THIS much can be said for Bill de Blasio’s inauguration, which featured a concentration of left-wing agitprop unseen since the last time Pete Seeger occupied a stage alone: If the waning years of Barack Obama’s presidency are going to be defined by a liberal crusade against income inequality, there’s no more fitting place to kick it off than New York City.

It’s fitting because a glance at New York’s ever-richer 1 percent, its priced-out middle class and its majestic skyscrapers soaring above pockets of squalor makes it easy enough to understand left-wing populism’s appeal.

But it’s fitting, as well, because New York also illustrates the tensions that make the war on inequality hard to wage, and suggests reasons to question whether it’s actually worth fighting in the first place.

Those tensions start with the fact that despite a run of non-Democratic mayors, the five boroughs have hardly been a laboratory for Social Darwinists in the last two decades. Instead, de Blasio’s “tale of two cities,” one ever-richer and one struggling to keep up, has been unspooling in a liberal metropolis in a liberal state surrounded by a mostly liberal region, where many obvious anti-inequality policy levers are already being pulled.

This doesn’t mean inequality is immune to policy responses, especially when you leap to the national level — a leap, of course, that liberal populists want to see de Blasio’s message make.

But the new mayor’s political coalition also provides a clue as to why a comprehensive policy response may never actually be tried. In his primary upset, de Blasio enjoyed strong backing from the city’s college-educated upper middle class. He even did slightly better among voters making between $100,000 and $200,000 than he did among the poor.

In a way, this shows the potential breadth of populism’s appeal. But while upper-middle-class voters are happy to support higher taxes on 1 percenters — not least because they’re tired of trying to compete with them for schools and real estate — they don’t necessarily want a program that would require their own taxes to rise substantially.

And this is a problem for the populist left, because to build the kind of welfare state — European, Scandinavian — that seems to really level incomes, you need lots of tax dollars from the non-rich. Yet the current Democratic coalition has been built on a promise to never raise taxes on anyone making under $250,000 ... or maybe $400,000 ... or possibly $500,000, the threshold de Blasio chose.

That promise has made it safe for many well-off voters, in New York and elsewhere, to cast votes for liberal populism. But it’s also made it impossible for the populist war on inequality to ever actually be won.

But should we even want that war to be fought? Here, too, New York’s experience raises difficult questions for egalitarians. Of all the arguments for reducing inequality, the most potent is the claim that a more unequal society is one with fewer opportunities to rise, and that a hardening of class lines in America is intimately connected to growing fortunes at the top.

This makes some intuitive sense, and there is international data — dubbed “the Great Gatsby curve” by the economist Alan Krueger — suggesting a link between inequality and immobility. But within the United States, that link turns out to be much less readily apparent.

Using data from an ambitious research project on social mobility, the Manhattan Institute’s Scott Winship and the Heritage Foundation’s Donald Schneider recently tried to recreate the “Gatsby curve” for U.S. job markets. Instead, they found little-to-no correlation between inequality and mobility across different regions of the country.

And New York illustrates their point, because the city’s extreme income inequality hasn’t led to extreme immobility. In fact, compared with nationwide trends, New Yorkers born into poverty have an above-average chance of rising into the middle class. (And New Yorkers born into affluence have an above-average chance of dropping to the bottom.)Now it’s true that whatever the link between mobility and equality, there are potential policy moves — an expansion of housing stock, for instance, to make expensive cities more affordable — that would probably address both issues at once.

De Blasio’s signature proposal, universal pre-K, is a more ambiguous case. Most research indicates that early childhood education doesn’t have the benefits to children’s prospects that its advocates suggest. But it’s possible the program could increase the mobility of parents, by lowering costs and stress for two-earner and single-parent households.

But there’s also a pessimistic scenario, in which the growing cost of New York’s existing welfare state means that de Blasio’s crusade ultimately just devolves into interest-group featherbedding, in which the rich are squeezed to benefit a well-compensated public sector and preserve bureaucracies that ought to be reformed.

And that outcome — a populism that marginally inconveniences the richest without meaningfully changing life for anyone else — would be less a model for the post-Obama Democrats than a cautionary tale.