South Korea is a country with a free market economy. Next to the Philippines, it is the most Christian country in Asia. In other words, South Korea is the polar opposite of Red China yet the birth rate is well under that of its huge neighbor. Though no longer Communist and under a regime that encourages families and Orthodox Christianity, Russia reports a birth rate of 1.42 children. There is something going on beyond government policy.
While there are other factors in play, one big problem is government policy. Much like in the US, the government, directly or otherwise, is penalizing and therefore discouraging marriage and having children through taxation, incentives, the ridiculous family court system, etc.
Some European countries have realized this and are now offering incentives for women to have children as well as removing penalties for men to marry.