Measuring Productivity Dispersion: A Parametric Approach Using the Lévy Alpha-Stable Distribution

Abstract

Productivity levels and growth are extremely heterogeneous among firms. A vast literature has developed to explain the origins of productivity shocks, their dispersion, evolution and their relationship to the business cycle. We examine in detail the distribution of labor productivity levels and growth, and observe that they exhibit heavy tails. We propose to model these distributions using the four parameter Lévy stable distribution, a natural candidate deriving from the generalised Central Limit Theorem. We show that it is a better fit than several standard alternatives, and is remarkably consistent over time, countries and sectors. In all samples considered, the tail parameter is such that the theoretical variance of the distribution is infinite, so that the sample standard deviation increases with sample size. We find a consistent positive skewness, a markedly different behaviour between the left and right tails, and a positive relationship between productivity and size. The distributional approach allows us to test different measures of dispersion and find that productivity dispersion has slightly decreased over the past decade.

Levels of structural change

“…Another branch of literature (Duschl and Peng 2015;Yu et al 2015) investigates the distribution of growth rates in China (computed using value added growth or sales growth) in the tradition of Bottazzi and Secchi's analysis (Bottazzi and Secchi 2006; where the distribution is conjectured to belong to the family of Asymmetric Exponential Power (AEP) distributions. However, there is also strong evidence that both growth rates and many other quantities at the firm level are heavy-tailed (Axtell 2001;Heinrich and Dai 2016;Yang et al 2019;, such that the tails could be severely underestimated by AEP fits.…”

“…Would this not be bad, even if there was an unavoidable baseline misallocation due to bounded rationality? However, labor productivities (and total factor productivities, TFPs) follow heavy-tailed distributions, which makes these dispersion measures unreliable (Yang et al 2019). Other dispersion measures, that are robust with respect to heavy tails, such as the scale parameter of Lévy alpha stable fits, show no sign of increasing, at least not relative to the median of the distribution (Yang et al 2019;.…”

“…Heinrich and Dai (2016)), for labor productivity (cf. Yang et al (2019)), and value added (see Section 4.5). While employment is strictly positive and the heavy tail therefore only occurs on one side, value added and labor productivity are two-sided.…”

“…We find that Lévy alpha-stable distributions (Nolan 1998; with tail indices α < 2 are good empirical models for these variables (cf. Yang et al (2019) and Section 4.5). First differences of heavytailed variables will also be heavy-tailed, but will always be two-sided.…”

“…It can be parametrized as S (0, α, β, γ, δ) where the four variables stand for the tail index (α), the skew parameter (β), the scale (γ ) and the location shift (δ). For details on the fitting, see Yang et al (2019) and .…”