Since the Depression-era efforts of Robert Gibrat, much research has been done into the distribution of firm size in capitalist economies. Josef Steindl’s postwar work on industrial concentration, firm growth and oligopoly, influenced as it was by the Polish Marxist Kalecki, was largely ignored by the economics mainstream. Herbert Simon produced a book on the topic during the 1970s, and it’s from that, and the recent trend for detecting power-law distributions in city sizes, etc. that most recent contributions (many of them by physicists and other non-economists, and published in *Physica A*) spring.

By all accounts, whether based on empirical observation or the behaviour hypothesized in models and simulations, the firm-size distribution conforms to a similar basic shape – skewed to the right (i.e. the mean size of firms is higher than the median), linear or slightly concave when plotted on a log-log scale, etc. Real-world results also show that the probability mass in the top end (the very largest businesses) is fatter, and that of intermediate-sized firms more slender, than would be expected from a log-normal distribution (Gibrat’s conjectured shape). These results are independent of the chosen metric of size: it holds whether firms are ranked by revenue, asset holdings or number of employees. This leads to a large number of small enterprises and a tiny number of huge companies.

Earlier this week *New Scientist *and other outlets described another paper, on a related topic and using an even trendier tool – graph theory or network analysis – to look at the direct and indirect ownership of equity between transnational corporations (TNCs). The firms making up the global capitalist economy are analysed as a structured population on a graph depicting ownership networks (a given firm being linked directly to its subsidiaries on the one hand and its shareholders on the other). The authors explore the number of connections which each firm (a vertex on the graph) has to its neighbours (other adjacent nodes) and the strength of these connections or edges, measured by the weight of shareholdings and the level of control they bestow (based on e.g. the operating revenue of the company). The relative ‘centrality’ of various companies is then compared by seeing how many vertices are directly reachable from each one and seeing for how many other vertices a given vertex lies indirectly on the shortest path via a chain of links.

The results are very interesting, though scarcely surprising:

The number of outgoing links of a node corresponds to the number of firms in which a shareholder owns shares. It is a rough measure of the portfolio diversification. The in-degree corresponds to the number of shareholders owning shares in a given firm. It can be thought of as a proxy for control fragmentation. In the TNC network, the out-degree can be approximated by a power law distribution with the exponent -2.15. The majority of the economic actors points to few others resulting in a low out-degree. At the same time, there are a few nodes with a very high out-degree (the maximum number of companies owned by a single economic actor exceeds 5000 for some financial companies).

A hub or cluster made up mostly of banks and financial institutions forms its own central sub-network or ‘clique.’ Thanks to cross-ownership of shares almost all the nodes in this core are reachable from all the other nodes, i.e. each pair of vertices is connected by an edge running in each direction:

The interest of this ranking is not that it exposes unsuspected powerful players. Instead, it shows that many of the top actors belong to the core. This means that they do not carry out their business in isolation but, on the contrary, they are tied together in an extremely entangled web of control.

As with the observed distribution of firm sizes and the observed distribution of income flows and wealth stocks between individuals, there is reason to think that this pattern is not alterable, in any meaningful sense, by regulatory reform (e.g. anti-trust law). Together all these arise as structural features of advanced market economies: with maturity, said Steindl, comes stagnation and centralization. Over time the increasing magnitude of fixed capital (buildings and equipment) required by production units sets a ceiling to the rate of firm entry; it also brings economies of scale and reliance on external borrowing to fund new investment; but exponential growth of the capital stock alongside a stabilising workforce brings a declining rate of return on investment. This in turn leads to an excess of savings flowing into the capital market relative to the amount withdrawn through equity issuance by industrial and commercial firms. Thanks to this, and over time, we would expect to see emerge, at the head of the social order, a small population of large rentiers (banks, pension and mutual funds, private equity and life insurance companies), a large number of small- and medium-sized firms with liquidity problems, etc.