# Cumulative emissions, right and wrong

During C-ROADS development, we explored several ways of accounting for cumulative per capita emissions. One practice that seems to be widespread is to accumulate (integrate) emissions divided by population, i.e.

 cumulative emissions per cap = INTEGRAL( emissions per capita(t) ) = INTEGRAL( emissions(t)/population(t) )

This is physically meaningless. Emissions per capita is an intensive variable, and you can’t average or accumulate intensive variables in this way. It’s like averaging the temperature of a duck and a supertanker without accounting for the tankers 100,000x greater mass.

A proper thing to do is integrate emissions, then divide by population:

 cumulative emissions per cap = INTEGRAL( emissions per capita(t) ) / population

That yields a physically meaningful number, interpreted as cumulative emissions of a nation per current inhabitant. That’s a bit like per capita national debt.

A big question is what population to use as a basis. If you use current population, that’s fine, but it’s important to recognize that a comparison of per caps in 2100 between countries with slow and rapid population growth in effect gives credit for creating more people – possibly not a desirable incentive. Here are cumulative per capita emissions for developed and developing countries on a current basis, along a hypothetical 2C path with stricter commitments for developed countries:

If you don’t want your metric to credit future population growth, you can freeze population in the denominator at current levels, which looks like this:

If you wanted to ponder history, you could use a 1900 population basis:

## 2 thoughts on “Cumulative emissions, right and wrong”

1. jon

Isn’t it more natural to use INTEGRAL( emissions(t) ) / INTEGRAL(population(t)) or some reweighting that multiplies the integrands by a function of time?

1. Tom Fid Post author

We experimented with INTEGRAL( emissions(t) ) / INTEGRAL(population(t)), but it’s effectively an average rate – gigations/(person*years). What people seem to be referring to with cumulative emissions is, how much carbon debt do you have per person?

A related number that’s harder to calculate is the carbon debt of an average individual. To do that you have to account for births/deaths/migration, and decide whether today’s individuals are responsible for the emissions of their ancestors (debtor’s prison for heirs, anyone?).