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, S_t+1 = S_t exp(λ_t) , 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(lnS_t) < 0. Moreover, if S_t 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:

(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(lsS_t< 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 S_t (thus leading to a 'pure' multiplicative process) and F → ∞ for S_t → 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 S_min (of magnitude 1), and that at each period ΔN = k ·(S*_t+1 - S*_t) new firms enter the market , with size S_min (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)→ ∞.

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

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:

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.

(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

    (10)

    where D_{1/2, 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
    

    (11)

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

    (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 D_{1/2, 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
	ENTRY	EXIT
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)

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.

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.

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

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:

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.

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.

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
Exit		Gibrat	NON-THRESHOLD		THRESHOLD
		No entry	Proportional Number	Proportional Dimension (*)	Excess Demand
Gibrat	No exit	implosion explosion	implosion explosion	-	R-distribution (*) explosion
NON-THRESHOLD	Minimum Size	implosion explosion	implosion explosion	implosion explosion	R-distribution (*) explosion
	Proportional Number	implosion monopoly	implosion explosion/monopoly	implosion explosion	R-distribution (*) monopoly
THRESHOLD	Excess Supply Affects Small	implosion monopoly	implosion monopoly	implosion explosion	R-distribution (*) monopoly
	Excess Supply Affects Large	implosion monopoly	implosion monopoly	implosion R-distribution	R-distribution (*) monopoly
	Excess Supply Affects All	implosion 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
time	meanGrowth	stdGrowth	minSize	k	meanSize	varSize	firmNumber
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
time	meanGrowth	stdGrowth	birthRate	meanSize	varSize	firmNumber
113	-0.0199	0.1407	0.0365	0.007073119	1.31E-03	10000
50	-0.0149	0.1055	0.0813	0.113285536	1.21E-02	10000
47	-0.0122	0.1023	0.0879	0.145749252	1.93E-02	10000
46	-0.0104	0.15	0.0881	0.094486148	4.30E-02	10000
44	-0.0076	0.1318	0.0922	0.144882328	3.96E-02	10000
366	-0.0075	0.1824	0.0122	2.87E-04	6.25E-05	10000
245	-0.0071	0.1565	0.0176	0.003809768	0.003164975	10000
5000	-0.0069	0.155	0.0005	4.53E-32	1.94E-61	100
194	-0.0066	0.103	0.0216	0.019709888	1.25E-02	10000
60	-0.0037	0.1231	0.0684	0.139585452	7.20E-02	10000
341	-0.0034	0.1077	0.0131	0.012854722	1.72E-02	10000
63	-0.0033	0.1413	0.0649	0.087967898	5.63E-02	10000
108	-0.0015	0.1143	0.038	0.108112195	2.02E-01	10000
375	-0.0013	0.1923	0.012	0.002633735	3.57E-03	10000
84	0.0014	0.1918	0.0484	0.060633785	7.06E-01	10000
46	0.0026	0.1277	0.0891	0.228038483	8.03E-02	10000
49	0.0042	0.1868	0.0837	0.106101557	2.05E-01	10000
60	0.0052	0.1288	0.0674	0.237871611	1.43E-01	10000
53	0.006	0.1116	0.0775	0.328486402	1.27E-01	10000
5000	0.0073	0.112	0.0035	8.99E+07	7.97E+17	100
72	0.0076	0.1992	0.0571	0.109641713	1.37E+00	10000
5000	0.0125	0.1221	0.0021	1.94E+19	3.32E+40	100
232	0.0145	0.1339	0.0185	1.465814644	232.1458876	10000
5000	0.0151	0.141	0.0032	1.15E+19	1.29E+40	100
5000	0.0169	0.1912	0.0092	9.33E+10	8.68E+23	100
301	0.0178	0.199	0.0146	2.610688082	4.66E+03	10000
86	0.0191	0.1142	0.0478	0.748169033	2.12225235	10000
49	0.0198	0.1643	0.0829	0.408440377	0.701304711	10000
61	0.022	0.1794	0.0667	0.348275489	9.23E+00	10000
5000	0.0227	0.1224	0.0064	4.23E+39	1.21E+81	100
43	0.0279	0.1405	0.0967	0.625611697	8.91E-01	10000
77	0.0288	0.1941	0.0529	0.7237275	4.55E+01	10000
66	0.0297	0.1019	0.0622	1.721070697	3.89E+00	10000
103	0.0328	0.1318	0.0398	2.926515182	1.34E+02	10000
5000	0.0348	0.1275	0.0012	4.62E+68	2.14E+139	100
160	0.0371	0.1456	0.0261	11.4366613	3287.954539	10000
54	0.0383	0.1751	0.0762	0.842794132	5.93E+00	10000
61	0.0391	0.186	0.0669	0.787607046	1.75E+01	10000
159	0.0392	0.1347	0.0262	25.84512331	3.52E+04	10000
104	0.0418	0.1236	0.0394	8.122864781	5.67E+02	10000
155	0.0451	0.1125	0.0269	89.0935399	2.18E+05	10000
66	0.0453	0.1369	0.0621	2.609594454	4.61E+01	10000
46	0.0476	0.1498	0.0887	1.557484775	7.07E+00	10000
162	0.0479	0.1416	0.0258	100.7326391	7.49E+05	10000
97	0.0514	0.1812	0.0421	6.102945117	1.65E+03	10000
76	0.0518	0.1758	0.0537	3.102340484	1.78E+02	10000
313	0.0546	0.1214	0.0142	457849.6943	3.54E+13	10000
62	0.0555	0.1078	0.0666	6.555210396	5.60E+01	10000
5000	0.0564	0.1998	0.0011	3.72E+90	1.10E+183	100
5000	0.0598	0.196	0.0052	4.42E+98	8.90E+198	100
5000	0.0649	0.1001	0.0034	2.19E+132	2.81E+266	100
173	0.0656	0.1831	0.0243	1515.77226	4.53E+08	10000
47	0.0657	0.1432	0.0876	3.507168094	2.79E+01	10000
177	0.0675	0.1008	0.0238	10640.51041	1.04E+09	10000
188	0.0679	0.1917	0.0224	10915.37818	1.84E+11	10000
5000	0.0729	0.151	0.0033	4.35E+139	6.93E+280	100
89	0.0734	0.1317	0.0462	67.69909378	2.99E+04	10000
219	0.0751	0.1405	0.0193	336439.7406	5.48E+12	10000
41	0.0756	0.1797	0.0995	3.437785585	3.76E+01	10000
41	0.0762	0.1268	0.0999	6.15519068	33.57587775	10000
46	0.0847	0.1147	0.0889	12.80794819	1.37E+02	10000
211	0.0864	0.1308	0.02	1674692.651	1.57E+14	10000
102	0.092	0.1782	0.04	508.8034846	1.10E+07	10000
48	0.0921	0.1583	0.086	11.96688222	502.5483125	10000
122	0.0934	0.1312	0.0338	5913.84004	4.94E+09	10000
169	0.0954	0.1291	0.0248	265109.8097	1.46E+12	10000
48	0.0956	0.1334	0.085	18.16608091	4.57E+02	10000
129	0.0968	0.1843	0.0319	7027.572723	6.67E+09	10000
49	0.0969	0.178	0.0825	15.14853015	1.54E+03	10000
101	0.0985	0.1596	0.0405	1205.134053	2.13E+07	10000
54	0.0996	0.1986	0.0756	16.87269573	4.31E+03	10000

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
time	meanGrowth	stdGrowth	minSize	meanSize	varSize	firmNumber
86	- 0.017	0.183390165	0.00	0.0	8.32E-01	10000
102	- 0.016	0.117801847	0.00	0.0	7.01E-03	10000
88	- 0.015	0.169921587	0.00	0.0	5.64E-02	10000
105	- 0.014	0.15744653	0.00	0.0	7.38E-02	10000
140	- 0.007	0.135999785	0.00	0.0	5.42E-02	10000
185	- 0.004	0.134036494	0.00	0.0	8.10E-02	10000
276	- 0.002	0.145608349	0.00	0.0	3.31E-01	10000
521	0.001	0.173816895	0.00	0.0	1.35E+00	10000
1971	0.003	0.136858843	0.00	0.0	2.53E+00	10000
407	0.005	0.180479	0.00	0.0	8.44E-01	10000
878	0.005	0.162632365	0.00	0.0	7.59E-01	10000
774	0.005	0.187372841	0.00	0.0	8.06E-01	10000
5000	0.007	0.132841394	0.00	986.1	8.71E+07	326
1983	0.008	0.110686137	0.00	1060388.9	2.30E+14	326
1102	0.010	0.137889921	0.00	1048316.0	2.55E+14	319
1708	0.012	0.140865172	0.00	1066578.4	2.23E+14	311
1052	0.014	0.105100224	0.00	1008348.9	7.31E+13	305
1423	0.015	0.197298795	0.00	1022229.7	3.32E+14	318
1209	0.017	0.166114255	0.00	1060477.9	6.89E+13	305
786	0.019	0.107521161	0.00	1009923.2	1.46E+13	306
635	0.021	0.100813935	0.00	1047619.5	6.55E+13	318
603	0.024	0.136851006	0.00	1006363.1	3.58E+13	320
596	0.026	0.146662735	0.00	1016119.3	4.47E+13	309
545	0.030	0.177885087	0.00	1075796.0	9.51E+13	299
500	0.030	0.136751732	0.00	1040911.1	2.40E+13	323
460	0.034	0.178484828	0.00	1001656.5	4.87E+13	302
379	0.037	0.143598034	0.00	1106669.3	7.84E+13	304
353	0.038	0.199081417	0.00	1027488.1	2.69E+14	304
381	0.038	0.147913122	0.00	1005277.6	4.38E+13	296
375	0.038	0.152318634	0.00	1096690.3	7.07E+13	305
369	0.039	0.108664023	0.00	1041130.8	9.67E+12	300
338	0.042	0.101374458	0.00	1009610.9	8.70E+12	295
297	0.046	0.186618452	0.00	1000945.5	1.38E+14	301
305	0.048	0.136860176	0.00	1019298.9	1.20E+13	297
283	0.051	0.126433815	0.00	1037268.2	5.67E+12	287
241	0.061	0.176564235	0.00	1041227.7	2.03E+13	297
232	0.062	0.160566786	0.00	1010452.3	1.34E+13	295
213	0.066	0.122797088	0.00	1022044.0	1.20E+13	289
211	0.068	0.190573817	0.00	1015530.1	3.06E+13	300
202	0.069	0.115458192	0.00	1011232.1	1.24E+13	281
199	0.071	0.140121704	0.00	1007669.2	9.08E+12	278
198	0.073	0.166998349	0.00	1011510.1	1.72E+13	286
199	0.073	0.168185148	0.00	1007171.7	1.95E+13	291
191	0.078	0.173193092	0.00	1143654.1	3.83E+13	284
183	0.079	0.179915633	0.00	1047905.9	1.55E+13	281
174	0.081	0.143656232	0.00	1044834.6	1.02E+13	277
172	0.085	0.196514968	0.00	1024234.7	3.81E+13	283
164	0.087	0.158574015	0.00	1133015.1	2.23E+13	277
161	0.092	0.155774591	0.00	1014843.3	6.52E+12	283
148	0.095	0.121072573	0.00	1001135.8	3.11E+13	270

time	meanGrowth	stdGrowth	minSize	meanSize	varSize	firmNumber
5000	- 0.017	0.174937733	0.46	1.2	1.53E+00	236
5000	- 0.017	0.157630062	0.29	0.5	9.15E-02	489
5000	- 0.015	0.167134407	0.05	0.1	1.16E-02	1647
5000	- 0.014	0.146423363	0.35	0.7	2.62E-01	393
5000	- 0.012	0.151384946	0.42	1.0	8.42E-01	297
5000	- 0.009	0.191627284	0.24	0.7	5.56E-01	421
5000	- 0.005	0.175235791	0.03	0.1	7.46E-02	1972
5000	- 0.005	0.135787397	0.36	1.5	1.75E+01	205
5000	- 0.001	0.157757404	0.25	1.1	2.82E+00	280
5000	0.005	0.180028131	0.15	4.9	5.40E+02	107
5000	0.007	0.167542556	0.46	5.9	2.80E+02	59
5000	0.008	0.193803277	0.15	4.2	1.05E+02	74
1349	0.010	0.10404251	0.30	1063150.0	1.42E+14	184
2741	0.010	0.149947068	0.16	1043546.1	9.14E+12	25
2006	0.011	0.145633403	0.09	1081066.5	4.45E+13	103
1094	0.012	0.143763984	0.04	1065803.0	1.98E+14	184
1355	0.012	0.183116142	0.22	1064620.7	1.52E+13	14
606	0.024	0.102594689	0.36	1013401.6	9.70E+12	294
527	0.028	0.15830852	0.40	1054555.2	9.98E+13	209
551	0.029	0.163590133	0.21	1000182.7	3.79E+13	268
479	0.031	0.134963877	0.42	1017652.7	2.13E+13	275
485	0.031	0.131929416	0.09	1007541.0	1.81E+13	303
398	0.036	0.11595775	0.47	1037959.6	1.26E+13	284
391	0.037	0.115712528	0.49	1020198.6	1.15E+13	289
374	0.039	0.141035212	0.32	1029554.1	2.19E+13	295
336	0.045	0.177942767	0.16	1001148.2	3.00E+13	289
328	0.045	0.159032862	0.04	1021096.0	2.67E+13	289
308	0.048	0.156564901	0.34	1017922.3	1.68E+13	297
287	0.049	0.157177996	0.27	1068906.5	8.38E+13	290
297	0.051	0.160824264	0.49	1012151.3	1.04E+13	252
278	0.051	0.126710825	0.26	1017803.5	9.96E+12	292
272	0.054	0.189621796	0.35	1115571.8	1.26E+14	260
265	0.055	0.129066632	0.10	1056510.1	8.15E+12	286
263	0.055	0.118855622	0.26	1028302.6	7.02E+12	290
244	0.060	0.182138986	0.01	1090163.4	4.97E+13	306
237	0.061	0.184915454	0.38	1091542.1	2.52E+13	281
209	0.069	0.136618117	0.16	1048682.2	1.94E+13	288
197	0.072	0.113550509	0.21	1007903.6	6.25E+12	278
200	0.074	0.19942245	0.04	1027767.6	1.92E+13	295
181	0.079	0.10256811	0.12	1047373.4	3.52E+12	281
190	0.080	0.185604695	0.06	1050727.6	1.73E+13	309
176	0.082	0.17173219	0.20	1068564.5	9.74E+12	300
171	0.084	0.146215918	0.36	1028333.9	1.19E+13	286
161	0.090	0.169057392	0.24	1014307.3	2.46E+13	288
158	0.090	0.118332839	0.12	1070007.8	3.20E+12	280
157	0.091	0.160123217	0.41	1032706.9	1.88E+13	286
162	0.092	0.186103018	0.44	1068732.7	9.20E+12	279
153	0.094	0.153381295	0.05	1063409.1	1.05E+13	277
151	0.096	0.123931071	0.04	1095804.8	4.45E+12	284
145	0.097	0.174438297	0.49	1091289.8	4.47E+13	266

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
time	entry	meanGrowth	stdGrowth	minSize	firmNumber
5000	1	0.011	0.134	0.00	1
853	1	0.016	0.136	0.00	1
1577	1	0.019	0.167	0.00	1
1074	1	0.020	0.176	0.00	1
639	1	0.023	0.132	0.00	1
693	1	0.027	0.174	0.00	1
306	1	0.028	0.151	0.00	1
409	1	0.030	0.101	0.00	1
440	1	0.032	0.152	0.00	1
296	1	0.033	0.129	0.00	1
373	1	0.034	0.192	0.00	1
293	1	0.038	0.194	0.00	1
455	1	0.039	0.107	0.00	1
373	1	0.045	0.106	0.00	1
488	1	0.046	0.198	0.00	1
292	1	0.050	0.153	0.00	1
252	1	0.054	0.122	0.00	1
223	1	0.058	0.157	0.00	1
229	1	0.061	0.138	0.00	1
183	1	0.068	0.114	0.00	1
228	1	0.071	0.121	0.00	1
234	1	0.073	0.196	0.00	1
147	1	0.074	0.159	0.00	1
180	1	0.076	0.103	0.00	1
168	1	0.080	0.124	0.00	1
180	1	0.082	0.164	0.00	1
136	1	0.084	0.161	0.00	1
135	1	0.085	0.139	0.00	1
163	1	0.090	0.182	0.00	1
138	1	0.092	0.147	0.00	1
156	1	0.093	0.125	0.00	1
118	1	0.095	0.127	0.00	1
104	1	0.097	0.185	0.00	1
133	1	0.100	0.121	0.00	1
159	1	0.100	0.133	0.00	1
1360	2	0.015	0.106	0.00	1
422	2	0.031	0.183	0.00	1
463	2	0.037	0.148	0.00	1
312	2	0.039	0.126	0.00	1
366	2	0.043	0.138	0.00	1
314	2	0.050	0.153	0.00	1
337	2	0.053	0.197	0.00	1
189	2	0.058	0.175	0.00	1
192	2	0.061	0.121	0.00	1
184	2	0.065	0.105	0.00	1
183	2	0.071	0.109	0.00	1
151	2	0.075	0.100	0.00	1
181	2	0.075	0.195	0.00	1
191	2	0.075	0.176	0.00	1
159	2	0.080	0.153	0.00	1
190	2	0.084	0.174	0.00	1
170	2	0.086	0.106	0.00	1
181	2	0.093	0.154	0.00	1
144	2	0.098	0.109	0.00	1
142	2	0.098	0.122	0.00	1
134	2	0.099	0.113	0.00	1

time	entry	meanGrowth	stdGrowth	minSize	l	k	firmNumber
2287	3	0.010	0.175	0.29	1	0.033	10000
721	3	0.015	0.118	0.01	1	0.057	10000
254	3	0.018	0.184	0.08	1	0.040	10000
595	3	0.019	0.121	0.40	1	0.033	10000
320	3	0.021	0.106	0.18	1	0.032	10000
280	3	0.026	0.162	0.16	1	0.085	10000
354	3	0.026	0.146	0.06	1	0.004	10000
313	3	0.034	0.103	0.30	1	0.061	10000
321	3	0.036	0.112	0.43	1	0.019	10000
342	3	0.040	0.164	0.32	1	0.005	10000
337	3	0.040	0.148	0.13	1	0.057	10000
225	3	0.042	0.122	0.00	1	0.058	10000
332	3	0.043	0.157	0.06	1	0.007	10000
181	3	0.044	0.119	0.05	1	0.090	10000
203	3	0.044	0.110	0.36	1	0.005	10000
137	3	0.046	0.195	0.42	1	0.008	10000
189	3	0.047	0.193	0.38	1	0.017	10000
210	3	0.048	0.133	0.42	1	0.093	10000
154	3	0.048	0.182	0.23	1	0.044	10000
174	3	0.049	0.172	0.28	1	0.008	10000
155	3	0.050	0.191	0.01	1	0.009	10000
181	3	0.053	0.144	0.42	1	0.093	10000
196	3	0.054	0.124	0.20	1	0.017	10000
218	3	0.055	0.113	0.26	1	0.057	10000
224	3	0.058	0.109	0.28	1	0.087	10000
111	3	0.059	0.156	0.12	1	0.004	10000
161	3	0.060	0.104	0.46	1	0.075	10000
124	3	0.062	0.142	0.36	1	0.072	10000
169	3	0.062	0.102	0.29	1	0.020	10000
206	3	0.063	0.149	0.32	1	0.096	10000
205	3	0.066	0.184	0.50	1	0.088	10000
124	3	0.068	0.168	0.39	1	0.004	10000
135	3	0.069	0.177	0.10	1	0.003	10000
151	3	0.071	0.111	0.23	1	0.007	10000
170	3	0.071	0.181	0.36	1	0.029	10000
150	3	0.072	0.102	0.46	1	0.024	10000
107	3	0.077	0.192	0.08	1	0.007	10000
152	3	0.078	0.158	0.05	1	0.001	10000
142	3	0.078	0.179	0.43	1	0.004	10000
120	3	0.079	0.128	0.25	1	0.005	10000
92	3	0.080	0.128	0.47	1	0.042	10000
138	3	0.083	0.198	0.29	1	0.076	10000
134	3	0.085	0.184	0.16	1	0.009	10000
131	3	0.086	0.123	0.42	1	0.022	10000
139	3	0.086	0.112	0.50	1	0.055	10000
100	3	0.087	0.165	0.11	1	0.019	10000
133	3	0.087	0.171	0.08	1	0.005	10000
106	3	0.087	0.134	0.47	1	0.081	10000
72	3	0.093	0.184	0.24	1	0.079	10000
95	3	0.093	0.131	0.34	1	0.020	10000
107	3	0.094	0.164	0.20	1	0.008	10000
93	3	0.094	0.196	0.10	1	0.032	10000
102	3	0.095	0.138	0.19	1	0.001	10000
101	3	0.095	0.104	0.41	1	0.058	10000
126	3	0.096	0.186	0.40	1	0.048	10000
113	3	0.098	0.172	0.47	1	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
time	entry	meanGrowth	stdGrowth	minSize	meanSize	varSize	firmNumber
5000	1	0.002	0.137	0.37	1.2	9.90E-01	244
5000	1	0.004	0.164	0.01	0.1	5.39E-02	2722
5000	1	0.006	0.200	0.29	1.3	1.34E+00	232
5000	1	0.007	0.105	0.35	1.2	4.61E-01	241
5000	1	0.007	0.126	0.45	1.6	9.97E-01	195
5000	1	0.009	0.158	0.11	0.7	3.89E-01	453
5000	1	0.009	0.120	0.03	0.4	2.58E-01	709
5000	1	0.010	0.103	0.03	0.4	2.20E-01	706
5000	1	0.011	0.123	0.28	1.5	1.04E+00	209
5000	1	0.016	0.162	0.07	0.8	7.21E-01	378
5000	1	0.016	0.152	0.36	2.0	1.87E+00	156
5000	1	0.018	0.158	0.13	1.2	1.12E+00	257
5000	1	0.023	0.195	0.02	0.5	5.85E-01	542
5000	1	0.026	0.173	0.32	2.6	3.58E+00	117
5000	1	0.027	0.183	0.38	2.4	3.75E+00	127
5000	1	0.032	0.182	0.37	2.9	3.83E+00	106
5000	1	0.032	0.178	0.20	2.6	5.97E+00	116
5000	1	0.035	0.144	0.24	5.6	1.97E+01	55
5000	1	0.039	0.133	0.32	12.5	3.49E+01	25
5000	1	0.041	0.102	0.20	19,969.8	0.00E+00	1
5000	1	0.041	0.116	0.29	14.3	2.70E+01	21
5000	1	0.042	0.159	0.02	4.7	1.66E+01	68
5000	1	0.043	0.130	0.46	9.2	3.98E+01	34
5000	1	0.045	0.162	0.15	5.0	1.57E+01	56
5000	1	0.049	0.142	0.41	12.6	5.01E+01	26
5000	1	0.052	0.159	0.09	10.7	5.15E+01	30
5000	1	0.055	0.167	0.16	4.7	2.03E+01	60
5000	1	0.055	0.151	0.46	21.1	1.51E+02	16
5000	1	0.057	0.162	0.05	18.6	8.21E+01	16
5000	1	0.057	0.168	0.37	8.2	2.71E+01	35
5000	1	0.060	0.183	0.05	5.4	1.94E+01	58
5000	1	0.062	0.154	0.19	20.4	3.48E+01	15
541	1	0.062	0.138	0.40	1,014,142.4	0.00E+00	1
5000	1	0.063	0.160	0.22	15.2	1.28E+02	19
5000	1	0.068	0.173	0.18	138.7	3.86E+02	2
476	1	0.069	0.130	0.49	1,055,520.1	0.00E+00	1
3615	1	0.071	0.163	0.00	1,123,354.2	0.00E+00	1
1085	1	0.078	0.165	0.29	1,157,512.8	0.00E+00	1
313	1	0.078	0.121	0.32	1,063,971.3	0.00E+00	1
291	1	0.082	0.134	0.21	1,058,231.0	0.00E+00	1
5000	1	0.083	0.189	0.14	81.9	1.11E+03	3
1177	1	0.085	0.183	0.19	1,100,311.1	0.00E+00	1
276	1	0.086	0.140	0.38	1,064,183.5	0.00E+00	1
232	1	0.088	0.121	0.15	1,150,826.4	0.00E+00	1
232	1	0.091	0.114	0.36	1,037,486.7	0.00E+00	1
492	1	0.092	0.187	0.46	1,173,068.8	0.00E+00	1
367	1	0.094	0.167	0.37	1,058,030.9	0.00E+00	1
204	1	0.096	0.126	0.41	1,047,706.9	0.00E+00	1
268	1	0.096	0.127	0.41	1,174,664.4	0.00E+00	1
216	1	0.097	0.113	0.10	1,033,523.4	0.00E+00	1
394	1	0.100	0.185	0.32	1,366,950.0	0.00E+00	1

time	entry	meanGrowth	stdGrowth	meanSize	varSize	firmNumber
48	2	0.008	0.155	0.06	1.48E-03	10000
49	2	0.025	0.189	0.06	1.66E-03	10000
69	2	0.035	0.149	0.07	8.09E-04	10000
57	2	0.041	0.192	0.06	1.39E-03	10000
90	2	0.046	0.145	0.07	5.92E-04	10000
62	2	0.050	0.189	0.07	1.17E-03	10000
67	2	0.052	0.182	0.07	1.08E-03	10000
92	2	0.054	0.153	0.07	6.62E-04	10000
291	2	0.057	0.112	0.07	2.94E-04	10000
188	2	0.059	0.128	0.07	6.08E-04	10000
644	2	0.059	0.102	0.07	2.70E-04	10000
269	2	0.060	0.114	0.07	3.55E-04	10000
189	2	0.060	0.120	0.07	3.41E-04	10000
126	2	0.060	0.142	0.07	6.36E-04	10000
185	2	0.062	0.130	0.07	4.16E-04	10000
326	2	0.063	0.114	0.07	3.48E-04	10000
100	2	0.064	0.163	0.07	6.68E-04	10000
383	2	0.064	0.116	0.07	3.55E-04	10000
139	2	0.070	0.146	0.07	5.56E-04	10000
412	2	0.071	0.101	1,018,888.28	0.00E+00	1
488	2	0.075	0.106	1,081,347.00	0.00E+00	1
358	2	0.081	0.101	1,250,319.52	0.00E+00	1
483	2	0.082	0.114	1,056,847.05	0.00E+00	1
98	2	0.082	0.184	0.07	1.07E-03	10000
177	2	0.082	0.161	0.07	7.45E-04	10000
111	2	0.083	0.184	0.07	9.32E-04	10000
361	2	0.083	0.143	0.07	5.04E-04	10000
331	2	0.086	0.109	1,057,224.16	0.00E+00	1
1022	2	0.089	0.143	0.07	4.67E-04	10000
144	2	0.093	0.187	0.07	8.62E-04	10000
185	2	0.097	0.172	0.07	8.14E-04	10000
231	2	0.100	0.107	1284366.505	0.00E+00	1
213	2	0.114	0.121	1058135.665	0.00E+00	1
199	2	0.122	0.135	1074552.236	0.00E+00	1
348	2	0.127	0.178	1.34E+06	0.00E+00	1
168	2	0.138	0.150	1698748.278	0.00E+00	1
139	2	0.142	0.111	1114288.407	0.00E+00	1
173	2	0.143	0.165	1015582.173	0.00E+00	1
144	2	0.147	0.152	1.28E+06	0.00E+00	1
207	2	0.153	0.194	1188511.326	0.00E+00	1
130	2	0.155	0.126	1.01E+06	0.00E+00	1
119	2	0.158	0.104	1.01E+06	0.00E+00	1
182	2	0.162	0.179	1.26E+06	0.00E+00	1
108	2	0.168	0.106	1.00E+06	0.00E+00	1
177	2	0.173	0.198	1151604.912	0.00E+00	1
116	2	0.177	0.144	1178586.303	0.00E+00	1
135	2	0.178	0.162	1028319.37	0.00E+00	1
112	2	0.179	0.122	1039013.602	0.00E+00	1
102	2	0.185	0.118	1140938.126	0.00E+00	1
112	2	0.188	0.141	1.00E+06	0.00E+00	1
107	2	0.189	0.161	1220154.975	0.00E+00	1

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
time	meanGrowth	stdGrowth	minSize	k	mean	var	firmNumber
5000	0.001	0.155	0.36	0.001	1.84	3.33E+00	37
5000	0.006	0.161	0.07	0.001	0.97	4.40E+00	7
5000	0.006	0.129	0.12	0.006	2.47	1.84E+01	124
5000	0.009	0.124	0.33	0.005	2.66	1.22E+01	109
5000	0.010	0.182	0.34	0.007	2.29	2.44E+01	93
5000	0.015	0.123	0.31	0.003	2.10	4.69E+00	140
5000	0.018	0.167	0.11	0.005	1.89	1.01E+01	156
5000	0.018	0.101	0.25	0.007	2.17	5.32E+00	141
5000	0.019	0.185	0.30	0.009	2.36	1.07E+01	109
5000	0.022	0.128	0.10	0.004	2.06	5.49E+00	142
5000	0.025	0.135	0.14	0.006	2.09	4.19E+00	146
5000	0.025	0.114	0.03	0.001	1.39	3.66E+00	212
5000	0.028	0.159	0.34	0.003	2.21	5.90E+00	136
5000	0.029	0.121	0.40	0.004	2.58	5.25E+00	119
5000	0.032	0.139	0.28	0.002	2.79	6.88E+00	109
5000	0.035	0.187	0.42	0.005	2.82	9.81E+00	117
5000	0.035	0.193	0.25	0.010	2.15	6.64E+00	134
5000	0.046	0.146	0.42	0.003	2.81	8.08E+00	111
5000	0.048	0.163	0.40	0.005	2.95	6.23E+00	103
5000	0.049	0.103	0.11	0.006	2.15	5.44E+00	143
5000	0.049	0.192	0.03	0.010	1.59	5.63E+00	197
5000	0.052	0.133	0.08	0.002	1.83	5.91E+00	170
5000	0.052	0.106	0.37	0.009	2.98	5.81E+00	105
5000	0.053	0.116	0.06	0.008	1.56	6.78E+00	188
5000	0.054	0.173	0.26	0.006	3.01	9.74E+00	102
5000	0.054	0.107	0.16	0.002	2.14	4.49E+00	146
5000	0.054	0.129	0.18	0.008	2.25	6.87E+00	137
5000	0.056	0.163	0.02	0.005	1.52	5.03E+00	201
5000	0.062	0.195	0.39	0.006	2.63	6.74E+00	117
5000	0.063	0.195	0.25	0.006	2.48	6.90E+00	123
5000	0.064	0.107	0.08	0.002	2.31	7.72E+00	139
5000	0.066	0.124	0.09	0.006	2.00	7.37E+00	161
5000	0.066	0.197	0.15	0.007	2.21	6.27E+00	145
5000	0.066	0.133	0.20	0.004	2.39	6.86E+00	132
5000	0.068	0.192	0.13	0.002	1.70	4.32E+00	189
5000	0.069	0.168	0.08	0.001	1.78	6.32E+00	179
5000	0.070	0.145	0.23	0.010	2.76	9.43E+00	112
5000	0.072	0.136	0.02	0.002	1.39	4.07E+00	231
5000	0.078	0.133	0.13	0.001	2.15	6.73E+00	143
5000	0.078	0.101	0.34	0.001	2.71	8.68E+00	119
5000	0.084	0.101	0.08	0.005	2.39	8.37E+00	133
5000	0.085	0.165	0.01	0.002	1.73	7.56E+00	192
5000	0.086	0.181	0.07	0.002	2.03	1.07E+01	165
5000	0.087	0.154	0.34	0.002	2.80	8.55E+00	117
5000	0.087	0.189	0.29	0.006	3.07	8.02E+00	103
5000	0.090	0.107	0.17	0.003	2.45	9.27E+00	130
5000	0.091	0.184	0.05	0.004	1.64	3.75E+00	201
5000	0.093	0.133	0.10	0.002	2.47	8.12E+00	131
5000	0.095	0.162	0.07	0.007	2.03	7.06E+00	158
5000	0.098	0.140	0.45	0.006	3.06	5.83E+00	105
5000	0.098	0.137	0.09	0.005	2.21	7.31E+00	145

Table A.7: Simulation results for Excess Supply Affects All exit, mean and variance homoskedasticity
time	entry	meanGrowth	stdGrowth	minSize	birthRate	meanSize	firmNumber
5000	1	0.008	0.129	0.24	0.05	1.37	213
5000	1	0.011	0.107	0.24	0.08	1.18	261
5000	1	0.011	0.117	0.34	0.08	1.36	223
5000	1	0.014	0.172	0.30	0.02	1.53	200
5000	1	0.015	0.186	0.26	0.04	1.43	225
5000	1	0.038	0.195	0.21	0.04	2.19	135
5000	1	0.051	0.170	0.41	0.02	4.10	74
5000	1	0.078	0.163	0.03	0.09	0.95	367
5000	1	0.081	0.121	0.20	0.02	2.22	153
5000	1	0.084	0.195	0.09	0.09	1.35	233
5000	1	0.090	0.114	0.37	0.10	4.54	72
5000	1	0.104	0.102	0.04	0.04	2.49	140
5000	1	0.106	0.159	0.19	0.03	2.84	119
5000	1	0.107	0.154	0.24	0.05	2.15	145
5000	1	0.117	0.158	0.48	0.10	10.16	35
5000	1	0.119	0.153	0.30	0.08	13.29	25
5000	1	0.130	0.188	0.48	0.01	24.23	12
5000	1	0.135	0.118	0.02	0.01	0.50	643
5000	1	0.154	0.194	0.18	0.08	6.07	58
5000	1	0.158	0.163	0.03	0.08	0.94	347
5000	1	0.159	0.179	0.38	0.03	4.88	61
5000	1	0.159	0.182	0.10	0.05	7.28	55
5000	1	0.162	0.185	0.26	0.03	12.53	28
5000	1	0.165	0.147	0.22	0.10	6.92	52
5000	1	0.167	0.176	0.39	0.06	12.65	26
5000	1	0.189	0.196	0.49	0.01	12.83	26
5000	1	0.193	0.114	0.14	0.04	9.77	35
5000	1	0.198	0.145	0.28	0.06	15.60	26
5000	2	0.005	0.175	0.34	0.08	0.87	377
5000	2	0.009	0.143	0.39	0.06	0.81	371
5000	2	0.020	0.141	0.35	0.09	0.60	514
5000	2	0.025	0.107	0.32	0.07	0.52	599
5000	2	0.050	0.128	0.08	0.08	0.14	2331
5000	2	0.099	0.149	0.27	0.06	0.55	598
5000	2	0.102	0.101	0.14	0.05	0.21	1571
5000	2	0.116	0.193	0.45	0.08	1.16	303
5000	2	0.134	0.144	0.41	0.09	0.60	578
5000	2	0.139	0.112	0.37	0.01	335.72	1
5000	2	0.139	0.177	0.09	0.07	0.17	2052
5000	2	0.145	0.159	0.25	0.07	0.42	829
5000	2	0.153	0.180	0.29	0.07	0.54	655
5000	2	0.156	0.129	0.13	0.03	0.37	958
5000	2	0.171	0.200	0.43	0.05	1.08	331
5000	2	0.175	0.147	0.38	0.08	0.52	694
5000	2	0.176	0.161	0.45	0.08	0.68	523
64	2	0.180	0.136	0.03	0.07	0.07	10000
5000	2	0.181	0.112	0.30	0.06	0.40	900
5000	2	0.192	0.181	0.39	0.06	0.78	470
5000	2	0.195	0.180	0.27	0.07	0.52	705
5000	2	0.197	0.191	0.14	0.00	262.04	1
5000	2	0.198	0.162	0.12	0.09	0.16	2247
5000	3	0.033	0.176	0.15	0.09	1.64	162
5000	3	0.044	0.173	0.47	0.08	2.97	104
5000	3	0.050	0.166	0.08	0.08	1.65	200
5000	3	0.080	0.113	0.41	0.08	2.34	141
5000	3	0.102	0.128	0.38	0.09	2.62	128
5000	3	0.103	0.166	0.03	0.09	0.76	375
5000	3	0.112	0.102	0.10	0.02	2.07	144
5000	3	0.116	0.147	0.29	0.01	2.38	139
5000	3	0.120	0.107	0.14	0.03	1.55	212
5000	3	0.125	0.199	0.47	0.04	3.17	103
5000	3	0.125	0.192	0.44	0.10	2.49	140
5000	3	0.130	0.177	0.28	0.02	2.48	124
5000	3	0.136	0.181	0.49	0.06	2.56	139
5000	3	0.146	0.142	0.06	0.03	1.10	312
5000	3	0.146	0.185	0.38	0.10	1.90	172
5000	3	0.152	0.159	0.21	0.08	1.98	178
5000	3	0.173	0.106	0.42	0.05	2.23	159
5000	3	0.173	0.163	0.10	0.03	1.17	269
5000	3	0.181	0.179	0.14	0.05	1.70	204
5000	3	0.189	0.159	0.22	0.08	2.25	157
592	1	0.106	0.129	0.00	0.01	0.05	10000
506	1	0.111	0.172	0.00	0.08	0.06	10000
1112	1	0.116	0.112	0.00	0.02	0.04	10000
336	1	0.117	0.193	0.00	0.02	0.05	10000
567	1	0.123	0.134	0.00	0.07	0.05	10000
434	1	0.126	0.165	0.00	0.08	0.06	10000
223	1	0.130	0.196	0.00	0.04	0.06	10000
1411	1	0.133	0.121	0.00	0.03	0.05	10000
465	1	0.136	0.147	0.00	0.00	0.04	10000
2349	1	0.138	0.102	0.00	0.04	0.05	10000
722	1	0.144	0.158	0.00	0.07	0.08	10000
817	1	0.148	0.158	0.00	0.08	0.05	10000
800	1	0.158	0.146	0.00	0.02	0.04	10000
2236	1	0.164	0.115	0.00	0.04	0.05	10000
1199	1	0.175	0.129	0.00	0.07	0.05	10000
556	1	0.190	0.188	0.00	0.00	0.08	10000
1185	1	0.193	0.119	0.00	0.06	0.03	10000
1163	1	0.195	0.171	0.00	0.04	0.04	10000
2336	1	0.196	0.140	0.00	0.02	0.04	10000
118	2	0.101	0.121	0.00	0.04	0.06	10000
147	2	0.105	0.136	0.00	0.03	0.07	10000
179	2	0.107	0.117	0.00	0.02	0.07	10000
119	2	0.117	0.142	0.00	0.03	0.07	10000
69	2	0.131	0.131	0.00	0.06	0.07	10000
42	2	0.133	0.137	0.00	0.10	0.07	10000
130	2	0.134	0.162	0.00	0.03	0.07	10000
47	2	0.144	0.187	0.00	0.09	0.07	10000
47	2	0.145	0.157	0.00	0.09	0.07	10000
58	2	0.145	0.199	0.00	0.07	0.07	10000
5000	2	0.146	0.155	0.00	0.01	3.88	100
43	2	0.151	0.177	0.00	0.09	0.07	10000
41	2	0.151	0.122	0.00	0.10	0.07	10000
108	2	0.158	0.124	0.00	0.04	0.07	10000
284	2	0.168	0.182	0.00	0.02	0.07	10000
75	2	0.178	0.169	0.00	0.05	0.07	10000
91	2	0.190	0.141	0.00	0.05	0.07	10000
60	2	0.192	0.165	0.00	0.07	0.07	10000
102	2	0.194	0.157	0.00	0.04	0.07	10000
187	2	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.

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