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We study empirically and analytically growth and fluctuation of firm size distribution. An empirical analysis is carried out on a US data set on firm size, with emphasis on one-time distribution as well as growth-rate probability distribution. Both Pareto's law and Gibrat's law are often used to study firm size distribution. Their theoretical relationship is discussed, and it is shown how they are complementable with a bimodal distribution of firm size. We introduce economic mechanisms that suggest a bimodal distribution of firm size in the long run. The mechanisms we study have been known in the economic literature since long. Yet, they have not been studied in the context of a dynamic decision problem of the firm. Allowing for these mechanism thus will give rise to heterogeneity of firms with respect to certain characteristics. We then present different types of tests on US data on firm size which indicate a bimodal distribution of firm size.

Recent statistical studies of firm size distribution in different countries and time periods (see Gibrat [

Looking at the empirical data, one observes that firms grow unevenly over time. Firm size changes or fluctuates. One quantitative way to look at this is to look at the probability distribution function (pdf) of the growth rate, which is defined as the ratio of the second year's value and the first year's value. Several studies on this aspect [

The fact that these two laws, Pareto's law and Gibrat's law, were observed to coexist suggested some deep relations between them. In fact, just such a relation was found when the law of detailed balance, which states that the simultaneous pdf of the first year's value and the second year's value is symmetric under the exchange of the two variables. This also leads to a theoretical relationship of the positive-growth side of the growth-rate pdf and the negative-growth side of the same, which were confirmed by the data.

Overall recent empirical research has found that, first, for the upper tail of the size distribution of firms the Pareto law holds (which may also be consistent with Gibrat's law that above a certain minimum size the growth rate of firms is independent of size), second, the variance of firms' growth rates is independent of size over and above a certain minimum size and, third, the frequency of the moving up and down in size class is also roughly the same above certain size classes.

Yet, all three results appear to be consistent with the fact that the upper size classes behave distinctively from lower size classes and that the middle size classes are “thinned out.” Thus, there might be a tendency toward a twin-peak distribution with different behavior of the upper and lower size classes. Empirical work may have to deal with the fact that groups of firm size classes behave differently and that there is a bimodal distribution of size classes in the long run. (Other recent literature has also discovered bimodal firm size distribution; see Bottazzi and Secchi [

Since the statistical study of the size distribution of firms does not provide us with much insight into underlying mechanisms, we also focus on the dynamics of firm size distribution to give the statistical results some theoretical underpinning. We introduce and study economic mechanisms that may explain the dynamics of firm size and firm growth over time. We show that there is a long tradition in economic theory and, in particular, in the industrial organization literature of the last fifty years that has pointed out some major mechanisms why one indeed would expect a bimodal size distribution of firms in the long run. In Sections

We want to exclude some arbitrary behavior of firms, and thus presume that all firms pursue the same dynamic investment strategy by aiming at maximizing the present value of their future pay off. Of course, there are also other influences on firm growth such as growth of overall demand, industry demand, cost and technology shocks and elaborate financing practices. Yet we want to focus only on some major mechanisms that have been studied in economic theory since long and confront their implied predictions for firm size dynamics with the results of our empirical study.

The remainder of the paper is organized as follows. In Sections

We introduce here some mechanisms that may give rise to some dynamics toward a twin-peak size distribution of firms. The first important mechanism that we want to study is based on locally increasing returns to scale. This idea of locally increasing returns to scale exist in the literature since Marshall [

Our second mechanism is based on the adjustment cost of capital which has been employed in investment theory since Eisner and Stroz [

A third mechanism can be seen to originate in cost and ease (or tightness) of credit. Important contribution to this line of research can be found in Townsend [

It is presumed that the risk premium drives a wedge between the expected return of the borrower and the risk-free interest rate and becomes zero, in the limit, when debt approaches zero. A suitable function for this premium, will be introduced below. Yet, even if the credit cost spread is endogenized, there might be borrowing constraints for the firm that finances investment externally.

We may specify a general model of a firm

The decision problem of the firm is to maximize its present value

Allowing for adjustment cost,

As to the production function,

Since net income in (

Note that in the above general case of adjustment cost

For representing our third mechanism, we assume that the credit cost

In general, we define the limit of

If the interest rate

The case, however, when the credit cost is endogenous, thus, when we have

In the context of the model that explores the role of risk premiums and credit spread as a cause for unequal firm size, we can also study the impact of “ceilings” in debt contracts and their impact on firm size. Indeed, credit restrictions may affect the investment decisions. (Suppose the “ceiling” is of the form

In all three cases—locally increasing returns to scale, nonlinearity in adjustment costs of capital, and credit spread and credit constraints (the latter two arising from credit risk)—the optimal investment strategy and the growth of the firm depend on the initial size of the firm. We permit firms to be heterogeneous with respect to the way how they are exposed to the above three mechanisms.

We will show that there can be thresholds that separate the solution paths for

Next, we present numerical results obtained for our different specification of a production function, adjustment costs of capital, and imperfect capital markets. Throughout this section, we specify the parameter

As for the numerical procedure, we use two algorithms. The first is a dynamic programming algorithm as presented in Grüne et al. [

For a second algorithm that computes domains of attraction of the problem (

We start our numerical examples first with the common case of quadratic adjustment costs of investment and constant returns to scale. This gives us the usual case of a unique (positive) steady state and about which firm size is predicted to be normally distributed. In the next examples, we have built in the above stated three mechanisms giving rise to multiple domains of attraction and thus multimodal distribution of firm size.

For the common case usually found in the literature, and as represented in (

To compute the optimal investment strategy and firm value, we can use our first procedure, our dynamic programming (DP) algorithm. The firm value is given by the optimal value function in Figure

Quadratic adjustment cost of capital.

Next, we numerically solve for the case of increasing returns to scale and adjustment costs with size effects. Although, as shown in Grüne et al. [

Optimal value function and optimal investment.

For our parameters, the model does not necessarily have an unique positive steady state equilibrium. There can be multiple domains of attraction for firms that are exposed to either increasing returns and/or adjustment costs with size effects. The fate of a particular firm, when it is exposed to increasing returns to scale and/or to adjustment costs with size effects, then will depend on the initial capital stock size,

Thus, the dynamic decision problem of the firm faces a discontinuity. For firms with initial values of the capital stock

A similar result can be obtained if there is a distinct credit spread, and firms have to pay idiosyncratic risk premiums, or there may be credit constraints on firms' investment. This may also result in multiple domains of attraction. Let us presume that credit cost

Before, we had postulated that a risk premium is positively related to the default cost and inversely related to the borrowers net worth. Net worth is defined as the firm's collateral value of the (illiquid) capital stock less the agent's outstanding obligations. Following Bernanke et al. [

In general, it is not possible to transform the above problem into a standard infinite horizon optimal control problem for our prototype of firm; hence, we will use our second procedure, the algorithm for the computation of domains of attractions from Grüne et al. [

For the endogenous credit cost function (

Figure

Present value curve

For

The latter is illustrated for a discrete value of

On the other hand, often one has to impose for the investment decision of the firm a debt ceiling, defined as a fraction of the capital stock. This is the case when a firm's borrowing is constrained. For a particular

Present value curve

The potentially high-value curve can be reached with no credit constraint with

This means credit constrained firms are not able to undertake sufficiently high investments and will thus grow at a lower rate. They will not be able to realize their growth potentials, and thus their potential present value that they would be able to obtain without credit constraint. We can observe that with borrowing constraints, even for an optimal investment strategy, firms of different size classes would be expected if there are strict borrowing limits on firms.

The above mechanisms predict that one might expect in the long-run movements of growth rates and a size distribution of firms characterized by a clustering of firms in the upper region of size classes as well as at the lower end of size classes. Moreover, one would expect that the middle size classes are “thinned out,” possibly giving rise to a long-run twin-peak distribution of firm size.

We want to present some empirical evidence that may confirm some dynamics toward a long-run twin-peak distribution of firm size which is implied by the above model variants. To address this issue empirically, we concentrate on firm size distribution in the US manufacturing industry.

The data of the following empirical study are taken from the

The pstar data set gives us a set of observed data points or capital stocks, respectively, which can be interpreted as a sample of an unknown probability density function for several years. To analyze certain characteristics of this density, one has to determine the unknown density. If, for example, the density function has changed from being unimodal to a bimodal one, it can be regarded as a hint that the middle size classes have been thinned out, supporting the above stated theoretical ideas. The graphs in Figure

PDF of the (normalized) netcap from year 1960 to 1991. Normalization is done so that average netcap is one for each year. Data for different years are plotted in different hue and dashing as in the legend.

In order to discuss this in a more quantitative manner, we first fit the probability density function (pdf) with a log-normal distribution.

Figure

Rank-size plots with the log-normal fits for the years 1970, 1975, 1980, 1985, and 1990 in log-log scale. In all the cases the over-all fit is good, except for the small systematic differences, which is a sign of the thinning out.

Difference between the rank size and the log-normal fits for the same years as in Figure

This means that they are not random fluctuations around the log-normal. Most notable feature is that that they dip between −1 and 0, which means that since the derivative of cumulative probability is the negative of pdf, the actual pdf is given by log-normal pdf plus a twin peak pdf with its first peak at around −1 and second peak at a little less than 1. The strength of the twin-peak can be measured by fitting these differences with a function such as

The resulting amplitude

Plot of the “amplitude”

In this paper, we have first studied the two scaling laws, Pareto's law and Gibrat's law. Both seem to hold for large firms. Their relations under the law of detailed balance was also studied. This discussion is done purely kinematically, that is, independent from any models one might put forward. Yet, this kind of study helps one to isolate dynamical features, which remains to be explained. This is the strength and the power of this kind of analysis. As a result, we now have a specific shape of the growth-rate pdf in the framework of Gibrat’s law, which should be the target of the modern analysis of firm size dynamics. The analysis of the upper and lower tail of the firm size distribution via Pareto’s law and Gibrat`law is empirically complemented by a statistical study of what happens in the middle. Our study empirically addresses the issue of the “thinning out” of the middle and the issue of a bimodal distribution of firm size.

In order to theoretical motivate such an enlarged study, namely, a study of the bimodal size distribution of firm classes in the long run, we have introduced a dynamic model of firm behavior, where firms might be exposed to locally increasing returns, nonlinear adjustment costs of capital, credit spread, determined by risk premiums, and credit constraints. Using dynamic programming, we compute the dynamics of firm size and their long run size distribution. A bimodal distribution is predicted. Empirically, this means that over time, there is a “thinning out” of firm size classes in the middle. Our statistical analysis then has supported the theoretical model's predictions.

Finally, we want to note that, of course, the evolution of firm size distribution in industries is presumably more complex than characterized by our above statistical and analytical studies. Numerous empirical studies on the dynamics of the firm size distribution over the life cycle of an industry have documented the complexity of the forces affecting the size distribution of firms such as growth of overall demand, industry demand, cost and technology shocks, financing practices, and industry regulation. (For a survey of the variety of forces of growth in certain stages of the life cycle of the industry, see Mazzucato and Semmler [

We, here, give an analysis of Pareto’s law and the Gibrat's law, which are often observed in the data. We present a theoretical discussion of the relationship between these two laws. They are also called scaling laws.

The cumulative probability distribution function

The growth rate

In the following, we discuss the relationship between these two laws.

All the data mentioned above satisfy the law of detailed balance to a certain degree. In terms of the simultaneous pdf

The following can be proven under the law of detailed balance [

If Gibrat's law holds, Pareto's law is satisfied.

Under the same condition as the above, the “Reflection law,”

The proof goes as follows: we denote the joint pdf of

We define conditional probability

The empirical facts corresponding to this are observed in Fujiwara et al. [

The joint pdf

The pdf

The conditional probability

In passing, it should be noted that (

Let us prove that the properties (A) and (C) lead to (B). By substituting Gibrat's law (

Let us rewrite (

From (

The result (

In the general case of (

Note that in the limit case, where there is no borrowing and

In this case,

The HJB-equation (

The equilibria of the HJB equation (

If the HJB equation (

Here, again, for

The authors would like to thank Dr. Masanao Aoki, Dr. Mauro Gallegati, Dr. Corrado Di Guilmi, and Dr. Taisei Kaizouji for collaborative works at various stages of this research. They would like to thank two referees of the journal for their extensive comments on the paper.