La realidad: la naturaleza de lo Absoluto

of these results: Is the measured value of the exponent β due to some averaging over the different industries, or is it due to a universal behavior valid across all industries? As a “robustness check”, Amaral et al. split the entire company sample into two distinct intervals of SIC codes. It is visually apparent in Fig. 4.14 that the same be- havior holds for the different industries. Thus, it can be concluded that these results are indeed universal across different manufacturing industries in the US. In statisti- cal physics, scaling phenomena are sometimes represented graphically by plotting a suitably “scaled” dependent variable as a function of a suitably “scaled” independent variable. If scaling holds, then the data for a wide range of parameter values are said to “collapse” upon a single curve. From the following empirical results, the data collapse upon the single straight line shows small but consistent deviations for large growth rates from the exponential distribution in Eq. (4.5). Thus, Eq. (4.5) can be regarded only as a first-order approximation to reality.

4.1

Previous Models

All of the empirical results presented above not only provide a justification for developing a model to explain the statistical properties of firm growth, but also have implications for what statistical properties are most important to explain. In 1931, Gibrat presented striking evidence [81, 82] that the distribution of firm size at different times and within different “populations” was approximately log-normal; and showed the log-normality could be generated by a process in which the distribution of growth rates is independent of initial size. Gibrat’s stochastic model is an early example of the use of statistical physics in economics. His model is one of a random stochastic process, and the objective of his model was to explain the shape of a distribution that emerged from it.

The assumptions Gibrat proposed are : (1) the growth rate R of a company is independent of its size (this assumption is usually referred to by economists as the law

of proportionate effect), (2) the successive growth rates of a company are uncorrelated

in time, and (3) the companies do not interact.

In mathematics, Gibrat’s model is expressed by the stochastic process:

St+∆t= St(1 + ²t), (4.9)

where St+∆t and St are, respectively, the size of the company at times (t + ∆t) and

t, and ²t is an uncorrelated random number with some bounded distribution and

variance much smaller than one (usually assumed to be Gaussian). Hence,

St= S0(1 + ²1)(1 + ²2) · · · (1 + ²t). (4.10)

If it is assumed that all companies are born at approximately the same time and have approximately the same initial size, then the distribution of company sizes is also log-normal. This prediction from the Gibrat model is approximately correct.

However, under Gibrat’s assumptions, it can be easily derived the expression of growth rate g as follows: at time T (T is much larger than t, for example a year)

log S(t = T ) = log S(t = 0) +

M

X

t0₌₁

log(ηt0), (4.11)

where M is the total number of time steps. Because of g(T ) = log(S(T )/S(0)), we get g = M X t0₌₁ log(ηt0). (4.12)

Since M is large, and ηtis a random variable following a certain distribution, using

the Central Limit Theorem the distribution of g is Gaussian. Now we know that this prediction is wrong and this model needs some changes.

Buldyrev [33] et al. (1997) present a model of organizational hierarchy in which decisions get passed down through successive layers. At each decision point in the hierarchy, a manager can either accept the decision of his immediate superior or reject it and make his own independent decision. Amaral [4] et al. (1998) describe a model of Gibrat-type growth processes at the business unit level and firms consisting of multiple units with uncorrelated growth processes. This model predicts the key empirical findings about firm growth for a wide variety of parameters. The Buldyrev and Amaral et al. models qualitatively justify why the distribution of firm growth rates shows a “tent” shape, and also numerically give the size-variance relation.

Sutton [194] postulates that all partitions of a company of size S into smaller sub- pieces are equiprobable . This is similar to the corresponding hypothesis in statistical physics that all microstates which a physical system can attain are equiprobable. More precisely Sutton assumes that S is a large integer, and uses known mathematical results on the number of partitions to compute σ(S). Finally, Sutton analytically gives β ≈ 0.24 as a universal power-law exponent. Following Sutton, Bouchaud et al. (2002) present a model in which firms consist of independent, divisions that are varying groupings of a basic unit size [206]. He assumes that all partitions of the firm are equally likely. For example, a firm of size 4 could consist of four one-unit divisions, two one-unit divisions and one two-unit division, a two two-unit division, one three-unit and one one-unit division, or one four-unit division. This model is similar to the Amaral et. al. model in that it views firms as consisting of units whose growth rates are independent of each other. But, Sutton model discusses little about the distribution of firm growth rates, and its relationship with firm size.

Axtell (1999) presents a model in which firms are teams of individuals who select effort levels and share the output [12]. Holding effort levels constant, adding indi- viduals (or collections of individuals) to a firm increases output. As firms become larger, however, each individual has more of an incentive to shirk because of the shar- ing rule. The growth dynamics come from having individuals randomly joining and leaving firms. This model implies a Pareto distribution of firm size and the scaling relationships in growth dynamics described above.

4.2 The distribution of unit number 83 Given that there are already four models that predict the same empirical findings, one might question the need for a fifth. However, there is no reason to believe that these findings sufficiently identify the “true” model of the firm. Models that are not consistent with empirical findings can be rejected, but a model just because it is consistent with them cannot be accepted. The primary feature that distinguishes the preferential attachment is that it is explicitly designed to address the question of which activities are organized within a given firm.