08 September 2007

"exponential" decay of iPhone prices?

In yesterday's New York Times: IPhone Owners Crying Foul Over Price Cut, by Katie Hafner and Brad Stone.

As you may know, the iPhone originally cost $599 for the 8-gigabyte model and $499 for the 4-gigabyte model. The 4GB model has been discontinued, and the 8GB model has been cut to $399.

People who bought the iPhone when it came out are complaining. Why? Because they are silly. Prices on consumer electronics almost always go down. Anyone who's bought two computers in their lifetime knows this, because the second one was almost certainly cheaper and had more computing power. (I hesitate to say "faster" because newer software uses computing resources less efficiently than older software.)

The article comments on this:
The price of consumer electronics is always going down thanks to intense competition and the steady decrease of the cost of electronic parts. The pricing is largely determined by Moore’s Law, the observation made by Intel’s co-founder Gordon Moore that the number of transistors on a silicon chip doubles roughly every 18 months. Because this rate of change is described by an exponential curve, it dictates not only that prices fall, but also that they fall at an increasing rate.


Okay, I buy Moore's Law, although perhaps not with the "18 months" coefficient there; depending on who you ask you'll hear anything from 12 to 24 months. And does Moore's law apply to flash memory, which is clearly one of the larger costs involved in the iPhone? Still, let's give them Moore's law. That doesn't mean prices fall at an increasing rate. Let's say that everything involved in the iPhone is subject to Moore's law, so the price decays exponentially; then the price falls at a decreasing rate. For example, let's say that iPhone prices fall by one-third per quarter (as they have so far, although I don't see that holding up) -- then they fall from $600 to $400 in the first three months, then $400 to $267 in the next three, $267 to $178 in the next three, and so on. Each of these decreases is smaller than the one before. If you take the logarithm of the prices and then differentiate -- which has the effect of measuring the rate of price change in units of the current price -- you get a constant. It's not accelerating. More generally, the price of the object is given by P(t) = P0e-kt for some constants P and k; P'(t) is a decreasing function, and d/dt log P(t) is the constant -k.

But some parts of the iPhone are not subject to Moore's Law. The advertising budget, for instance -- there are a lot of ads. I doubt the case that it comes in is made up of transistors, either. The pay of the people in [third-world country] who put it together probably isn't decreasing.

So a better model for the price of the iPhone -- or any such object -- is probably a constant plus a decaying exponential. (The "constant" is measured in real, i. e. inflation-adjusted dollars; if you wanted to do this in nominal dollars it would be a rising exponential.) So we have

P(t) = P1 + P0e-kt

and P'(t) is still decreasing. In this case, even d/dt log P(t) is decreasing:

{d \over dt} \log P(t) = {{-P_0 k e^{-kt}} \over P_1 + P_0 e^{-kt}}

{d \over dt} \log P(t) = {{-P_0 k e^{-kt}} \over P_1 + P_0 e^{-kt}}
which isn't obviously decreasing; we could differentiate again, but it's easier to rewrite it as

{d \over dt} \log P(t) = {{-P_0 k} \over P_1 e^{kt} + P_0}

which has a constant numerator and a clearly increasing denominator; thus the rate of price decrease is actually slowing.

At least they used the word "exponential" correctly; usually the media doesn't even do that.


Patrick said...

Moore's Law was actually original formulated with a 24 month doubling period.

The doubling also applies to storage of all kinds (the most important ones being flash and hard drives); the doubling rate is faster than 18 months.

Isabel said...

I knew I'd heard that hard drives doubled at a different rate than transistor-things, but I couldn't remember which one increased faster. And flash is transistor-based, so I wasn't sure which doubling rate would apply. It doesn't matter, though; what matters is that the trend is exponential.

In terms of hard drives, if memory serves me correctly my family bought a computer with a 20 MB hard drive in 1991 (plus or minus a year). My current machine has a 120 GB hard drive and I bought it last year. That's an increase of 6,000 in 15 years, which works out to a doubling time of 14 months or so. I'm confident that the doubling time is not as long as eighteen months; that would allow for only ten doublings in that period, giving me a 20 GB hard drive currently.

John said...

People shouldn't complain that the price of the iPhone is dropping. They should complain that the iPhone's battery isn't user-replaceable. Sure, you can send it back to them, and they'll replace it in 3 days for $79.95 plus $6.95 shipping and handling. They'll even rent you a replacement for $29.95 in the meantime! But, all your data will be wiped out. Fun, fun!

Anonymous said...

People who bought the iPhone when it came out are complaining. Why? Because they are silly.

I've heard a better explanation. Why does someone buy an iPhone? There are cheaper PDA phones with better features (such as much faster internet access), so the main reason to buy an iPhone is because it is cool. A few people are content to admire its coolness in private, but many also care about the image it projects for them. They are seen (and can view themselves) as trendsetters with the hottest phone on the market. A lot of Apple advertising campaigns directly push the image that cool people use Apple products, so you should demonstrate your coolness by doing so.

What's the upshot regarding the price change? The day before the price reduction, iPhone owners looked fashionable and were widely envied. After the sudden $200 drop, they looked like suckers who overpaid by 50% to get their hands on the phone a few months before the hoi polloi. This dramatically undercuts the whole reason to have an iPhone by making the early adopters look foolish.

Of course price decreases were guaranteed to come, but Apple should have decreased them more gradually. It would also have been better to tie the decreases to the introduction of new iPhone models.