Price's law is not intensive

Price's law was proposed in the context of scientific publishing: 

The square root of the total number of authors contribute half of the total number of publications.

It is a more extreme version of Pareto's 20/80 rule, which would state that 20% of authors contribute 80% of the total number of publications (see next post for the relation between the two). Unlike Pareto's rule, however, Price's law is not stable under extension. This is a trivial observation, but I have not yet seen it in the literature, just like I have not seen much empirical evidence for Price's law.

Let us denote by \(N\) the total number of authors and by \(N_p\) the number of "productive" authors (the top authors that provide half of all publications). As the ratio of two extensive quantities, \(p\) should be independent of the system size \(N\): consider ten systems (e.g. the research communities in different countries, different subjects, etc.), each of size \(N\), with the same publication distribution and hence the same \(N_p\). Half of the total number of publications is published by \(10 \, N_p\) contributors, so the overall productivity is \(p' = \dfrac{10 N_p}{10 N} = p\). According to Price's law, it should however be \(p' = p/\sqrt{10}\) ! The situation is similar to having ten identical vessels, all under the same pressure \(p\). If we connect them all together the pressure does not change, although both the volume and energy increase by a factor of ten.

Price's law does have a "convenient" feature: simply by selecting the representative size \(N\) one can obtain any productivity, since \(p = 1/\sqrt{N}\). For instance, the same Pareto distribution that yields the 20/80 rule predicts that 0.7% of causes yield half of the effects. This result is reproduced by Price's law with \(N \simeq 23000\).

Outside of bibliometry, Price's law has been invoked in economics, for instance by Jordan Peterson in (at least) one of his videos. What I find amusing is that it seems to contradict the principle of economies of scale: if there is a connection between the productivity \(p\) and the economic efficiency (and this is the more likely the higher the personnel costs are) then an increase in the size of a company decreases its efficiency. For instance, a chain of ten supermarkets would be less effective than ten independent units, which would be less effective than many small shops. Since the market is supposed to select for efficiency, we should witness fragmentation, rather than consolidation. 

References:

https://subversion.american.edu/aisaac/notes/pareto-distribution.pdf Clear derivation of the 20/80 principle from the general Pareto distribution.