Would New York's Economy Collapse If Nannies Were Paid On The Books?

I-banker blogger Mijka Samora has done the math, and realized that hiring an illegal immigrant nanny, paying her lower wages, and paying her off the books "puts a cool $16,000 in after-tax dollars in your pocket every year." This "nanny arbitrage" is epidemic because, Mijka says, of "the widespread, if misguided, notion that 'everyone else does it', and in the conflicted satisfaction of helping an illegal make a living." But even though Mijka is obviously correct that monies saved by using cheap illegal labor are funneled back into the economy, his other arguments for why the city's fiscal infrastructure might come crumbling down if labor laws were actually enforced are less buyable.

The New York economy is dependent on the perpetuation of the nanny arbitrage. If every parent decided to respect the law and to only hire a legal nanny on the books, all earners with an income between $48,000 and $72,000 would forgo the hiring of a nanny as it would be more financially advantageous to stay home with the children instead. Because this income range ($48,000 to $72,000) probably includes a very large number of parents, the stay-home decision could lead to a shortage of workers in New York City.
Not to be all Amy Sohn here or anything, but there must be some reasons besides simple cost/benefit analysis why people have jobs rather than staying home with their children, mustn't there?