How much would fertility rate need to drop for the world to stabilise at 2 billion people in 200 years time? Would the world with this amount of people be vastly more peaceful and prosperous?
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Wealth and Prosperity Cannot be Predicted This Way
I can kinda see your logic "if there were less people that means more resources for everyone!" Problem is, that simply isn't the case. As a population shrinks GDP and industrial output shrink with it proportionally. In addition, what is considered valuable and what measure by which prosperity is determined changes with cultures over time. Ultimately though, it is simply not possible to determine how wealthy or prosperous a culture will be by simply knowing it's population.
Propensity for War is Not Intrinsically Related to Population
Again, I see your logic here. "More space, more resources, less people, less reason to fight!" If I pointed out that the last time we had a World War the population was about 2 billion it might hamper this idea. Additionally, despite our population exploding like never before in human history, wars are becoming both less numerous and less violent. The last decades since WW2 ended have actually been far less violent and warlike than at any time in the last several centuries despite our population explosion. But Again, much like economics, We cannot possibly predict how warlike or peaceful a culture will be by simply knowing the population.
I'm too Dumb To Calculate Population Decline
Frankly, calculating birth rates, death rates, and how long it would take is some math that is a bit beyond me, so I will have to pass on that section of the answer. I do not math goodly. Not even a little bit.