# Distribution in the Geometrically Growing System and Its Evolution

Kim Chol-jun
Department of Astronomy, Faculty of Physics, Kim Il Sung University, DPR Korea
postal code:+850
email address<EMAIL_ADDRESS>
###### Abstract

Recently, we developed a theory of a geometrically growing system. Here we
show that the theory can explain some phenomena of power-law distribution
including classical demographic and economic and novel pandemic instances,
without introduction of delicate economic models but only on the statistical
way. A convexity in the low-size part of the distribution is one peculiarity
of the theory, which is absent in the power-law distribution. We found that
the distribution of the geometrically growing system could have a trend to
flatten in the evolution of the system so that the relative ratio of size
within the system increases. The system can act as a reverse machine to covert
a diffusion in parametric space to a concentration in the size distribution.

### keywords:

power-law; firm size distribution; the COVID-19 pandemic ; population in city;
spectral hardening;

### JEL code:

C11; O1

Significance: Most economic systems that seem to show the power law
distribution are analyzed by Gibrat’s model, alias a geometrically growing
system, which seems to give the log-normal distribution. We showed that the
system can give an asymptotic power law if the correlation between parameters
is considered. In this paper we show the system can lead to the spectral
hardening provided the diffusion, or the increment of variances, along with
the growth of the system.

##  1 Introduction

First, we explain the problem and some terminology. A system is composed of
members and we call each member an item. The item has a measurable property,
which is called a size. The population in city and firm size can be regarded
as sizes while city and firm stand for item within country, which is in turn
the system. The power law, alias Zipf’s law or the Pareto distribution, states
that the probability of an item is inversely proportional of a power of the
size of the item: $p(Z)=\frac{M}{Z^{\gamma}}$, where $p(Z)$ stands for the
frequency of item of size $x$, $\gamma$ for exponent of power and $M$ for the
normalization constant.

Historically, Pareto (1896) showed that the distribution for income follows
the power law. Estoup (1916) and Zipf (1932) observed the power law in word
frequency in a novel and Auerbach (1913) and Zipf (1949) indicated the law for
the population size of city. Much diverse things show the power-law
distribution, for reviewing which we can refer to many works (e.g. see
Mitzenmacher, 2004; Newman, 2005). In fact, the author was interested in the
cosmic ray spectrum, which seems a typical power-law. Salpeter (1955) had
found that the mass distribution of stars follows the power law.

Several generative models for the power-law distribution are proposed, which
we can categorize into some groups. The first models are based on a
preferential attachment or “rich-get-richer” process (Yule, 1924; Simon, 1955;
Barabasi & Albert, 1999). The second ones pursue the scale invariance, which
is a peculiarity of the power law distribution (Bak, Tang & Wiesenfeld, 1987;
Sneppen et al., 1995). The third ones begin with demanded optimization
(Mandelbrot, 1953). And others composite models derive the power-law
distribution from specially assumed elementary distributions of the parameters
(Gibrat, 1931; Miller, 1957; Gabaix, 1999; Reed & Jorgensen, 2004). Those
models show many possibilities generating the power-law distribution. However,
all those are based on special assumptions. Though a postulation is the start
of logic, but it should better have generality. And the logic should better
cover wider range of size in data.

##  2 The formalism for the distribution in the geometrically growing system

Recently, we developed a theory of a geometrically growing system (GGS) (Chol-
jun, 2022) on the basis of statistically maximally plausible assumptions, i.e.
the normality of distribution of parameters. If the size of each item in an
system grows geometrically or proportionately, we call the system
geometrically growing. A GGS can be modeled by

$\displaystyle Z=(1+\alpha)^{t}Z_{0},$ (1)

where $Z$ is the size of an item in system, $\alpha$ is the growth rate
(hereafter simply, growth), $t$ stands for the the age of growth111In Chol-jun
(2022) the age was denoted by $n$. and $Z_{0}$ is an initial size of the item.

We can assumed the normal distribution for not only $\alpha$ but also $t$,
which is statistically maximally plausible. Here we can introduce a
correlation $R$ between $\alpha$ and $t$ without loss of generality because
the correlation can be given even in completely arbitrary configuration of
$\alpha$ and $t$.222 The instance of the COVID-19 pandemic in Chol-jun (2022)
showed a systematic correlation: the countries that had later outbreak of the
pandemic show relatively lower growth, i.e. the positive correlation is
obtained, which might be because they could have a warning or preparation.
Gabaix & Ioannides (2004) indicated that in some decades large cities grow
faster, but in other decades small cities grow faster. This implies the sign
of the correlation between the age (assuming that older cities are greater)
and the growth flips over casually.  If the correlation is positive ($R>0$),
then the log-size ($Y=\log{Z}$)at the upper limit can be approximate by

$\displaystyle Y_{3}=(Ax_{3}+B)^{2}+C,$ (2)

while if the correlation is zero or negative ($R\leq 0$), it is approximated
by333$F$ and $G$ are interchanged in comparison with Chol-jun (2022).

$\displaystyle Y_{4}=Fx_{4}+G,$ (3)

where $x_{3},x_{4}$ are variables following the standard normal distribution
$N(0,1)$ and, if $\mu_{t},\sigma_{t},\mu_{\alpha},\sigma_{\alpha},\mu_{i}$ and
$\sigma_{i}$ stand for the means and standard deviations of $t,\alpha$ and
$Y_{0}=\log{Z_{0}}$, the parameters are given as follows:

$\displaystyle A=a,\qquad B=\dfrac{\sqrt{4a^{2}b^{2}+d^{2}}}{2a},\qquad
C=c-\texttt{sgn}(R)\frac{d^{2}}{4a^{2}},$ (4) $\displaystyle
F=\texttt{sgn}(R)(a^{2}+b^{2})+c,\qquad G=\sqrt{2a^{4}+4a^{2}b^{2}+d^{2}},$
(5)

where

$\displaystyle a=\sqrt{\sigma_{\alpha}\sigma_{t}|R|},\qquad
b=\dfrac{|\mu_{\alpha}\sigma_{t}+\texttt{sgn}(R)\mu_{t}\sigma_{\alpha}|}{2\sqrt{\sigma_{\alpha}\sigma_{t}}},\qquad
c=\mu_{i}+\mu_{t}\mu_{\alpha}-\texttt{sgn}(R)b^{2},$ $\displaystyle
d=\sqrt{\sigma_{i}^{2}+(\mu_{t}^{2}\sigma_{\alpha}^{2}+\mu_{\alpha}^{2}\sigma_{t}^{2})(1-|R|)+\sigma_{t}^{2}\sigma_{\alpha}^{2}(1-R^{2})},$
(6)

and $\texttt{sgn}(R)$ stands for the sign of $R$ and $|\cdot|$ for the
absolute value.

Thus, if $R>0$, the log-size behaves such as a $\chi^{2}$ variable while for
the case of $R\leq 0$, as a normal variable. We can derive the probability
density function (PDF) of size for both case:

$\displaystyle p_{Z_{3}}(Z)$
$\displaystyle=\frac{\exp\left[-\dfrac{1}{2}\left(\frac{\sqrt{(\log
Z-C)}-B}{A}\right)^{2}\right]}{2\sqrt{2\pi}AZ\sqrt{(\log Z-C)}}\qquad$
$\displaystyle\text{for }R>0,$ (7) $\displaystyle p_{Z_{4}}(Z)$
$\displaystyle=\frac{1}{\sqrt{2\pi}FZ}\exp\left[-\dfrac{\left(\log
Z-G\right)^{2}}{2F^{2}}\right]\qquad$ $\displaystyle\text{for }R\leq 0.$ (8)

We call $p_{Z_{3}}(Z)$ the log-completely squared chi ($\chi$) distribution
with 1 degree of freedom (shortly, log-CS$\chi_{1}$ or log-CS) while
$p_{Z_{4}}(Z)$ is well-known log-normal distribution. What is interesting is
that the asymptotic exponent, i.e. the asymptotic slope in log-log scale
diagram of the PDF, of the log-CS tends toward a constant:

$\displaystyle\gamma_{3\infty}=\lim_{\footnotesize\begin{subarray}{1}Y\to+\infty(Z\to+\infty)\text{
in }R>0\end{subarray}}\frac{d(\log(p_{Z_{3}}(Z)))}{d(\log
Z)}=-\left(1+\dfrac{1}{2A^{2}}\right)=-\left(1+\dfrac{1}{2\sigma_{\alpha}\sigma_{t}R}\right),$
(9)

which says that the log-CS has asymptotic power-law
$p(Z)=\frac{M}{Z^{\gamma_{3\infty}}}$ behavior. Especially, the asymptotic
exponent depends only on the variances of the age and the growth. By the way,
the exponent is negative so that we usually consider only its absolute value.

For the log-normal distribution, local slope is determined by the variance
(more exactly, the standard deviation) $F$, which in turn depends on the means
and variances of the parameters.

##  3 Statistics of the COVID-19 pandemic and its evolution

The propagation of the pandemic can be considered as a typical example of the
geometrically growing system. In spite of seasonal rise and falls, the lock-
down measures, administration of vaccines and appearance of variants, the
propagation of the pandemic has been accelerated over 2 years since outbreak.
The power-law distribution of infected in countries were reported in the early
stage when the pandemic was propagating between countries (Blasius, 2020;
Beare & Toda, 2020). However, once propagation between countries had been
saturated, the distribution should be deviated from the power-law.

Chol-jun (2022) showed that the distribution of accumulated infected in
countries in May 2021 could be approximated by the log-CS excellently. Note
that the approximation is not a best-fitting but derived from the history of
the pandemic. In fact, distributions of the age and growth are similar to the
normal and their correlation turned out to be
positive.${}^{\ref{ftnt:covidcor}}$ Figure 1 shows the consistency between
observation and the log-CS approximation at several stages of the pandemic: as
examples, in late February 2020 (the early stage), late July 2020 (the
saturation of propagation between countries) and early February 2022 (the
propagation of the Omicron variant).444 Data for COVID-19 propagation is
available, for example, at the website of Our World in Data
https://ourworldindata.org/coronavirus/. Especially for July 2020, the
observation curve is weaving around the log-CS approximation. Distribution in
early February 2022 seems to be getting distorted by the unprecedented quick
propagation of the Omicron variant.

Figure 1:  The distribution of the accumulated infected cases of the COVID-19
pandemic in countries and its evolution. The distribution at several stages of
propagation and the log-CS$\chi_{1}$ approximation in (a) count histogram and
(b) the probability density. The evolution of (c) the tail exponent of the
distribution in probability (see (b)) and (d) the variance in log-normal
approximation.

In the above count and probability histograms we could hardly sense a change
of the slope of the probability curve. If we try to use a maximum likelihood
(ML) estimation of the power-law exponent (Newman, 2005)

$\displaystyle\hat{\gamma}=1+N\left(\sum_{i=1}^{N}\log\frac{z_{i}}{z_{\text{min}}}\right)^{-1},$
(10)

where $z_{i}$ stands for the size for data items and $N$ for the number of the
items, we should set $z_{\text{min}}$, i.e. the lowest allowable size or
truncation size. However, the distribution is not the power-low over all
domain of size but only in the tail part of big size. We could set
$z_{\text{min}}$ as the modal (most probable) size in the probability
histogram (Fig. 1) and determine the tail exponent. Figure 1 shows that the
tail exponent appears to decrease over all the past time in spite of local
rises.

We could propose another proxy for the tail exponent: the variance of the log-
normal distribution. In fact, the distribution can be approximated by the log-
normal (Eq. 8) as well. This distribution has not an asymptotic exponent and
local slope is determined by the variance $F$ (Eq. 5). We can estimate this
variance also in the maximum likelihood approach:

$\displaystyle\hat{F}^{2}=\frac{1}{N}\sum_{i=1}^{N}(\log
z_{i}-\hat{G})^{2},\qquad\hat{G}=\frac{1}{N}\sum_{i=1}^{N}\log z_{i}.$ (11)

This approach has advantage over the above evaluation of the tail exponent
because the selection of an optimal truncation size or $z_{\text{min}}$ does
not matter. In both the log-CS and log-normal distributions the slope
increases for bigger size so that the tail exponent could be evaluated greater
for greater size and vice versa even though the distribution still remains the
same. The greater variance corresponds to the smaller local slope or the tail
exponent. In the evolution of the COVID-19 pandemic, the variance seems to
increase (Fig. 1), which is coincident with decreasing the tail exponent as
aforementioned.

##  4 Statistics for population in city and in country

The distribution of population in city is a classical instance of the power-
law. The growth of population over centuries shows an exponential or
geometrical profile though sometimes was so saturated that expressed by the
logistic function. Therefore, we can express the evolution of population by
the geometrically growing system (GGS).

First, we analyzed the population in city within a country using data in
stellarium-0.21.1.555We use the dataset for cities compiled as observation
locations on the globe in stellarium-0.21.1, which is an open-source
astronomical software. The software is available at Stellarium Github webpage
https://github.com/stellarium/stellarium/releases/. The dataset covers
$\sim$24,000 cities with their location, population and other information
gathered between 2006 and 2019. Because of discontinuity in lower population,
we limit cities to the population over 20,000. Analyzing the population in
cities for U.S., Gabaix (1999) indicated that the populations in biggest
cities follow the power-law distribution and Zipf exponent is almost unity. We
obtained Zipf exponent 1.29 for U.S. (Fig. 2), which differs from 1 according
to Gabaix (1999) and 0.639 for Iraq, for example. We perform the best-fitting
analysis for population in cities of U.S. with various approximations: the
log-CS, the log-normal and power-law (Fig. 2). We infer the best-fit
parameters by a Markov Chain Monte Carlo (MCMC) method, especially making use
of the Metropolis-Hastings algorithm (Metropolis et al., 1953; Hastings,
1970). $R^{2}$ with best-fit parameters is evaluated: $R^{2}=0.9956$ for the
log-CS, $R^{2}=0.9905$ for the power-law and $R^{2}=0.9833$ for the log-normal
approximation. Therefore, we can prefer the log-CS as the closest
approximation in considering population in city within country. Ioannides &
Skouras (2013) claimed that most cities in the U.S. obeys a log-normal, but
the upper tail and therefore most of the population obeys a power-law. In
fact, the log-CS and log-normal distributions are almost indistinguishable
except for in the infinity (Chol-jun, 2022).

Figure 2:  The distribution of population in city and country and its
evolution. (a) A plot of log(Rank) vs. log(Population) for cities of U.S. The
linear regression gives log(Rank) = 6.1946 - 0.6387 log(Population). (b) The
bet-fitting to the distribution of population in cities of U.S. The log-CS
appears the closest. (c) The population in countries and areas over the world
in 2015. The evolution of (d) the tail exponent and (e) the variance in the
log-normal approximation for population in countries and areas over the world.

Next, we perform the best-fitting analysis for the distribution of population
in countries and areas over the world, making use of World Population
Prospects (WPP) 2015 dataset666The data is available at the website of World
Population Prospects https://population.un.org/wpp/. with the approximations
(Fig. 2). With the best-fit parameters inferred by the MCMC method, $R^{2}$ is
evaluated: $R^{2}=0.9851$ for the log-CS, $R^{2}=0.9380$ for the power-law and
$R^{2}=0.9920$ for the log-normal approximation. Therefore, in this case we
can prefer the log-normal as the closest approximation.

On the other hand, we try to apply our approach of the GSS. We expect the
correlation between the age and the growth${}^{\ref{ftnt:covidcor}}$. For
that, we have inspected the age of countries in the Korean Great
Encyclopedia777the Korean Great Encyclopedia, Pyongyang, Encyclopedia Press,
2001. We date the starting epoch of country by appearance of the first
administration, e.g. the first dynasty or city-state, the independence and so
on. However, for many African and American countries, we should consider that
the establishment of the colony had changed greatly the composition of
population in those countries. We are afraid that we might take missed or
distorted official record of real history for many countries and areas in
extracting typical dates. The growth rate is so evaluated for each country
that the population was originated from a couple of Adam and Eve, which
approximation was applied to COVID-19 pandemic in Chol-jun (2022).
Surprisingly, the growth rate and age show a exactly inverse relation,
furthermore its exponent is almost unity (Fig. 3). This gives a negative
correlation between the age and growth and, of course, we could expect that
the world population should follow the log-normal distribution.

Also from Britanica888Encyclop$\ae$dia Britannica Ultimate. Reference Suite.
Chicago: Encyclop$\ae$dia Britannica, 2014. we extracted the date of the first
habitation of the tribe or immigration. This kind of age that could be called
“habitation age” are much older than the previous “administration age.” But
the negative correlation is obtained still (Fig. 3). We also considered
another kind of age which is extrapolated from the current growth of
population: the origin of age is set so that the initial population was also a
couple. We call such an age “extrapolation age.” Figure 3 shows relation
between the age and the growth those are obtained by the extrapolation from
the period 1950-2015 of WPP dataset. Also a negative correlation is given.
And, for any kind of age, the inverse relation between the age and the growth
still holds and their exponent are near unity.

Figure 3:  The relation of the age and the growth for population in countries
and areas over the world assuming the extreme initial condition. (a) The
relation between a “administration age” and the corresponding growth. The
slope is $-0.9943$ in log-log scale and the correlation between the age and
the growth is $R=-0.6191$. (b) For a “habitation age,” the slope is $-0.9281$
and the correlation is $R=-0.3599$. (c) For an “extrapolation age,” the slope
is $-0.9771$ and the correlation is $R=-0.3911$. (d) The difference of size
forms a stripe which corresponds to a negative correlation.

We analyze this fact. From Eq. (1) we can derive

$\displaystyle\log{z}-\log{z_{0}}=t\log(1+\alpha)\approx t\cdot\alpha.$ (12)

If $t\alpha=b$ and $b$ is determined to extent of $k$ times, then $t$ or
$\alpha$ are also determined to extent of $k$ times and in log-log diagram of
$t$ vs. $\alpha$ appears a stripe of width $\log k$ (Fig. 3). This stripe has
slope of $-1$ surely so that a negative correlation between $\alpha$ and $t$
is obtained. In order that a positive correlation gets, this stripe must cover
all the range of $t$ in dataset. Therefore, it must hold that
$k\geqslant\sqrt{\frac{t_{\text{max}}}{t_{\text{min}}}}$, which we can rewrite
from Eq. (12):

$\displaystyle\frac{\left(\log z-\log z_{0}\right)_{\text{max}}}{\left(\log
z-\log
z_{0}\right)_{\text{min}}}\geqslant\frac{t_{\text{max}}}{t_{\text{min}}},$
(13)

where $t_{\text{max}}$ and $t_{\text{min}}$ could be appropriate extremes of
the dataset, e.g. of $3\sigma$ region. This might be a necessary condition for
positive correlation between $\alpha$ and $t$. To summarize, we can claim a
theorem.

######  Theorem 1.

The geometrically growing system can have positive correlation between the
growth and the age only if Eq. (13) satisfies.

From the theorem we could derive another conclusion:

######  Corollary 1.

If new items with the lowest size are continuously born within the
geometrically growing system, the system should be approximated by the log-
normal. On the other hand, the system could be approximated by the log-CS
after the creation of new items with lowest size has been stopped.

For the countries over the world, if we take an initial condition $z_{0}=2$ (a
couple) and consider that the maximum and minimum population in countries are
now $z_{\text{max}}=10^{9},z_{\text{min}}=10^{3}$ and $t_{\text{max}}=5000$
and $t_{\text{min}}=50$, then we obtain $100$ in r.h.s and only $3$ in l.h.s.
of Eq. (13). However, if we would consider a non-constant initial condition
(in real circumstance), we might gain so much greater value of l.h.s that we
could perform log-CS approximation. For the case of the COVID-19 pandemic,
$t_{\text{max}}-t_{\text{min}}\approx 120$ and $z_{\text{min}}=1,z_{0}=1$, so
we can apply the log-CS soon after saturation of propagation between
countries. Note that Eq. (13) is not a sufficient condition.

We inspect the evolution of the population distribution over the world. WPP
2015 dataset provides the population in countries and areas in time 1950-2015.
As aforementioned, the upper slope of the distribution can be evaluated by
both methods: the tail exponent and the variance in the log-normal
approximation. Figures 2 and 2 show the similar trend of the flattening slope
as in case of the COVID-19 pandemic. This means that the variance of the
parameters such as the age and the growth is so growing that the variance of
the size distribution grows, and the tail exponent decreases.

The tail exponent for population in city within a country also evolves. Citing
previous works, Gabaix & Ioannides (2004) indicated that the tail exponent for
the U.S. decreased in the period from 1900 to 1990 to imply a greater
concentration. Gonzáles-Val (2010) assured a monotonically decreasing of the
tail exponent with time, provided the truncation number of cities keeps as
10,000. Interestingly, observing data for dynamics of cities in the central
and eastern Europe (CEE) countries during 1970-2007 (Necula et al., 2010), we
can find that in most countries the exponent has almost a negative relation
with the population itself: if the population increases, the exponent
decreases and vice versa (Fig. 4). The exponent for European cities in the
middle ages seems to decrease after 1500 (Bairoch, Batou & Chèvre, 1987;
Gonzáles-Val, 2019), and only then Zipf’s law was reported to emerge for
cities in Europe (Dittmar, 2011).

Figure 4:  Evolution of the population and the tail exponent for CEE
countries. (a) Belarus, (b) Bulgaria, (c) Hungary, (d) Poland, (e) Romania,
(f) Russia, (g) Ukraine, (h) Baltic states (Estonia, Latvia and Lithuania).
The data is extracted in Necula et al. (2010) where the exponent is calculated
by two methods: ordinary least squares (OLS) estimation for linear regression
and maximum likelihood estimation (MLE). For some countries one of either
estimations of the exponent shows abrupt rise and fall so that we neglect. For
most countries except Bulgaria and Poland we can observe inverse relation
between the population and the tail exponent in evolution.

Considering countries, quite different with respect to wealth, size and
geography, Pinto, Lopes & Machado (2012) claimed that the countries presenting
higher wealth levelsreveal higher values of the exponent while most African
countries unveil smaller values of the exponent. In our simulation, oldest
Asian countries such as Iraq and China appear to have smallest exponent.
However, their reasoning was so not obvious: if it was right, the exponent
over the world should increase as the world economy proceeds but the exponent
surely is decreasing. Our approach can give an obvious reasoning: younger
countries might have greater exponent and vice versa. In fact, the most more
developed countries are younger while the older countries are underdeveloped
so that more wealthy countries could appear to have higher exponent, which,
however, is casual but not inevitable.

Gabaix & Ioannides (2004) related the urbanization with the economic factor,
e.g. the economic integration and the international trade. Then why is the
exponent increasing in some countries in spite of economic progress? Necula et
al. (2010) proposed political factor to determine the urbanization. We could
give a statistical analysis ahead of or including all the economic or
political or any other factors. For example, the exponent depends on the
variances of the parameters.

##  5 Statistics of firm size

The power-law distribution has appeared widely in economic and financial
phenomena (e.g. see Farmer & Geanakoplos, 2008). The power-law distribution in
firm size that could be measured by diverse properties have anounced long ago
(Zipf, 1949; Ijiri & Simon, 1977). Making use of Economic Census 1997, Axtell
(2001) showed the power-law distribution in firm size measured by employees
and revenue. The firm size is a quantity which could be apt to grow
geometrically. In fact, we commonly evaluate the growth of firm in terms of
proportionality but not additivity.

We analyse the data in Axtell (2001), where numerical data for the size of
firm expressed by the number of employees (the employment size) were shown
explicitly. In fact, the distribution has a convex form in low-size part,
which is in more favor of the log-CS rather than the pure power-law modeling.
Giovanni, Levchenko & Rancière (2010) analyzed French firms, and obtained a
similar convex profile of distribution. We neglect 0-size firms as Axtell did.
We perform the best-fitting with the various approximations by the MCMC method
(Fig. 5). For the power-law fitting $R^{2}$ is obtained the same as Axtell:
$R^{2}=0.9932$. The log-CS fitting gives a greater value: $R^{2}=0.9987$. We
should expect that the dataset could be approximated by the log-normal:
$R^{2}=0.9952$. This says that the log-CS could be the closest to the real
dataset.

Figure 5:  Distribution of the employment size of firm and evolution of the
tail exponent. (a) The log-CS$\chi_{1}$, log-normal and power-law fitting for
distribution of the employment size of firm in Census/SBA 1997 dataset. The
evolution of (b) histogram and (c) the tail exponent for firm’s employment
size in BDS 1977-2014 dataset. The tail exponent is evaluated not in count
histogram such as (b) but in probability histogram such as (a) by linear
regression.

As aforementioned, the geometrically growing system can be modeled
alternatively by either log-CS and log-normal depending on the correlation
between the growth and age. Data for employment dynamics by firm age,
1987-2005, from the Census Bureau Business Dynamics Statistic and Longitudinal
Business Database, showed that young firms have higher employment growth
rates, if they survive, than older firms (Haltiwanger, Jarmin & Miranda, 2009,
2010). This might be because the growth of older or greater firms seems to be
saturated due to market limitation while this limitation does not affect
younger firms so the latter appears to have higher growth in spite of higher
establishment exit. Analyzing data from the EFIGE survey that sampled French,
Italian and Spanish firms in the period from 2001 to 2008, Navaretti,
Castellani & Pieri (2012) showed that younger firms have a highly probability
of experiencing high growth rates both in the short-run (e.g. for 1-year) and
in the long-run (i.e. for existing age). Therefore, we could expect a negative
correlation between firm age and firm growth. This should lead to the log-
normal fitting to the distribution, though the real dataset seems closer to
the log-CS. This might be originated from non-normal distribution of the age
and growth.

We trace the evolution of the distribution. We inspect data from the Census
Bureau Business Dynamics Statistic (BDS), 1977-2014.999The data is available
at website of Small Business Administration
https://www.sba.gov/sites/default/files/advocacy/%bds_firm_size.xlsx Though
the data shows a lowering of exponent in lower-size part in contrast to
higher-size part which could stand for a convexity of the distribution (Fig.
5), we could evaluate the tail exponent by linear regression, excluding both
the lowest- and highest-size bins, because the highest bin has inappropriate
upper limit for infinity. Figure 5 shows clearly that the exponent decreases
as time goes. Therefore, we can see that the tail exponent is evolving to
lower, i.e. the distribution is flattening. We can give a reason in our
approach as aforementioned: the variances of the age or the growth so increase
that the variance of the size increases and the distribution of size flattens.

##  6 Conclusion and discussion

In this paper, we consider some special properties of distribution for the
geometrically growing system (GSS) with pandemic, demographic and economic
phenomena.

First, the distribution has a convexity in the lower-size part. It is not
surprising, it represents only the modal (most probable) size, which is
popular in almost distributions but absent in the power-law. In fact, the log-
CS has additional concavity and singularity. In the above approximations the
log-CS or log-normal, both or either, dominate over the power-law. This means
that the demographic, pandemic and economic distributions can be explained by
the GSS properly. In fact, difference between the log-CS and the log-normal is
not great in most cases and not important.What matters is that both of them
represent the distribution of GSS and the convexity appears commonly in both
them. However, profile of those distributions may be changed if the
distribution of parameters is deviated from the normal. For the log-CS or log-
normal, there does never appear a divergence problem which happens for the
pure power law with a certain exponent.

If new items are born with low size continuously flourishing, this convexity
will get fainter and the distribution seems to be closer to the power-law.
However, once the number of items in the system is saturated, the number of
low-size items decreases in evolution, unless they are isolated from the
ensemble and never grow, and a kind of roll-over in lower-size part grows.
Until early stage of such a period, the distribution of GGS should be
represented only by the log-normal while long after the saturation of the
number of items, the log-CS fitting can become possible.

Second, while the parameters such as the age and the growth diffuse, the tail
exponent lowers and the distribution is flattening in the evolution of system,
which is often called the spectral hardening. As aforementioned, the slope of
distribution depends on the variances of the parameters fully or partly. In
many systems such as Brownian motion, the variance of parameters are growing
with time. The diversity in economical actions and variance in economical
growth are accelerating with the time. The second law of the thermodynamics
dictates only that the matter should spread out by the diffusion. However, the
matter is collecting and agglomerating over the universe. Though, in the
physical view, it can be explained by the gravitation and so on, but, on the
statistical way, the geometrically growing system can act as a reverse machine
that converts the diffusion in parametric space to the concentration in the
distribution of size.

The flattening distribution in turn implies the enlargement of the relative
ratio in size between the highest- and lowest-size items within the system.
This could explain the urbanization, monopolization and so on. The
urbanization proceeded in ancient times such as in ancient Rome. The
urbanization can occur not only by migration due to economic and political
reasons, but also by the stochastic nature of the growth itself, e.g., by
different birth (or death) rate or involving all the former factors. If we
would apply this property to wealth distribution, which has geometrically
growing trend and follows a power-law, we could expect aggravation of the
“rich-get-richer” process and the monopolization in the economic regime by
nature if money begets money.

It is interesting that the concentration might be compatible with or even
driven by the diffusion. The growth in GSS could give rise to the diffusion in
parametric space, which in turn leads to the centralization in matter space.
The “rich-get-richer” phenomenon does never imply that the rank in system
should be fixed, that is, the richest or biggest one could keep their first
rank naturally. The rank could be determined by the growth rate, the diversity
of which can be changed with time. The richest job or biggest city have been
alternating with era, as we have seen.

We can find properties of the geometrically growing system in many other
phenomena. We wish that our approach could contribute to analyze the problems.

## Conflict of interest

The author has no conflicts to disclose.

## Data availability

Data used in this paper are available at the website addresses indicated or by
corresponding with the author.

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