If my understanding is correct, the $4K is paid after taxes and the ex-wife pays no taxes on the money received. That is, he is going to pay $48K per year in child support but has to earn maybe $70K pre tax.
Not correct, the taxes are actually paid by the receiver and not the payer. If the receiver is in a lower tax bracket, then they pay taxes at that rate. This is basically all factored in to the process of setting alimony by the courts, so pretax and post-tax don't really make any sense here unless the tax brackets are different. Note, IANAL or an accountant.
Actually, alimony is taxable to the recipient but child support is taxable to the payer.
That is, if the $4K/month is actually $3K child support and $1K alimony, the father's taxable income will be reduced by the $1K, but he's paying taxes on the rest. The ex-wife will pay taxes on the alimony received but not on the child support received.
This is one of the several reasons why women seek child support rather than alimony in divorce proceedings.