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

Alexander Saichev (), Yannick Malevergne and Didier Sornette ()
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Alexander Saichev: Nizhni Novgorod State University
Didier Sornette: EMLYON Business School – Cefra

Chapter Chapter 6 in Theory of Zipf's Law and Beyond, 2010, pp 73-95 from Springer

Abstract: Abstract 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 s 1 > 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 s 1, the possibility of touching s 1 arises, and the rate R(t, Δ) given by (2.1) significantly depends on s 1. 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 6.1 $$d S (t) = a [S(t)]dt + b[S(t)]dW (t), \qquad S(t = 0) = s_0,$$ 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.

Keywords: Brownian Motion; Diffusion Equation; Mutual Fund; Stochastic Equation; Geometric Brownian Motion (search for similar items in EconPapers)
Date: 2010
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Persistent link: https://EconPapers.repec.org/RePEc:spr:lnechp:978-3-642-02946-2_6

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DOI: 10.1007/978-3-642-02946-2_6

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