Napster's Math Doesn't Account for the Extinction Factor

- 2005.02.15

I did the math Napster suggests, but like all poorly trained high school students, they have set down the arithmetic without showing the formulas that explain how you know what to multiply and divide. Upon further consideration, we here at the Lite Side have done a sophisticated mathematical model based on algebra rather than on arithmetic, which has generated a nice set of equations we like to call

Napster's Math Doesn't Account for the Extinction Factor

Consider the following. A Napster user, having finally found a $100 Napster-to-go player that actually, you know, functions, spends $14.95 per month to download and listen to as many songs as they like. This could potentially be several thousand songs if they never listen to anything twice and listen to music eight hours a day (as we all know that young people are wont to do).

iTunes, on the other hand, has a higher up front cost because iPods are more expensive (let's say $295, for example) and users must pay $0.99 for songs they keep forever. A person could purchase a certain number of songs per month (call this the song rate, R) and after a while, they could accumulate enough songs to listen to a song no more frequently than once per month.

It would take a while, though, depending on your assumptions it might take as long as five to ten years of steady purchases to have enough songs on your iPod that you wouldn't need to listen to anything twice in a month.

The factor that is missing is the extinction factor, Ex. The extinction factor is a flag that equals 1 when a company is solvent and 0 when it is not.

Apple's demise has been predicted frequently for a long time now, but let's suppose it applies to Napster as well.

When Ex for Napster goes to zero (let's say in about three years), the total tunes you will have available for listening is . . . well, zero.

When Apple's Ex goes to zero (we all hope it won't, but you never can tell about the future) the total number of songs you will have is . . . not zero.

Mathematically, that looks like the following:

Consider this graph, where neither company goes out of business in the next four years:

chart 1

But suppose both companies go bankrupt and out of business in three years. Look what happens:

chart 2

Which is a fancy way of saying what everyone else has been saying: When Napster goes belly-up (not if - when) your investment is toast. Whereas even if Apple gets sold to Disney or something, you will still have your iTunes as long as your equipment holds out.

And we're not even getting into probability here - the odds of Apple going bankrupt in three years are pretty slim, whereas Napster, as a company name, has already tanked once. (Not with the same people, but, hey, when you buy damaged goods, you get the good with the bad.)

I hope this mathematical model helps you understand that your iTunes expenditure is a really good return on investment because you actually have a lot more freedom to do your own thing. You could even use iTunes for a while, then switch to another system, and you'd still own your iTunes, DRM restrictions notwithstanding (that's true for Napster, too, y'know), whereas once you're in Napster you're trapped forever unless you want to start over with another vendor.

Tonight's homework: Assume the rate of music buying is inversely proportional to the number of songs already owned (you've bought everything you like in the genre already) and directly proportional to the rate of new high-quality music production by the music industry (which must be a small, positive number . . . right? Right?) and rebuild this simulation making reasonable assumptions about what a typical person already owns.

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