© Copyright JASSS

  JASSS logo ----

Matteo Richiardi (2004)

Generalizing Gibrat: Reasonable Multiplicative Models of Firm Dynamics

Journal of Artificial Societies and Social Simulation vol. 7, no. 1
<http://jasss.soc.surrey.ac.uk/7/1/2.html>

To cite articles published in the Journal of Artificial Societies and Social Simulation, please reference the above information and include paragraph numbers if necessary

Received: 24-Mar-2003      Accepted: 23-Jul-2003      Published: 31-Jan-2004


* Abstract

Multiplicative models of firm dynamics á la Gibrat have become a standard reference in industrial organization. However, some unpleasant properties of their implied dynamics - namely, their explosive or implosive behaviour (firm size and number collapsing to zero or increasing indefinitely) - have been given only very little attention. In this paper I investigate using simulations which modifications to the standard multiplicative model of firm dynamics lead to stable (and reasonable) distributions of firm size. I show that in order to obtain stable systems for a wide range of average growth rate, either heteroskedasticity in the growth rates has to be assumed, or entry and exit mechanisms included. In particular I show that combining the broad class of threshold entry mechanisms and the more restricted class of threshold exit mechanisms with overcapacity penalizing all firms (where entry and exit are determined with reference to an exogenously defined total capacity of the market), lead to stable distributions even in the case of growth rate homoskedasticity, given a non-zero minimum threshold for firm size.

Keywords:
Firm growth, Gibrat's Law, Entry, Exit

* Introduction

1.1
Multiplicative models of firm dynamics 'à la Gibrat' (Gibrat 1930, 1931), where the evolution of any given firm is thought of as a sequence of stochastic multiplicative shocks, have become a standard reference in the industrial organization and economic geography literature. However, some unpleasant properties of their implied dynamics – namely, their explosive or implosive behaviour (firm size and number collapsing to zero or increasing indefinitely) - have been given only very little attention. Overall, these models may be well suited to study particular urban population dynamics in non-mature economies, where 'explosive' dynamics may also be welcome, but they seem to offer very little to the study of firm growth.

1.2
It is thus surprising that such a big strand in the literature of applied industrial economics is devoted to trying to confirm or reject Gibrat assumptions of mean and variance homoskedasticity in the rate of firm growth[1]. Most papers don't pay attention at all to the fact that the growth rates they found in the data are incompatible with any definition of equilibrium in the industry, given the multiplicative model they assume: apply the empirical growth process these papers claim to have characterized for just a few periods and the industry will dry out, or expand without limits. Here, two issues are at stake. First, we may actually face an out-of-equilibrium situation, but then it seems to make little sense to characterize the industry at that stage. Second, changes in the theoretical model could be more appropriate, but this could in turn imply the need to focus also on other things than the growth rate of existing firms. Entry and exit dynamics are first candidate, as it will be shown below. Interaction models, where each firm is affected by what happens to the others, are also naturally called for. However, these models move away from a purely stochastic description of firm demography, to adopt some kind of maximising framework (Sutton 1997).

1.3
Multiplicative models have nourished because they offer a simple but realistic description of what happens at an individual firm level. After all, it is not that easy to predict whether a given firm will grow or shrink in the future. Success and failure contain without any doubts some stochastic elements, and it appears reasonable to think of them as multiplicative shocks, where big firms variations in size are in absolute value larger than small firms variations.

1.4
Moreover, multiplicative models lead very easily to nice aggregate distributions of firm size. Lognormal distributions (the probability distribution function of the logarithm of size f (log S) is Normal) and power-law distributions (in a log-log plot, the complementary cumulative distribution function of size 1 - F (S) is a straight line) are often obtained. They both imply a very large number of small firms, and a small number of very large firms, a feature observed in the real world, even if the empirical literature has stressed that both sectoral factors (such as size of the market, capital intensity, R&D intensity) and country factors (such as level of human capital, judicial efficiency and accounting standards) affect the actual distribution (Bartelsman et al. 2003).

1.5
However, the focus of the theoretical literature on the exact shape of the resulting firm size distribution is surprising, compared with the little attention its anchorage has received. We should not pay too much attention to a model that predicts a power-law distribution of firm size, as long as this distribution degenerates rapidly to zero, or to infinity.

1.6
In this paper, I want both less and more. While remaining in the stochastic framework of multiplicative models, I don't go in search of a particular functional form for the size distribution. Instead, I look for 'reasonable' distributions of firm size, where I define a Reasonable distribution (R-distribution) as:
  • a left skewed and truncated distribution
  • with fat right tail, in order to span empirically over some orders of magnitude
  • with finite stationary mean and variance,
  • of a finite stationary number of firms.

1.7
In the next sections, I will look at which conditions a multiplicative model of firm growth must satisfy in order to show an R-distribution of firm size. In particular, I will consider different entry and exit mechanisms. The paper is structured as follows. Section 2 is devoted to a brief survey of the use of multiplicative models in the literature. Section 3 describes a general multiplicative model with entry and exit mechanisms embedding most models described in section 1 as particular cases. The case for a simulation study is put forward in section4. Section 5 deals with the simulation set-up, while in the following section, Section 6, I deal with the issue of recognizing the long-term equilibrium of each simulation run. Section 7 contains the results, both for models without and for models with entry and exit dynamics. Section 8 offers my conclusions.

* The literature

2.1
It is a well known fact that firm size distribution – as well as many other aggregate phenomena like city size, web graph and file size, word frequency, average weight of different species... - exhibits common features across time and space. In particular, it is recognized to be highly skewed to the left, and to span over several order of magnitude, i.e. to show 'fat' tails. Moreover, many studies have found that this distribution can be very well approximated by a power-law, given by the probability distribution function:

P[X = x] ~ x-(k+1) = x-a (1)

2.2
The first appearance of this - now very fashionable - distribution dates back to Pareto in the late XIX century. He showed that income distribution follows what since then has been called a Pareto distribution, with a cumulative distribution function given by:

P[X > x] ~ x-k (2)

2.3
More than 30 years later, George Kingsley Zipf, a Harvard linguistics professor, sought to determine the 'size' (or frequency of use in English text) y of the ith most common word. He found that this frequency is inversely proportional to it's rank r. This regularity has of course been baptized after him and with the name of Zipf's Law (Zipf 1932)

yi ~ ri-b, with b close to unity (3)

has found many applications, in particular to the study of city size distribution, where it is known to be surprisingly robust and stable.

2.4
Even if a long time had to pass before the connection was recognized, Zipf, Pareto and power-law distributions are actually the same thing, Pareto being the c.d.f. of a power-law and Zipf its expression in terms of rank, in the particular case of a Pareto coefficient equal to 1 (Adamic 2000).

2.5
Once these regularities were noted, the challenge became finding simple statistical models to explain them[2]. Here, the benchmark is still Gibrat's 1930 Law of Proportionate Effect, stating that if growth rates of firms in a fixed population (i.e. abstracting from entry and exit dynamics) are independent of size and uncorrelated, the resulting distribution is lognormal. He thus introduced the first multiplicative model of firm dynamics

St+1 = λ t St (4)

Taking logs, this model reduces to

lnSt+1 = Σ ln λt (5)

2.6
The model is sometimes expressed in terms of instantaneous growth rate, St+1 = St exp(λt) , but it leads to the same conclusions. If the growth rates are independently distributed, by the law of great numbers each firm's logarithm of size at any time sufficiently far from the start is a random extraction from a normal distribution. Thus, in this very simple model with no interaction between firms, firm size follows a lognormal distribution.

2.7
Many variations of the Gibrat model have been developed in the literature, while remaining within a purely statistical description of firm dynamics. The main results of this strand of research was to show that even small variations from Gibrat Law lead to power-law distributions[3].

2.8
Simon and Bonini (1958) introduced very simple entry dynamics, by assuming that:
  • only a fixed number of independent opportunities arise in the market at each time,
  • the probability of an existing firm taking up each opportunity is proportional to its size (Gibrat Law)
  • the probability of a new firm taking up each opportunity is constant
With this model of 'preferential attachment' they showed that firm size distribution follows a power-law, although only in the upper tail.

2.9
Kesten (1973) added to Gibrat model an additive term

St+1 = λtSt + ρ t (6)

This model defines a stationary process if E(lnSt) < 0. Moreover, if St sometimes takes values larger than one (intermittent amplifications) and the (constant or stochastic) additive term is not null, the process leads to a power-law pdf.

2.10
More recently, Levy and Solomon (1996a) show that a power-law can also be obtained by adding a reflection condition to the Gibrat model, i.e. by assuming that firm size is bounded from below to a threshold proportional to the average firm size:

Eqn 7 (7)

2.11
Manrubia and Zanette (1999) study a stochastic multiplicative process with reset events, and show that the model develops a stationary power-law probability distribution. Nirei and Souma (2002), disliking the mobile minimum threshold feature of Solomon and Levy's model, show that adding a fixed lower reflective barrier to the model of Manrubia and Zanette basically preserves their results.

2.12
An interesting and general link between multiplicative models and Markovian processes is found by Cordoba (2002). He refers to city growth, and shows that, if size follows a stationary Markov continuous diffusion process and total urban population keeps growing, in order to produce a power-law distribution cities "must exhibit (i) an expected growth rate that is independent of their size; and (ii) a growth variance that is proportional to size δ-1, where δ is the Pareto exponent found in the data". This means that Zipf's Law (the case δ = 1) must result from Gibrat Law, if the hypothesis about the diffusion process are correct. Cordoba's result also allows for the emergence of new city, as far as the emergence rate is lower than the growth rate of existing cities.

2.13
In another paper, Levy and Solomon (1996b) show that power-law distributions arise very naturally from stochastic multiplicative dynamics[4]. Sornette & Cont (1997) and Sornette (1998) further generalize this strand of literature. They show that the Kesten model is only one among many convergent E(lsSt< 0 multiplicative processes with repulsion from the origin of the form

St+1 = exp( F(x), {λt, ρt,...}λt xt (8)

such that F → 0 for large St (thus leading to a 'pure' multiplicative process) and F → ∞ for St → 0 (repulsion from the origin). With some additional constraints on F, this class of processes has a power-law pdf.

2.14
Finally, Blank and Solomon (2000) incorporate both entry and exit dynamics by assuming that firms disappear if they fall below a certain threshold Smin (of magnitude 1), and that at each period ΔN = k ·(S*t+1 - S*t) new firms enter the market , with size Smin (S* is the sum of all firms size, i.e. the total dimension of the industry). This model also leads to a power-law distribution of firm size.

2.15
However, most of these models are not able to solve the basic problem of multiplicative processes, namely their implosive or explosive behaviour: in the 'pure' Gibrat model, if the growth rate is too low, firm size rapidly drifts towards zero, while if it is too high, it degenerates to infinity.

2.16
In particular, even in the case when the average growth rate is zero ( E (λ) = 1), the distribution degenerates. The expected value of firm size remains constant, but this is due to a combination of a large number of realizations collapsing towards zero, and a small number of cases in which S becomes extremely large[5]. This is shown in the simulation results presented in figure 1, where 1000 firms of unity size are evolved according to a pure multiplicative model with normally distributed rate of growth (mean = 0, standard deviation = 0.15). In the logarithmic formulation of equation [5], E (ln S) → -∞, since E (lnλ) < ln ( E (λ)) = 0 and all terms in the right hand side of equation [5] have negative expected value. One could argue that the mean of the distribution of firm size is of little interest, since it can be heavily influenced by a limited number of outliers. This is in particular true for random multiplicative processes, characterized by a large discrepancy between expected value and most probable value. For instance, if we consider a random variable λ such that ln&lamda; is normally distributed with 0 mean, we obtain a more constant trend for the median of the firm size distribution, while the mean trends upwards (figure 2). Here, of course, E (λ) > 1, and thus E (S)→ ∞.

Fig 1
Figure 1. Evolution of firm size distribution with λ ~ N (1, 0.15)

Fig 2
Figure 2. Evolution of firm size mean and median with ln λ ~ N (1, 0.20)

2.17
In general, in random multiplicative processes, "the distribution of the product and the behaviour of the moments are crucially sensitive to extreme events. Consequently, there is no analog of a central limit theorem, as in the case of random additive processes, in which typical events are sufficient to determine the statistical properties of the sum of a large number of random variables. [...] [T]he log-normal approximation fails to adequately represent the statistical properties of the product in the continuum limit. [...] [T]he correct continuum limit [...] can be viewed as a log-normal function, but one whose precise form depends on the order of the moment being considered" (Redner 1990).

2.18
At this point, it could be argued that 'pure' multiplicative processes remain of little interest for the analysis of firm demography. Whatever the growth rate, the variance of size keeps increasing indefinitely (the more the higher the variance of the growth rate distribution), thus flattening out the shape of the resulting size distribution.

2.19
Now, let's define the implosion set of a random multiplicative process to include all distributions of growth rates such that actual realizations are in general too low to empirically avoid the collapse of most firms. More precisely, we could say that, given a sample of n firms and a time horizon of T periods, the median firm size decreases in say 95% of the simulation runs. Correspondingly, the explosion set includes all distributions of growth rates such that actual realizations lead to a general 'run away' of firms (i.e. the median firm increases in size in 95% of the runs). For gaussian growth rates, the boundary between the two sets depends on the mean and variance of the distribution, for given n and T. An example is depicted in figure 3:

Fig 3
Figure 3. Median stationarity in a pure Gibrat model

2.20
Since considering multiplicative shocks seem a reasonable way of modelling from a statistical point of view the evolution of firm size, the challenge becomes adding some feature to the basic Gibrat model to avoid its degenerative behaviour.

2.21
Simon and Bonini variation does not help much: in their model, the number of firms in the market keeps rising; thus, existing firms face on average negative returns, since business opportunities are fixed. This implies the model implodes, no matter which value of the growth rate variance.

2.22
By adding a (stochastic) term ρ Kesten preserves the model from the risk of implosion, but not from the risk of explosion. Moreover, for combinations of the parameters that guarantee an implosion in Gibrat model, equilibrium firm size distribution in Kesten model is nothing else than that of ρ.

2.23
In Levy and Solomon's (1996a) model, low values of the minimum threshold still imply implosion, while high values lead to an explosion of the system. Moreover, the existence of such a mobile threshold is difficult to justify from an economic point of view.

2.24
Blank and Solomon's (2000) model faces the problem of implosion with the assumption that firms disappear if they fall below a minimum (constant) size. However, the model produces reasonable dynamics only for a very narrow range of the relevant parameters, when the average growth rate is close to zero. Implosion is avoided because the model provides an absolute anchorage to firm size, i.e. new firms size is independent of the existing firm size distribution. However, if the average growth rate is too small (negative, for instance) nothing can save the system from extinction, while if it is too high (where 'too high' means just slightly above 0) the number of firms keep rising indefinitely, and the convergence to a finite non-zero mean, equal to 1/k, is guaranteed only by more and more new firms of minimum size contrasting the explosion of the existing firms. Again, not a reasonable dynamics, even if characterized by a power-law distribution of firm size[6].

2.25
Cordoba's paper is the more general we have treated so far. However, it does not solve our problem of finding multiplicative models with 'stable' distributions, since total population in his model keeps also growing indefinitely, by assumption. Moreover, the characteristics of a continuous diffusion process do not fit properly an industrial dynamics perspective, since it is well known that firms can face drastic change in size, especially downwards. I won't investigate further relationships between other kind of Markovian processes and multiplicative models in this paper, although this could be of interest.

2.26
Among the models surveyed here, the most reasonable dynamics are generated by Manrubia and Zanette. However, it is difficult to justify any particular distribution for the reset dimension, without referring to a firm's death and subsequent birth of a new one. This way of modelling entry and exit implies of course that the total number of firms in the market is kept constant: a valid assumption only in license-regulated markets! Moreover, as it will be shown below, if the size distribution of new firms is related to the size distribution of the existing firms (a reasonable assumption, after all), the model is not able anymore to generate stable distributions, and firm size either implodes or explodes.

2.27
My paper follows a radically different approach. It investigates which kind of modifications to the pure Gibrat model lead to stable long-run equilibria. It is closer in spirit and in methods to McCloughan (1995). McCloughan considers a modified Gibrat model, which takes into account some of the main violations of Gibrat's Law (namely, the mean of the growth rate is allowed to be dependent on the size of the firm, and the growth rates are allowed to be serially correlated), together with a specific mechanism for entry and exit (entry is modelled as a learning game about the market opportunity between incumbents and potential entrants, which leads to a Poisson entry process; exit entails a distribution of critical sizes below which firms disappear). Calibration of the model yields thirteen empirical growth scenarios, six entry scenarios and three exit scenarios. The simulated data are then used to investigate the importance of growth, entry and exit in shaping concentration development.

2.28
My paper also simulates the effects of violations of Gibrat's Law, entry and exit on industry dynamics. However, it does not deal with real data. Rather, it is aimed at characterizing which processes lead to a distribution of firms size with some desired properties. A number of entry and exit mechanisms are considered, rather than just one. Which version of the general model better fits the data is left for future research.

* The model

3.1
McCloughan discerns five types of violation of Gibrat's Law:
  1. mean growth rate decreases with size of firm;
  2. mean growth rate increases with size of firm;
  3. growth variability decreases with size of firm;
  4. growth variability decreases with age of firm;
  5. growth rates exhibit first-order positive autocorrelation.

3.2
I'm concerned only with violations 1-3, i.e. with the issue of heteroskedasticity with respect to size. My starting point is a general multiplicative model, where growth rates are drawn from a Normal distribution, with mean and variance that are allowed to depend on the size of the firm (they can decrease to 0 with size at exponential speed, governed by two parameters, m and s; when m and s equal 0, mean and variance are homoskedastic). Firms with zero or negative dimension are cleaned up and exit the market.

Eqn 9 (9)

3.3
Different entry and exits dynamics are considered, representing two broad classes of mechanisms: threshold mechanisms and non-threshold ones.

3.4
Non-threshold mechanisms imply there is no reference to an exogenous threshold in determining entry and exit of firms, i.e. the model has no anchorage, except for the 'natural' floor at 0 for firms' size. Among this class, I consider:
  • Proportional Number entry:
    • a constant share of total population of firms is added at each period
    • initial size is drawn from the lower half of the existing firms size distribution

      Eqn 10 (10)

      where D1/2, t is the lower half of the (empirical) firm size distribution at time t.

  • Proportional Number exit:
    • a constant share of total population of firms exits the market at each period
    Note that the combination of Proportional Number entry and Proportional Number exit leads to Manrubia and Zanette model, but for the fact that the new firm size distribution is endogenously determined by the existing firms size.
  • Proportional Dimension entry (the mechanism postulated by Blank & Solomon in their 2000 paper):
    • new firms are added in proportion of the increase in the total sector dimension
    • initial size is the exogenously determined minimum size allowed
      Eqn 11 (11)

  • Minimum Size exit (the mechanism postulated by Blank & Solomon 2000):
    • firms below minimum size Smin exit the market

3.5
Among threshold mechanisms, where entry and exit are determined with reference to an exogenously defined demand (i.e. to a maximum capacity of the market, that could change exogenously from period to period), I consider:
  • Excess Demand entry:
    • if total size of the market is smaller than the optimal dimension, (a part of) the gap is filled with new firms;
    • initial size is drawn from the lower half of the existing firms size distribution

      Eqn 12 (12)

      where S* is the optimal size of the market, i.e. the dimension that keep supply and (exogenous) demand in equilibrium, given (exogenous) prices, and D1/2, t is the lower half of the (empirical) firm size distribution at time t.

  • Excess Supply Affects All exit:
    • if total size of the market is bigger than the optimal dimension, first all firms are reduced proportionally. Then, firms smaller than a minimum size exit the market;
  • Excess Supply Affects Small exit:
    • if total size of the market is bigger than the optimal dimension, smaller ones exit the market;
  • Excess Supply Affects Large exit:
    • if total size of the market is bigger than the optimal dimension, larger ones exit the market;

Table 1: Entry and exit mechanisms

ENTRYEXIT
NON-THRESHOLD· Proportional Number
· Proportional Dimension
· Minimum Size
· Proportional Number
THRESHOLD· Excess Demand· Excess Supply Affects All
· Excess Supply Affects Small
· Excess Supply Affects Large

3.6
Most of these mechanisms seem plausible, and altogether they characterize a fairly general class of entry and exit dynamics. Of course, the list could (and should) be extended, but it can work as an initial set of analysis.

3.7
It must be noted that I have not specified how size is measured: it could be both an input variable (employment) or an output variable (turnover). Of course, in case size employment is used as the target variable, the additional hypothesis of constant returns to scale must be made in order to justify threshold mechanisms.

* Methodology

4.1
One reason why the literature has focused so far only on more restrictive models, with single entry and/or exit mechanisms, is that it becomes quite hard to deal analytically with more general models. Since my goal is studying more generally the interaction between multiplicative models and entry and exit mechanisms, I have to give something, and abandon the purity of analytical models. I will simulate my models.

4.2
Simulation is often thought to be less general than analytical models. This is because analytical results are conditional on the specific hypothesis made about the model only, while simulation results are conditional both on the specific hypothesis of the model and the specific values of the parameters used in the simulation runs.

4.3
This is partly reversed by the fact that simulation allows fairly less restrictive hypothesis about the model, since the results are computed and need not to be solved analytically. However, the problem to state general propositions about the dynamics of the model starting only from point observations remains.

4.4
Another way of stating more or less the same thing is the follow. Both an analytical model and a simulation model are expressed in their structural form, although in the simulation model more flexible rules may be specified instead of equations (one example is the following, hypothetical, rule for determining firm growth: "first look at the particular firm which is closest in size; if it exits the market, do the same with a probability p, otherwise grow accordingly to a function of some other parameters" – try to express this with a formula!). By solving an analytical model, if possible, we find the only one reduced form corresponding to the structural form of the model. This is impossible to do in a simulation model. The reduced form (the data generating process) remains unknown. But we may estimate the reduced form (better: the local data generating process) on the artificial data resulting from a number of (somehow designed) artificial experiments. Of course, it is always possible that as soon as we move to other values of the parameters, the local data generating process will change dramatically, for example exhibiting singularities. But if the design of the experiments is sufficiently accurate, this problem becomes marginal, since we're not really interested in what happens only with an infinitesimal probability. Moreover, critical values of the parameters can often be guessed, and thus included in the experiments.

4.5
Before moving on, one last issue has to be addressed. In estimating the local data generating process, one functional form must be chosen. Having specified the micro-rules of the artificial world, the researcher generally knows which variables affect the outcome variable of interest, even if sometimes, in complicated models, the causal link between inputs and outputs may be quite indirect, and thus remain at first unnoticed. However, there are methodologies to reconstruct the causal structure from statistical data, as well as software applications that do it automatically (see, for instance, the Tetrad project at Carnegie Mellon University http://www.phil.cmu.edu/projects/tetrad/). Of course, the final choice of a functional form remains to a certain extent arbitrary, and may lead to very different specifications of the aggregate laws of the system. But as long as two different specifications provide the same description of the dynamics of the model in the relevant range of the parameters, we should not bother too much about which one is closest to the 'true' data generating process. Differently from estimation on real world data, the problem of a mis-specified model unable to make good predictions in out-of-sample data is not important here, since we're not constrained to particular ranges of the parameters in the design of the artificial experiments.

4.6
In what follows, no attempt to estimate any reduced form is made. This is because the focus is more on the exploration of a general class of models, and the descriptive analysis of their implied dynamics, rather than on the analysis of any particular model. Further investigations of this issue are left for future research.

* The simulation

5.1
The simulation model is written in Java code, using JAS libraries, developed by Michele Sonnessa at the University of Torino (http://sourceforge.net/projects/jaslibrary/). Simulations can be monitored graphically. Here is a typical simulation output with parameters:

Entry mechanism: Excess Demand
Exit mechanism: Excess Supply Affects All
Initial number of firms: 100
Maximum sector dimension: 300
α:1.0 (Take-up rate)
μ:0.1
σ: 0.1
m:1.0 (mean heteroskedasticity)
s: 0.0 (variance homoskedasticity)

Fig 4.1
Fig 4.2
Figure 4. Simulation output

5.1
Here, parameters have to be edited by hand at each run. Clearly, this reduces the feasibility of a big number of experiments, and compromises the possibility of replication. JAS Multi-Run feature has thus been used, which allows batch runs of the simulation with parameters changing according to a pre-defined algorithm at each run[7].

* Long run equilibrium

6.1
The R-distribution we're interested in is obviously an equilibrium distribution. Thus, the issue of when the equilibrium is reached becomes relevant. In other words, we need to decide when to stop each simulation run. Moreover, since simulations are performed in a batch mode, the 'human eye' criterion cannot be applied. The fact that we want to focus only on equilibrium relationships between the variables doesn't mean we're not interested in the out-of equilibrium dynamics. After all, one of the strengths of simulations is exactly the possibility of studying the disequilibrium dynamics of a system: even when a system can be characterized analytically in equilibrium, its out-of-the-equilibrium behaviour can often be investigated only through numerical simulations. Moreover, it is quite relevant to know how long it takes to reach the equilibrium: a model could be empirically relevant also when it explodes or implodes, as long as it takes long enough.

6.2
Thus, the wait-for-long-enough criterion is also not appropriate. We cannot wait – say – 10.000 periods before stopping a simulation and analysing its results because we want to know whether the system reached approximately the same state much earlier! Moreover, waiting too much reduces the number of experiments that can be done.

6.3
So, I developed a few algorithms to determine when the system becomes approximately stable. The first algorithm looks at whether a moving average of the median size and a moving average of the firm number remain constant, i.e. the first differences of the moving average remain around zero (0 + 0.1 times the moving standard error) for a sufficiently long period of time (if n is the moving average windows, the first differences must remain in the range in 95% of the last n periods of the simulation). The median is considered, rather than the mean, for robustness concerns.

6.4
The second algorithm looks directly at the mean of the size distribution and at the firm number, and checks whether these series remain in a given range for long enough. Every 5n, periods, where n is the moving average window used in the first algorithm, the two ranges are computed again. The first one, centred on the average dimension at that moment in time, is 0.2 times the standard deviation of firm size wide, while the second one goes from 1/2 to 3/2 of the firm number at that moment in time. The two series must remain bounded within these ranges for 5n periods.

6.5
Altogether, these two algorithms work pretty well (which one is invoked first depending on the values of the parameters), and save on average around 75% of the time compared to an appropriate constant length criterion.

6.6
Simulation runs are also stopped when the firm number becomes too high (of course, entry dynamics are needed), in order to prevent the simulation to go slower and slower. Firm size in this case implodes, and the system is considered to exhibit 'non-reasonable' behaviour.

6.7
If no other stop mechanisms have already become binding, the wait-for-long-enough criterion is invoked, and the simulation is stopped after 2,000, 5,000 or 10,000 periods (depending on the experiment). If the outcome looks 'reasonable', the simulation is re-run in the graphical mode, in order to investigate more in depth the dynamics of this system. Often, its long run dynamics are simply more volatile than allowed by the stop algorithms, as shown for example in figure 5.

Fig 5
Figure 5. Example of firm size distribution when long-run stopping mechanisms don't work (single simulation run)

* Results

7.1
As expected, the homoskedastic system (mean and variance of the growth rate independent of size) with no entry and exit exhibits very fast diverging dynamics, as soon as we move away from the line of figure 1.

Reasonable dynamics with growth rate depending on size

Mean heteroskedasticity

7.2
The only way of obtaining stable systems within a 'pure' multiplicative model is to give away with the homoskedasticity assumption. In particular, heteroskedasticity of the mean of the growth rate is needed. However, as it will be shown, this heteroskedasticity doesn't have to be assumed all the times, since it can be found in the data when variance heteroskedasticity instead is assumed.

7.3
Figure 6 presents the outcomes of many simulation runs, where all parameters are kept constant except for the mean of the growth rate. Figure 6 shows the case of mean heteroskedasticity: the mean of the growth rate keeps declining with size with an exponential speed, governed by the parameter m, which remains fixed at 1.0 (see equation 9). It plots the different values of long-run (equilibrium) mean firm size, for different values of the mean of the growth rate (the latter are referred to a size 1 firm). Standard deviation of the growth rate remains fixed at 0.1. This may seem unrealistic, but it is useful to investigate the effect of mean heteroskedasticity alone. Variance heteroskedasticity will be considered below. This value of the standard deviation is low enough to guarantee that the number of firms remains constant, since it is very unlikely that any firm would pick up a growth rate lower than –1. However, in an infinite amount of time all firms will eventually experience such a casualty, and thus disappear.

7.4
The reason why the system reaches a 'reasonable' steady state, for values of μ (1) big enough, is the balance between the explosive tendency of smaller firms and the implosive tendency of bigger firms. Thus, all we need to have R-distributions is that the average of the growth rate at initial size is in the explosive region of figure 1. One typical resulting firms size distribution is reported in figure 7.

Fig 6
Figure 6. Long-run dynamics, mean heteroskedasticity (multiple simulation runs)

Fig 7
Figure 7. Example of firm size distribution, mean heteroskedasticity (single simulation run)

7.5
Note that the asymmetry due to the threshold at 0 induce per se – a negative correlation between the mean of the observed growth rate and size, the stronger the bigger the growth rate. However, in the case of mean homoskedasticity as defined above, this is not enough to preserve the system from implosion. In this case, we actually face a sort of trade-off between implosion and extinction: if we raise the growth rate variance, we observe more stable distributions, but only until all firms disappear. Moreover, the system still explodes for big enough growth rates.
Variance heteroskedasticity

7.6
Things change – if only slightly – in the mean homoskedasticity case, if we consider variance heteroskedasticity, i.e. the standard deviation of the growth rate keeps declining with size with an exponential speed, governed by the parameter s. Here, the tendency towards implosion (but not towards explosion) can be contrasted a little more. Bigger variance for small firms imply some firms will grow, while others will shrink considerably. When a firm shrinks to zero, it exits the market. This implies the average size does not decline steadily to zero, as in the standard Gibrat version. Of course, the number of active firms becomes smaller and smaller. However, some firms will pick up high growth rate at their initial stage. As they grow bigger, the variance of their growth rate becomes smaller. This induces a sort of lock-in effect: once a firm has become big, it can't move backwards as easily. However the tendency to implode due to the zero growth rate mean assumption is still at work: slowly, big firms keep shrinking. But the smaller they become, the bigger the variance of the growth rate: again, some will disappear and others will grow. Empirically, after a short time the number of active firms becomes quite stable, i.e. 'unlucky' firms who got negative returns in their initial stage disappear, while 'lucky' firms become big enough to avoid this risk. Only running the simulation for a very long time the system will eventually reach its final, empty, state. To give an idea, with a mean growth rate μ = 0, s = 1.0 and a standard deviation for a size-1 firm σ(1) = 0.5 (a high but not unrealistic value in order to magnify the effects of this parameter), starting from a population of 100 firms the system adjusts to around 30 firms in 700 periods, and then becomes quite stable. After 20,000 periods there are still more than 20 firms in the system.

Reasonable dynamics with entry and exit mechanisms

7.7
So far, I have showed that in order to get 'reasonable' dynamics in a multiplicative model, at least some form of mean or variance heteroskedasticity in the growth rate of firms must be assumed. However, models without entry or exit dynamics are not very interesting in themselves, but for benchmarking purposes. I now turn to investigating extensions of the basic multiplicative model, which take into consideration the fact that the market could be characterized by a turnover of firms. As it will be shown, weaker conditions are necessary in order to get 'reasonable' dynamics in such a market.

7.8
When entry or exit mechanisms are specified within the context of mean or variance heteroskedasticity, R-distributions are again obtained. Moreover, the risk of the market vanishing because all firms dry out is avoided either by specifying a threshold entry (like Excess Demand entry) or by specifying a non-threshold entry guaranteeing a sufficiently high number of new firms. The result is quite obvious, thus no simulation output is reported in order to show it.

7.9
More interestingly, by considering entry and exit R-distributions can also be obtained in the case of homoskedasticity. Here, however, a threshold mechanism must be specified both for entry and for exit.

7.10
This is an explanation of the underlying dynamics. We have already seen that multiplicative models suffer from two contrasting tendencies, either to implosive or to explosive behaviour. To reach R-distributions, these two tendencies have to be balanced. To contrast implosion, a natural threshold arises from the fact that no firm can shrink below size 0. Removing 0-sized firms from the market (or any firm below a minimum size, such as in Blank & Solomon 2000) induces, as I showed above, a negative correlation between size and growth rate. All we need then is that enough new firms are created, in order to preserve the system from extinction. Of course, if the growth rate variance is too small, it becomes too rare for any firm to actually fall below the minimum size. Firm dimension would then slowly deflate towards the minimum size.

7.11
Thus, given a high enough growth rate variance, the problem is to specify a mechanism that lets an appropriate number of new firms to enter the market, i.e. a number 'in the long run' approximately equal to that of disappearing old firms (falling below the minimum size threshold). Threshold entry mechanisms guarantee this condition is always met. Non-threshold mechanisms do not normally work, unless they're carefully ad hoc tuned. In the appendix simulation results for Proportional Dimension (Blank & Solomon 2000) and Proportional Number mechanisms are reported (tables A.1 and A.2)[8].

7.12
By specifying an Excess Demand entry mechanism, interesting dynamics as the one depicted in figure 8 for the total number of firms in the market can be obtained:

Fig 8
Figure 8. Total firm number with positive minimum size, zero average growth and Excess Demand entry (single simulation run)

7.13
One final note. Threshold mechanisms, as I have defined them, work by comparing total dimension (capacity) with an exogenously given demand. New firms are added if supply falls short of demand, and new firm size is drawn from the lower part of the existing firm size distributions. If the lower threshold is fixed at 0, these mechanisms keep introducing new firms as the existing ones shrink, since the effect of new capacity on total capacity gets smaller and smaller (adding a 0-sized firm doesn't help in increasing supply!). Simulation results are reported in the appendix (table A.3).

7.14
Thus, my conclusion is that in order to contrast implosion and get R-distributions, threshold entry mechanisms are enough, given a non-zero minimum size. It is important to note that – as long as the danger remains on the implosion side – non-threshold exit mechanisms could well be assumed, in conjunction with threshold entry mechanisms.

7.15
Let's now turn to explosive behaviour, where things get slightly more complicated. Clearly, non-threshold mechanisms cannot work, in general, because they do not pose any limit to firm growth. But even if a threshold exit mechanism is specified, the exogenously defined total capacity available is generally monopolized by a single firm. In Excess Supply Affects Small total overcapacity hits only small firms, while in Excess Supply Affects Large it hits only large firms. However, both mechanisms generally lead to a monopoly. This is obviously the case if no entry mechanism is specified, because no new firms enter the market, while old firms keep exiting. It may also seem trivial for Excess Supply Affects Small, regardless to the entry mechanism specified[9] (see table A.4 in the Appendix). But the result holds true also with Excess Supply Affects Large, with most entry mechanisms (see table A.5 in the Appendix). The only way out, in this case, is specifying an entry mechanism where new firms enter the market at every period, with size independent of the existing firm size distribution and small. The number of firms to enter the market should be either independent of the number of existing firms, or negatively dependent. This is exactly what Proportional Dimension entry provides: new firm size is small and exogenously defined, and new firm number is dependent on the increase in total dimension of the industry, not on the number of active firms. Simulation results are reported in table A.6 in the Appendix.

7.16
There is only one class of threshold exit mechanisms which preserve the system from moving towards a monopoly, no matter which entry mechanism is specified[10]: exit mechanisms where overcapacity affects all firms in the market, reducing their growth rate, like in the Excess Supply Affects All one. This ultimately provides a solution to the explosive behaviour caused by a high average growth rate, by reducing it! However, even when such a mechanism is specified, if the minimum size is set to 0 the system moves towards an oligopolistic or monopolistic situation, with all very small firms but very few. Simulation results are reported in table A.7 in the Appendix.

Fig 9
Figure 9. Firm size distribution with Excess Supply Affects All exit mechanism, 0 minimum size and μ = 0.2

With positive minimum size this risk is avoided.

7.17
In conclusion, I have shown that in order to contrast explosion and get R-distributions, threshold exit mechanisms that penalize all firms are enough, given a non-zero minimum size. At this point, it can be noted that the two problems of implosion and explosion are disjoint. This is a general characteristic of multiplicative models. To avoid implosion, one should care about entry, and to avoid explosion one should care about exit. Thus, in order to obtain a stable system for a wide range of (homoskedastic) mean growth rates, we can combine the two results presented above.

7.18
Figure 10 shows the long-run relation between mean firm size and mean growth rate when Excess Demand entry and Excess Supply Affects All exit mechanisms are included, with a minimum size of 0.5 and a standard deviation of the growth rate of 0.1.

Fig 10.1 Fig 10.2
Figure 10. Long run dynamics, Excess Demand entry and Excess Supply Affects All exit

7.19
Finally, table 2 summarizes the implications of all combinations of different entry and exit mechanisms examined in the paper. Within each cell, the first line refers to what happens with 'low' average growth rate, while the second line to what happens with 'high' average growth rate.

Table 2: Model outcomes


Entry
ExitGibratNON-THRESHOLDTHRESHOLD

No entryProportional NumberProportional Dimension (*)Excess Demand
GibratNo exitimplosion
explosion
implosion
explosion
-R-distribution (*)
explosion

NON-THRESHOLDMinimum Sizeimplosion
explosion
implosion
explosion
implosion
explosion
R-distribution (*)
explosion
Proportional Numberimplosion
monopoly
implosion
explosion/monopoly
implosion
explosion
R-distribution (*)
monopoly

THRESHOLDExcess Supply Affects Smallimplosion
monopoly
implosion
monopoly
implosion
explosion
R-distribution (*)
monopoly
Excess Supply Affects Largeimplosion
monopoly
implosion
monopoly
implosion
R-distribution
R-distribution (*)
monopoly
Excess Supply Affects Allimplosion
monopoly
implosion
R-distribution (*)
implosion
R-distribution
R-distribution (*)
R-distribution (*)

(*) with positive minimum size

Implosion means firm size converging to zero, or firm number converging to zero given non-increasing firm size

Explosion means firm size growing indefinitely, or firm number growing indefinitely given non-collapsing firm size

When a Monopoly is reached, the remaining firm keeps growing indefinitely

All combinations were simulated. Results not reported in the Appendix are available upon request.

* Concluding remarks

8.1
In this paper I have extended, through simulation, previous results on multiplicative stochastic models. In particular I have shown that – in order for a multiplicative model of firm growth to exhibit reasonable dynamics for a wide range of average growth rate, we have either to assume heteroskedasticity in the growth rates, or to include entry and exit mechanisms. While other particular, ad hoc, entry and exit mechanisms could be imagined, I have shown that combining the broad class of threshold entry mechanisms and the more restricted class of threshold exit mechanisms where overcapacity penalizes all firms, lead to R-distributions even in the case of growth rate homoskedasticity, given a non-zero minimum threshold for firm size. Threshold mechanisms are defined as rules where entry and exit are determined with reference to an exogenously defined total capacity of the market.

8.2
It is interesting to note that real data seem to confirm the existence of many of the above examined features leading to R-distributions at work at the same time. Caves (1998) in his enumeration of stylized facts on the turnover and mobility of firms, indicates that "1. The variance of firms' proportional growth rates is not independent of their size but diminishes with it [...] 2. Mean growth rates of surviving firms are not independent of their sizes but tend to decline with size and also with the unit's age (given size) [...] 3. Entry and exit are intimately involved in growth-size relations. Entry is more likely to occur into smaller size classes, and the likelihood of a unit's exit declines with its size". A positive minimum size is obviously at work, since it is difficult to imagine a firm asking for no more than a very small fraction of the entrepreneur's time.

8.3
However, in order to preserve the skewness of the firm size distribution, the amount of variance heteroskedasticity (the rate of decline of the standard deviation of the growth rate, s) must be small, in order to maintain some variance in the growth rates when firm size approaches its long-run distribution. Thus, the smaller the mean growth rate (or the higher the amount of mean heteroskedasticity), the more shifted to the left will be the long-run distribution of firm size, and the smaller the amount of variance heteroskedasticity that can be supported.

8.4
Finally, to my judgement future research on these issue could go in three directions. First, generalized Gibrat models could be calibrated on real data, by choosing the appropriate entry and exit mechanisms and the values of the relevant parameters in order to characterize different industries. Second, the limit distribution of firm size could be computed using these calibrated models, thus providing a measure whether any particular market is in equilibrium (close to its implied long-run distribution). Third, economic models of firms or employees behaviour could be developed, with the aim to reconstruct 'from the inside out' or 'from the bottom up' the aggregate behaviour implied by these 'reasonable' multiplicative models.


* Notes

1 First empirical tests date back to the work of Hart and Prais (1956), Hart (1962), Mansfield (1962), Hymer and Pashigian (1962) and Samuels (1965). Other related work can be found in Singh and Whittington (1975), Chesher (1979), Kumar (1985), Leonard (1986), Evans (1987a, 1987b), Hall (1987), Boeri (1989), Contini and Revelli (1989), FitzRoy and Kraft (1991), Variyam and Kraybill (1992), Wagner (1992), Amirkhalkhali and Mukhopadhyay (1993), Bianco and Sestito (1993), Dunne and Hughes (1994), Tschoegl (1996), Amaral, Buldyrev, Havlin, Leschhorn, Maass, Salinger, Stanley and Stanley (1997), Harhoff, Stahl and Woywode (1998), Hardwick and Adams (1999), Hart and Oulton (1999), Fariñas and Moreno (2000), Geroski, Lazarova, Urga and Walters (2000), Machado and Mata (2000), Acs and Armington (2001), Vander Vennet (2001), Audretsch, Klomp, Santarelli and Thurik (2002), Delmar, Davidsson and Gartner (2002), Goddard, Wilson and Blandon (2002). For a survey of the main results see Audretsch, Klomp, Santarelli and Thurik (2002). Stylized facts are worked out in Caves (1998).

2 See Sutton (1997) for a survey on the developments of these models

3 Indeed, if the variance of a lognormal distribution is large, it may appear like a power-law distribution for several orders of magnitude (Mitzenmacher 2001).

4 In particular, they show that the appearance of the scaling power-laws is as generic in multiplicative stochastic systems as the Boltzmann law is in additive stochastic systems

5 Redner (1990) notes that <<[a] crucial feature of the process is that extreme events, although exponentially rare in n, are exponentially different from the typical, or most probable value of the product>>.

6 Simulation results for Blank and Solomon model are presented in section 6

7 This feature also allows model calibration, if needed.

8 Since Proportional Dimension entry mechanism requires new firms size to be equal to the minimum size, this cannot be 0. So, in considering Proportional Dimension entry the minimum size has to be raised, and Minimum Size exit also considered (firms below minimum size exit the market), thus reproducing Blank & Solomon model.

9 The only exception, among the entry mechanisms considered here, is Proportional Dimension (Blank & Solomon 2000). In this case however, the combination of entry and exit mechanisms is simply incoherent, because threshold exit mechanisms imply a reduction in the number of firms in the market when capacity grows above the threshold, while Proportional Dimension entry implies an increase in the number of firms as total capacity increases. Which one predominates depends on the particular event schedule implemented in the simulation.

10 but as long as some new firms are allowed to enter the market. Otherwise, with exit but without entry, the system obviously moves towards a monopoly.


* Appendix - Simulation results

Legend:

  • 'entry' code 1 is Excess Demand entry; 'entry' code 2 is Proportional Number Entry; 'entry' code 3 is Proportional Dimension entry. For the sake of simplicity, in Excess Demand entry it is assumed that the gap between exogenous demand and supply is filled in just one period; in Proportional Number entry, the rate of entry is fixed at 10% of existing firms.
  • 'minGrowth' and 'stdGrowth' are respectively the mean and standard deviation of the growth rate
  • 'minSize' is the minimum size
  • 'k' is the coefficient governing the entry of new firms in Proportional Dimension entry
  • 'birthRate' is the exogenous birth rate in Proportional Number entry
  • 'meanSize' and 'varSize' are the mean and variance of firm size

Parameter values were drawn randomly within the range of interest. In general, the growth rate mean goes from –0.02 to 0.2; the growth rate standard deviation from 0.1 to 0.2; the minimum size from 0 to 0.5; the birth rate from 0 to 0.1 (when not specified, it is fixed at 0.1). Particularly interesting subsets of the parameter space were over-investigated.

Each simulation can stop because:

  • automatic stop algorithms signalled a long run equilibrium was reached;
  • there were no more firms in the system, or there were too many, i.e. more than 10,000, or the average size exceeded 1x106 (implosion, explosion or extinction);
  • the simulation reached the end of time, i.e. period 10,000.

Table A.1: Simulation results for Minimum Size exit and Proportional Dimension entry (Blank & Solomon 2000), mean and variance homoskedasticity

timemeanGrowthstdGrowthminSizekmeanSizevarSizefirmNumber
144- 0.019 0.100 0.369 0.088 - - -
190- 0.017 0.162 0.244 0.001 - - -
190- 0.016 0.155 0.410 0.026 - - -
184- 0.016 0.130 0.205 0.039 - - -
429- 0.015 0.114 0.187 0.013 - - -
566- 0.010 0.115 0.042 0.084 - - -
582- 0.009 0.101 0.331 0.057 - - -
811- 0.007 0.166 0.048 0.073 - - -
718- 0.004 0.130 0.446 0.030 - - -
654- 0.002 0.121 0.184 0.036 - - -
3209 0.004 0.164 0.239 0.077 3.208 20,788.608 10,000
736 0.007 0.128 0.046 0.048 2.391 15,597.058 10,000
541 0.010 0.128 0.090 0.020 3.466 3,854.679 10,000
595 0.013 0.193 0.225 0.001 4.357 16,559.634 10,000
304 0.017 0.140 0.013 0.036 3.233 14,043.853 10,000
242 0.021 0.159 0.494 0.097 4.241 3,634.481 10,000
254 0.021 0.110 0.129 0.033 3.350 1,779.483 10,000
212 0.024 0.180 0.343 0.091 4.075 3,653.648 10,000
156 0.027 0.104 0.289 0.093 2.986 570.287 10,000
157 0.031 0.139 0.219 0.072 3.387 692.511 10,000
119 0.032 0.178 0.483 0.029 3.477 1,642.138 10,000
125 0.033 0.113 0.334 0.093 2.647 210.094 10,000
120 0.038 0.133 0.312 0.056 2.695 395.503 10,000
119 0.039 0.118 0.038 0.033 2.324 708.301 10,000
95 0.040 0.127 0.390 0.045 2.509 322.448 10,000
101 0.041 0.181 0.454 0.030 2.981 395.599 10,000
101 0.041 0.162 0.363 0.046 2.895 652.151 10,000
109 0.042 0.112 0.085 0.035 2.170 820.554 10,000
91 0.048 0.101 0.026 0.053 1.948 457.778 10,000
93 0.050 0.190 0.181 0.092 2.703 4,772.303 10,000
82 0.057 0.106 0.023 0.068 1.823 292.299 10,000
61 0.057 0.107 0.315 0.038 1.778 69.647 10,000
61 0.065 0.114 0.186 0.062 1.753 119.713 10,000
63 0.066 0.132 0.167 0.058 1.810 206.126 10,000
60 0.067 0.132 0.190 0.066 1.863 111.348 10,000
69 0.067 0.122 0.010 0.028 1.764 431.344 10,000
52 0.067 0.157 0.401 0.095 2.025 81.031 10,000
55 0.069 0.123 0.262 0.049 1.715 55.592 10,000
57 0.071 0.183 0.263 0.100 2.163 228.262 10,000
48 0.081 0.145 0.247 0.026 1.758 76.322 10,000
50 0.084 0.193 0.152 0.065 1.978 254.309 10,000
48 0.089 0.191 0.179 0.084 1.966 184.449 10,000
42 0.090 0.193 0.392 0.071 1.935 46.714 10,000
38 0.092 0.128 0.285 0.020 1.532 34.855 10,000
39 0.093 0.128 0.280 0.094 1.523 31.653 10,000
48 0.093 0.165 0.064 0.093 1.754 272.310 10,000
37 0.094 0.113 0.256 0.096 1.418 25.878 10,000
43 0.095 0.117 0.098 0.074 1.442 74.375 10,000
46 0.095 0.125 0.037 0.061 1.477 115.729 10,000
29 0.099 0.131 0.481 0.060 1.506 10.518 10,000
38 0.100 0.149 0.240 0.056 1.610 50.363 10,000

The system generally either becomes extinct, or explodes in the number of firms. Only by appropriately choosing values for the average growth rate very close to zero, does it look stationary


Table A.2: Simulation results for Proportional Number entry, mean and variance homoskedasticity

timemeanGrowthstdGrowthbirthRatemeanSizevarSizefirmNumber
113-0.01990.14070.03650.0070731191.31E-0310000
50-0.01490.10550.08130.1132855361.21E-0210000
47-0.01220.10230.08790.1457492521.93E-0210000
46-0.01040.150.08810.0944861484.30E-0210000
44-0.00760.13180.09220.1448823283.96E-0210000
366-0.00750.18240.01222.87E-046.25E-0510000
245-0.00710.15650.01760.0038097680.00316497510000
5000-0.00690.1550.00054.53E-321.94E-61100
194-0.00660.1030.02160.0197098881.25E-0210000
60-0.00370.12310.06840.1395854527.20E-0210000
341-0.00340.10770.01310.0128547221.72E-0210000
63-0.00330.14130.06490.0879678985.63E-0210000
108-0.00150.11430.0380.1081121952.02E-0110000
375-0.00130.19230.0120.0026337353.57E-0310000
840.00140.19180.04840.0606337857.06E-0110000
460.00260.12770.08910.2280384838.03E-0210000
490.00420.18680.08370.1061015572.05E-0110000
600.00520.12880.06740.2378716111.43E-0110000
530.0060.11160.07750.3284864021.27E-0110000
50000.00730.1120.00358.99E+077.97E+17100
720.00760.19920.05710.1096417131.37E+0010000
50000.01250.12210.00211.94E+193.32E+40100
2320.01450.13390.01851.465814644232.145887610000
50000.01510.1410.00321.15E+191.29E+40100
50000.01690.19120.00929.33E+108.68E+23100
3010.01780.1990.01462.6106880824.66E+0310000
860.01910.11420.04780.7481690332.1222523510000
490.01980.16430.08290.4084403770.70130471110000
610.0220.17940.06670.3482754899.23E+0010000
50000.02270.12240.00644.23E+391.21E+81100
430.02790.14050.09670.6256116978.91E-0110000
770.02880.19410.05290.72372754.55E+0110000
660.02970.10190.06221.7210706973.89E+0010000
1030.03280.13180.03982.9265151821.34E+0210000
50000.03480.12750.00124.62E+682.14E+139100
1600.03710.14560.026111.43666133287.95453910000
540.03830.17510.07620.8427941325.93E+0010000
610.03910.1860.06690.7876070461.75E+0110000
1590.03920.13470.026225.845123313.52E+0410000
1040.04180.12360.03948.1228647815.67E+0210000
1550.04510.11250.026989.09353992.18E+0510000
660.04530.13690.06212.6095944544.61E+0110000
460.04760.14980.08871.5574847757.07E+0010000
1620.04790.14160.0258100.73263917.49E+0510000
970.05140.18120.04216.1029451171.65E+0310000
760.05180.17580.05373.1023404841.78E+0210000
3130.05460.12140.0142457849.69433.54E+1310000
620.05550.10780.06666.5552103965.60E+0110000
50000.05640.19980.00113.72E+901.10E+183100
50000.05980.1960.00524.42E+988.90E+198100
50000.06490.10010.00342.19E+1322.81E+266100
1730.06560.18310.02431515.772264.53E+0810000
470.06570.14320.08763.5071680942.79E+0110000
1770.06750.10080.023810640.510411.04E+0910000
1880.06790.19170.022410915.378181.84E+1110000
50000.07290.1510.00334.35E+1396.93E+280100
890.07340.13170.046267.699093782.99E+0410000
2190.07510.14050.0193336439.74065.48E+1210000
410.07560.17970.09953.4377855853.76E+0110000
410.07620.12680.09996.1551906833.5758777510000
460.08470.11470.088912.807948191.37E+0210000
2110.08640.13080.021674692.6511.57E+1410000
1020.0920.17820.04508.80348461.10E+0710000
480.09210.15830.08611.96688222502.548312510000
1220.09340.13120.03385913.840044.94E+0910000
1690.09540.12910.0248265109.80971.46E+1210000
480.09560.13340.08518.166080914.57E+0210000
1290.09680.18430.03197027.5727236.67E+0910000
490.09690.1780.082515.148530151.54E+0310000
1010.09850.15960.04051205.1340532.13E+0710000
540.09960.19860.075616.872695734.31E+0310000

The system either implodes in size, or explodes in size or in the number of firms


Table A.3: Simulation results for Excess Demand entry, mean and variance homoskedasticity

timemeanGrowthstdGrowthminSizemeanSizevarSizefirmNumber
86- 0.017 0.1833901650.000.08.32E-0110000
102- 0.016 0.1178018470.000.07.01E-0310000
88- 0.015 0.1699215870.000.05.64E-0210000
105- 0.014 0.157446530.000.07.38E-0210000
140- 0.007 0.1359997850.000.05.42E-0210000
185- 0.004 0.1340364940.000.08.10E-0210000
276- 0.002 0.1456083490.000.03.31E-0110000
521 0.001 0.1738168950.000.01.35E+0010000
1971 0.003 0.1368588430.000.02.53E+0010000
407 0.005 0.1804790.000.08.44E-0110000
878 0.005 0.1626323650.000.07.59E-0110000
774 0.005 0.1873728410.000.08.06E-0110000
5000 0.007 0.1328413940.00986.18.71E+07326
1983 0.008 0.1106861370.001060388.92.30E+14326
1102 0.010 0.1378899210.001048316.02.55E+14319
1708 0.012 0.1408651720.001066578.42.23E+14311
1052 0.014 0.1051002240.001008348.97.31E+13305
1423 0.015 0.1972987950.001022229.73.32E+14318
1209 0.017 0.1661142550.001060477.96.89E+13305
786 0.019 0.1075211610.001009923.21.46E+13306
635 0.021 0.1008139350.001047619.56.55E+13318
603 0.024 0.1368510060.001006363.13.58E+13320
596 0.026 0.1466627350.001016119.34.47E+13309
545 0.030 0.1778850870.001075796.09.51E+13299
500 0.030 0.1367517320.001040911.12.40E+13323
460 0.034 0.1784848280.001001656.54.87E+13302
379 0.037 0.1435980340.001106669.37.84E+13304
353 0.038 0.1990814170.001027488.12.69E+14304
381 0.038 0.1479131220.001005277.64.38E+13296
375 0.038 0.1523186340.001096690.37.07E+13305
369 0.039 0.1086640230.001041130.89.67E+12300
338 0.042 0.1013744580.001009610.98.70E+12295
297 0.046 0.1866184520.001000945.51.38E+14301
305 0.048 0.1368601760.001019298.91.20E+13297
283 0.051 0.1264338150.001037268.25.67E+12287
241 0.061 0.1765642350.001041227.72.03E+13297
232 0.062 0.1605667860.001010452.31.34E+13295
213 0.066 0.1227970880.001022044.01.20E+13289
211 0.068 0.1905738170.001015530.13.06E+13300
202 0.069 0.1154581920.001011232.11.24E+13281
199 0.071 0.1401217040.001007669.29.08E+12278
198 0.073 0.1669983490.001011510.11.72E+13286
199 0.073 0.1681851480.001007171.71.95E+13291
191 0.078 0.1731930920.001143654.13.83E+13284
183 0.079 0.1799156330.001047905.91.55E+13281
174 0.081 0.1436562320.001044834.61.02E+13277
172 0.085 0.1965149680.001024234.73.81E+13283
164 0.087 0.1585740150.001133015.12.23E+13277
161 0.092 0.1557745910.001014843.36.52E+12283
148 0.095 0.1210725730.001001135.83.11E+13270


timemeanGrowthstdGrowthminSizemeanSizevarSizefirmNumber
5000- 0.017 0.1749377330.461.21.53E+00236
5000- 0.017 0.1576300620.290.59.15E-02489
5000- 0.015 0.1671344070.050.11.16E-021647
5000- 0.014 0.1464233630.350.72.62E-01393
5000- 0.012 0.1513849460.421.08.42E-01297
5000- 0.009 0.1916272840.240.75.56E-01421
5000- 0.005 0.1752357910.030.17.46E-021972
5000- 0.005 0.1357873970.361.51.75E+01205
5000- 0.001 0.1577574040.251.12.82E+00280
5000 0.005 0.1800281310.154.95.40E+02107
5000 0.007 0.1675425560.465.92.80E+0259
5000 0.008 0.1938032770.154.21.05E+0274
1349 0.010 0.104042510.301063150.01.42E+14184
2741 0.010 0.1499470680.161043546.19.14E+1225
2006 0.011 0.1456334030.091081066.54.45E+13103
1094 0.012 0.1437639840.041065803.01.98E+14184
1355 0.012 0.1831161420.221064620.71.52E+1314
606 0.024 0.1025946890.361013401.69.70E+12294
527 0.028 0.158308520.401054555.29.98E+13209
551 0.029 0.1635901330.211000182.73.79E+13268
479 0.031 0.1349638770.421017652.72.13E+13275
485 0.031 0.1319294160.091007541.01.81E+13303
398 0.036 0.115957750.471037959.61.26E+13284
391 0.037 0.1157125280.491020198.61.15E+13289
374 0.039 0.1410352120.321029554.12.19E+13295
336 0.045 0.1779427670.161001148.23.00E+13289
328 0.045 0.1590328620.041021096.02.67E+13289
308 0.048 0.1565649010.341017922.31.68E+13297
287 0.049 0.1571779960.271068906.58.38E+13290
297 0.051 0.1608242640.491012151.31.04E+13252
278 0.051 0.1267108250.261017803.59.96E+12292
272 0.054 0.1896217960.351115571.81.26E+14260
265 0.055 0.1290666320.101056510.18.15E+12286
263 0.055 0.1188556220.261028302.67.02E+12290
244 0.060 0.1821389860.011090163.44.97E+13306
237 0.061 0.1849154540.381091542.12.52E+13281
209 0.069 0.1366181170.161048682.21.94E+13288
197 0.072 0.1135505090.211007903.66.25E+12278
200 0.074 0.199422450.041027767.61.92E+13295
181 0.079 0.102568110.121047373.43.52E+12281
190 0.080 0.1856046950.061050727.61.73E+13309
176 0.082 0.171732190.201068564.59.74E+12300
171 0.084 0.1462159180.361028333.91.19E+13286
161 0.090 0.1690573920.241014307.32.46E+13288
158 0.090 0.1183328390.121070007.83.20E+12280
157 0.091 0.1601232170.411032706.91.88E+13286
162 0.092 0.1861030180.441068732.79.20E+12279
153 0.094 0.1533812950.051063409.11.05E+13277
151 0.096 0.1239310710.041095804.84.45E+12284
145 0.097 0.1744382970.491091289.84.47E+13266

With 'low' average growth rates, threshold entry mechanisms are enough to guarantee R-distributions, given a non-zero minimum size (grey area above).


Table A.4: Simulation results for Excess Supply Affects Small exit, mean and variance homoskedasticity

timeentry meanGrowth stdGrowth minSizefirmNumber
50001 0.011 0.134 0.001
8531 0.016 0.136 0.001
15771 0.019 0.167 0.001
10741 0.020 0.176 0.001
6391 0.023 0.132 0.001
6931 0.027 0.174 0.001
3061 0.028 0.151 0.001
4091 0.030 0.101 0.001
4401 0.032 0.152 0.001
2961 0.033 0.129 0.001
3731 0.034 0.192 0.001
2931 0.038 0.194 0.001
4551 0.039 0.107 0.001
3731 0.045 0.106 0.001
4881 0.046 0.198 0.001
2921 0.050 0.153 0.001
2521 0.054 0.122 0.001
2231 0.058 0.157 0.001
2291 0.061 0.138 0.001
1831 0.068 0.114 0.001
2281 0.071 0.121 0.001
2341 0.073 0.196 0.001
1471 0.074 0.159 0.001
1801 0.076 0.103 0.001
1681 0.080 0.124 0.001
1801 0.082 0.164 0.001
1361 0.084 0.161 0.001
1351 0.085 0.139 0.001
1631 0.090 0.182 0.001
1381 0.092 0.147 0.001
1561 0.093 0.125 0.001
1181 0.095 0.127 0.001
1041 0.097 0.185 0.001
1331 0.100 0.121 0.001
1591 0.100 0.133 0.001
13602 0.015 0.106 0.001
4222 0.031 0.183 0.001
4632 0.037 0.148 0.001
3122 0.039 0.126 0.001
3662 0.043 0.138 0.001
3142 0.050 0.153 0.001
3372 0.053 0.197 0.001
1892 0.058 0.175 0.001
1922 0.061 0.121 0.001
1842 0.065 0.105 0.001
1832 0.071 0.109 0.001
1512 0.075 0.100 0.001
1812 0.075 0.195 0.001
1912 0.075 0.176 0.001
1592 0.080 0.153 0.001
1902 0.084 0.174 0.001
1702 0.086 0.106 0.001
1812 0.093 0.154 0.001
1442 0.098 0.109 0.001
1422 0.098 0.122 0.001
1342 0.099 0.113 0.001


timeentry meanGrowth stdGrowth minSizel k firmNumber
22873 0.010 0.175 0.291 0.033 10000
7213 0.015 0.118 0.011 0.057 10000
2543 0.018 0.184 0.081 0.040 10000
5953 0.019 0.121 0.401 0.033 10000
3203 0.021 0.106 0.181 0.032 10000
2803 0.026 0.162 0.161 0.085 10000
3543 0.026 0.146 0.061 0.004 10000
3133 0.034 0.103 0.301 0.061 10000
3213 0.036 0.112 0.431 0.019 10000
3423 0.040 0.164 0.321 0.005 10000
3373 0.040 0.148 0.131 0.057 10000
2253 0.042 0.122 0.001 0.058 10000
3323 0.043 0.157 0.061 0.007 10000
1813 0.044 0.119 0.051 0.090 10000
2033 0.044 0.110 0.361 0.005 10000
1373 0.046 0.195 0.421 0.008 10000
1893 0.047 0.193 0.381 0.017 10000
2103 0.048 0.133 0.421 0.093 10000
1543 0.048 0.182 0.231 0.044 10000
1743 0.049 0.172 0.281 0.008 10000
1553 0.050 0.191 0.011 0.009 10000
1813 0.053 0.144 0.421 0.093 10000
1963 0.054 0.124 0.201 0.017 10000
2183 0.055 0.113 0.261 0.057 10000
2243 0.058 0.109 0.281 0.087 10000
1113 0.059 0.156 0.121 0.004 10000
1613 0.060 0.104 0.461 0.075 10000
1243 0.062 0.142 0.361 0.072 10000
1693 0.062 0.102 0.291 0.020 10000
2063 0.063 0.149 0.321 0.096 10000
2053 0.066 0.184 0.501 0.088 10000
1243 0.068 0.168 0.391 0.004 10000
1353 0.069 0.177 0.101 0.003 10000
1513 0.071 0.111 0.231 0.007 10000
1703 0.071 0.181 0.361 0.029 10000
1503 0.072 0.102 0.461 0.024 10000
1073 0.077 0.192 0.081 0.007 10000
1523 0.078 0.158 0.051 0.001 10000
1423 0.078 0.179 0.431 0.004 10000
1203 0.079 0.128 0.251 0.005 10000
923 0.080 0.128 0.471 0.042 10000
1383 0.083 0.198 0.291 0.076 10000
1343 0.085 0.184 0.161 0.009 10000
1313 0.086 0.123 0.421 0.022 10000
1393 0.086 0.112 0.501 0.055 10000
1003 0.087 0.165 0.111 0.019 10000
1333 0.087 0.171 0.081 0.005 10000
1063 0.087 0.134 0.471 0.081 10000
723 0.093 0.184 0.241 0.079 10000
953 0.093 0.131 0.341 0.020 10000
1073 0.094 0.164 0.201 0.008 10000
933 0.094 0.196 0.101 0.032 10000
1023 0.095 0.138 0.191 0.001 10000
1013 0.095 0.104 0.411 0.058 10000
126 3 0.096 0.186 0.401 0.048 10000
1133 0.098 0.172 0.471 0.002 10000

With 'high' average growth rates, the system leads to a monopoly (degenerates when Proportional Number entry is considered). Simulation for low values of mean growth rates are not reported.


Table A.5: Simulation results for Excess Supply Affects Large exit, mean and variance homoskedasticity

timeentrymeanGrowthstdGrowthminSizemeanSizevarSizefirmNumber
50001 0.002 0.137 0.37 1.2 9.90E-01244
50001 0.004 0.164 0.01 0.1 5.39E-022722
50001 0.006 0.200 0.29 1.3 1.34E+00232
50001 0.007 0.105 0.35 1.2 4.61E-01241
50001 0.007 0.126 0.45 1.6 9.97E-01195
50001 0.009 0.158 0.11 0.7 3.89E-01453
50001 0.009 0.120 0.03 0.4 2.58E-01709
50001 0.010 0.103 0.03 0.4 2.20E-01706
50001 0.011 0.123 0.28 1.5 1.04E+00209
50001 0.016 0.162 0.07 0.8 7.21E-01378
50001 0.016 0.152 0.36 2.0 1.87E+00156
50001 0.018 0.158 0.13 1.2 1.12E+00257
50001 0.023 0.195 0.02 0.5 5.85E-01542
50001 0.026 0.173 0.32 2.6 3.58E+00117
50001 0.027 0.183 0.38 2.4 3.75E+00127
50001 0.032 0.182 0.37 2.9 3.83E+00106
50001 0.032 0.178 0.20 2.6 5.97E+00116
50001 0.035 0.144 0.24 5.6 1.97E+0155
50001 0.039 0.133 0.32 12.5 3.49E+0125
50001 0.041 0.102 0.20 19,969.8 0.00E+001
50001 0.041 0.116 0.29 14.3 2.70E+0121
50001 0.042 0.159 0.02 4.7 1.66E+0168
50001 0.043 0.130 0.46 9.2 3.98E+0134
50001 0.045 0.162 0.15 5.0 1.57E+0156
50001 0.049 0.142 0.41 12.6 5.01E+0126
50001 0.052 0.159 0.09 10.7 5.15E+0130
50001 0.055 0.167 0.16 4.7 2.03E+0160
50001 0.055 0.151 0.46 21.1 1.51E+0216
50001 0.057 0.162 0.05 18.6 8.21E+0116
50001 0.057 0.168 0.37 8.2 2.71E+0135
50001 0.060 0.183 0.05 5.4 1.94E+0158
50001 0.062 0.154 0.19 20.4 3.48E+0115
5411 0.062 0.138 0.40 1,014,142.4 0.00E+001
50001 0.063 0.160 0.22 15.2 1.28E+0219
50001 0.068 0.173 0.18 138.7 3.86E+022
4761 0.069 0.130 0.49 1,055,520.1 0.00E+001
36151 0.071 0.163 0.00 1,123,354.2 0.00E+001
10851 0.078 0.165 0.29 1,157,512.8 0.00E+001
3131 0.078 0.121 0.32 1,063,971.3 0.00E+001
2911 0.082 0.134 0.21 1,058,231.0 0.00E+001
50001 0.083 0.189 0.14 81.9 1.11E+033
11771 0.085 0.183 0.19 1,100,311.1 0.00E+001
2761 0.086 0.140 0.38 1,064,183.5 0.00E+001
2321 0.088 0.121 0.15 1,150,826.4 0.00E+001
2321 0.091 0.114 0.36 1,037,486.7 0.00E+001
4921 0.092 0.187 0.46 1,173,068.8 0.00E+001
3671 0.094 0.167 0.37 1,058,030.9 0.00E+001
2041 0.096 0.126 0.41 1,047,706.9 0.00E+001
2681 0.096 0.127 0.41 1,174,664.4 0.00E+001
2161 0.097 0.113 0.10 1,033,523.4 0.00E+001
3941 0.100 0.185 0.32 1,366,950.0 0.00E+001


timeentrymeanGrowthstdGrowthmeanSizevarSizefirmNumber
482 0.008 0.155 0.06 1.48E-0310000
492 0.025 0.189 0.06 1.66E-0310000
692 0.035 0.149 0.07 8.09E-0410000
572 0.041 0.192 0.06 1.39E-0310000
902 0.046 0.145 0.07 5.92E-0410000
622 0.050 0.189 0.07 1.17E-0310000
672 0.052 0.182 0.07 1.08E-0310000
922 0.054 0.153 0.07 6.62E-0410000
2912 0.057 0.112 0.07 2.94E-0410000
1882 0.059 0.128 0.07 6.08E-0410000
6442 0.059 0.102 0.07 2.70E-0410000
2692 0.060 0.114 0.07 3.55E-0410000
1892 0.060 0.120 0.07 3.41E-0410000
1262 0.060 0.142 0.07 6.36E-0410000
1852 0.062 0.130 0.07 4.16E-0410000
3262 0.063 0.114 0.07 3.48E-0410000
1002 0.064 0.163 0.07 6.68E-0410000
3832 0.064 0.116 0.07 3.55E-0410000
1392 0.070 0.146 0.07 5.56E-0410000
4122 0.071 0.101 1,018,888.28 0.00E+001
4882 0.075 0.106 1,081,347.00 0.00E+001
3582 0.081 0.101 1,250,319.52 0.00E+001
4832 0.082 0.114 1,056,847.05 0.00E+001
982 0.082 0.184 0.07 1.07E-0310000
1772 0.082 0.161 0.07 7.45E-0410000
1112 0.083 0.184 0.07 9.32E-0410000
3612 0.083 0.143 0.07 5.04E-0410000
3312 0.086 0.109 1,057,224.16 0.00E+001
10222 0.089 0.143 0.07 4.67E-0410000
1442 0.093 0.187 0.07 8.62E-0410000
1852 0.097 0.172 0.07 8.14E-0410000
2312 0.100 0.107 1284366.5050.00E+001
2132 0.114 0.121 1058135.6650.00E+001
1992 0.122 0.135 1074552.2360.00E+001
3482 0.127 0.178 1.34E+060.00E+001
1682 0.138 0.150 1698748.2780.00E+001
1392 0.142 0.111 1114288.4070.00E+001
1732 0.143 0.165 1015582.1730.00E+001
1442 0.147 0.152 1.28E+060.00E+001
2072 0.153 0.194 1188511.3260.00E+001
1302 0.155 0.126 1.01E+060.00E+001
1192 0.158 0.104 1.01E+060.00E+001
1822 0.162 0.179 1.26E+060.00E+001
1082 0.168 0.106 1.00E+060.00E+001
1772 0.173 0.198 1151604.9120.00E+001
1162 0.177 0.144 1178586.3030.00E+001
1352 0.178 0.162 1028319.370.00E+001
1122 0.179 0.122 1039013.6020.00E+001
1022 0.185 0.118 1140938.1260.00E+001
1122 0.188 0.141 1.00E+060.00E+001
1072 0.189 0.161 1220154.9750.00E+001

With Excess Demand entry (upper table), stability in the implosion set is obtained, due to positive minimum size. However, in the explosion set the system leads to a monopoly. With Proportional Number entry (lower table), the system implodes for low growth rates, and leads to a monopoly for high growth rates.


Table A.6: Simulation results for Excess Supply Affects Large exit and Proportional Dimension Entry, mean and variance homoskedasticity

timemeanGrowthstdGrowthminSizekmeanvarfirmNumber
5000 0.001 0.155 0.36 0.001 1.84 3.33E+0037
5000 0.006 0.161 0.07 0.001 0.97 4.40E+007
5000 0.006 0.129 0.12 0.006 2.47 1.84E+01124
5000 0.009 0.124 0.33 0.005 2.66 1.22E+01109
5000 0.010 0.182 0.34 0.007 2.29 2.44E+0193
5000 0.015 0.123 0.31 0.003 2.10 4.69E+00140
5000 0.018 0.167 0.11 0.005 1.89 1.01E+01156
5000 0.018 0.101 0.25 0.007 2.17 5.32E+00141
5000 0.019 0.185 0.30 0.009 2.36 1.07E+01109
5000 0.022 0.128 0.10 0.004 2.06 5.49E+00142
5000 0.025 0.135 0.14 0.006 2.09 4.19E+00146
5000 0.025 0.114 0.03 0.001 1.39 3.66E+00212
5000 0.028 0.159 0.34 0.003 2.21 5.90E+00136
5000 0.029 0.121 0.40 0.004 2.58 5.25E+00119
5000 0.032 0.139 0.28 0.002 2.79 6.88E+00109
5000 0.035 0.187 0.42 0.005 2.82 9.81E+00117
5000 0.035 0.193 0.25 0.010 2.15 6.64E+00134
5000 0.046 0.146 0.42 0.003 2.81 8.08E+00111
5000 0.048 0.163 0.40 0.005 2.95 6.23E+00103
5000 0.049 0.103 0.11 0.006 2.15 5.44E+00143
5000 0.049 0.192 0.03 0.010 1.59 5.63E+00197
5000 0.052 0.133 0.08 0.002 1.83 5.91E+00170
5000 0.052 0.106 0.37 0.009 2.98 5.81E+00105
5000 0.053 0.116 0.06 0.008 1.56 6.78E+00188
5000 0.054 0.173 0.26 0.006 3.01 9.74E+00102
5000 0.054 0.107 0.16 0.002 2.14 4.49E+00146
5000 0.054 0.129 0.18 0.008 2.25 6.87E+00137
5000 0.056 0.163 0.02 0.005 1.52 5.03E+00201
5000 0.062 0.195 0.39 0.006 2.63 6.74E+00117
5000 0.063 0.195 0.25 0.006 2.48 6.90E+00123
5000 0.064 0.107 0.08 0.002 2.31 7.72E+00139
5000 0.066 0.124 0.09 0.006 2.00 7.37E+00161
5000 0.066 0.197 0.15 0.007 2.21 6.27E+00145
5000 0.066 0.133 0.20 0.004 2.39 6.86E+00132
5000 0.068 0.192 0.13 0.002 1.70 4.32E+00189
5000 0.069 0.168 0.08 0.001 1.78 6.32E+00179
5000 0.070 0.145 0.23 0.010 2.76 9.43E+00112
5000 0.072 0.136 0.02 0.002 1.39 4.07E+00231
5000 0.078 0.133 0.13 0.001 2.15 6.73E+00143
5000 0.078 0.101 0.34 0.001 2.71 8.68E+00119
5000 0.084 0.101 0.08 0.005 2.39 8.37E+00133
5000 0.085 0.165 0.01 0.002 1.73 7.56E+00192
5000 0.086 0.181 0.07 0.002 2.03 1.07E+01165
5000 0.087 0.154 0.34 0.002 2.80 8.55E+00117
5000 0.087 0.189 0.29 0.006 3.07 8.02E+00103
5000 0.090 0.107 0.17 0.003 2.45 9.27E+00130
5000 0.091 0.184 0.05 0.004 1.64 3.75E+00201
5000 0.093 0.133 0.10 0.002 2.47 8.12E+00131
5000 0.095 0.162 0.07 0.007 2.03 7.06E+00158
5000 0.098 0.140 0.45 0.006 3.06 5.83E+00105
5000 0.098 0.137 0.09 0.005 2.21 7.31E+00145


Table A.7: Simulation results for Excess Supply Affects All exit, mean and variance homoskedasticity

timeentrymeanGrowthstdGrowthminSizebirthRatemeanSizefirmNumber
50001 0.008 0.129 0.24 0.05 1.37 213
50001 0.011 0.107 0.24 0.08 1.18 261
50001 0.011 0.117 0.34 0.08 1.36 223
50001 0.014 0.172 0.30 0.02 1.53 200
50001 0.015 0.186 0.26 0.04 1.43 225
50001 0.038 0.195 0.21 0.04 2.19 135
50001 0.051 0.170 0.41 0.02 4.10 74
50001 0.078 0.163 0.03 0.09 0.95 367
50001 0.081 0.121 0.20 0.02 2.22 153
50001 0.084 0.195 0.09 0.09 1.35 233
50001 0.090 0.114 0.37 0.10 4.54 72
50001 0.104 0.102 0.04 0.04 2.49 140
50001 0.106 0.159 0.19 0.03 2.84 119
50001 0.107 0.154 0.24 0.05 2.15 145
50001 0.117 0.158 0.48 0.10 10.16 35
50001 0.119 0.153 0.30 0.08 13.29 25
50001 0.130 0.188 0.48 0.01 24.23 12
50001 0.135 0.118 0.02 0.01 0.50 643
50001 0.154 0.194 0.18 0.08 6.07 58
50001 0.158 0.163 0.03 0.08 0.94 347
50001 0.159 0.179 0.38 0.03 4.88 61
50001 0.159 0.182 0.10 0.05 7.28 55
50001 0.162 0.185 0.26 0.03 12.53 28
50001 0.165 0.147 0.22 0.10 6.92 52
50001 0.167 0.176 0.39 0.06 12.65 26
50001 0.189 0.196 0.49 0.01 12.83 26
50001 0.193 0.114 0.14 0.04 9.77 35
50001 0.198 0.145 0.28 0.06 15.60 26
50002 0.005 0.175 0.34 0.08 0.87 377
50002 0.009 0.143 0.39 0.06 0.81 371
50002 0.020 0.141 0.35 0.09 0.60 514
50002 0.025 0.107 0.32 0.07 0.52 599
50002 0.050 0.128 0.08 0.08 0.14 2331
50002 0.099 0.149 0.27 0.06 0.55 598
50002 0.102 0.101 0.14 0.05 0.21 1571
50002 0.116 0.193 0.45 0.08 1.16 303
50002 0.134 0.144 0.41 0.09 0.60 578
50002 0.139 0.112 0.37 0.01 335.72 1
50002 0.139 0.177 0.09 0.07 0.17 2052
50002 0.145 0.159 0.25 0.07 0.42 829
50002 0.153 0.180 0.29 0.07 0.54 655
50002 0.156 0.129 0.13 0.03 0.37 958
50002 0.171 0.200 0.43 0.05 1.08 331
50002 0.175 0.147 0.38 0.08 0.52 694
50002 0.176 0.161 0.45 0.08 0.68 523
642 0.180 0.136 0.03 0.07 0.07 10000
50002 0.181 0.112 0.30 0.06 0.40 900
50002 0.192 0.181 0.39 0.06 0.78 470
50002 0.195 0.180 0.27 0.07 0.52 705
50002 0.197 0.191 0.14 0.00 262.04 1
50002 0.198 0.162 0.12 0.09 0.16 2247
50003 0.033 0.176 0.15 0.09 1.64 162
50003 0.044 0.173 0.47 0.08 2.97 104
50003 0.050 0.166 0.08 0.08 1.65 200
50003 0.080 0.113 0.41 0.08 2.34 141
50003 0.102 0.128 0.38 0.09 2.62 128
50003 0.103 0.166 0.03 0.09 0.76 375
50003 0.112 0.102 0.10 0.02 2.07 144
50003 0.116 0.147 0.29 0.01 2.38 139
50003 0.120 0.107 0.14 0.03 1.55 212
50003 0.125 0.199 0.47 0.04 3.17 103
50003 0.125 0.192 0.44 0.10 2.49 140
50003 0.130 0.177 0.28 0.02 2.48 124
50003 0.136 0.181 0.49 0.06 2.56 139
50003 0.146 0.142 0.06 0.03 1.10 312
50003 0.146 0.185 0.38 0.10 1.90 172
50003 0.152 0.159 0.21 0.08 1.98 178
50003 0.173 0.106 0.42 0.05 2.23 159
50003 0.173 0.163 0.10 0.03 1.17 269
50003 0.181 0.179 0.14 0.05 1.70 204
50003 0.189 0.159 0.22 0.08 2.25 157
5921 0.106 0.129 0.00 0.01 0.05 10000
5061 0.111 0.172 0.00 0.08 0.06 10000
11121 0.116 0.112 0.00 0.02 0.04 10000
3361 0.117 0.193 0.00 0.02 0.05 10000
5671 0.123 0.134 0.00 0.07 0.05 10000
4341 0.126 0.165 0.00 0.08 0.06 10000
2231 0.130 0.196 0.00 0.04 0.06 10000
14111 0.133 0.121 0.00 0.03 0.05 10000
4651 0.136 0.147 0.00 0.00 0.04 10000
23491 0.138 0.102 0.00 0.04 0.05 10000
7221 0.144 0.158 0.00 0.07 0.08 10000
8171 0.148 0.158 0.00 0.08 0.05 10000
8001 0.158 0.146 0.00 0.02 0.04 10000
22361 0.164 0.115 0.00 0.04 0.05 10000
11991 0.175 0.129 0.00 0.07 0.05 10000
5561 0.190 0.188 0.00 0.00 0.08 10000
11851 0.193 0.119 0.00 0.06 0.03 10000
11631 0.195 0.171 0.00 0.04 0.04 10000
23361 0.196 0.140 0.00 0.02 0.04 10000
1182 0.101 0.121 0.00 0.04 0.06 10000
1472 0.105 0.136 0.00 0.03 0.07 10000
1792 0.107 0.117 0.00 0.02 0.07 10000
1192 0.117 0.142 0.00 0.03 0.07 10000
692 0.131 0.131 0.00 0.06 0.07 10000
422 0.133 0.137 0.00 0.10 0.07 10000
1302 0.134 0.162 0.00 0.03 0.07 10000
472 0.144 0.187 0.00 0.09 0.07 10000
472 0.145 0.157 0.00 0.09 0.07 10000
582 0.145 0.199 0.00 0.07 0.07 10000
50002 0.146 0.155 0.00 0.01 3.88 100
432 0.151 0.177 0.00 0.09 0.07 10000
412 0.151 0.122 0.00 0.10 0.07 10000
1082 0.158 0.124 0.00 0.04 0.07 10000
2842 0.168 0.182 0.00 0.02 0.07 10000
752 0.178 0.169 0.00 0.05 0.07 10000
912 0.190 0.141 0.00 0.05 0.07 10000
602 0.192 0.165 0.00 0.07 0.07 10000
1022 0.194 0.157 0.00 0.04 0.07 10000
1872 0.197 0.150 0.00 0.02 0.07 10000

Stability is reached if minimum size is different from 0 (left columns).


* Acknowledgements

This work benefited from very useful comments by three anonymous referees.

* References

ACS Z.J. and Armington C. (2001), "Gibrat's Law Reconsidered: The Relationship Between Firm Growth, Establishment Age, Establishment Size and Firm Size", Working Paper of the Regional Entrepreneurship Catalyst Kauffman Center for Entrepreneurial Leadership, Kansas City, MO.

ADAMIC L.A. (2000) "Zipf, power-laws, and Pareto - a ranking tutorial", http://ginger.hpl.hp.com/shl/papers/ranking/ranking.html

AMARAL L.A.N, Buldyrev S.W., Havlin S., Leschhorn H., Maass P., Salinger M.A., Stanley H. E. and Stanley M.H.R. (1997), "Scaling Behavior in Economics: Empirical Results for Company Growth", Journal de Physique I, 7, 621-633

AMIRKHALKHALI S. and Mukhopadhyay A.K. (1993), "The Influence of Size and R&D on the Growth of Firms in the U.S.", Eastern Economic Journal, 19, 223-233.

AUDRETSCH D. B., Klomp L., Santarelli E., and Thurik A.R. (2002), "Gibrat's Law: Are the Services Different?", mimeo

BARTELSMAN E., Scarpetta S. & Schivardi F. (2003), "Comparative Analysis of Firm Demographics and Survival: Micro-level Evidence for the OECD Countries", OECD Economics Department Working Paper no. 348

BLANK A. and Solomon S. (2000), "Power-laws in cities population, financial markets and internet sites (scaling in systems with a variable number of components)", Physica A, 287, 279-288

BOERI T. (1989), "Does Firm Size Matter?" Giornale degli Economisti e Annali di Economia, 48(9-lo), 477-95.

BIANCO M. and Sestito P. (1993), "Entry and Growth of Firms: Evidence for the Italian Case", Unpublished manuscript, Banca d'Italia, Rome.

CAVES R.E. (1998), "Industrial Organization and New Findings on the Turnover and Mobility of Firms", Journal of Economic Literature, 36, 1947-1983

CHESHER A. (1979), "Testing the Law of Proportionate Effect", Journal of Industrial Economics, 27, 403-411.

CONTINI B. and Revelli R. (1989), "The Relationship between Firm Growth and Labor Demand", Small Business Economics, 1, 309-314.

CORDOBA J.C. (2002), "Balanced City Growth and Zipf's Law", RICE Economics Working Papers 2002-03

DELMAR F., Davidsson P. and Gartner W.B. (2002), "Arriving at the High-growth Firm", Journal of Business Venturing, 17, forthcoming.

DUNNE P. and Hughes A. (1994), "Age, Size, Growth and Survival: UK Companies in the 1980s", Journal of Industrial Economics, 42, 115-140.

EVANS D.S. (1987a), "The Relationship between Firm Growth, Size, and Age: Estimates for 100 Manufacturing Industries", Journal of Industrial Economics, 35, 567-581.

EVANS D.S. (1987b), "Tests of Alternative Theories of Firm Growth", Journal of Political Economy, 95, 657-674.

FARIÑAS J.C. and Moreno L. (2000), "Firms' Growth, Size and Age: A Nonparametric Approach", Review of Industrial Organization, 17, 249-265.

FITZROY F.R. and Kraft K. (1991), "Firm Size, Growth and Innovation: Some Evidence from West Germany", in Z.J. Acs and D.B. Audretsch (eds.), Innovation and Technological Change: An International Comparison, Harvester Wheatsheaf, London, 152-159.

GIBRAT R. (1930), "Une loi des réparations économiques: l'effet proportionnel", Bull. Statist. gén Fr. 19, 469.

GIBRAT R. (1931), "Les Inégalités Économiques", Librairie du Recueil Sirey, Paris.

GEROSKI P.A., Lazarova S., Urga G. and Walters C.F. (2000), "Are Differences in Firm Size Transitory or Permanent?", London Business School, Department of Economics, mimeo, August, forthcoming in Journal of Applied Econometrics.

GODDARD J., Wilson J. and Blandon P. (2002), "Panel Tests of Gibrat's Law for Japanese Manufacturing", International Journal of Industrial Organization, 20, 415-433.

HALL B.H. (1987), "The Relationship between Firm Size and Firm Growth in the US Manufacturing Sector", Journal of Industrial Economics, 35, 583-606.

HARDWICK P. and Adams M. (1999), "Firm Size and Growth in the United Kingdom Life Insurance Industry", University of Bournemouth, School of Finance and Law Working Paper Series, No. 16.

HARHOFF D., Stahl K. and Woywode M. (1998), "Legal form, Growth and Exit of West German Firms- Empirical Results for Manufacturing, Construction, Trade and Service Industries", Journal of Industrial Economics, 46, 453-488.

HART P.E. (1962), "The Size and Growth of Firms", Economica, 29, 29-39.

HART P.E. and Prais S.J. (1956), "The Analysis of Business Concentration: A Statistical Approach", Journal of the Royal Statistical Society, 119 (part 2, series A), 150-191.

HART P.E. and Oulton N. (1999), "Gibrat, Galton and Job Generation", International Journal of the Economics of Business, 6, 149-164.

HYMER S. and Pashigian P. (1962), "Firm Size and Rate of Growth", Journal of Political Economy, 70, 556- 569.

KESTEN H. (1973), "Random Difference Equations and Renewal Theory for Products of Random Matrices", Acta Mathematica, 131, 207-248.

KUMAR M.S. (1985), "Growth, Acquisition Activity and Firm Size: Evidence from the United Kingdom", Journal of Industrial Economics, 33, 327-338.

LEONARD J.S. (1986), "On the Size Distribution of Employment and Establishments," Working Paper No. 1951, National Bureau of Economic Research.

LEVY M. and Solomon S. (1996 a), "Spontaneous Scaling Emergence in Generic Stochastic Systems", International Journal of Modern Physics C, 7, 745

LEVY M. and Solomon S. (1996 b), "Power-laws are Logarithmic Boltzmann Laws", International Journal of Modern Physics C, 7, 595

MACHADO J.A.F. and Mata J. (2000), "Box-Cox Quantile Regression and the Distribution of Firm Sizes", Journal of Applied Econometrics, 15, 253-274.

MANRUBIA S.C. and Zanette D. H. (1999), "Stochastic multiplicative processes with reset events", Physical Review E 59, 4945-4948.

MANSFIELD E. (1962), "Entry, Gibrat's Law, Innovation, and the Growth of Firms", American Economic Review, 52, 1023-1051.

MCCLOUGHAN P. (1995), "Simulation of Concentration Development from Modified Gibrat Growth-Entry-Exit Processes", Journal of Industrial Economics, 43 (4), 405-432

MITZENMACHER M. (2001), "A Brief History of Generative Models for Power Laws and Lognormal Distributions", Computer Science Group, Harvard University Working Paper TR-08-01

NIREI M. and Souma W. (2002), "Income distribution and stochastic multiplicative process with reset events", Santa Fe Institute Working Papers.

Pareto V. (1896), Cours d'Economie Politique, Droz, Geneva.

REDNER S. (1990), "Random multiplicative processes: An elementary tutorial", American Journal of Physics, 58, 267-273.

SAMUELS J.M. (1965), "Size and Growth of Firms", Review of Economic Studies, 32, 105-112.

SIMON H. and Bonini C. (1958), "The Size Distribution of BusinessFirms" American Economic Review, 48, 607-617.

SINGH A. and Whittington G. (1975), "The Size and Growth of Firms", Review of Economic Studies, 42, 15-26.

SORNETTE D. and Cont R. (1997), "Convergent multiplicative processes repelled from zero: power-laws and truncated power-laws", J. Phys. I France, 7, 431-444.

SORNETTE D. (1998), "Multiplicative processes and power-laws", Physics Review E, 57, 4811-4813.

SUTTON J. (1997), "Gibrat's Legacy", Journal of Economic Literature, 35, 40-59.

TSCHOEGL A. (1996), "Managerial Dis(economies) of Scale: The Case of Regional Banks in Japan", Reginald H. Jones Center for Management and Policy, Strategy and Organization, The Wharton School of the University of Pennsylvania Working Paper No. 96-04.

VANDER Vennet R. (2001), "The Law of Proportionate Effect and OECD Bank Sectors", Applied Economics, 33, 539-546.

VARIYAM J.N. and Kraybill D.S. (1992), "Empirical Evidence on Determinants of Firm Growth", Economics Letters, 38, 31-36.

WAGNER J. (1992), "Firm Size, Firm Growth, and Persistence of Chance: Testing Gibrat's Law with Establishment Data from Lower Saxony, 1978-1989", Small Business Economics, 4, 125-131.

ZIPF G. (1932), Selective Studies and the Principle of Relative Frequency in Language, Harvard University Press, Cambridge, MA.

----

ButtonReturn to Contents of this issue

© Copyright Journal of Artificial Societies and Social Simulation, [2003]