Now the real question - if Japan is the closest thing to our high-providing benevolent future government, what do Japan's emigration rates look like?
According to the CIA World Factbook, Japan was has 0/1000 net migration. That is, people leave and arrive in precisely equal numbers.
https://www.cia.gov/library/publications/the-world-factbook/geos/ja.html
Looking at correlation of birth rate, GDP per capita in PPP, and migration rates, at the top and bottom of the world league tables, with honorable mention for Afghanistan as the highest birth rate country outside Sub-Saharan Africa
Japan Birth Rate 8.23 (220th) GDP $35,900 (36th) Migration Rate: 0.00/1000 (91st)
Singapore Birth Rate 7.91 (221th) GDP $60,800 (7th) Migration Rate: +15.80/1000 (5th)
Hong Kong Birth Rate 7.58 (224th) GDP $50,900 (14th) Migration Rate: +3.75/1000 (28th)
Monaco Birth Rate 6.79 (225th) GDP $70,600 (6th) Migration Rate: +1.93/1000 (44th)
Niger Birth Rate 46.84 (1st) GDP $800 (221st) Migration Rate: -0.60/1000 (139th)
Mali Birth Rate 46.06 (2nd) GDP $1,100 (214th) Migration Rate: -2.41/1000 (170th)
Uganda Birth Rate 44.50 (3rd) GDP $1,400 (205th) Migration Rate: -0.01/1000 (110th)
Burkina Faso Birth Rate 42.81 (4th) GDP $1,400 (204th) Migration Rate: 0.00/1000 (105th)
Zambia Birth Rate 42.79 (5th) GDP $1,700 (198th) Migration Rate: -0.75/1000 (144th)
Afghanistan Birth Rate 39.05 (12th) GDP $1,100 (216th) Migration Rate: -2.16/1000 (166th)
So rich countries with low birth rates tend to pull in extra population from less developed countries.
-- People go where the money is, unless they aren't allowed to (Japan).
-- Rich countries like Singapore are growing due to immigration, not birth rates.
If the whole world were highly developed, I think we'd have shrinking population until it reached equilibrium, probably with government encouragement to have more kids. Singapore actually has a number of policies to encourage growth of their native-born population, like housing and tax benefits, but as you can see they've had little effect on a small crowded island with lots of highly educated people.
Of course, with mass cloning and "live forever" medical technology in the far future, all bets are off, I suppose.
For those playing at home, it's interesting also to note that the low birth rate countries are extremely stable and peaceful places (no wars since WWII, low crime rates), whereas the high birth countries often have civil wars and Al-Qaeda problems now (Niger, Mali, Afghanistan) or had serious problems in the previous generation (Uganda under Idi Amin).
Of the low birth rate countries, 2 don't control their own politics (St. Pierre & Miquelon, Hong Kong), 1 is a monarchy with substantial foreign control by a neighbor (Monaco), 1 is a "semi-democracy" with elections, strong rule of law, and a very strong, competent, and successful bureaucracy, but only one party ever winning elections because it's genuinely popular (Singapore), and 1 used to be like a slightly-less organized Singapore and now has multiple parties winning elections after decades of economic stagnation (Japan).
So, yes, the future might resemble Singapore, if we're lucky. Not that different from the Federation, I think.
Or might be dystopian and resemble South Africa crossed with California, as seen in Elysium, if capitalism goes seriously wrong.