Democratic domestic product

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.

Fig.1 Situation: each blob represents one person’s wealth, which changes over time. Analysis: on the left we carry out a GDP analysis (plutocratic) and find growth of the aggregate. On the right we carry out a DDP analysis (democratic) and find that the average over each individual’s growth is negative: except for one person, everyone loses. Summary: a graph of what just happened.


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.

Fig.2 Tweeted by Gabriel Zucman. Here, growth rates of quantiles of the income distribution are averaged. On short time scales, like a year, that’s statistically similar to identifying individuals, and people’s growth will approximate DDP growth. On time scales like 1980-2018 the correspondence is not exact because individuals move around in the distribution (people’s growth is “anonymized”).

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.

Fig.3 Inequality measure J compared to the top 1% national income share in the US (data from world inequality data base), analysis by Yonatan Berman. The two measures of inequality are in approximate agreement.

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.


24 thoughts on “Democratic domestic product

  1. Excellent explanation. I would only recommend adding a link to the definition of the Theil’s second inequality index. It is not easy to find it on the web.


    1. Thank you. Thinking in terms of growth rates is very intuitive for me, whereas regarding Theil I agree with Sen: it’s difficult to get an intuition from his angle. Is there a link you’d recommend?


  2. I did both pleasant exercises. I would also like to match your derivation and the Theil’s ones. Information theory is another powerful perspective to achieve an intuitive interpretation and a way to link the problem with other phenomena. But I’m not about if this one [1] is the correct definition of the Theil’s second inequality index.



  3. James Galbraith of U Texas austin uses Theil index. (he is the son of J Kenneth Galbraith who was at harvard–alot of academics have sort of hereditary dynasties—‘ like father, like son.). I view theil index as a form of entropy measure.

    There is alot of stuff on Theil both on the web and in books—eg Galbraith’s books on inequality. (Other people use different entropy or related measures for inequality. Gini coefficient is not sufficient–its mostly just an average. ) Renyi entropy may be better or even Tsallis,

    I use something like your equation 6 for DPP (though i haven’t published it and may not).

    (There are similar ideas to DPP like ‘ISEW’ (index of sustainable social welfare ) and GPI (genuine progress index). These ideas have been around for decades as alternatives to GDP. . Amartya Sen worked on some of these—but my impression is Sen is not very knowledgeable about math formalism. He is correct i think that these formulas are not intuitive. Also, modern measures use ‘quality of life’ or some analog–thats what i use—its a multidimensional measure. (possibly like ‘IQ’—you can be ‘smart’ in some dimensions and ‘stupid’ in others.).

    (i’ve known people who are ‘land rich’ –have title to like 400 or 3000 acres of land. but they live in shacks and have no running water or electricity,)


  4. I think you found the correct source. You may want to look at the Wikipedia article on Theil index which cites the same source by Theil (though it appears they use T_t and T_L for the 2 indexes (rather than 1 and 2) which are closely related ).


    1. I’d reccomend googling ‘A young person’s guide to the Theil Index’ (from 2000 , which is on SSRN as well as co-written by people at MIT and U Texas Austin—part of James Galbraith’s group) .

      Unfortunately some of the mathematical equations in the paper appear to be garbled by some computer error (I am not familiar with things like using Latex to write math equations using a standard comoputer, and it appears that whoever wrote that paper wasn’t either, so half the equations look like ‘ancient greek’–but you can sort of figure them out—somehow they were unable to get the various sum and logarithmic formulas used in the Theil index to be portrayed as they usually are–instead its alot of weird symbols and letters, sometimes undefined).

      That paper also goes through the difference between using the Gini Coefficient (GC )as opposed to the theil index. From a ‘growth and inequality’ perspective, both can be used to derive the dynamics of an economic systems which are described by both factors. The problem with using GC is very different unequal distributions of income in a society can have the same GC; while the Theil index can capture these differences. (Also Theil index can be used to describe dynamics of income differences and growth within a particular society—say UK (Wilkinson’s book ‘the Spirit Level’ would be an example) — as well used to describe differences between different societies (eg UK vs Germany vs Romania vs Greece–ie would it be better to have all these countries (as well as ones outide the EU) all ‘rise together’ and even ‘close the income gaps ‘ between them, or is it just as good to have some countries become very rich while others stagnate .

      In USA top 1% of incomes have increased by large percentages over last 20 or more years (Piketty) —gone from a few million $/year to many—while places in Africa incomes have also grown but from maybe 400$ /year to 800$/year.

      This does leave some things out of the picture–in Africa some societies still use alot of subsistance agriculture, and also don’t have to pay for alot of energy (eg oil) because they are in warm climates, and also do not have alot of transportation costs—some have few cars, dont do plane travel, and may use farm animals for their transportation and farmwork instead of using machinery. So, while their cash incomes are low, they have alot of ‘natural capital’. (Some people are ‘cash poor but land rich’.)

      (I’m interested in this field and spent some time trying to figure out all the various inequality indices (Gini, Theil, etc.–there are many) but I haven’t been able to find people working much on this.)


  5. Hello, Professor Peters.

    I’m amazed by ergodic economics – I’ve been studying it independently for the last 6 months. I’m doing my Master’s degree in Statistics in Brazil (using MCMC – so I love ergodicity) and I’m wondering: what is one supposed to do to pursue this area? I’m thinking about a doctorate program in Economics, but I have no idea what one should do to work, research and develop a carrer in ergodic eco.


  6. Hello, Professor Peters.

    I’m amazed by ergodic economics – I’ve been studying it independently for the last 6 months. I’m doing my Master’s degree in Statistics in Brazil (using MCMC – so I love ergodicity) and I’m wondering: what is one supposed to do to pursue this area? I’m thinking about a doctorate program in Economics, but I have no idea how can one pursue a carrer in such a quaint (and revolutionary) field as ergodic eco.


  7. Is the “mean-logarithmic deviation” basically the same as the cost of capital in finance? I am trying to wrap my head around this.


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