Archive for the ‘Maths’ Category

It’s a small world

October 21, 2011

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.

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Girls + Maths ≠ Boys

November 15, 2010

One of the most embarrassing things I have to admit to on a regular basis is that I struggle with maths. I struggle so badly that I suspect I have a learning disability, although I’ve never been diagnosed and am a bit reluctant to categorize my lack of maths skills as such, mainly because it doesn’t really affect my day to day life in the same way that – for example – dyslexia or ADHD affect people living with those conditions.

A recent study I read about in today’s Age has shown that Victorian girls are consistently outperformed by boys in maths at VCE level – in fact, boys achieve better results in all mathematics subjects offered. I can only speculate as to why boys are more successful than girls in these subjects, and I certainly believe the reasons are cultural and not biological or “evolutionary,” as many might argue.  But having completed my entire education in Victorian public schools, and being one of those students who ‘slipped through the cracks’ in every one of my maths classes, I can certainly see how anyone, whether a boy or a girl, might feel completely disinclined to try and learn maths if they constantly lag behind the rest of the class and never seem to be able to catch up.

From Prep to Year 11, I was one of those students who could not catch up. This is despite being reasonably good at everything else – I usually achieved excellent marks in English and aside from maths, I never seriously struggled with any other subjects. If I didn’t have the most coherent understanding of science, I had a keen interest in it and (secretly) loved going to science classes. Mind you, once the science got super mathsy, I lost the thread entirely. Thankyou, Year 10 chemistry!

This is what constantly confuses me about my maths skills – I was a decent student up until Year 11, and an excellent student by the end of Year 12, achieving marks that put me close to the top of my year. How can a student excel at every subject she takes, but not be able to solve basic maths problems? And even now, at the age of 24 and having graduated from uni?

My experience of maths classes in primary and high school was so depressing it still upsets me to think about it. I remember crying during a maths test in Grade 2 because everyone around me was finished and I was still stuck on the third sum. I remember being repeatedly humiliated in Grade 4 while playing a ridiculous game, which involved competing with another student to be the first to answer a sum in front of the class. I was always beaten. One year,when I was about eight or nine years old, I was shunted off to the remedial class with all of the kids who struggled with everything at school (reading, writing, maths, socializing). That might have been alright if it was just a remedial maths class, but it covered everything – I could read at a very high level, had little trouble socializing and could write coherently. I didn’t understand why I had to read books with one word printed on every page when I had a ‘chapter book’ in my schoolbag. Adding insult to injury was the fact that the maths component of those remedial classes made no impact on my maths skills. I continued to struggle in the regular classroom.

Once I got to high school, I was fairly used to being slow at maths, and adopted a sullen, ‘quitter’ style attitude that was apt considering I was a teen. I continued to flounder at the bottom of the maths class, although I compensated by frequently talking back to the teacher and openly laughing at him when he came to school with an eyepatch, after what I can only assume was a cataract operation. Then he got sick and was replaced by a teacher who is now a convicted paedophile. Awesome.

In Year 10, something bizarre happened and I was put in the advanced maths class with all the girls who were good at it. I can only assume that I was placed there because I hung out with girls who were good at maths, or because I was good at everything else and they just assumed my maths marks were an anomaly. Whatever the reason, my teacher soon realized that placing me in the advanced class was a mistake, and I got relocated to the simple maths class. I still couldn’t do the work.

I try to avoid apportioning blame to schools and teachers when students underperform, or have behavioural or social problems. Unfortunately, I don’t think there is any other explanation in my instance. Half of my issues with maths stem from the way I was taught – not through discussion, practical and collective problem solving or logical analysis – but by opening a book full of numbers and symbols that meant nothing to me, and were not adequately explained in the book or by the teacher, unless you count a rehash of the same phenomenon with more bewildering numbers and symbols. If it wasn’t the textbook, it was the teacher who drew some graphs and/or triangles on the whiteboard, with the best explanation they could manage in five minutes, then problem solving from the textbook. And if it wasn’t the teacher trying her best with limited time and resources, it was my best friend writing solutions on her eraser and throwing it at me when I was nearly having a break down on the other side of the table during a test.

Once, my Year 10 maths teacher held my hand before she handed back an exam that I’d failed. I guess she felt sorry for me because she knew that I just didn’t get it, but also knew that I wasn’t stupid. This particular teacher did try really hard to help me, but to no avail – I gave up on maths after Year 11. As well intentioned as she was – and she really gave me a lot of her time – my problems were so far advanced by that stage that honestly, I needed to start again from the beginning.

Had teaching methods been more flexible and constructive when I was in primary school, rather than being based on exclusion and humiliation if you made a mistake, and had my maths teachers given a crap before Year 10, perhaps maths wouldn’t be such an issue for me now. I have no doubt that I was a ‘problem child’ with ‘special needs’ when it came to maths. I accept that I will never be as skilled with numbers and logic problems as I am with words and analysis. But it would be nice not to feel so ashamed when I can’t add up a bill in my head, or work out whether the change I’ve been given in the milk bar is correct. The culture of maths education needs to change if students with the same problems as me are to be given a chance at understanding the very basics, and it’s this same culture that needs to be adjusted if the outcomes of boys and girls are to be equalized at VCE level.