Last week I thought of a more general form of the relationship between population and the consumption of resources. Preliminary testing suggests that it's much more robust than previous versions of my population-consumption model.

You may recall that I had identified what I called “transactions” as the only mechanism determining how much mass people will convert into waste each year. People extract resources from wherever they are, process them into useful forms, and exchange the results with other people. If everyone in a population conducts one transaction per year with everyone else in the population, the total number of such transactions is one-half the square of the total number of people. The average mass for each such transaction is what I call the “transaction mass.” With transactions accounting for all consumption, transaction mass included both the mass of stuff exchanged and whatever was used to do the exchange (such as fuel used in transportation).

In the new version of the model, I've redefined transaction mass as only the average amount used to perform an exchange. The majority of the total consumption is what is actually consumed by people. Each person in the population, on average, consumes an amount of mass which I call “extraction mass” (because in the simplest case each person could extract resources on their own). Total consumption is the sum of the transaction mass and the extraction mass, and per capita consumption as a function of population ends up being a straight line.

When I was assuming that total consumption varied with the square of the population in an isolated population like the Earth, all that was required to determine how it changed over time was a set of historical population numbers and a value for consumption at some point in time. As a proxy for consumption, I used the global ecological footprint, which measures the per capita ecological impact of humanity on a global scale, and the starting value was assumed to be the minimum reported for countries in 2006. I then did an elaborate curve fit of consumption, constrained so that when projected to the present, it matched the most recent measured value. Projecting consumption into the future showed that it would peak and then drop; and since it was interdependent on population, population would likewise peak and crash.

The new version of the model was inspired by an attempt to simply describe and justify the elements of the previous one, including some inconsistencies with current data that couldn't be easily explained. Specifically, recent estimates of ecological footprint show very little change over the past fifty years; and per capita world energy consumption shows the same pattern, even though population more than doubled over the same period. In contrast, the previous version of my model shows a steep change in the equivalent per capita consumption. If the trend of the data was consistent over all time, early civilization should have been consuming almost as much as we are today, and living just as long, which was clearly wrong. It was natural to assume that the flatness of the data was a historical fluke, but as I was testing my assumptions, I realized that such an explanation was unsatisfactory. Unfortunately, even with the new version the problem remained.

Then I realized that I had a way to measure per capita consumption going back much further in time than the footprint and energy data. In addition to historical estimates of population, there are also estimates of life expectancy, and I've known for a while how life expectancy and footprint are empirically related. I could therefore use life expectancy as a way of calculating footprint, just as I have used it to convert my projections of footprint into life expectancy (and, similarly, happiness).

The results were astonishing. For one thing, ten thousand years ago, the ecological footprint was one-fifth of my previous estimate of the minimum footprint. The transaction and extraction masses stayed effectively constant right until the middle of the last century. From 1950 until 1960 (the decade I was born), the transaction mass jumped by a factor of nearly 30, and then stopped changing; meanwhile, the transaction mass remained what it had always been. The footprint (and presumably all per capita consumption) looked just like a mathematical “step function,” corresponding to an almost doubling of life expectancy. The reasons for this near-discontinuity in the historical trend likely involve a combination of major advances in medicine (such as the development of antibiotics) and the widespread availability of fossil fuels and oil derivatives for nearly every purpose, not the least of which being the creation of artificial fertilizers that could immensely increase food production.

Perhaps the most important prediction of the previous version of my model was the impending crash of the world's population. I have so far been unable to find evidence for such a crash in the new version. The closest I can come to justifying such an expectation now is the existence of the step function itself, an understanding that the oil that powered it is becoming much harder to get, and the clear evidence that we have exceeded the ecological carrying capacity of the Earth and may soon reach a tipping point in Nature's ability to support us. To the extent that the previous version does an excellent job of curve-fitting population over time, and population is the main variable in the new version, it may yet prove to be accurate in at least that one regard.

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My apologies about all the back-and-forth about the statement of per capita consumption. It turns out that the first version of my post was correct (it wasn't a derivative, simply the consumption divided by population). I've removed the offending phrase and corrected my Web site.

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