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Size Distribution of Portuguese Firms between 2006 and 2012
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1  University of Coimbra

Abstract: This study aims to describe the size distribution of Portuguese firms, as measured by annual sales and total assets, between 2006 and 2012, giving an economic interpretation for the time evolution of the distribution. Three distributions are fitted to data: the lognormal, the Zipf and the Simplified Canonical Law (SCL). Methods of estimation include Maximum Likelihood, modified Ordinary Least Squares in log-log scale and Nonlinear Least Squares considering the Levenberg-Marquardt algorithm. Lognormal and Zipf distributions can be justified from the Gibrat's law and, in Zipf case a constraint must be added, for instance, on the minimum size of firms. As for SCL, firstly presented by Mandelbrot in the context of linguistics, the argument used to deduce the distribution of words frequency in texts is adapted to the distribution of firms. This is done by defining production units which can be glued to constitute firms so that the network of firms can be used flexibly to satisfy the most diverse needs in the economy in the less costly way and reducing at the maximum the delay in achieving it. For this distribution, different interpretations of the estimated parameters found in the literature are confronted and discussed in the light of data. Using this interpretation, we attempt to find a characterization of the Portuguese economic activity rhythm, diversity of activity sectors and competition. Diversity is captured by an entropy measure at different levels of aggregation and not just at the higher level as is usually done in literature. The interrelations between these economic characteristics and others, such as the level of concentration of firm size distribution, are reassembled from literature and confronted. Analyzing entropy at different levels of aggregation allows verifying how the evolution of activity rhythm can have a distinct impact in diversity at those different levels.
Keywords: Firms size; lognormal law; Zipf law; simplified canonical law; Shannon entropy