Korea is adding a $275/month allowance for the first year of a baby's life, increasing the pregnancy medical benefit by $400 or so, and providing an $1800 baby bonus. There's also some unspecified benefits related to leave and high-parity families.
The thing to understand is that these efforts are not *actually* huge. Quoting things in won gives you big numbers so it's easy to lose sight of the fact that we're not talking about enough money to actually offset a large share of wages.
I'm a fan of baby bonuses and first-year benefits. They're probably the most bang for the buck.

But this isn't actually all that many bucks!
These benefits add up to about $5,500 in benefits. A similar program compared to average incomes in the US would be about $8,700.
That sounds like a lot!

But it's actuarially equivalent to about a $575 increase in the U.S. child tax credit. A non-trivial increase to be sure, but not like a sea-change in family policy regimes.
Frontloading it probably makes it more effective, but still this isn't the kind of thing that turns 0.9 TFR into 1.3 or whatever the government's current goal is.
Okay, I got some details on the other benefits.

They're expanding the parental leave benefit to a cap of $14,000 (combined, I think?) in parental wage replacement benefits received across 3 months. I'm not sure what the prior replacement rate was.
They're also waiving university fees for low-income families with 3 kids.
It's key to understand that the average OECD country spends 2.4% of GDP on family benefits.

Korea spends 1.5%. This program will increase it to 1.6-1.7%.

It's an improvement but it's not a radical change.
So since this plan would increase per-child benefits by about 14% of GDP per capita, we'd expect it to increase TFR, assuming a stable-birth-rate-counterfactual, from TFR of 0.92 to TFR of.... 0.93-0.98.
And that's assuming a stable counterfactual. If you assume a linear future trend, this program doesn't even necessarily fully offset the underlying trend!
Meanwhile in China, the official propagandists are worried that public officials coming out in increasingly voluble critique of family planning restraints may be going to far, so did this puff piece on the new "inclusive" approach: https://www.globaltimes.cn/content/1210053.shtml
It's pretty clear that there's a debate going on within the party elite about family policy. There are both diehards and reformists who want to keep limits, and the current regime is strongly associated with imposing new limits on minorities.
At the same time, the current regime is also clearly concerned about long-term power, and is skeptical of the reformist view that that power can be achieved purely through market reform, technological advancement, etc.
Hence the tension: they want to crack down on fertility but also raise fertility. The issue is that the diehard bureaucrats don't have any ideas on how to raise fertility; only reformists actually have a policy program here.
So you get a situation where a very conservative government is sort of eyebrow-raisingly teasing the possibility of subsidized surrogacy for gay couples, removal of benefit limitations for married people, uncapped fertility, liberalized divorce, etc.
How this is going to get settled next year is anyone's guess. Some moderate divorce reforms seem likely, and I suspect we'll see advisory guidance to localities to let unmarried women get at least some benefits.
Whether we'll get a new three child cap, or no cap at all, is anyone's guess. I'm a little skeptical of the no cap idea given what we're seeing in Xinjiang. And of course province level policy can vary widely. Maybe we get no cap nationally but a local option to retain caps.
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