> But why would you control for population?
Because the atmosphere does not care about arbitrary political boundaries. If we take whatever total amount of CO2 emissions per year we decide the world needs to stay under and divide that up giving each country an equal share you get an unstable result.
For example let's say we agree on 4 x 10^10 tons of CO2/year. There are around 200 countries, so each country gets 2 x 10^8 tons per year.
Under that each person in Liechtenstein could live a lifestyle that required 5 000 tons of CO2/year.
In the US each person could only live a 0.6 ton CO2 lifestyle, which would require each person on average to cut their CO2 by a factor of 25. That's around what they use in Zimbabwe today.
In China each person could only live a live a 0.14 ton CO2 lifestyle, and would have to cut emissions by a factor of almost 60 to do that. That's around what Ugandans use today.
But wait...what if the US holds a constitutional convention and reorganizes? It could dissolve and each of its 50 states could become an independent country, and those countries could form a strong trade and defense alliance by treaty to make it so that they effectively share the CO2 shares.
Under that the world has around 250 countries instead of 200, so the per country share drops to 1.6 x 10^8 tons per year, but the American Union collection of 50 countries together has shares totaling nearly 8 x 10^9 tons. The people of the American Union can now enjoy a 24 ton lifestyle which is about 60% more than they have now.
China could do the same with their 33 provinces. In that world the share per country is 1.4 x 10^8 tons and the 33 countries that form the Chinese Union would have 33 shares giving them 4.6 x 10^9 tons, or 3.3 tons per person.
But why stop there? China is pretty good at tackling big things, so instead of turning each province into a separate country and putting together a 33 country Chinese Union, why not make each prefecture a country and put together a 333 country Chinese Union?
In that world, with ~200 countries not in the American Union or Chinese Union, 50 American Union countries, and 333 Chinese Union countries the per country share is 6.3 x 10^7 tons. The Chinese Union would have 333 of those shares giving them 2.1 x 10^10 tons or 15 tons per person which is a current US-level amount.
The American Union would have 50 shares giving 3 x 10^9 tons or 9.6 tons per person.
The EU would have 3.8 tons per person.
Liechtenstein would have to make do with 1600 tons per person.
In this system any country can get a bigger share of the world's CO2 emissions budget by splitting and forming alliances among the newly formed countries.
The limit of this would be a world of thousands or even millions of micro-countries, and where the CO2 allowance per person is about the same in all of them.
That's why mathematically per capita is the correct way to allocate.