Didier Sornette

Properties of the Realization Dependent Distribution of Firm Sizes

This chapter discuss the properties of the realization-dependent density of firm’s sizes. We put together the different ingredients introduced in the previous chapters to analyze the extent to which the mean density of firm’s sizes, which has been the main topic of this book, is representative of the realized density of firm’s sizes in a given economy (i.e., a single realization).

Firm’s Sudden Deaths

There are a priori two exit mechanisms for firms: Firms disappear when their asset values become smaller than some minimum level. This is based on the standard idea, justified by the existence of a minimum efficient size, that there is a minimum firm size below which the firm cannot exist. This idea has been considered in several models of firm growth (see, e.g., de Wit, 2005 and references therein). An alternative approach suggested for instance by Gabaix (1999), considers that firms cannot decline below a minimum size and remain in business at this size until they start growing up again. In addition to the exit of a firm resulting from its value decreasing below a certain level, it sometimes happens that a firm encounters financial troubles while its asset value is still fairly high. One could cite the striking examples of Enron Corp. and Worldcom, whose market capitalization were supposedly high (actually the result of inflated total asset value of about

Future Directions and Conclusions

11 billion for Worldcom and probably much higher for Enron) when they went bankrupt. Beyond these anecdotic examples, there is a large empirical literature on firm entries and exits, that suggests the need for taking into account the existence of failure of large firms. For example, while it has been established that a first-order characterization for firm death involves lower failure rates for larger firms (Dunne et al., 1988, 1989), Bartelsman et al. (2003) also state that, for sufficiently old firms, there seems to be no difference in the firm failure rate across size categories. In previous chapters, we have examined the consequences and impact on Zipf’s law of the first exit mechanism. The present chapter is devoted to the study of the second mechanism.

Deviations from Gibrat’s Law and Implications for Generalized Zipf’s Laws

The set of mechanisms that we have considered up to now explains quite well both ends of the distribution of firm’s sizes. Indeed, while the average growth rate of firm’s asset values, the intensity of firm’s births and the hazard rate of a firm’s sudden death accurately explain the behavior of the population of large firms, the introduction of a lower threshold below which firms disappear (by lack of efficiency, for instance) explains the behavior of the population of small firms. Everything is not perfect however. In many countries, like France and India amongst others, the population of medium size firms exhibits an anomalous behavior. One usually speaks of the “missing middle” phenomenon to describe the fact there is a deficit of firms of intermediate sizes. Our model, while already quite versatile, does not account for this country-specific stylized fact. The reason is in fact rather simple. To a large extent, the “missing middle” phenomenon could be explained by the strong propensity of large firms to merge with firms which are still small but present a promising potential. This should yield a depletion of the population of medium size firms. In addition to the mechanisms in terms of birth, death and random growth which have been considered in the previous chapters, we envision that the next level of development of a complete mathematical theory of firms needs to take into account the mechanism of mergers between firms (referred to as MA Leyvraz, 2003).

Flow of Firm Creation

The introduction of a mechanism in which firms die introduces already a deviation from Gibrat’s law for small s-values. Killing firms upon first touching the level s1 > 0 actually means that the corresponding firm’s asset values S(t) do not obey strictly Gibrat’s law of proportionate growth. Indeed, when S(t) becomes close to s1, the possibility of touching s1 arises, and the rate R(t, Δ) given by (2.1) significantly depends on s1. In the present chapter, we will discuss in detail another general class of models in which the stochastic growth process deviates from Gibrat’s law in different ways. Specifically, we will suppose that S(t) is a diffusion process, obeying the stochastic equation n n

Continuous Gibrat’s Law and Gabaix’s Derivation of Zipf’s Law

d S (t) = a [S(t)]dt + b[S(t)]dW (t), qquad S(t = 0) = s_0,

Non-stationary Mean Birth Rate

n n(6.1) n nso that the corresponding pdf f(s; t) satisfies the diffusion equation (2.39) and the initial condition (2.40). Recall that Gibrat’s law of proportionate growth implies in particular that the coefficients a(s) and b(s) of the stochastic equation (6.1) are given by relations (2.41), i.e., are proportional to s. However, there is a wide and recent empirical literature, that suggests that Gibrat’s law does not hold, in particular for small firms (Reid, 1992; Audretsch, 1995; Harhoff et al., 1998; Weiss, 1998; Audretsch et al., 1999; Almus and Nerlinger, 2000; Calvo, 2006) See however Lotti et al. (2003, 2007) for a dissenting view.

Recurrence Statistics of Earthquakes Explained by the ETAS Model

The failure of the approach based solely on Gibrat’s principle stems, at least in part, from the fact that it attempts to derive the distribution of firm sizes directly from the distribution of the asset value of a single firm. Indeed, many models start with the implicit or explicit assumption that the set of firms under consideration was born at the same origin of time. This approach is equivalent to considering that the economy is made of only one firm. Therefore, the distribution of firm sizes can reach a steady-state if and only if the distribution of the asset value of a single firm reaches a steady state, which seems rather counterfactual. An alternative approach to model a stationary distribution of firm sizes is to account for the fact that firms do not appear at the same time but are born according to a more or less regular flow of newly created firms. For instance, Bonaccorsi Di Patti and Dell’Ariccia (2004) report a yearly average (over all industry branches) rate of birth for the period 1996–1998 equal to 5.6% for Italian firms with a maximum of 32% in some industry branches. Reynolds et al. (1994) give the regional average firm birth rates (annual firm births per 100 firms) of several advanced countries in different time periods: 10.4% (France; 1981–1991), 8.6% (Germany; 1986), 9.3% (Italy; 1987–1991), 14.3% (United Kingdom; 1980–1990), 15.7% (Sweden; 1985–1990), 6.9% (United States; 1986–1988). They also document a large variability from one industrial sector to another.