Abstract
The paper explores the frequency and size distributions of firm-size in a novel dataset for the mid-Victorian era from a recent extraction of the England and Wales population censuses of 1851, 1861, 1871, and 1881. The paper contrasts the hypothesis of the Power Laws against the Lognormal model for the tails of the distributions using maximum likelihood estimation, log likelihood ratio, clipped sample coefficient of variation UMPU-Wilks test, Kolmogorov–Smirnov statistic, among other state-of-the-art statistical methods. Our results show that the Power Law hypothesis is accepted for the size distribution for the years 1851 and 1861, while 1871 is marginally non-significant, but for 1881 the test is inconclusive. The paper discusses the process that generates these distributions citing recent literature that shows how after adding an i.i.d. noise to the Gibrat’s multiplicative model one can recreate a Power Law behaviour. Overall, the paper provides, describes and statistically tests for the very first time a unique historical dataset confirming that the tails of the distributions at least for 1851 and 1861 follow a Pareto model and that the Lognormal model is firmly rejected.
Highlights
Fujita et al in their classic textbook observe that often “theory gives simple, sharp-edged predictions, whereas the real world throws up complicated and messy outcomes”.[17]
This paper explores two distributions of firm data during the midVictorian era to test the Power Law compared to the Lognormal hypotheses at the tails: that is, a tale of two tails
We show that the Pareto distribution fits the tails of the size data better than a Lognormal one for at least the years 1851 and 1861, and with marginally non-significant values in 1871, but for the frequency distributions the test is inconclusive
Summary
Fujita et al in their classic textbook observe that often “theory gives simple, sharp-edged predictions, whereas the real world throws up complicated and messy outcomes”.[17]. The unique data examined in this paper permits analysis of both the distributions of the frequency, and the size, measured by number of total employees of firms in England and Wales between 1851 and 1881. We are interested in which of either a Power Law or Lognormal better describes the distributions of our data, but to aid interpretation it is important to clarify why firm size follows these distributions? We call the cut-off parameter u as in Malevergne et al and we use Clauset et al α so the Zipf’s law will be attained for a value of α = 2 They add that, as in the normal distribution, an additive 68–95–99.7 rule applies (i.e., the arithmetic value plus and minus one, two, or three standard deviations), so in the Lognormal distribution a multiplicative rule 68–95–99.7 rule emerges for iterative multiplication and division by standard deviations
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