Over the past few years, the arithmetic behind Amazon Prime has become one of online shopping's most familiar math problems: Do I buy enough from Amazon to justify paying $79 per year for unlimited two-day shipping?

But this calculus could soon change. Amazon makes so much money off Prime customers, according to a new report, that the company could drop the fee by dozens of dollars and still come out ahead.

As heavily as it promotes Prime, which also comes with free Netflix-style streaming video and access to the Kindle lending library, Amazon is equally circumspect about how well the program performs. In the heated debate over whether a company with profits as meager as Amazon's deserves such a high-flying stock, that information gap leads partisans both pro and con to play Prime as a wildcard in support of their claims.

Bullish analysts at Morningstar teamed with Consumer Intelligence Research Partners (CIRP) to dig into what is known about Amazon to come up with some reasonable estimates of the numbers behind Prime. The results are startling.

Amazon started its fiscal 2012 with a little fewer than 7 million Prime members and ended with nearly 10 million, largely thanks to the free Prime promotion that comes with the purchase of the company's bestselling Kindle Fire. That increase alone represents a huge coup for Amazon, an awesome display of locking in customer loyalty. As the report points out, those millions of people spending $79 each are all incurring a major "switching cost" — in other words, since they've shelled out so much to enjoy special privileges for shopping at Amazon, they're less likely to shop elsewhere.

The growth in 2012 memberships yields an average of 7.6 million Prime members over the course of the year. In all, Amazon had about 182 million customers for the year, the report estimates. By those figures, Prime members made up about 4 percent of Amazon's total customer base.

At the same time, CIRP's market research found those Prime members annually spent more than twice as much on average ($1,224) than non-Prime customers ($505). Adding in the $79 membership fee, Prime members bring nearly $800 more in sales per customer to Amazon. After subtracting all the additional costs of doing business with Prime customers — mainly greater expenses for sales, shipping and video streaming — Morningstar analyst R.J. Hottovy found that Prime members accounted for $78 more in profit before interest and taxes per customer than non-Prime customers. That's almost exactly the cost of a Prime membership.

To put that figure in context: Amazon's average operating income last year per each of its 182 million total customers came to less than $10. In other words, every Prime member is about eight times as valuable to Amazon as a non-Prime member. Put yet another way: More than one-third of Amazon's profits before interest and taxes came from fewer than four percent of the people who buy stuff on Amazon.

"They have hit on a means of creating some very, very valuable customers," says Michael Levin, a partner at CIRP.

What's more, the report says the value of those Prime members will only increase as the efficiencies built up by Amazon through its heavy spending on distribution centers and technological infrastructure start to take hold. By 2017, the report predicts Prime will have 25 million members. As Amazon's costs fall and customer loyalty is locked in, the report's authors anticipate Prime customers will spend even more as they start thinking of Amazon not just as the place to buy higher-ticket items like electronics but their main destination for everyday goods.

"They're starting to encroach on the territory of general merchants and grocery stores," says Hottovy. "I think that's the next frontier."

If it's true that Prime threatens not just Best Buy and Barnes & Noble but also Target, Kroger and Wal-Mart, the logic of lowering Prime's price makes even more sense. The more Prime members, the more purchases made on Amazon. The trick for Amazon is finding the sweet spot between how much a Prime membership increases spending versus how much cutting the price of Prime eats into the increased profit margin. Yet even by last year's estimated figure of $78 in extra profits from Prime members, Amazon could cut the price by $50 and still make far more than what the company makes off non-Prime customers.

Already, Amazon offers Prime to college students at a discounted rate of $39 per year. That discount does double-duty by catching customers just as they're developing their adult consumer loyalties while at the same time getting them so used to Prime that they'll miss it if they give it up after graduation. It's possible Morningstar's numbers are off and Amazon is losing money on student Prime members. But such a discount also gives Amazon a chance to beta-test consumer behavior when Prime is offered for less than half its regular price. If they see increased rates of sign-ups and spending, and Morningstar's numbers are anywhere close, Amazon is just as likely to be figuring out a way to push Prime's price point down.

If Prime members are anywhere near as valuable a catch as the Morningstar report's figures make them out to be, Amazon should be trying to make that price as low as possible. Or as CIRP's Levin says: "They should be able to give Prime away for free."