The algebra of iTunes by Philip Elmer-DeWitt @FortuneMagazine June 15, 2013, 4:30 PM EDT E-mail Tweet Facebook Google Plus Linkedin Share icons Source: Asymco.com FORTUNE — In a post titled “What’s an iTunes user worth?” Asymco’s Horace Dediu used the number of iTunes accounts Apple AAPL CEO Tim Cook announced Monday — 575 million — to draw the graph at right. It shows user accounts growing nearly six fold since 2009 — a growth rate of nearly half a million per day, or 44% compounded annually. He also calculated iTunes revenue per user over the same four years and found it had fallen from nearly $74 per year to less than $40. “One would expect such a decline as a user base expands,” he wrote, “and this approximate 50% decline in revenues per account seems reasonable for a six-fold base increase.” Really? I’m pretty sure that I’m spending a lot more on Apple stuff these days than I was four years ago. So I asked Dediu the dumb question: Why does a 50% decline in revenues per account seem reasonable for a six-fold base increase? Click to enlarge. “I thought it might have been obvious,” he replied. “but the latest 500 million users are not the same as the first 100 million. Namely, they are less likely to spend on media (or anything else) because they are less wealthy or live in countries which don’t have the same patterns of consumption. The first 100 million iTunes users were in the U.S. and Europe. The latest 500 million are more likely to be in China, Asia the Middle East and Eastern Europe.” OK. That makes sense. What makes less sense is the precipitous decline in another chart in the same post, reproduced below. Source: Asymco.com “It shows that for a number of years,” Dediu explains, “Apple’s users were valued (implicitly by the stock market) as likely to create a net present value of about $1200 in earnings. The current value is about a third of that, or $440 in earnings. Today’s expectation is therefore that each current customer will buy the equivalent of 1.8 iPhones. And nothing more, ever. A few months ago it was expected that each customer would buy three times as much. “Although the rate of spending of each customer is decreasing, the number of new customers is increasing more rapidly. As this is happening the company’s equity is being priced such that an Apple user is considered less than half as valuable as she used to be—from 3.2x revenues to 1.5x revenues. This change in user value has been quite abrupt, happening only within the last nine months. What could justify a drop in user value would be a drop in customer satisfaction or loyalty. So far, I have seen no evidence of this happening.” For a lively discussion about why the market values iTunes users the way it does, go to Asymco.com.