Gov. Andrew Cuomo and Mayor Bill de Blasio remain locked in a political and legislative tug-of-war over how to pay for expanded public pre-K education.

Cuomo wants to phase in a statewide pre-K program, but de Blasio prefers a much more expansive initiative supported by a dedicated stream of higher income taxes on NYC residents earning more than $500,000. He called this a “stable, consistent and reliable funding mechanism.”

In fact, high-income New Yorkers are probably the least stable, consistent and reliable segment of the city’s income tax base. According to state tax data, there were 49,000 city households with adjusted gross income of more than $500,000 as of 2007. They earned $135 billion that year.

Then came the Great Recession. Over the next four years, the incomes of New Yorkers earning less than $500,000 dropped by 5 percent, then bounced to pre-recession levels. But adjusted gross income in the $500,000-and-above bracket sunk nine times faster, to $75 billion by 2009. As of 2011, the most recent state data show, the adjusted gross income for this group remained $42 billion below the pre-recession level.

So, if de Blasio’s pre-K program had been in effect as of 2007, it would have lost a huge chunk of its funding in the next few years.

As it is, the highest-earning 1 percent of residents and workers generates more than 40 percent of both state and city income taxes. Cuomo himself has doubled down on the wealthy, twice extending a state income tax hike on taxpayers earning $1 million or more ($2 million for married couples filing joint returns).

Politicians can’t say they haven’t been warned about the risks of balancing budgets on a tiny pinnacle of wealthy households. In 2009, for example, a state Assembly report pointed out that New York had become heavily reliant on “inherently unstable” and “unsustainable” revenues generated by its richest residents.

The report warned that changes in compensation practices in the finance sector would slow the growth of taxable income in top brackets, which is, more or less, what happened.

That report was issued by the staff of the Assembly Ways & Means Committee, headed at the time by Dean Fuleihan — who is now de Blasio’s budget director.