
1−θ
γ
θ
T
1−θ
− fN wγ T ≤ 0.
Hence s(z, wγ) − s(z, w) is positive. This shows that the surplus increases in every
unsuccessful innovation. We know that for z and w such that T (z, w) = 0, we have
θ
(1 − θ)θ 1−θ
zη
w
θ
θ
1−θ
w 1−θ − fN w ≤ 0
BN 0 = 0, DN 0 = 0
therefore
θ
VI (z, w) = (1 − θ)θ 1−θ
zη
w
θ
1−θ
−c
φτ z η
θ
w 1−θ
Therefore, for T = 0 the surplus is positive. Moreover,
limT →∞ DN T = DI
40
− fN w < 0.
and
BN T < BI
so that there is large enough T such that the surplus is negative. This shows that there
exists t̂ < ∞ such that an I-firm sells the technology to an entrant N -firm.
Proposition 2 (Proof ).
We first argue that BI > BN t̂ .
θ
BI = max(1 − θ)θ 1−θ − cφτ +
φ≥1
1−δ
1
1
[qBI φη θ + (1 − q)BI θ ]
R
γ 1−θ
γ 1−θ
θ
≥ (1 − θ)θ 1−θ − c +
1−δ
1
BI θ
R
γ 1−θ
Meanwhile,
θ
BN t̂ < (1 − θ)θ 1−θ +
1−δ 1
θ BN t̂
R γ 1−θ
Hence for c small enough,
θ
(1 − θ)θ 1−θ − c
BI ≥
> BN t̂ .
1
1 − 1−δ
θ
R
γ 1−θ
Under the condition that fN − fI is not too large, we also know that DI > DN T since
DI − DN T
t
t
T −1 ∞ X
X
1−δ
1−δ
= fI
γ − fN
γ .
R
R
0
0
Next, we use the last findings to show that the surplus is increasing unsuccessful innovations, in other words it decreases with T (z, w). Define the surplus function:
s(z, w) ≡ −VI (z, w) + VN (z, w) = −(BI − BN T )
41
zη
θ
w 1−θ
+ (DI − DN T )w
Then in one unsuccessful innovation the surplus becomes:
s(z, wγ) = −(BI − BN T −1 )
= −(BI −(BN T −(1−θ)θ
θ
1−θ
θ
1 − δ − 1−θ
γ
R
s(z, wγ) − s(z, w) = −(BI − BN T )
−
zη
θ
(1 − θ)θ
θ
1−θ
(wγ) 1−θ
−
1−δ
R
+ (DI − DN T −1 )wγ
θ
(wγ) 1−θ
T −1
zη
))
θ
(wγ) 1−θ
zη
w
zη
w
T −1
))wγ
(γ − 1−θ − 1) + (DI − DN T )w(γ − 1)
T −1
+ fN
1−δ
γ
R
T −1
wγ
θ
θ
1−θ
T −1
+(DI −(DN T −f
1−δ
γ
R
θ
θ
1−θ
θ
1 − δ − 1−θ
γ
R
s(z, wγ) − s(z, w) = −(BI − BN T )
zη
(γ − 1−θ − 1) + (DI − DN T )w(γ − 1)
zη
θ
[(1 − θ)θ 1−θ
w
θ
1−θ
γ
θ
T
1−θ
− fN wγ T ]
We know γ > 1, BI > BN T and DI > DN T . Hence, the first line in the equation below
is positive. Moreover, by definition of T ,
zη
θ
(1 − θ)θ 1−θ
w
θ
1−θ
γ
θ
T
1−θ
− fN wγ T ≤ 0.
Hence s(z, wγ) − s(z, w) is positive. This shows that the surplus increases in every
unsuccessful innovation. We know that for z and w such that T (z, w) = 0, we have
θ
(1 − θ)θ 1−θ
zη
w
θ
θ
1−θ
w 1−θ − fN w ≤ 0
BN 0 = 0, DN 0 = 0
therefore
θ
VI (z, w) = (1 − θ)θ 1−θ
zη
θ
w 1−θ
42
−c
φτ z η
θ
w 1−θ
− fN w < 0.
Therefore, for T = 0 the surplus is positive. Moreover,
limT →∞ DN T = DI
and
BN T < BI
so that there is large enough T such that the surplus is negative. This shows that there
exists t̂ < ∞ such that an I-firm sells the technology to an entrant N -firm.
Proposition 3 (Proof ).
Define:
XI ≡ EI (z η )
XN ≡ EN (z η )
Also define the cumulative distribution function for the group I as:

 1−
FI (z) =
 0,
ẑ λ
z
, if z ≥ ẑ;
o/w.
Moreover, let MI and MN be the number of firms in each group, which will be
MI
. To show that φ =
MI +MN
max{ẑ, µφ ẑ}, and a ≡ z̄ẑ . Then
constant in the BGP, let α ≡
along the BGP. Define z̄ ≡
XI0
Z
= (1 − δ)q
z̄
∞
µ we first show that µ ≤ φ
λẑ λ
(φz) λ+1 dz + (1 − δ)(1 − q)
z
η
Z
∞
zη
ẑ 0
+[δ + (1 − δ)qFI (z̄) + (1 − δ)(1 − q)FI (ẑ 0 )]XI0
R∞
R ∞ η λẑλ
λ
q z̄ (φz)η zλẑ
λ+1 dz + (1 − q) ẑ 0 z z λ+1 dz
=
1 − qFI (z̄) − (1 − q)FI (ẑ 0 )
43
λẑ λ
dz
z λ+1
q
=
η
φ
µ
aλ−η + (1 − q)µ−λ
qaλ + (1 − q)µ−λ
µη
λ η
ẑ
λ−η
For N-firms we need to keep track of all the endogenous exits from group-I for the
last t̂ periods. Define:
Z
k(ž) ≡ q
ž
ž/a
λž λ
(φz) λ+1 dz + (1 − q)
z
η
µž
Z
zη
ž
λž λ
dz
z λ+1
λ
λ
φη [1 − aλ−η ]ž η + (1 − q)
[1 − µη−λ ]ž η
= q
λ−η
λ−η
Then:
ẑ
XN = (1 − δ)t̂ k(ẑµ−t̂ ) + (1 − δ)t̂−1 k(ẑµ−t̂+1 ) + .. + (1 − δ)k( )
µ


X
t̂−1
λ
λ
=
q
φη [1 − aλ−η ] + (1 − q)
[1 − µη−λ ]  (1 − δ)t+1 µ−η(t+1)  ẑ η
λ−η
λ−η
t=0
λ
λ
1 − (1 − δ)t̂ µ−ηt̂ η
=
q
φη [1 − aλ−η ] + (1 − q)
[1 − µη−λ ] (1 − δ)µ−η
ẑ
λ−η
λ−η
1 − (1 − δ)µ−η
Moreover:

α
=
w0
=
γ=
w
αXI0 + (1 − α)XN0
αXI + (1 − α)XN
1−θ
=
1−θ
Pt̂−1
t+1 −ηt
η t̂
η
λ−η
η−λ
(1
−
δ)
µ
µ
+
(1
−
α)
qφ
[1
−
a
]
+
(1
−
q)[1
−
µ
]
t=0
qaλ +(1−q)µ−λ

P
µη 
t̂−1
t+1 µ−ηt
αµηt̂ + (1 − α) (qφη [1 − aλ−η ] + (1 − q)[1 − µη−λ ])
t=0 (1 − δ)
η
φ
q( µ
) aλ−η +(1−q)µ−λ
Since γ = µη(1−θ) :
η
φ
q
aλ−η + (1 − q)µ−λ = qaλ + (1 − q)µ−λ
µ
Which gives
φ
=a≤1⇒φ≤µ
µ
44
We prove µ ≤ φ by contradiction. In particular, suppose that µ > φ, i.e., threshold productivity grows at a faster rate than the productivity growth of the successful
innovative firms. Then, a measure FI (ẑ µφ ; t) − FI (ẑ; t) innovative firms will not be able
to stay in the market even if they successfully increase their productivity. This implies
that these firms would not invest in productivity growth, since the increments for the
exiting firms are nullified by assumption.
Notice that then the value of a firm with productivity z ∈ [ẑ, ẑ µφ ] the value is:
zη
θ
V I (z, w) = (1 − θ)θ 1−θ
w
− fw − c
θ
1−θ
zη
w
θ
1−θ
+
1−δ I
V (z, wµ)
R
Since the firm endogenously exits, we have:
V I (z, wµ) ≤ V N (z, wµ)
Then:
θ
V N (z, w) = (1 − θ)θ 1−θ
> (1 − θ)θ
θ
1−θ
zη
θ
1−θ
w
zη
θ
w 1−θ
1−δ N
V (z, wµ)
R
zη
1−δ I
− fw − c θ +
V (z, wµ)
R
w 1−θ
− fw +
= V I (z, w)
This contradicts ẑ < z being the threshold productivity level for innovating firms;
hence, shows that φ = µ.
Proposition 3 (Proof ).
Define:
XI ≡ EI (z η )
XN ≡ EN (z η )
45
Also define the cumulative distribution function for the group I as:

 1−
FI (z) =
 0,
ẑ λ
z
, if z ≥ ẑ;
o/w.
Moreover, let MI and MN be the number of firms in each group, which will be
MI
. To show that φ =
MI +MN
max{ẑ, µφ ẑ}, and a ≡ z̄ẑ . Then
constant in the BGP, let α ≡
along the BGP. Define z̄ ≡
XI0
Z
∞
= (1 − δ)q
z̄
µ we first show that µ ≤ φ
λẑ λ
(φz) λ+1 dz + (1 − δ)(1 − q)
z
η
Z
∞
ẑ 0
λẑ λ
z λ+1 dz
z
η
+[δ + (1 − δ)qFI (z̄) + (1 − δ)(1 − q)FI (ẑ 0 )]XI0
R ∞ η λẑλ
R∞
λ
q z̄ (φz)η zλẑ
λ+1 dz + (1 − q) ẑ 0 z z λ+1 dz
=
1 − qFI (z̄) − (1 − q)FI (ẑ 0 )
η
q µφ aλ−η + (1 − q)µ−λ
λ η
=
µη
ẑ
λ
−λ
qa + (1 − q)µ
λ−η
For N-firms we need to keep track of all the endogenous exits from group-I for the
last t̂ periods. Define:
Z
k(ž) ≡ q
ž
ž/a
λž λ
(φz) λ+1 dz + (1 − q)
z
η
Z
ž
µž
zη
λž λ
dz
z λ+1
λ
λ
φη [1 − aλ−η ]ž η + (1 − q)
[1 − µη−λ ]ž η
= q
λ−η
λ−η
46
Then:
ẑ
XN = (1 − δ)t̂ k(ẑµ−t̂ ) + (1 − δ)t̂−1 k(ẑµ−t̂+1 ) + .. + (1 − δ)k( )
µ


X
t̂−1
λ
λ
φη [1 − aλ−η ] + (1 − q)
[1 − µη−λ ]  (1 − δ)t+1 µ−η(t+1)  ẑ η
=
q
λ−η
λ−η
t=0
λ
λ
1 − (1 − δ)t̂ µ−ηt̂ η
=
q
φη [1 − aλ−η ] + (1 − q)
[1 − µη−λ ] (1 − δ)µ−η
ẑ
λ−η
λ−η
1 − (1 − δ)µ−η
Moreover:

α
=
w0
=
γ=
w
αXI0 + (1 − α)XN0
αXI + (1 − α)XN
1−θ
=
η
φ
q( µ
) aλ−η +(1−q)µ−λ
η t̂
η
λ−η
η−λ
Pt̂−1
] + (1 − q)[1 − µ ]
t=0 (1 − δ)
P
t̂−1
η
t̂
η
λ−η
η−λ
t+1
−ηt
αµ + (1 − α) (qφ [1 − a ] + (1 − q)[1 − µ ])
µ
t=0 (1 − δ)
qaλ +(1−q)µ−λ
µ + (1 − α) qφ [1 − a
t+1 −ηt
Since γ = µη(1−θ) :
η
φ
q
aλ−η + (1 − q)µ−λ = qaλ + (1 − q)µ−λ
µ
Which gives
φ
=a≤1⇒φ≤µ
µ
We prove µ ≤ φ by contradiction. In particular, suppose that µ > φ, i.e., threshold productivity grows at a faster rate than the productivity growth of the successful
innovative firms. Then, a measure FI (ẑ µφ ; t) − FI (ẑ; t) innovative firms will not be able
to stay in the market even if they successfully increase their productivity. This implies
that these firms would not invest in productivity growth, since the increments for the
exiting firms are nullified by assumption.
Notice that then the value of a firm with productivity z ∈ [ẑ, ẑ µφ ] the value is:
47
µ
1−θ

µη 
zη
θ
V I (z, w) = (1 − θ)θ 1−θ
w
− fw − c
θ
1−θ
zη
w
θ
1−θ
+
1−δ I
V (z, wµ)
R
Since the firm endogenously exits, we have:
V I (z, wµ) ≤ V N (z, wµ)
Then:
θ
V N (z, w) = (1 − θ)θ 1−θ
> (1 − θ)θ
θ
1−θ
zη
θ
1−θ
w
zη
θ
w 1−θ
1−δ N
V (z, wµ)
R
zη
1−δ I
V (z, wµ)
− fw − c θ +
R
w 1−θ
− fw +
= V I (z, w)
This contradicts ẑ < z being the threshold productivity level for innovating firms;
hence, shows that φ = µ.
Corollary From the previous result it follows that:
XI =
XN = (1 − q)
λ η
ẑ
λ−η
(2)
λ
1 − (1 − δ)t̂ µ−ηt̂ η
[1 − µη−λ ](1 − δ)µ−η
ẑ
λ−η
1 − (1 − δ)µ−η
From the exit condition of group N , we know that:
θ
1
(1 − θ)θ 1−θ z̃ η = f w 1−θ
where
w=
θ
(αM XI + (1 − α)M XN )1−θ
(1 − f M )1−θ
48
(3)
w=
=
t̂ µ−η t̂ 1−θ
α + (1 − α)(1 − q)[1 − µη−λ ](1 − δ)µ−η 1−(1−δ)
ẑ η(1−θ)
−η
1−(1−δ)µ
1−θ t̂ −η t̂ 1−θ
λ
η−λ
−η 1−(1−δ) µ
α
+
(1
−
α)(1
−
q)[1
−
µ
](1
−
δ)µ
z̃ η(1−θ) µη(1−θ)t̂
−η
λ−η
1−(1−δ)µ
θM 1−θ
(1−f M )1−θ
θM 1−θ
(1−f M )1−θ
λ
λ−η
1−θ Hence we get
θ
(1−θ)θ 1−θ z̃ η
f
(1 − θ)θ
θ
1−θ
1
=
θ 1−θ M λ
(1−f M ) λ−η
1
=
θ 1−θ f M λ
(1−f M ) λ−η
t̂ −η t̂
µ
α + (1 − α)(1 − q)[1 − µη−λ ](1 − δ)µ−η 1−(1−δ)
1−(1−δ)µ−η
α + (1 − α)(1 − q)[1 − µ
η−λ
](1 −
t̂ −η t̂
µ
δ)µ−η 1−(1−δ)
1−(1−δ)µ−η
ẑ η µ−ηt̂
t̂
µ−η(4)
This gives us M .11 In what follows, we show how to get MN (or equivalently, how
to find α).
Proposition 4 (Proof ).
Define
µž
λž λ
dzMI
z λ+1
ž
= (1 − q)[1 − µ−λ ]MI
Z
m(ž) ≡ (1 − q)
then
ẑ
MN = (1 − δ)t̂ m(ẑµ−t̂ ) + (1 − δ)t̂−1 m(ẑµ−t̂+1 ) + .. + (1 − δ)m( )
µ


t̂−1
X
−λ 
MN = (1 − q)[1 − µ ]
(1 − δ)t+1  MI
t=0
11
Notice that this is in line with strong and positive correlation between population size and firm
population illustrated in Bollard, Klenow and Li (2013).
49
= (1 − q)[1 − µ−λ ]
1−δ
[1 − (1 − δ)t̂ ]MI
δ
(1 − α) = α(1 − q)[1 − µ−λ ]
1−δ
[1 − (1 − δ)t̂ ]
δ
(5)
which proves the claim.
Proposition 4 (Proof ).
Define
µž
λž λ
dzMI
z λ+1
ž
= (1 − q)[1 − µ−λ ]MI
Z
m(ž) ≡ (1 − q)
then
ẑ
MN = (1 − δ)t̂ m(ẑµ−t̂ ) + (1 − δ)t̂−1 m(ẑµ−t̂+1 ) + .. + (1 − δ)m( )
µ


t̂−1
X
MN = (1 − q)[1 − µ−λ ]  (1 − δ)t+1  MI
t=0
= (1 − q)[1 − µ−λ ]
1−δ
[1 − (1 − δ)t̂ ]MI
δ
(1 − α) = α(1 − q)[1 − µ−λ ]
which proves the claim.
50
1−δ
[1 − (1 − δ)t̂ ]
δ
(6)