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

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 $$d S (t) = a [S(t)]dt + b[S(t)]dW (t), \qquad S(t = 0) = s_0,$$ (6.1) so 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.