Postage stamps and other big numbers

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A continuing theme of this blog is that humans have a difficult time grasping some basic math, even if their futures depend on it. A combination of “scare quote” stories I have read recently about the size of the national debt as well as discovering the new price of first-class postage stamps (now 55 cents) set my mind on the relationship between the two.

The universal postage stamp

My first observation is that quoting a very large number about something affecting the entire country can be much scarier than it needs to be if we were to place the number in perspective. For this perspective, consider what it would cost just for the first class postage stamps required to send a letter to every person in the United States. You can look up the math, but by my calculation this number is just a bit under $200 million. So we will round this and call the amount of $200 million a universal postage stamp.

Now of course nobody is ever going to send a first-class letter to every person in the country, but my point is that whenever you attempt any governmental action that effects a large proportion of the country’s population, no matter how trivial, you can’t do much of anything for under $200 million. And thus five “universal postage stamps” worth of services affecting everyone in the United States already gets us to $1 billion.

We are a big country, and anything we do for each other as part of “the common good,” or any amount that we tax each other to pay for it, no matter how small, quickly adds up to big bucks in the billion-dollar-range. If you think you can do government for the common good on the cheap then you don’t know math. [1]

We have trouble thinking straight when millions turn into trillions

Somewhat conversely, humans also tend to “compress” differences between numbers at the high end, perceiving less of a difference when the amounts are, say, in the billions of dollars than when they are in the millions, despite there being a 1000-to-1 difference between the two for starters, and another 1000-to-1 on top of that to get to trillions.

For instance, roughly one-half of the households in the country (just over 60 million households) have a net worth below $100,000. Just over 10% of households have a net worth of 10 times that, or $1,000,000. That is still a lot of households (about 14 million of them), but in many parts of the country the ownership of a middle-class house and an insufficient 401(k) retirement account will get the household to that million-dollar level. When we go up to the top 1% of American households, we reach the $10,000,000 net worth level.

When we see this in graph form, the “orders of magnitude” (powers of ten) differences come more into focus. Here is that same data, showing millions of cumulative U.S. households on the horizontal axis. That bottom half of households barely register when we look at the difference between millionaires and billionaires:

Household Net Worth 2016

On the very edge of that graph, Amazon head Jeff Bezos has personal wealth that has likely exceeded $120 billion. At the scale the above graph is likely appearing on your screen, that would place Bezos’ “dot” over 1600 feet in the air above your roof (about 1/3rd of a mile). To mentally grasp that difference in wealth, it would take a city of millionaires the size of Columbia, Missouri, to equal Bezos’ wealth.

A likely reason why these huge numbers, such as the national debt going up to $22 trillion (that’s 22 million millionaires) are so hard to grasp is that humans naturally think linearly in terms of size and distance when the differences are instead often exponential. If we were to take that graph above and display net worth in “logarithmic” units (that is, adjusting for that natural exponential curve), the graph looks much more linear. Note that every line on the vertical axis here is a power of ten:

Household Net Worth Logarithmic

Let me suggest that one reason we have problems with “really big” numbers is that our brain is more likely “calculating” using the above graph, as it does something similar with the pitch and loudness of sound, as well as the intensity of light. [2]

While, as noted above, a billion dollars might not be as much as you think it is when it comes to total spending in the U.S, $22 trillion dollars of debt is more like the high end of our graph, with our “little” billions getting out of control.

And also when things get very small

Finally, we have a similar problem of understanding differences when distances and probabilities get very small. For instance, going back to our “universal postage stamp” example, what if we did send a letter to every one of the over 300 million people currently living in the United States, but just one of those letters contained a check for $100 million? How much are you willing to spend on the faint hope that this one letter with a check in it just might come to your house?

These are the approximate odds of winning either of the multi-state Mega Millions or Powerball lotteries. But we pay a hundreds of millions of dollars weekly to support these lotteries because just maybe we might be the one to receive that letter. And because of a mathematical principle called “the law of large numbers,” somebody likely will receive that winning “letter.” It just won’t be you. [3]


Notes:

  1. I looked at the effect of millions of very small business transactions, and how they also add up to big dollars, in an earlier series of posts about “penny-sucking economics.”
  2. I wrote a series last year about how, while we perceive sound and light differences as linear in nature even when differences are really exponential in nature. Our brains are really using logarithmic math to make the calculations.
  3. For more on this concept, see “Why there is always a winner, but it’s probably not you.”

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