The big three cell phone carriers have a paradoxical relationship with the iPhone: each carrier needs the device to survive, but the more iPhones they sell, the more money they seem to lose. This morning Sprint released its fourth quarter results; Verizon and AT&T quarterly announcements came out last month. The reports all tell a similar story: Each company sold a lot of iPhones; each company lost money.

Here's how the numbers break down, Sprint, Verizon and AT&T each signed up 1.8 million, 4.2 million and 7.6 million iPhone users during the holiday season, respectively. And for the quarter, the companies each lost $1.3 billion, $2.02 billion and $6.68 billion, respectively. And a big part of these losses come from iPhone sales. In order to offer the phones at reasonable and competitive prices, these companies subsidize the price of phone, losing money on each sale. The idea is that they'll make up offering the devices at a lost by locking in subscribers into their services. But the front-end costs are enormous. Verizon, for example, loses an estimated $3 to $5 billion on this buyer-support system.

Phone salespeople like subscribers, and they especially like smartphone subscribers who spend more money on data plans. And life sans iPhone is worse. Take Sprint, for example, which posted a smaller loss than expected because it signed up fewer iPhone subscribers than expected, notes Reuters. But the company is not celebrating, since that means lower revenues from data plans going forward. "It's still unbelievably depressed," Roe Equity Research analyst Kevin Roe told Reuters.

Of course, the hope is that this product investment will eventually pay off. It's the cost of doing business with Apple, explained Verizon chief financial officer Fran Shammo during a conference call. Before Sprint started offering the iPhone it cited the hole in its inventory as the main reason its customers were leaving. "The No. 1 reason why customers churn off our network is 'no iPhone,' and we believe the reason why most don't try Sprint is no iPhone," said Sprint CEO Dan Hesse. Happier to have the offering, Sprint says it plans on losing money on the iPhone until 2015. Until then, these companies will continue seeing losses. Or, to make up for it, will raise cell phone plan rates, as all three companies have threatened.