JPMorgan Chase created a sensation last year with a new credit card, the Chase Sapphire Reserve, crafted from “a proprietary mix of embedded metals.” Also sensational, and pretty crafty, was the annual fee of $450. A turnoff? So many people signed up, Chase ran out of embedded metals, and the card cost the company some profits in the fourth quarter.

Fighting back, American Express sweetened some of the perks on its own, 33-year-old premium card, the Amex Platinum. Included in the company’s latest enticement? An increase in Platinum’s annual fee from $450 to $550.

There’s something strange about the credit card business.

It’s technically possible to justify paying $450, maybe even $550, for the privilege of using a credit card. Look at the long list of perks, the points and credits toward trips and hotels, the freebies and upgrades.

Then do the math. Will you actually use enough to justify the fee -- or have the time and patience to obsess over the reward program’s complicated rules and requirements? If so, go ahead and sign up. If not, you might be better off with a simple, no-fee card, maybe one with a generous cash-back bonus or an especially low interest rate.

But that’s no fun. Part of the appeal of a premium credit card is that it’s a premium card. It might not save you a dime, but it will get you on a plane earlier or, even better, in the lounge sipping cocktails while everyone else on your flight lines up to board. It makes you feel special, just holding that solid metal rectangle in your hand. When you flash it, everyone knows you can afford it.

Of the nearly 350 million U.S. credit card accounts (up from about 275 million since 2010, according to the American Bankers Association), the vast majority of cards don’t charge any annual fee at all -- though they still bring in plenty of revenue from penalties, interest, and the interchange fees paid by merchants.

Will American Express customers pay an extra $100 to keep carrying around their Platinum cards? Each customer will have to make his or her own decision, based on rational, or irrational, considerations.

Recently, Mark Jackson was thinking about canceling his Platinum. The 27-year-old resident of Scottsdale, Ariz., realized he probably wasn’t getting the full benefit from the card’s $450 annual fee. Now that the card is an extra $100 ... he’s planning to stick with it. In the end it was hard to imagine giving up access to American Express’s airport lounges. Jackson has stopped in the Centurion lounges in Miami and San Francisco on layovers. At the lounge in Dallas, he got a free 15-minute massage.

“It’s kind of nice to pamper yourself,” said Jackson, who writes for a deals website. “It’s nice to have a place, if you have an hour to kill, to get a nice cup of coffee or a cocktail.”

American Express is betting that most of its customers act like Jackson, and not like coldly rational bargain hunters. Charlotte Fuller, a spokeswoman, points out that the Platinum fee increase is the first in 10 years. The company is hoping customers pay more attention to the new perks, which include $200 a year toward Uber rides and extra rewards for hotel and airline purchases. Platinum customers also have exclusive access to events and dinner reservations at famous restaurants. “This means more tables held every night for our card members,” Fuller said.

Premium cards don’t always make it easy to get the perks. Jackson, who doesn’t own a car, is excited about the Amex Platinum’s new $200 Uber credit, but one feature annoys him. The $200 is doled out at $15 a month (with an extra $20 in December). You use that credit each month or lose it. If Jackson doesn’t hail an Uber in April, there’s no way to claim that $15 in May or June.

“It really is going to depend on what the customer values,” said Tiffani Montez, senior analyst at Aite Group. “People are going to have to dissect the rewards structures and see if it makes sense for them.”

After a long day of travel, a $550 massage might just make sense for you.

This article was provided by Bloomberg News.