A look at the relationship between industrial dynamics and aggregate fluctuations
ABSTRACT The firmly established evidence of rightskewness of the firms’ size distribution is generally modelled recurring to some variant of the Gibrat’s Law of Proportional Effects. In spite of its empirical success, this approach has been harshly criticized on a theoretical ground due to its lack of economic contents and its unpleasant longrun implications. In this chapter we show that a rightskewed firms’ size distribution, with its upper tail scaling down as a power law, arises naturally from a simple choicetheoretic model based on financial market imperfections and a wage setting relationship. Our results rest on a multiagent generalization of the preypredator model, firstly introduced into economics by Richard Goodwin forty years ago.
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ABSTRACT: I modify the uniformprice auction rules in allowing the seller to ration bidders. This allows me to provide a strategic foundation for underpricing when the seller has an interest in ownership dispersion. Moreover, many of the socalled "collusiveseeming" equilibria disappear.National Bureau of Economic Research, Inc, NBER Working Papers. 01/1988;  [Show abstract] [Hide abstract]
ABSTRACT: The market for unseasoned equity has the unusual and distinguishing feature of periods of concentrated activity in terms of both volume and underpricing. This paper formally documents the existence of such periods using a regimeswitching model that dates transitions between hot and cold states. A number of hot periods are identified over a 20year period using a variety of IPO activity measures that capture different aspects of new issue volume, proceeds and underpricing. The study further documents a leading relationship between underpricing and IPO volume of up to six months. This relationship supports the contention that the decision to issue is a function of current underpricing. Various reasons are hypothesised for this result and the paper finds supportive evidence through a VAR analysis that reveals the influence of stock market and business conditions. The results have implications for the information contained in current market conditions and the role of issuers, underwriters and investors (JEL G12, G14, G32).01/2000;
Page 1
DIPARTIMENTO DI ECONOMIA
_____________________________________________________________________________
A look at the relationship between
industrial dynamics and aggregate
fluctuations
Domenico Delli Gatti
Edoardo Gaffeo
Mauro Gallegati
_________________________________________________________
Discussion Paper No. 3, 2008
Page 2
The Discussion Paper series provides a means for circulating preliminary research results by staff of or
visitors to the Department. Its purpose is to stimulate discussion prior to the publication of papers.
Requests for copies of Discussion Papers and address changes should be sent to:
Dott. Luciano Andreozzi
E.mail luciano.andreozzi@economia.unitn.it
Dipartimento di Economia
Università degli Studi di Trento
Via Inama 5
38100 TRENTO ITALIA
Page 3
1
A look at the relationship between industrial
dynamics and aggregate fluctuations*
Domenico Delli Gatti,1 Edoardo Gaffeo,2 Mauro Gallegati,3
1 Catholic University of Milan, Italy
2 University of Trento, Italy
3 Polytechnic University of Marche, Ancona, Italy
Abstract
The firmly established evidence of rightskewness of the firms’ size
distribution is generally modelled recurring to some variant of the
Gibrat’s Law of Proportional Effects. In spite of its empirical success,
this approach has been harshly criticized on a theoretical ground
due to its lack of economic contents and its unpleasant longrun
implications. In this chapter we show that a rightskewed firms’ size
distribution, with its upper tail scaling down as a power law, arises
naturally from a simple choicetheoretic model based on financial
market imperfections and a wage setting relationship. Our results
rest on a multiagent generalization of the preypredator model,
firstly introduced into economics by Richard Goodwin forty years
ago.
JEL classification: L11, D92, E32
Keywords: Firm size; Preypredator model; Business Fluctuations.
To appear in
Faggini, M. and T. Lux (Eds), Economics: From Tradition to
Complexity. Berlin: Springer.
* We would like to thank an anonymous referee, Richard Day, Corrado Di Guilmi, Nicolàs
Garrido, Sorin Solomon and seminar participants at the Universities of Leiden and Salerno for
useful comments on earlier drafts. Emiliano Santoro provided excellent research assistance.
Responsibility for remaining errors remains with us.
Page 4
2
1. Introduction
Starting with the pioneering work of Gibrat (1931), the study of the
determinants and the shape of the steadystate distribution of firms’ size has
long fascinated economists. While the conventional view received from the
seminal work of e.g. Hart and Prais (1956), Hart (1962) and Mansfield (1962)
holds that the firms’ size distribution is significantly rightskewed and
approximately lognormal, recent empirical research has lent support to the
view suggested by H. Simon and his coauthor (Ijiri and Simon, 1977),
according to whom a ParetoLevy (or power law) distribution seems to return
a better fit to the data for the whole distribution (Axtell, 2001), or at least for
its upper tail (Ramsden and KissHaypal, 2000; Gaffeo et al., 2003).1
Regardless of the different outcomes obtained from distributionfitting
exercises, the most popular explanation for rightskewness emerged so far in
the literature rests on stochastic growth processes, basically because of their
satisfactory performance in empirical modelling.2 From a theoretical point of
view, however, random growth models of firms’ dynamics have been
generally seen as far less satisfactory. On the one hand, several authors have
simply discarded purely stochastic models as adhoc and uninformative, given
that a proper theory of firms’ growth should be grounded on richer economic
contents and maximizing rational behaviour (Sutton, 1997).3 Alas, the
introduction of stochastic elements in standard maximizing, gametheoretic
models4 has shown that their implications as regards the steadystate firms’
size distribution are highly dependent on initial assumptions and modelling
choices, to the point that we cannot find “[…] any reason to expect the size
distribution of firms to take any particular form for the general run of industries”
1 Another challenging stylized fact on the drivers of corporate growth and the resulting
industrial structure is the ubiquitous exponential shape of the growth rates density (Bottazzi
and Secchi, 2003; Bottazzi et al., 2007).
2 For recent evidence, see Geroski et al. (2001), who point towards a pure random walk model
for firms’ growth, and Hart and Oulton (2001), who instead suggest a Galtonian, reversionto
themean growth process.
3 In spite of being very popular among economists, Sutton’s critique is not properly
established. In fact, the Gibrat’s Law may be perfectly consistent with the behaviour of rational,
profitmaximizing firms. In a nutshell, consider a model in which firms, à la Penrose (1959), are
constrained in their growth opportunities only by their internal resources. Since firms’ size, at
the optimum, depends on current expectations on future conditions, if firms form rational
expectations changes in expectations will be unpredictable. This implies that growth rates are
realizations of pure random processes. See e.g. Klette and Kortum (2004).
4 See, for example, Jovanovic (1982) and Ericson and Pakes (1995).
Page 5
3
(Sutton, 1997, p. 43). On the other hand, the original Gibrat’s random
multiplicative model (also known as the Law of proportionate effect) and its
numerous extensions5 all share the unpleasant property of possessing either
an implosive or an explosive behaviour under rather general conditions, so
that their crosssection dynamics tend alternatively towards a degenerate
firms’ size distribution with zero mean and variance or a degenerate
distribution with infinite mean and variance (Richiardi, 2004).6
In this chapter, we propose a model which embeds idiosyncratic stochastic
influences in a simple imperfect information, rational expectations
framework. We show that a rightskewed firms’ size distribution emerges as
a natural feature of the endogenous crosssection dynamics, and that the
upper tail is Pareto distributed. This results from the interplay of the cross
sectional dispersion (i.e., heterogeneity) of firms and the feedback exerted on
it by the competitive pressure that individual actions determine through the
labour and the equity markets.7 Furthermore, certain properties of the steady
state distribution are in some sense universal, i.e. they are independent of
some of the model’s parameters. In other terms, our approach does not
possess any tendency towards a longrun degenerate behaviour, typical of
Gibrat’s processes, and it grants the modeller more degrees of freedom than
the stochasticgametheoretic models referred to above. From an analytical
point of view, our results are based on the possibility of describing the
economy as a Generalized LotkaVolterra system, that is a multiagent
extension of the preypredator framework formally introduced to economists
by Richard M. Goodwin (19131996) back in the 1960s (Goodwin, 1967). While
inside the Econophysics community such a formalism has been already
successfully employed to explain puzzling statistical regularities regarding
5 Among the many variations of the Gibrat’s model, one can cite the preferential attachment
mechanism introduced by Simon and Bonini (1958), the multiplicative plus additive random
process due to Kesten (1973), or the lower reflection barrier by Levy and Solomon (1996).
6 To grasp the argument, consider this simple example reported by Richiardi (2004). Let us
presume that the size of a firm increases or decreases, with equal probability, by 10% in each
period. Now suppose that, starting from a size of 1, the firm first shrinks and then bounces
back. In the first period the firm’s size is 0.9, while in the second one it is only 0.99. If we let the
dynamics be inverted, so that the firm first grows and then shrinks, then the firms’ size is 1.1 in
the first period, and 0.99 in the second one. Clearly, this effect is stronger the bigger the
variance of the stochastic growth process, and can be contrasted only by increasing the average
growth rate. Richiardi (2004), resorting to simulations, shows that a degenerate longrun
distribution can be prevented only by a limited number of mean/variance pairs.
7 Another model in which financing constraints help to explain the skewness of the firms’ size
distribution is the one recently proposed by Cabral and Mata (2003).
Page 6
4
stock market returns and the personal wealth distribution (Solomon and
Richmond, 2001), the application of this Generalized LotkaVolterra approach
to the relation between the firms’ size distribution and aggregate fluctuations
is new.
The organization of this chapter is as follows. In Section 2 a simple
macroeconomic model resting on financial market imperfections, maximizing
behaviour, rational expectations and heterogeneity is presented. Section 3
contains the characterization of the associated firms’ size crosssection
dynamics and its longrun attractor or, in other terms, the equilibrium firms’
size distribution. Section 4 concludes.
2. The model
The purpose of this section is to create a simple macroeconomic model that,
besides explaining aggregate fluctuations, replicates the main features of
industrial dynamics and the firm size distribution. At the centre of our
approach there are three basic ideas:
i) The firms’ financial position matters. Firms display heterogeneous
unobservable characteristics, so that lenders may not be perfectly
informed on firms’ ability or willingness to pay back. As a result, the
various financial instruments whereby firms can raise means of
payment are not perfect substitutes. Decisions about employment and
production are conditional on the cost and contractual terms of the
financial instruments available to firms (Gertler, 1988). From an
empirical viewpoint, it appears that firms tend to be rationed on the
market for equity due to both adverse selection and moral hazard
effects (Fazzari et al., 1988).
ii) Agents are heterogeneous as regards how they perceive risk
associated to economic decisions. Numerous experimental studies
indicate that heterogeneity in risk perception is pervasive. Such a
finding holds both for consumers and insurees (Hammar and
JohanssonStenman, 2004; Lundborg and Lindgren, 2002; Filkestein
and Poterba, 2004), and for entrepreneurs (Pennings and Garcia, 2004;
van Garderen et al., 2005). While popular in mathematical psychology,
the exploitation of these findings is far less common in theoretical
economics, the random utility model discussed in Anderson et al.
(1992) being a relevant exception.
Page 7
5
iii) Firms interact through the labour and equity markets. While the
feedbacks occurring in the labour market between hiring firms are
clear enough not to deserve particular emphasis, a few words are in
order to discuss interactions in the equity market. In particular, we are
referring to the evidence suggesting that the number of initial public
offerings (IPO), of additional new stock issues, as well as the
proportion of external financing accounted for by private equity, all
tend to increase as the aggregate activity expands and the equity
market is bullish, and viceversa (Choe et al., 1993; Brailsford et al.,
2000).
All other aspects of the model are kept as simple as possible. 8
We follow the literature on information imperfections in financial markets,
in particular Greenwald and Stiglitz (1993), in assuming that finitely many
competitive firms indexed by i = 1,…, I operate in an uncertain environment,
in which futures markets do not exist.9
Each firm has a constant returns to scale technology which uses only
labour as an input, yit = φnit, where yit is the time t output of firm i, nit its
employment, and φ the labour productivity, constant and common to all. The
production cycle takes one period regardless of the scale of output, implying
that firms have to pay for inputs before being able to sell output.
Since capital markets are characterized by informational imperfections,
firms’ ability to raise risk capital on external stock markets is suboptimal,
and restricted to depend on the average capitalization in the economy. As a
result, firms must generally rely upon bank loans to pay for production costs
(i.e., the wage bill). In real terms the demand for credit of the ith firm is dit =
wtnit – ait, where wt is the real wage, determined on an aggregate labour
market, while ait is the firm i’s real equity position. For simplicity we assume
that firms can borrow from banks as much as they want at the market
expected real return r, and debt is totally repaid in one period.
The individual demand faced by firms is affected by idiosyncratic real
shocks. The individual selling price of the ith firm is the random outcome of
a market process around the average market price of output Pt, according to
the law Pit = uitPt, with expected value E(uit) = 1 and finite variance. Let the I
8 For a more complete, agentbased model where aggregate (meanfield) interactions occur
through the credit market, see Delli Gatti et al. (2005).
9 The analytical details of the framework used in this paper are discussed at length in Delli
Gatti (1999).
Page 8
6
random variables {ui} be uniformly, but nonidentically, distributed. In
particular, support of the ith firm relative price shock uit is given by [zit, 2−zit],
with {zi} ∈ (0,1) being a iid random variable with finite mean and variance. It
follows that individual price shocks are characterized by a common and
constant expected value equal to 1, but the variance, ( )
()
3
1
2
it
it
z
uV
−
=
, evolves
stochastically. This aims to capture the idea that people usually perceive the
same signal in different ways or, alternatively, that people differ as regards
the degree of risk they attach to random events to be forecasted.
If a firm cannot meet its debt obligations (in real terms, πit+1 < 0, where πit+1
are real profits at the beginning of period t+1), it goes bankrupt. From the
assumptions above, it follows that this event happens as the relative price is
below a threshold given by:
⎛
=
it
Ru
φ
where R = (1 + r).
Going bankrupt is costly, not only because of direct legal and
administrative costs (Altman, 1984; White, 1989), but also because of the
indirect costs of bankruptcyinduced disruptions, like asset disappearance,
loss of key employees and investment opportunities, and managerial stigma
and loss of reputation (Gilson, 1990; Kaplan and Reishus, 1990). Estimates
suggests that the costs associated to bankruptcy may be very large (White,
1983; Weiss, 1990). For simplicity, we assume that the real bankruptcy cost Cit
is an increasing quadratic function of output, so that the expected bankruptcy
cost becomes:
⎟⎟
⎠
⎞
⎜⎜
⎝
−
it
itt
y
aw
(1)
()()
(
(
1
)
)
()
it
it
it
itt
it
it
itit
−
it
2
itit itit
z
z
y
aw
φ
R cy
z
zucy
uucyCE
−
⎥
⎦
⎤
⎢
⎣
⎡
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
=
−
=<=
12
Pr
2
2
2
(2)
where c > 0 is a measure of the aversion to bankruptcy on the part of the
firm’s owners and managers. As recalled above, the magnitude of the
parameter measuring aversion to bankruptcy depends on a number of
factors, and it can vary with the institutional framework and the economic
Page 9
7
conditions faced by firms. Clearly, the higher is c, the lower is the incentive to
recur to debt in financing production and — as we will see presently — the
lower is the level of output maximizing real profits.
While the model could be expanded to more general cases with similar
qualitative results  although at the cost of additional remarkable analytical
complications – for the sake of tractability we limit the dynamics of the
system in a region in which the two following conditions holds true:
C1: The individual demand for credit is always positive, i.e. di > 0, ∀ i.
C2: Expected profits net of expected bankruptcy costs are always positive,
Rw
so that we are assuming that
1
<
φ
t
.
The problem of firm i consists in maximizing the expected value of real
profits net of real bankruptcy costs:
()
−
⎟⎟
⎠
⎝
φ
From the first order condition it follows that as c grows large10 the
individual supply can be approximated to the linear function:
R
y
⎜⎜
⎝
φ
where hit, being a function of the random variable zit, is a random variable as
well, which we assume follows a distribution Π(hi) with finite mean and
variance. Individual supply is an increasing linear function of net worth, as a
)
⎟⎟
⎠⎝
Rw
c
φ
this could happen well before c goes to infinity, since both terms at the numerator are lower
than 1. In particular, for c ≤ 10,000 the expression above becomes lower than 0.01 but for a
negligible number of cases.
()
⎥
⎦
⎤
⎢
⎣
⎡
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−
⎞
⎜⎜
⎛
−−=−
itit itit
t
it
it
itt
it itit
y
yRayz
Rw
φ
z
c
a
yw
RyCE
2
12
max
π
. (3)
ititit
it
it
aha
z
Rw
=
⎟⎟
⎠
⎞
⎛
−
≅
2
(4)
10 Such that the term (
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
⎞
⎜⎜
⎛−
1
−
it
t
t
it
z
Rw
φ
z
1
tends to approach 0. It seem worthwhile to note that
Page 10
8
higher net worth reduces the marginal bankruptcy costs. In turn, for any
given level of net worth higher uncertainty (a realization of the random
variable zit close to 0) on the relative price makes production more risky. As
firms trying to maximize the concave profit function (3) behaves in a risk
averse manner, higher uncertainty means lower production.
The dynamics of production can be tracked by the evolution of the equity
base, which in real terms reads as:
(
ititit
Ryua
−=
+1
where
The last term measures the amount of new equity firm i can raise on the stock
market, assumed to be proportional to the average capitalization of the
economy at time t. The value of γi depends on the level of development of the
financial system, that is on the presence of financial instruments, markets and
institutions aimed at mitigating the effects of information and transaction
costs. Firms operating in economies with larger, more active and more liquid
stock markets with a large number of private equity funds are characterized
by a higher γ than economies with poorly developed financial systems. From
the point of view of each individual firm, however, the hot market effect
represents an externality arising from a meanfield interaction.
Finally, we assume that at any time t the real wage is determined on the
labour market according to an aggregate wage setting function, which for
simplicity we assume to be linear:
wt = bnt
where nt is total employment, and b > 0. Such a positive relationship between
aggregate employment (at given labour supply) and real wage can be
alternatively derived from union models, insideroutsider models or
efficiency wage models (Lindbeck, 1992).
The parameters γi and b then capture the interdependence between
aggregate outcomes on the financial and the labour markets, respectively, and
individual decisions. Their values vary with the institutional context
governing both market transactions and nonmarket interactions. Such an
institutionallyconstrained microeconomic analysis calls for the search of
appropriate macrofoundations of microeconomics (Colander, 1996). The
importance of this issue will be further discussed below.
)
tiit itt
aanw
γ+−
(5)
ta is the average capitalization of firms operating at time t, and γi > 0.
(6)
Page 11
9
Due to the constant returns technology and the supply function (4),
knowledge of ait immediately translates into knowledge, at least in
expectational terms, of other traditional measures of firms’ size, that is total
sales (yit) and employment (nit). By inserting (4) and (6) into (5), and assuming
rational expectations so that uit = 1 for any i and t, we obtain:
ait+1 = (hit + R)ait +
Heterogeneity enters the model along two margins. First, because of
cumulative differences in individual equity bases, which are influenced by
past idiosyncratic shocks to relative prices. Second, because of differences in
the way agents perceive risk associated to future profits. The dynamical
system (7) for i ∈ (1, I) and its predictions for the firms’ size distribution will
be analyzed in the next Section.
For the time being we just want to highlight that this economy can display
aggregate — i.e., per capita —endogenous fluctuations under rather mild
conditions. To do this, let us take the crosssectional average in both members
of (7) to get:
(
ttt
aRha
+=
+
1
tia
γ
− Rbntnit. (7)
)
t
tt
a
ah
φ
Ib
R
γ+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
2
22
. (8)
where to simplify calculations we have assumed that the real shock to
individual aggregate demand is always equal to its average value, so that
hh =
, ∀ i. Simple algebra and a suitable change of variable, i.e.
h Ib
Rx
2
φ
logistic map (Iooss, 1979):
t
xx
Γ=
+
1
where
γ++=Γ
Rhtt
is the control parameter. It is well known that such a
firstorder nonlinear map can display deterministic cycles if 3 < Γ < 3.57, and
nonperiodic — i.e., chaotic — behaviour if 3.57 < Γ < 4 (Baumol and
Benhabib, 1989). In this latter case, the time series generated by the one
tit
t
t
t
a
2
=
, allows us to show that equation (8) can be reduced to a
()
tt
x
−
1
. (9)
Page 12
10
dimensional deterministic difference equation (9) are characterized by
irregular fluctuations which can mimic actual data. In particular, the dynamic
path of the economy enter a chaotic regions for particular combinations of the
average slope of the individual offer functions (itself parameterized by the
interest rate and the real product wage), of the interest rate on loans, and of
the parameter measuring spillovers in the equity market.
The literature modelling endogenous business fluctuations in terms of
nonlinear dynamic systems and chaotic attractors is large. See Day (1996) for
a nice introduction to methods and applications. In frameworks strictly
related to the one we employ here, Gallegati (1994) shows how to derive
sudden shifts between the periodic and nonperiodic regimes due to
stochastic influences on the tuning parameter, while Delli Gatti and Gallegati
(1996) introduce technological progress to obtain fluctuating growth.
3. The firms’ size distribution
This Section is devoted to analyzing the crosssection dynamics associated to
system (7), which amounts to studying the firms’ size distribution and its
evolution. A natural question is whether such a distribution converges
towards a longrun stable (i.e., invariant) distribution, or if it is bound to
fluctuate in a randomlike manner.
It must be stressed from the start that the aggregate dynamics generated
by the aggregate model (8)(9) resembles the Richard Goodwin’s prey
predator growthcycle model (Goodwin, 1967), which represents the first
attempt to adapt the mathematical description of biological evolution due to
Alfred Lotka and Vito Volterra to an investigation of the way a modern
economy works. In this masterpiece of nonlinear economic dynamics,
Goodwin recast in a new, analytically elegant guise11 the Marxian analysis of
capitalism and its inherent instability. The two classes of capitalist and
workers compete for the national income, which at any time period is divided
between profits and wages according to their relative strength. During an
expansion phase, growing investment opportunities call for an increasing
demand for labour. Rising wages and rising employment, in turn, determine
an increase in the share of national income going to workers. As the profit
11 The Nobel Laureate Robert Solow used the following words to comment the 1967
Goodwin’s paper: “[It] is five pages long. It does its business clearly and forcefully and stops. It
contains no empty calories” (Veluppillai, 1990, p.34).
Page 13
11
share declines, however, gross investments  and therefore total output  fall.
As a result, new job creation shrinks and the unemployed labour force (the
socalled reserve army of the unemployed) is enlarged. The growth in the
unemployed puts downward competitive pressure on wages, and thus
provides greater profit opportunities. Larger profits stimulates additional
capital accumulation, and economic activity and labour demand increase. The
business cycle can begin again.
The story we are telling is basically the same as the Goodwin’s one. During
an upswing, the increase of output induces higher profits and more private
equity funds. Higher production means also rising employment and higher
wages, however. The increased wage bill calls for more bank loans which,
when repaid, will depress profits and, through equations (4) and (5), the
production and the equity level as well. The labour requirement thus
decreases, along with the real wage, while profits rise. This restores
profitability and the cycle can start again.
Moving from an aggregate to a cross sectional perspective, it seems
worthwhile to stress that the predatorprey analogy can be extended to the
nonlinear dynamical system (7), as it represents a Generalized LotkaVolterra
(GLV) system (Solomon, 2000), whose solution describes the limit (longrun)
behaviour of the firms’ size distribution as measured by their equity,
employment or sales. Such a system is completely defined by: i) a stochastic
autocatalytic term representing production and how it impacts on equity; ii) a
drift term representing the influence played − via the hot market effect − by
aggregate capitalization on the financial position of each firm; and iii) a time
dependent saturation term capturing the competitive pressure exerted by the
labour market.
In what follows we will borrow from the work of Solomon and Richmond
(2001), to show that the cross sectional predictions of the GLV system (7) are
consistent with the available empirical evidence on how firms are distributed.
In particular, the right tail of the firms’ size distribution turns out to exhibit a
Pareto distribution of the form:
P
over several orders of magnitude. Interesting enough, the GLV model ensures
a stable exponent α even in the presence of large fluctuations of the terms
parameterizing the economy, namely the random equity productivity term h
and the aggregate employment level n.
( ) a
∼
α−−1
a
(10)
Page 14
12
To see how this result can be obtained, we transform the system (7) from
discrete to continuous time to write the time evolution of the equity base of
firm i as:
( )()( )( )
[
ttatat da
iiii
βτ=−+=
where τ is a (continuous) time interval, βi = hi + R, and (
] ( )
a
i
1
( )
t
() ( ) ta
t
taat
i
,
δγ−+−
(11)
)
( )( ) ta
th
φ
Rb
ta
2
2
,
δ
=
.
To ensure the limit τ → ∞ to be meaningful, let γi(t), (
averages
( ) 1
−=
ts
 equal for all i  to be of order τ. Let us define the
variance of the random terms β:
ββσ−=
i
) ta,
δ
and the time
i β
222
∼ ()
( )
t
[
ε
]2
2
ii
ββ=−
(12)
where
( )
t
( ) β−
t
βε=
ii
represents the stochastic fluctuations of the
autocatalytic term. Without any loss of generality, we assume
Note that β i(t) – 1 = ε i(t) + s, so we can write:
( ) ( )
[
tt da
ii
ε=
( )
t
0
=
iε
.
] ( )
a
i
( )
t
() ( ) ta
i
taats
i
,
δγ−++
. (13)
If we introduce the change of variable
( )
t
( )
( ) ta
ta
i
i
=ϕ
, which represents the
relative equity of the ith firm, and apply the chain rule for differentials, we
obtain:
( )( )
[
ii
ttd
εϕ=
The system (14) shows that the stochastic dynamics of the relative equity
base and, due to the linearity assumptions, of the relative sales and
employment as well, reduces to a set of independent, linear equations. If we
assume that the variance of the uncertainty associated to the relative price is
common to all, so that
σσ =
i
, and the same is true for the hot market
]
ϕ
iii
γγ+−
. (14)
22
Page 15
13
influence on individual equity,
2001):
γγ =
i
, we obtain (Solomon and Richmond,
( )
ϕ
P
∼
⎥⎦
⎤
⎢⎣
⎡−
σ
−−
ϕ
γ
2
ϕ
α
2
1
exp
(15)
with
2
2
σ
1
γ
α+=
. The distribution P(ϕ) is unimodal, as it peaks at
γ
σ
ϕ
2
0
1
1
+
=
. Above ϕ0, that is on its upper tail, it behaves like a power law
with scaling exponent α, while below ϕ0 it vanishes very fast.
The theoretical value of the scaling exponent α is bounded from below to
1,12 it increases with γ and it decreases with σ2. In other terms, α depends
exclusively on: i) how much firms are rationed in issuing new risk capital or,
in other terms, on how much capital markets are affected by adverse selection
and moral hazard phenomena; ii) how much individuals are heterogeneous
as regards the perceived risk associated to their final demand. Given that both
the degree with which these two factors bites and the array of institutions set
up to confront them differ widely among countries, besides predicting that
the firms’ size distribution scales down as a power law, our model suggests
that the degree of industrial concentration should be countryspecific.13
Both the shape of the distribution P(ϕ)  and therefore of P(a), P(y) and P(n)
 and the heterogeneity of scaling among countries are consistent with the
empirical evidence. Axtell (2001) reports estimates for the whole universe of
the U.S. firms as derived from the Census of Manufactures, suggesting that
the firms’ size distribution is Pareto distributed with α = 1. Other studies
concentrate on large firms in international samples, largely confirming
Axtell’s results. Gaffeo et al. (2003), for instance, show that the upper tail of
the firms’ size distribution as derived for a pool of quoted large companies
from the G7 countries is Pareto, and that the scaling exponent is comprised
between 1 (total sales) and 1.15 (total assets) as one changes the proxy used to
12 The case α = 1 is also known as the Zipf’s law.
13 Naldi (2003) derives analytical relationships between the scaling exponent of a power law
distribution and several major concentration indices, like the HirschmanHerfindahl and the
Gini indices.
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