As prosperity expands birthrate declines. End of discussion.
TLDR: Worrying about populations anywhere in the low billions is silly. And that’s just one planet.
Malthus was wrong objectively and qualitatively.
The food supply will increase exponentially over time along with everything else subject to the leveraging of computing power, which food is thanks to genetic engineering, robots, and fission.
Consumption of food is the hilariously inefficient process of swapping molecules to export entropy. To get some idea just how efficient, imagine if you could eat uranium for fuel. The idea that the only way to accomplish powering a human permitted by the laws of physics is a 1900s concept of farming is hilariously wrong. The myth of overpopulation is nothing but cover for the pathological greed of the 0.1% ownership caste.
But even assuming no changes to people themselves, it’s still not even that big of a problem the moment we grow a spine, or find ourselves choiceless. Trigger one good sized volcano somewhere and the volcanic winter will counter the runaway methane reaction. (CO2 is a wimp.) It may not perfectly balance but we can make a pretty safe calculation about which we’d prefer.
Or we could literally do the Matrix revolutions approach minus the permanent nano magic and create our own particulate winter on the installment plan. And that’s just one class of responses available the moment humanity decides to geoengineer willfully, as opposed to incidentally.
Humanity only needs to deploy three (or four depending on definitions) tools to turn this planet into a fully people serving ecosystem. Fission, Software, and Robotics/GE. Fission to power it, software to guide it, and Robotics/GE to actualize it.
Now, the population density of NYC, (because NYC and Texas were the base of the XKCD math) 28491 people per square mile.
So, 28491 people per square mile covering 200 million square miles leads to a population of 5 trillion 698 billion 200 million. And these are numbers generated without the downward pressure on birth rate caused by property. The final homeostatic number will likely be substantially lower.
Citations and alternate numbers.
http://message.snopes.com/showthread.php?t=51292
POP101: A primer on the future of planet Earth
Oh yeah and all this is assuming population rise isn’t already reversing. But it is.