Asymptotic Pricing: Unattainable Cell Phone Rates

"...Recipriversexcluson: a number whose existence can only be defined as being anything other than itself." This fictional entity, which was invented by Douglas Adams, is at the heart of Bistromathics -- also an Adams' invention -- as described in the third book of his five-book trilogy: The Hitchhiker's Guide to the Galaxy. Although entirely fictional, the idea of a recipriversexcluson is not entirely useless to the better understanding of cell phone service pricing, as we will soon see.

Cell phone plans and their pricing can be terribly confusing and opaque. In this post I will not be attempting to cut through all that nonsense; it needs doing, but not here. Instead I will focus on one core portion of the pricing model: voice minute rates. That is, the actual amount you are paying per minute to use your cell phone. So let's drill down into that a bit further. To start, I will consider a simpler model that will exemplify the concept I want to explain: which is -- to coin a term -- asymptotic pricing.

In the past -- and perhaps at the present, but I don't actually know -- Chapter's book stores offered a discount card. How it worked is that for $25 you get 10% off on all your book purchases. The term was one year (if I recollect correctly). To many book lovers that may at first blush seem like a good deal. The first time I was offered this card by the cashier it did momentarily peak my interest. However a few moments thought was enough for me to find the flaw in the offer, so I smiled knowingly and politely declined. Let's look at how it works, with the aid of a graph.

You pay $25 up front, so you are immediately in a losing position. To dig yourself out of this hole you must buy books, lots of books. How much? Well, to break even you must purchase $250 worth of books (before sales taxes) since 10% of $250 is $25. That is, you pay $225 for the books plus $25 for the discount card, which equals the undiscounted total of $250. Of course, beyond that you do begin to see real savings. As an aside we should note that for Chapter's there is a marketing advantage in that you are motivated to buy lots of books, perhaps more than you otherwise would. If you do, they win, and if you don't, they also win. Win-win for them and lose-sorta win for you.

Regardless of how much you spend on books the one total discount you will never see is 10%. You can get arbitrarily close to that number by buying ever more books, but even if you buy all the books in existence, and even all those that existed in the past and that will exist in the future, you will never save 10%. That is because the initial outlay of $25 doomed you as surely as the poor horse whose rider dangles a carrot in front of its nose to make it run faster; the horse will not get the carrot no matter how fast and far it gallops.

In mathematics that 10% figure is an asymptote: a number that can only be reached by a limiting function, in this case with the limit taken to infinity. (There is a pale similarity to the earlier Hitchhiker's reference.) With this simple example in hand we are now ready to tackle cell phone voice minutes pricing.

Forget about data, SMS, vertical services, premium phones and other add-ons to your cell phone plan; it's only voice minutes we care about here and now. Even so, the graph (below) is more complex than the discount card example. In this hypothetical though typical pricing plan we see that the basic plan costs $30 (you can add in other fees and taxes if you wish to make it more realistic) for 150 minutes of air time. Again, we're keeping it simple, so no free weekends and so on; we'll stick with minutes that are chargable.

If you give any thought at all to the price when you are considering signing the contract, it is very likely that the one figure you will calculate is your expected price per minute by dividing the monthly fee by the number of minutes. In the present case that is $30 divided by 150 minutes, which yields $0.20/minute. However, as you can see in the graph, that value is only attained if you use precisely 150 minutes. If your usage is any other amount you will pay more. If you use less, you might pay substantially more per minute. (I've assumed a typical value of $0.35/minute for every minute you use beyond the 150 limit.) This is a bit like coin flipping: flip a fair coin 1,000 times and the likelihood of getting exactly 500 each of heads and tails is much lower than you might guess. At least if you do the flipping experiment many times the average will tend toward 500, but this is not the case with cell phones since you are relying on your (poor) ability to estimate how much you will use your phone:

Three-part tariffs optimally exploit overconfident consumers because overconfident consumers both underestimate the likelihood of very high usage, and the need to pay high overage charges, and underestimate the likelihood of very low usage, and the likelihood of not getting a refund for included “free” minutes.

The first thing to realize is that the first estimate of the per-minute price is an unrealizable optimum. Second, unlike the book card case, the $0.20 rate is not an asymptote since you can (if you're very careful) actually get that rate. However (third point), there is an asymptote in the graph, except it's not where you might have expected it to appear. The asymptote, like in the book card case, is also a limit for minutes taken to infinity. The asymptote is at $0.35, which is the overage rate.

If you think about it, that makes perfect sense. With unbounded use of your phone -- which is still very finite, coming in at 44,640 minutes in a 31-day month -- you will pay as much as $15,601.50, or $0.349495967/minute. That's pretty close to $0.35 though not equal. This limit is approached when minutes of use (MOU) is much, much greater than the 150 minutes in the basic plan.

Well, that's it. You now understand cell phone pricing just a little bit better than before. Just don't bring any of this up at parties or you will shortly find yourself with no conversational partners.