Formal economics without parallel universes




Winner take all


Michael Mauboussin recently re-tweeted an article by Jason Zweig in the Wall Street Journal titled “Disturbing New Facts About American Capitalism”.

The article summarizes reports of an increasing concentration of economic power (market capitalization, profits etc) in ever smaller numbers of American companies, so-called super-star companies in their respective sectors. Well-known examples are Google (or Alphabet) and Apple. But the trend, Zweig says, is broader, also occurring in supermarkets and real estate services.

Why is this “disturbing” and what is it disturbing?

It is disturbing because the extrapolation of this trend will lead to the emergence of monopolies, which undermines competition and the functioning the market. Capitalism doesn’t work without competition.

The more interesting question from the theorists’ perspective is “what does it disturb”? Namely, what belief does it disturb? Zweig’s article summarizes this belief very nicely:

“Modern capitalism is built on the idea that as companies get big, they become fat and happy, opening themselves up to lean and hungry competitors who can underprice and overtake them.”

This means large entities grow systematically more slowly than small entities, which stabilizes the distribution of relative sizes of companies. In other words, relative company size in some mathematical model of this process would be an ergodic observable. Companies would dominate for a while and then be overtaken by competitors.

The ergodic hypothesis in economics

This belief in automatic stabilization is called the “ergodic hypothesis” in economics, and our research program questions it. To do that we have to build a mathematical model of the dynamics (in this case of company size). Then we ask two questions:

  1. Does the distribution of relative company sizes converge in the long run?
  2. What is the time scale for that convergence?

If the answer to 1. is “no” then the hypothesis is clearly invalid. If it’s yes to 1. but 100 years to 2. then the hypothesis is of little use in practice.

Personal wealth

In the study of personal wealth dynamics it is often assumed that convergence happens and is fast. This means that if no one interferes, relative wealth will settle into a stable distribution. Some people will be a little richer, some a little poorer, the wealth of some people will rise and that of others will fall, but the overall distribution won’t change.

If we believe this to be true, then laissez-fair policies seem the natural choice. Why interfere if the system will stabilize anyway? However, this is simply a belief, not some God-given truth.

In the context of personal wealth we’ve recently asked the question explicitly and addressed it empirically: are the dynamics of personal wealth well modeled by a process in which relative wealth is ergodic? The data told us no. We cannot safely assume that the distribution of personal wealth stabilizes.

Company sizes

Apparently, this is also true for the relative size of companies. The Wall Street Journal article suggests that the dynamics of relative company size are not well modeled by an ergodic process.

There are certainly many reasons for this. Zweig quotes Roni Michaely of Cornell University mentioning a decline in the enforcement of antitrust rules. This is an interesting point because it implies that without antitrust rules the system is believed to generate monopolies. Mathematically, that means the process that describes company sizes without antitrust rules is non-ergodic. More like a nuclear explosion than a gas in a box. Antitrust rules then are the policy-makers’ way of saying “we don’t believe in ergodic models of the economy.”

This may not come as a surprise: the most important model in mathematical finance, geometric Brownian motion (GBM), is non-ergodic in this sense: if lots of company sizes x_i(t) follow GBM, then the relative size of the largest x_i tends to 1, meaning the biggest company will end up with essentially all the wealth in the system, \lim_{t\to\infty}\frac{\max_i x_i(t)}{\sum_i x_i(t)}=1. A particularly important paper that introduces a helpful conceptual framework for discussing these issues is Bouchaud and Mezard 2000.

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2 responses to “Winner take all”

  1. Yonatan Berman avatar
    Yonatan Berman

    Interesting. For US commercial banks it is well supported by data that since the mid-80s their total number is gradually decreasing. The total national deposits that US community banks held declined from about 26% of all deposits in 1985 to 14% in 2003 and the total number of banks decreased from ~14000 to ~7000 during that period. Presumably a non-ergodic system. This is in line with the findings on personal wealth.
    Number of community banks declining

  2. Ole Peters avatar
    Ole Peters

    Thank you, Yonatan. This is a very suggestive figure. If I understand correctly, then the bottom panel shows the rate at which American personal wealth has been effectively reallocated in any given year (blue line) and the net change in the number of community banks (red line).

    The blue line says that total systemic wealth reallocation (including wages, dividends, investment income, taxes, benefits, inheritance… everything) has been negative since about 1982. Overall, the American economy has moved money from poor Americans to rich Americans. One effect of this is the concentration of wealth in the hands of fewer individuals.

    The red line shows a similar trend: the number of community banks has been decreasing since about 1985. Again, the effect is the concentration of resources in fewer entities.

    Both cases chime with Jason Zweig’s piece, and both cases are examples where models making the ergodic hypothesis are unrealistic.

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