Postage due

There was a line at the Post Office window, so I went to the self-service counter, plopped my letter on the scale, and found that it weighed a whisker under two ounces. I bought stamps from the machine and stuck on a 39-cent and a 24-cent. I was just about to drop the letter in the slot when a thought struck me. I went back to the scale. Sure enough: With the stamps affixed, I was over the two-ounce limit.

I’m not going to tell you what I did next—whether or not I put an extra stamp on the envelope. That’s between me and my postmaster, and until they repeal the Fifth Amendment I have nothing more to say about it. But I will concede that my conscience may have been troubling me, because last night I dreamed of postal reform.

In my dream, the nation finally scraps the whole bizarre congeries of ad hoc step functions that currently define U.S. postage rates. Postage becomes a continuous function of a letter’s weight. (The current rate structure for domestic first-class mail appears to be a feeble attempt to approximate a simple linear function: P = 24W + 15, with the weight W in ounces and the postage P in cents.)

In the new regime we also dispense with the baffling collection of arbitrary stamp denominations. (Currently on sale at 1, 2, 3, 4, 5, 10, 23, 24, 37, 39, 48, 60, 63, 70, 75, 83, 84, 87, 100, 385, 405, 500, 1440. (It’s not in the sequence server, and please don’t put it there.)) Sweeping away all this cruft, my dream Post Office sells postage in continuous strips and sheets with a defined value per unit area. You cut off a piece of the stuff—we can call it postage-tape, or maybe stampage—exactly as large as you need to pay the tariff on a letter of any given weight.

Better still, instead of measuring postage by the area of the stamp, we can measure it by the weight of the stampage stuff. The marvelous thing about this scheme is that the postage rate becomes a dimensionless quantity. Whether you express it in grams per gram or ounces per ounce, it comes out the same. The rate is a pure number. Let’s suppose it’s r = 1/10, just so we have something definite to talk about.

Now, when I take my letter to the Post Office, if it weighs, say, 50 grams, I know that I have to apply 5 grams of postage.

But wait. Now the letter-plus-stampage weighs 55 grams, and so the correct postage is 5.5 grams. When I add another half-gram of stamp stuff, the new weight is 55.5 grams, and the correct postage amount is 5.55 grams….

You may think that this endless series of adjustments to adjustments is a drawback of my new postal pricing model. Au contraire! It is the principal advantage. The benefits extend far beyond the Postal Service and promise to transform American life and culture, and especially education.

There is a scene that plays out every day in classrooms all across the country—or so I’m told. A high school kid, bored with a lesson on the summation of series, protests bitterly: “Why do I need to know this stuff? No way am I ever going to sum an infinite series in real life.” Now we have an answer for that young nihilist. Do you want to stand in the Post Office all day with cuticle scissors, cutting ever-smaller slivers of tape as you try to approximate the postage due on a letter? Or do you want to learn once and for all that

\\sum_{n=1}^\\infty r^n = \\frac{r}{1-r}

Here is America’s last best chance to be taken seriously as an educated and cultivated society. Nobody’s going to mess with a country where you need to know a little calculus just to mail a letter.

Addendum. Toward morning my dream took a darker turn. What if postal rates keep rising? Beyond, say, r = 1?

This entry was posted in mathematics, modern life.

7 Responses to Postage due

  1. Barry Cipra says:

    “Better still, instead of measuring postage by the area of the stamp, we can measure it by the weight of the stampage stuff.”

    But how can the post office tell that the right weight has been affixed, except by measuring the area?

  2. brian says:

    Indeed. They also have to solve an inverse problem just to work how how much postage should be on the letter.

  3. Seb says:

    Nice solution. Actually, there is no need for the rate r to ever increase beyond 1 since the sum gets arbitrarily large as r tends to 1. Sleep tight!

  4. nutbearer says:

    Good stuff.

    People have trouble grasping dimensionless quantities though. You can be sure that if your scheme were implemented there’d be a name for the unit, although the name would just be an alias for “1″!

    Also, the whole infinite series thing can be short-circuited by specifying the ratio of stamp weight to original package weight. You’d first weigh it, multiply by the postage rate, and cut out that quantity of tape.

  5. Octal says:

    I think nutbearer’s idea is on the money. And as long as the postage tape is uniform density and width, it’s a simple matter to calculate how much tape is on it, subtract that from the total weight, and see whether the postage is correct. It could even be printed with some kind of pattern (I’m thinking just dots) that could be scanned–1257 dots? That’s 4.3 grams.

    Also! I’d like to point out that in your scenario, it wouldn’t matter if the postage rates keep rising: r represents the fraction of the weight of the package that must be postage. So the solution is simple: just make the “postage tape” more expensive, and r can remain constant and the sum remains small. They could change the color or something when the rate was changed.

    PS: I think that’s the best spam screener I’ve ever seen.

  6. Andrew says:

    Brilliant. Except that you’ll never be able to send a letter from the comfort of your own home, except after buying a $5.00 mail scale.

    The advantage of the current system is that you can guess the weight in your hand, and usually can intuit how much postage to put on your envelope. (But not at the thresholds — which is what happened to me this lazy Saturday morning — forcing me to put double postage on an envelope “just to be safe.”)

    Two inconveniences arise from pricing postage as a function of weight: we’re penalized for a tiny increase in weight, and we’re generally inconvenienced for having to weigh our mail before putting on postage. Okay, you avoid the former by making postage a continuous function of weight, but that increases the latter inconvenience.

    But we can entirely avoid the latter by adding this optional element: Anyone can open an account with USPS and order personalized stamps with bar codes on them. (As before, we buy stamps from the post office with our credit card … but the credit card accesses our account). We slap exactly one onto each letter, regardless of the weight of the letter, and let USPS weigh our letters and charge our accounts. (They have to weigh every letter under any system, anyways.)

    Make the bar codes invisible except under ultra-violet, to reduce fraud. And we get emailed a statement at the end of the month with a record of our account activity.

    Problems: it really has to be optional, for obvious reasons. Maybe we can have all these systems concurrently, including the current system, and let people choose based on convenience.

  7. kevin says:

    Why not have a system where you have a prepay card, filled with money that is scanned after the post office determines the weight, and the card money (“postage”) is sent directly to a postage tax account for the government?