This challenge, however, seemed different. Inside Amex’s plushly carpeted New York offices and between rounds of squash, the company’s leaders began asking one another: Could it be that American Express, the card that had defined ostentatious luxury and capitalist striving since the 1980s, was on the brink of becoming passé? What kinds of hoops would Amex need to jump through to attract these new hoodie-wearing moguls and young tycoons?

Was it possible — and this really gave everyone the sweats — that millennials would never be convinced that income inequality was something they should aspire to?

For more than 30 years American Express has reaped enormous profits by telling its customers that they are successful, elite, the cream of the moneyed crop — and, with mottos like “Membership Has Its Privileges,” that there’s no better way to make certain everyone knows just how special you are than by pulling an Amex out of your wallet.

And for decades, that pitch worked marvelously. American Express has become a financial behemoth with more than 109 million cardholders around the world. Nearly every year — at least until recently — more and more people paid American Express up to $7,500 for the privilege of carrying cards that are very similar to the ones Visa and MasterCard give away free.

But in the last few years, Amex’s hold on our affluent fantasies has started to wobble. Last year, for instance, the number of American Express cards in use declined by almost 18 percent, according to industry analysts. Amex’s 2016 revenue was down more than $2 billion from two years before. And the company’s relationships with Costco and JetBlue — long big sources of new cardholders — summarily ended when those firms found alternative credit card partners.

One of the big reasons for those stumbles is that other financial companies, most notably Chase and Citibank, have started beating Amex at its own game, often by hiring the same executives who built Amex. The head of Citibank’s credit card division as well as the heads of its branded cards, global rewards, customer acquisition, proprietary products and analytics all came from American Express. The woman credited with creating Chase’s Sapphire Reserve is an Amex alum, as are her boss and two top colleagues.

But there is also the problem of simple demographics. Amex, like all credit card companies, desperately needs new customers each year, preferably ones with decades of spending ahead of them. Hence the focus on millennials.