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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

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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

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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 right-skewness 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 long-run

implications. In this chapter we show that a right-skewed firms’ size

distribution, with its upper tail scaling down as a power law, arises

naturally from a simple choice-theoretic model based on financial

market imperfections and a wage setting relationship. Our results

rest on a multi-agent generalization of the prey-predator model,

firstly introduced into economics by Richard Goodwin forty years

ago.

JEL classification: L11, D92, E32

Keywords: Firm size; Prey-predator 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.

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1. Introduction

Starting with the pioneering work of Gibrat (1931), the study of the

determinants and the shape of the steady-state 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 right-skewed and

approximately log-normal, recent empirical research has lent support to the

view suggested by H. Simon and his co-author (Ijiri and Simon, 1977),

according to whom a Pareto-Levy (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 Kiss-Haypal, 2000; Gaffeo et al., 2003).1

Regardless of the different outcomes obtained from distribution-fitting

exercises, the most popular explanation for right-skewness 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 ad-hoc 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, game-theoretic

models4 has shown that their implications as regards the steady-state 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, reversion-to-

the-mean 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,

profit-maximizing 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).

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(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 cross-section 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 right-skewed firms’ size distribution emerges as

a natural feature of the endogenous cross-section 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 long-run degenerate behaviour, typical of

Gibrat’s processes, and it grants the modeller more degrees of freedom than

the stochastic-game-theoretic models referred to above. From an analytical

point of view, our results are based on the possibility of describing the

economy as a Generalized Lotka-Volterra system, that is a multi-agent

extension of the prey-predator framework formally introduced to economists

by Richard M. Goodwin (1913-1996) 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 long-run

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).

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stock market returns and the personal wealth distribution (Solomon and

Richmond, 2001), the application of this Generalized Lotka-Volterra 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 cross-section

dynamics and its long-run 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

Johansson-Stenman, 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.

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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 vice-versa (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 sub-optimal,

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 i-th 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 i-th 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, agent-based model where aggregate (mean-field) 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).

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random variables {ui} be uniformly, but non-identically, distributed. In

particular, support of the i-th 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 bankruptcy-induced 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

it itit it

z

z

y

aw

φ

R cy

z

zu cy

uu cyCE

−

⎥

⎦

⎤

⎢

⎣

⎡

−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

=

−

=<=

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

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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.

()

⎥

⎦

⎤

⎢

⎣

⎡

−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−

⎞

⎜⎜

⎛

−−=−

ititit it

t

it

it

itt

itit it

y

y Rayz

Rw

φ

z

c

a

yw

RyCE

2

12

max

π

. (3)

it itit

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

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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:

(

it itit

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 mean-field 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, insider-outsider 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 non-market interactions. Such an

institutionally-constrained microeconomic analysis calls for the search of

appropriate macrofoundations of microeconomics (Colander, 1996). The

importance of this issue will be further discussed below.

)

ti ititt

aanw

γ+−

(5)

ta is the average capitalization of firms operating at time t, and γi > 0.

(6)

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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 cross-sectional 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

first-order nonlinear map can display deterministic cycles if 3 < Γ < 3.57, and

non-periodic — i.e., chaotic — behaviour if 3.57 < Γ < 4 (Baumol and

Benhabib, 1989). In this latter case, the time series generated by the one-

t it

t

t

t

a

2

=

, allows us to show that equation (8) can be reduced to a

()

tt

x

−

1

. (9)

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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

non-linear 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 non-periodic 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 cross-section 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 long-run stable (i.e., invariant) distribution, or if it is bound to

fluctuate in a random-like 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 growth-cycle 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 non-linear economic dynamics,

Goodwin re-cast 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).

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share declines, however, gross investments - and therefore total output - fall.

As a result, new job creation shrinks and the unemployed labour force (the

so-called 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 predator-prey analogy can be extended to the

non-linear dynamical system (7), as it represents a Generalized Lotka-Volterra

(GLV) system (Solomon, 2000), whose solution describes the limit (long-run)

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)

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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 i-th 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

σ σ =

i

, and the same is true for the hot market

]

ϕ

iii

γγ+−

. (14)

22

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influence on individual equity,

2001):

γ γ =

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 country-specific.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 Hirschman-Herfindahl and the

Gini indices.