Since the seminal work of Pareto, many empirical analyses suggested that the distribution of firms size is characterized by an asymptotic power like behavior. At the same time, recent investigations show that the distribution of annual growth rates of business firms displays a remarkable double-exponential shape. A recent letter propose a possible connection between these two empirical regularities. By assuming a bivariate Marshall-Olkin power-like distribution for the size of firms in subsequent time steps, and performing a qualitative asymptotic analysis, it is suggested that the implied growth rates distribution takes a Laplace shape. By performing a complete analytical investigation, I show that this statement is not correct. The implied distribution does in general possess a non-continuous component and becomes degenerate when perfect correlation is assumed between size levels at different time steps. Essentially, the approach is faulty as it treats firm size levels as stationary stochastic variables and neglects the integrated nature of the growth process.Firm Growth, Size distribution, Power law, Laplace distribution.

Giulio Bottazzi

English

Research Papers in Economics

￿￿￿￿￿￿￿￿￿￿￿
￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿
￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿
￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿
￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿
￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿ ￿￿￿￿￿￿￿￿￿￿
       
                                      
                                    
                                 ! ∀#∃%&  ∋   (∋    )
∗  + , −!.∀.!// ! 0   1 2 , −!.∀.!// ! 00
     3    4    5+    6       3    5377888+   +    5+  7
￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿A comment on the relationship between ﬁrms’
size and growth rate
Giulio Bottazzi∗
Center for the Analysis of Economic and Financial Dynamics
Scuola Superiore Sant’Anna
Abstract
Since the seminal work of Pareto, many empirical analyses suggested that the dis-
tribution of ﬁrms size is characterized by an asymptotic power like behavior. At the
same time, recent investigations show that the distribution of annual growth rates
of business ﬁrms displays a remarkable double-exponential shape. A recent letter
propose a possible connection between these two empirical regularities. By assuming
a bivariate Marshall-Olkin power-like distribution for the size of ﬁrms in subsequent
time steps, and performing a qualitative asymptotic analysis, it is suggested that
the implied growth rates distribution takes a Laplace shape. By performing a com-
plete analytical investigation, I show that this statement is not correct. The implied
distribution does in general possess a non-continuous component and becomes de-
generate when perfect correlation is assumed between size levels at diﬀerent time
steps. Essentially, the approach is faulty as it treats ﬁrm size levels as stationary
stochastic variables and neglects the integrated nature of the growth process.
Key words: Firm Growth, Size distribution, Power law, Laplace distribution.
JEL Codes: C16,D21,D30
∗ Scuola Superiore Sant’Anna, P.zza Martiri della Libert` a 33, 56127, Pisa, Italy
Tel.: +39-050-883365; fax: +39-050-883344;
Email address: bottazzi@sssup.it (Giulio Bottazzi).1 Introduction
Since the seminal work of Pareto on wealth distribution of households, many em-
pirical studies suggested that, at least above some minimal size threshold, the
distribution of a number of economic relevant variables (like wealth, income, rev-
enues or capital assets) follow a power like behavior (see Kleiber and Kotz (2003)
for an extended review). This statistical fact is commonly named, after his early
discoverer, “Pareto Law”. At the same time, recent investigations show that the
distribution of annual growth rates of business ﬁrms displays a remarkable double-
exponential shape, called “Laplace” distribution (see Bottazzi and Secchi (2006)
and reference therein). This result is robust to diﬀerent levels of aggregation and
to the use of diﬀerent variables (revenues, number of employees, value added, etc.)
to proxy ﬁrm size.
The question naturally arises if these two distinct “stylized facts” can be somehow
reconciled so that a more “unifying” theoretical view of the dynamics of ﬁrms is
recovered. This is certainly a relevant task, since, as vigorously pointed out in Brock
(1999), moving the analysis from “unconditional” quantities, like distributions of
random variables, to more “conditional” (if not causal) relationships between the
diﬀerent regularities will obviously improve our understanding of the underlying
drivers of the observed dynamics.
A recent letter (Palestrini, 2007) tries to directly address the question of unifying
diﬀerent regularities by proposing a possible relationship between the power-like
nature of the distribution of ﬁrm sizes and the Laplace character of the distribu-
tion of ﬁrm growth rates. The issue raised is relevant, and the letter by Professor
Palestrini represents, without any doubt, a valuable eﬀort. His approach falls how-
ever short in two respects. First, he treats the logarithm of ﬁrm size at diﬀerent
time steps like a stationary random process, while it is constantly observed on real
data that these variables are in general characterized by a so called “integrated”
nature: it is the diﬀerence 1 of these logarithms, rather than their levels, which
displays, at least some degree of, stationarity. Second, assuming a bivariate Pareto
distribution for the size of ﬁrms in two diﬀerent time steps, he derives the asymp-
totic behavior of the implied distribution of growth rates. In the hope to reconcile
the two stylized facts mentioned above, he wrongly concludes, from the asymp-
totic analysis, that this distribution possesses a Laplace shape. In what follows, I
will start by addressing the latter issue. Namely, I will show how to derive, under
1 Or some other more complicate manipulation of the original variables in the case of
fractional integration. The discussion of this point is outside the scope of the present
note.
2the same hypothesis, the exact distribution of growth rates. As my computation
reveals, the implied distribution is not, in general, Laplacian nor everywhere con-
tinuous, having a ﬁnite atomic contribution for null growth rates. Using this result,
I will show that the failure of the proposed model to reconcile Pareto and Laplace
Laws is ultimately due to the integrated nature of the growth process.
2 The Marshall-Olkin bivariate distribution
Abstracting from precise economic deﬁnitions, let S1 and S2 be the size of a ﬁrm in
two successive time steps. If the distribution of ﬁrm size is Paretian, the distribu-
tion of its logarithm follows an exponential distribution. Then, taking s1 = log(S1)
and s2 = log(S2), the validity of the Pareto laws implies that s1 and s2 are expo-
nentially distributed
log(Prob{si > x}) ∼ x i = 1 2  
Consider the joint distribution of the couple of random variables (s1 s2). For the
Pareto Law to be valid, this distribution should possesses exponential marginals.
A natural candidate for a distribution of this kind, and the one considered in
Palestrini (2007), is constituted by the multivariate exponential distribution pro-
posed in Marshall and Olkin (1967), which in the bivariate case reads
Prob{s1 > s1 s2 > s2}=1 − F(s1 s2) (1)
=exp{−λ1s1 − λ2s2 − λ12 max{s1 s2}}  
It is immediate to see that, according to (1), the marginal distribution for the
random variables s1 and s2 are exponential with parameter ¯ λ1 = λ1 + λ12 and
¯ λ2 = λ2+λ12, respectively. Hence, the expression for expected vale and the variance
of the two random variables immediately follow. The covariance between s1 and
s2 is given by
Cov(s1 s2) =
λ12
λ ¯ λ1¯ λ2
  (2)
where λ = λ1 + λ2 + λ12. Thus, the correlation coeﬃcient becomes equal to λ12 λ
and is diﬀerent from zero as long as λ12  = 0. As discussed in Marshall and Olkin
(1967), the distribution F deﬁned in (1) is not everywhere continuous. The atomic
contribution comes from the presence of the discontinuous “maximum” term. The
3presence of this term will play a major role in the subsequent analysis. For easy of
reference I report here the expression for the generating function (Laplace trans-
form) of the distribution F derived in Marshall and Olkin (1967). I will use it in
the ﬁrst Theorem of the next Section. It reads
ψ(ω1 ω2)=
+∞  
0
+∞  
0
e
−s1ω1−s2ω2dF(s1 s2) (3)
=
¯ λ1¯ λ2(λ + ω1 + ω2) + ω1ω2λ12
(λ + ω1 + ω2)(¯ λ1 + ω1)(¯ λ2 + ω2)
 
3 The distribution of growth rates
In this section we derive the analytical expression of the distribution of the (log-
arithmic) growth rate under the assumption that the log size of the ﬁrms is dis-
tributed, in two successive time steps, according to the bivariate Mashall-Olkin
distribution deﬁned in (1). The ﬁrm growth rate r over a given period of time
is, by deﬁnition, the diﬀerence of the logarithm of the size at the end and at
the beginning of said period. Then, using the notation introduced above, one has
r = s2 − s1. Let G(r) = Prob{r ≤ r} be the probability distribution of growth
rates and ˜ g(k) = E[expikr] the associated characteristic function. Using the gen-
erating function in (3) one can easily derive the expression for ˜ g(k). One has the
following
Theorem 1 If s1 and s2 follow a bivariate Marhsall-Olkin distribution the char-
acteristic function ˜ g(k) of their diﬀerence r = s2 − s1 is given by
˜ g(k) =
λ¯ λ1¯ λ2 + λ12k2
λ(¯ λ1 − ik)(¯ λ2 + ik)
  (4)
PROOF. Considering the distribution function G(r) one has
G(r)=Prob{r ≤ r} = Prob{s2 − s1 ≤ r}
=
+∞  
0
+∞  
0
θ(r − s1 + s2)dF(s1 s2)
4l2
~
−i
l
~
 i 1
r>0
r<0 l1
l2
x
x1
2
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0.8
 0.9
 1
−3 −2 −1  0  1  2  3
Fig. 1. Left: Complex plane for the anti-transform of the characteristic function. The
poles are shown together with the paths of the Cauchy integral l1 and l2 used when
r < 0 and r > 0, respectively. Right: Growth rates distribution function G for λ1 = 1,
λ2 = 1 and λ12 =  5.
where θ is the right continuous Heaviside theta function, which is equal to 1 if its
argument is positive or null and zero otherwise. From the deﬁnition of characteristic
function it is
˜ g(k) =
+∞  
−∞
e
ikr dG(r)  
substituting the previous expression and inverting the order of integrations one
has
˜ g(k) =
+∞  
0
+∞  
0
e
ik(s1−s2) dF(s1 s2)
which, remembering (3), gives ˜ g(k) = ψ(−ik ik). Finally, direct substitution of
(3) proves the assertion. 2
The term proportional to k2 in the numerator of ˜ g(k) in (4) is what makes the
implied growth rate distribution not Laplacian. Indeed, if λ12 = 0, then the ex-
pression in (4) would reduce to the characteristic function of an asymmetric (or
symmetric, if λ1=λ2) Laplace distribution (c.f. Kotz et al. (2003) p.141). As shown
in the next Theorem, the eﬀect of this term is to introduce a ﬁnite probability for
the occurrence of zero growth rates, that is an atomic component in the point
r = 0. In general, one has the following
5Theorem 2 If s1 and s2 follow a bivariate Marhsall-Olkin distribution the distri-
bution function G(r) of their diﬀerence r = s2 − s1 is given by
G(r) =

 
 
e
¯ λ1r λ2
λ r < 0
1 − e−¯ λ2r λ1
λ r ≥ 0
  (5)
PROOF. Consider the formal deﬁnition of the density g(r) = G′(r) as the anti-
Fourier transform of the characteristic function
g(r) =
1
2π
+∞  
−∞
dk e
−ikr ˜ g(k)  
Since limk→∞|˜ g(k)| = 0 on the whole complex plane, for Jordan’s lemma the
previous integral, and consequently the density function, exists for any r  = 0. The
characteristic function ˜ g(k) possess two simple poles on the imaginary axis, in i¯ λ1
and −i¯ λ2. Then, the previous expression can be written as a Cauchy integral on
the upper or lower half plane when the value of r is respectively lower or greater
then zero. The two closed curves are depicted in Fig. 3 (left panel). According to
Cauchy integral theorem, in each case the value of the integral is proportional to
the residue of the function computed in the internal pole. After a little algebra one
has
g(r) =

 
 
e
¯ λ1r ¯ λ1λ2
λ r < 0
e−¯ λ2r ¯ λ2λ1
λ r ≥ 0
 
Using the previous expression, the distribution function G can be computed as
G(r) =
r  
−∞
dr
′g(r
′) if r < 0
or
G(r) = 1 −
+∞  
r
dr
′g(r
′) if r > 0  
Since the distribution function is by deﬁnition right continuous, the assertion fol-
lows. 2
6An example of the shape of the distribution G isnreported in Fig. 3 (right panel).
Notice that the ﬁnite weight at r = 0 can be easily computed using (5). Indeed
one has
Prob{r = 0} = lim
δ→0+ G(δ) − G(−δ) =
λ12
λ
 
that is the discontinuity in zero of the distribution function is proportional to the
correlation coeﬃcient of the two variables s1 and s2.
4 Conclusions
I have explicitly obtained the distribution function of the growth rates of ﬁrms
under the assumption that their sizes, in two subsequent time steps, followa bi-
variate Marshall-Olkin distribution. This assumption was introduced in Palestrini
(2007) to account for the validity of the Pareto Law in both time steps. How-
ever, by extending the analysis there, I have shown that this assumption implies
a growth rates distribution which is not only incompatible with a Laplace (sym-
metric or asymmetric) shape but is not even everywhere continuous. Moreover,
the discontinuity at the origin is proportional to the cross correlation coeﬃcient
between the size levels s1 and s2 in two subsequent time steps. As many empiri-
cal studies have shown, this coeﬃcient is always very near (and often statistically
equal) to one. Consequently, in real cases, the growth rates distribution implied by
Marshall-Olkin distributed logarithmic sizes is almost degenerate. This fact greatly
reduces the possibility of this distribution to ever provide an eﬀective statistical
description of empirical data. On the theoretical side, it should be noted that the
ultimate responsible of the failure of the unifying approach proposed in Palestrini
(2007) is the fact that the integrated nature of the growth process of ﬁrms, even
if largely conﬁrmed by empirical data, was not taken into account. The lesson to
be learned is that in order to obtain a reliable phenomenological description of
the growth dynamics of ﬁrms, or, to that extent, of any economic process, one has
to start from data and carefully investigate their regularities while, at the same
time, abstaining himself, as far as possible, from introducing untested theoretical
hypothesis.
7Acknowledgements
I would like to thank Angelo Secchi for a number of helpful comments. Research
support by Scuola Superiore Sant’Anna (grant n. E6006GB) and by European
Commission (CO3 NEST Pathﬁnder project, 6-th Framework program) is thank-
fully acknowledged.
References
Bottazzi, G., A. Secchi, (2006) Explaining the Distribution of Firms Growth Rates,
Rand Journal of Economics, 37, pp. 234-263, 2006.
Brock W. (1999) Scaling in Economics: a Reader’s Guide, Industrial and Corporate
Change 8, 409-446.
Palestrini, A. (2006) Analysis of industrial dynamics: A note on the relationship
between ﬁrms’ size and growth rate, Economic letters, in press.
Gabaix, X. (1999) Zipf’s Law for Cities: an Explanation, The Quarterly Journal
of Economics, 114, 739-767.
Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and
Actuarial Sciences, New Jersey: John Wiley & Sons.
Kotz, S., Kuzubowski T.J. and Podg´ orski K. (2001) The Laplace Distribution and
Generalizations, Birkhauser, Boston.
Marshall, A. and Olkin, I. (1967) A Multivariate Exponential Distribution Journal
of the American Statistical Association , 62, 30-44.
8

A comment on the relationship between firms' size and growth rate

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics