Over the years, some words have established themselves at the London Mathematical Laboratory as a useful vocabulary. “Laplacing something” and “Weltschmerz” (p.32) are among these words. Another is “Democratic Domestic Product” or DDP — a humorous term, like all the others, that reflects the first response a student trained in ergodicity economics will have when confronted with Gross Domestic Product (GDP).

In 2016 Alex Adamou, and I published a paper in a journal. It went through a thorough peer-review process, as such papers do. At the end of the process it contained less of what we’d wanted to say, and more things we hadn’t wanted to say. You can read it here, or you can read on.

Well-known doubts about GDP

GDP has been criticized for many reasons. For instance, when a natural disaster strikes, or when a part of a city is destroyed with bombs, the re-building that (hopefully) follows is economic activity, which boosts GDP. In order to optimize GDP, we can just destroy stuff (up to a point).

Another angle of critique is that GDP just measures how much money people spend but not how they’re actually doing. We could all be quite “rich” in meaningless money-terms but spiritually empty, in a clockwork economy that has forgotten that money is at best a means to an end: it should enable well-being. But often it’s a simple irrelevance, and in many cases a destructive curse that distracts us from the human emptiness it helps create.

When economists are confronted with critiques of GDP, different responses are observed. Some say “yes, and we’ve been working on better measures, like the Human Development Index.” Others say “yes, GDP is a catastrophe — it was designed to measure how fast we could build tanks to help end Germany’s Nazi terror. But soon after that had been achieved, politicians started to use it as a measure of economic well-being, which it really isn’t.” This brings to mind the good advice “measure what you value because you will value what you measure.” Finally, another group of economists will insist that GDP is not a problem at all because no one uses it. I was quite surprised by this last response: love it or loathe it, as far as I can tell GDP is the headline figure of choice for the vast majority of politicians, journalists, and economic analysts.

So much for a nod to the sizeable debate around GDP. I won’t go into other people’s work any further because they’re the experts, of course. Instead, I’ll ask: what does ergodicity economics have to say about GDP?

A very simple model economy

You may recall that ergodicity economics asks the question whether what happens to the aggregate (the ensemble average) reflects what happens to the individual (over time). A powerful model for addressing this question in economics is geometric Brownian motion (something I’ve previously called the equation of life). In this model, a quantity, x, grows by a Gaussian-distributed factor, \mu dt + \sigma dW, in each small time step, dt,

(1) dx=x (\mu dt + \sigma dW).

For what follows, it’s not important whether this equation accurately describes income, but to give ourselves a concrete mental model, let’s assume that it does. Let’s also assume that there are a large number of people, N, who all receive an income that grows according to Eq. 1 — in good years it will go up for an individual, in bad years it can also go down. Finally, let’s assume that GDP at time t is just the sum of all these individual incomes at time t. We will work with GDP per capita and define

(2) \text{GDP}(t)=\frac{1}{N}\sum_i^N x_i(t).

What we read in the news so often is that GDP went up by 2.3%, and everyone cheers. But during one of the many recent British political campaigns, an observant member of the public explained to a journalist who had told him that one thing or another would be good for GDP: “yes, but that’s not my GDP.”

The journalist duly reported this, and there was much merriment about the lack of understanding of basic economics by the general public.

Ergodicity economics says: the man was not quite so wrong. The UK’s GDP is not his GDP. We’re working with GDP per capita, so this isn’t about the fact that the man doesn’t own the UK. His statement is true in a less trivial sense, too.

When we measure GDP growth (in our simple model, but essentially in real life too), we  compute the growth rate of Eq. 2,

(3) g_{\text{GDP}}=\frac{1}{\Delta t} \ln \frac{\text{GDP}(t+\Delta t)}{\text{GDP}(t)}.

GDP: one dollar, one vote

Eq.3 is the growth rate of the mean income. It has an interesting property: it’s invariant under redistribution. I can shuffle the income around the population any way I like — g_{\text{GDP}} is unaffected by that.

For example, let’s say at t=2020 everyone earns $50,000 per year, and at t=2021, everyone except one person earns nothing, with that one person earning N\times $51,500 per year, the exponential GDP growth rate would be 3% per year. The country is destroyed, cannibalism has broken out, the trees in the parks have been chopped down for fire wood. But GDP looks fine.

Why? The reason is that GDP is an ensemble average (over the finite ensemble that is the population). Ergodicity economics tells us that such averages don’t reflect what happens to the individual. What (typically) happens to the individual is reflected by the time-average growth rate, and that’s where DDP comes in.

Choosing how to average something means giving weights to chosen entities. Eq.3 gives equal weight to each dollar. It is a plutocratic average, meaning each dollar has the same power over the value of this measure.

DDP: one human, one vote

Now what if we had computed the time-average growth of income instead? In that measure, we imagine that an individual experiences in sequence all the changes in income that happen to each individual in the population.

(4) g_{\text{DDP}}=\frac{1}{N}\sum_i^N \underbrace{\frac{1}{\Delta t} \ln \frac{x_i(t+\Delta t)}{x_i(t)}}_{\text{growth of individual }i}

This procedure gives equal weight to each individual, not to each dollar. It’s the average of the individual income growth rates, not the growth rate of the average income. It has an interesting “no-person-left-behind” property. If even just one individual’s income drops to zero, the whole average is ruined, g_{\text{DDP}} \to - \infty. Clearly, this measure is not invariant under re-shuffling of income. And whereas GDP growth is a plutocratic measure of growth, DDP growth is a democratic measure: each member of the demos has the same power over the value of this measure. For a given value of GDP growth, DDP growth is higher when the less wealthy are catching up with the wealthy, and slower when the wealthy are pulling ahead.

The different statistics — GDP and DDP — are illustrated in Fig.1.

If I understood his tweet correctly, then Gabriel Zucman recently proposed to call DDP growth “people’s growth” — at least the basic idea is very similar, so I’ll post the figure from his tweet here.

As is often the case, with a little research we can relate concepts that arise naturally in ergodicity economics to concepts that exist somewhere in the economics literature. Let’s use g_{\text{DDP}} to define DDP. It is the rate at which something grows, so we can define DDP as the thing that grows at g_{\text{DDP}}:

(5) g_{\text{DDP}}=\frac{1}{\Delta t} \left[\ln \text{DDP}(t+\Delta t) - \ln \text{DDP}(t)\right]

Substituting from Eq.4 (do it — it’s a pleasing exercise!), we find that DDP is the geometric mean income,

(6) \text{DDP}=\left(\prod_i^N x_i\right)^{1/N}.

GDP, DDP, and intuitive sense for a well-known inequality measure

Under the income dynamics of Eq.1, GDP grows faster than DDP, meaning that the average income grows at a rate that’s greater than the time-average growth rate of income. Or put differently again: mean income grows faster than typical income.

That is only possible if income inequality increases: ever fewer a-typically income-rich individuals must account for the difference in growth rates as time goes by.

This in turn suggests a measure of inequality: the difference in growth rates is the growth rate of inequality:

(7) dJ=(g_{\text{GDP}}-g_{\text{DDP}})dt

Integrating and re-arranging (another satisfying exercise), we find that the inequality measure J is what’s called the mean-logarithmic deviation (MLD)

(8) J=\ln \text{GDP}-\ln \text{DDP}.

The economist Henri Theil identified this quantity as a good measure of income inequality, and it is also know as Theil’s second inequality index. Theil derived it on the basis of information theory, rather than using the dynamic arguments I have presented here.

Amartya Sen said of Theil’s work: “But the fact remains that [the Theil index] is an arbitrary formula, and [..] not a measure that is exactly overflowing with intuitive sense.”

(To be precise: Theil proposed two inequality measures; Eq.8 is his second index, whereas Sen commented on his first index, which is the same as the second except for a weighting that prevents the divergence for zero incomes).

Ergodicity economics can provide the lacking intuitive sense: this inequality measure is the difference between average and typical income. It does the right thing: Fig.3, produced by Yonatan Berman at the London Mathematical Laboratory, is a comparison between J and another commonly used measure of income inequality.

Ergodicity economics as a language

One key conflict in economic affairs is that between the individual and the collective — studying the relationship between these two perspectives means concerning oneself with the ergodicity problem. Some, it seems, believe that it is beneficial to let the individual tap into the strength of the collective; others, apparently, believe that the collective must not interfere with the uniqueness of the individual. Few believe that either extreme is desirable, and ergodicity economics provides a good language to speak about the trade-offs involved in moving towards greater or lesser collectivism.