The Law and Large Numbers

Human beings are not equipped for dealing with large numbers. Honestly, 7 thousand, 7 million, 7 billion and 7 trillion all register about the same in my mind, namely 7 big. Unfortunately, there is a world a different between each of these, three whole orders of magnitude, a thousand, the difference between lifting me and a US quarter.

This lack of respect for orders of magnitude has really been rearing its head recently with most of the political discussions surrounding the US budget.

Turns out the US Budget is really large. In 2010 it weighed in at $3.55 trillion. Thats big. Really big. So big that I can't fathom it.

Without getting too political, there has been a site going around recently; the You Cut program, which invites public suggestions for cuts to be made to the budget to try and fix the deficit. Now, personally, I believe we ought to do something about the deficit. To this end, I think it is useful to point out the scales involved. In particular, the link I gave above is to one of the suggested cuts: federal funding of NPR (Disclaimer alert: I love NPR), which weighs in at 7 million dollars.

Seven million dollars is a lot of money. A lot of money, more than I can imagine having personally. But to suggest that a 7 million dollar cut is any sort of progress towards solving a $1.2 trillion dollar deficit is a little amusing. As a fraction, this comes out to

\[ \frac{ 7 \text{ million} }{ 3.55 \text{ trillion} } = 2 \times 10^{-6} \]

Two parts in a million. To give a sense of scale to this, the gravitational influence of the moon on my weight is:

\[ \frac{ \frac{ G M_{\text{moon}} }{ R_{\text{earth-moon}}^2 } }{ 10 \text{ m/s}^2 } = 3 \times 10^{-6} \]
Three parts in a million. So, suggesting that you have made real gains in reducing the US budget by cutting federal funding for NPR is as silly as suggesting that if I want to lose weight, my first concern should be the current tides.

[I want to point out that I don't really mean to get too political, and that I've noticed both parties pulling these kinds of numbers tricks.]

So, wanting to get a little better understanding of the numbers at stake, I collected some data (all from the 2010 budget). My goal is to attempt to represent how the US government spends its money.

Before I begin I need to plug two great tools towards this end: Here the NYTimes graphically represents government spending, helping to give a sense of scale to different categories. Here the NYTimes lets you try and balance the budget, not only for next year but down the line, letting you choose from a wide array of proposed changes.


The Data

So, below is a list of some of the relevant numbers I could thing of, and some reference numbers (trillion, billion, million), as well as the US debt, and total deficit. In addition to just reporting the numbers (which you can find anywhere), in the second column I give the fraction of total spending in scientific units.

Name	$	Fraction
US Debt	13.8 trillion	3.9
Total Spending	3.55 trillion	1.0
Budget Deficit	1.17 trillion	3.3E-1
1 trillion	1 trillion	2.8E-1
SS / Def / MM	730 billion	2.1E-1
Education	93 billion	2.6E-2
Taxes (250G+)	54 billion	1.5E-2
Science	31 billion	8.7E-3
1 billion	1 billion	2.8E-4
NPR	7 million	2.0E-6
1 million	1 million	2.8E-7

So here, SS / Def / MM means each of Social Security, Defense spending and Medicare and Medicaid, which each comes in at about the same per program. We spend 730 billion on each of these. To first glance, these three programs are how the US spends money, each of these coming in at 21%. Notice also just how large the deficit is, coming in at 33% of total spending. The rather cryptic "Taxes (250G+)" is how much money would be saved by letting the Bush tax cuts expire for those making more than 250,000 a year, a currently hotly debated topic. Notice that it would only ease the burden by 15%. So this alone would only cut the budget deficit in half.

The above table shows the power of scientific notation. (Even though I had to use the ugly "E" notation). A number like "2.8E-4" is really 2.9 x 10^(-4).

But honestly, even a list like this doesn't really make an impact for me. So, I thought of a couple other ways to represent the same numbers, scaling them to some 'big' things I can conceive of:

Barry's Budget

Now, this might not be a fair comparison, but let's scale down the US budget to sizes that a person can understand. By scaling by a factor of $100 million, we end up with the story of my friend Barry.

Name	$	Fraction	Barry
US Debt	13.8 trillion	3.9	$138,495.50
Total Spending	3.55 trillion	1.0	$35,500.00
Budget Deficit	1.17 trillion	3.3E-1	$11,690.0
1 trillion	1 trillion	2.8E-1	$10,000.00
SS / Def / MM	730 billion	2.1E-1	$7,296.67
Education	93 billion	2.6E-2	$930.00
Taxes (250G+)	54 billion	1.5E-2	$540.00
Science	31 billion	8.7E-3	$310.00
1 billion	1 billion	2.8E-4	$10.00
NPR	7 million	2.0E-6	$0.07
1 million	1 million	2.8E-7	$0.01

As you'll notice, Barry makes $23,810.00 a year (total receipts)[he's a grad student], but spends $35,500 a year. This has created his current debt problem. Barry has $138,495 in credit card debt, and still overspends by $11,690 a year.

How does Barry spend his money? Well, every year Barry buys some $7,300 in guns, spends another $14,600 mostly taking care of his grandma. He pays a tuition of $930 a year at school, and $310 on science books. Every year he donates 7 cents to NPR.

Every million US dollars is 1 penny in Barry dollars.

Work Week

That might not have been fair. This time, lets scale the US government spending to a work week of 40 hours. I think a work week is a large amount of time that I still have a real grasp for.

Name	$	Work Time	Guess at time
US Debt	13.8 trillion	156 hours	Work month
Total Spending	3.55 trillion	40 hours	Work week
Budget Deficit	1.17 trillion	13 hours	All Mon and Tue morn
1 trillion	1 trillion	11.3 hours	two days of good work
SS / Def / MM	730 billion	8.2 hour	true work in a day
Education	93 billion	63 minutes	lunch break
Taxes (250G+)	54 billion	36 minutes	cooler chat
Science	31 billion	21 minutes	Show on Hulu
1 billion	1 billion	40 seconds	Stretching at desk
NPR	7 million	0.3 seconds	mouse click
1 million	1 million	0.04 seconds	blink

Cross Country

Now lets scale to a distance. The longest drive I've ever done is from Los Angeles, CA to Orlando, Fl, which was 2511 miles. In terms of this scale, the budget breaks down as:

Name	$	Dist
US Debt	13.8 trillion	9796 mi	NY - Sydney
Total Spending	3.55 trillion	2511 mi	LA - Orlando
Budget Deficit	1.17 trillion	827 mi	Texas
1 trillion	1 trillion	707 mi
SS / Def / MM	730 billion	516 mi	Orlando - Mobile
Education	93 billion	66 mi
Taxes (250G+)	54 billion	38 mi
Science	31 billion	22 mi	Cross town
1 billion	1 billion	0.7 mi	14 blocks
NPR	7 million	9 yards
1 million	1 million	4 feet

(So I just want to point out again, claiming that cutting NPR funding makes a dent in the US budget is similar to claiming you've moved closer to Orlando (while in LA) by crossing the room.)

The Lesson

So, without getting too political... I'd just like to point out that if our politicians are serious about solving the budget crisis, they need to stop talking about million dollar programs, and start taking about 100 billion dollar ones. The problem is that it's hard to either slash funding for large programs like defense, or social security, and it's even harder raise taxes (really at all). But if we never consider those options, we're never going to get out of the rut.

In my undergraduate physics lab, the instructor had a mantra: "A number without context is meaningless". Now, he originally meant the statement to be a lesson on how important it is to quote errors on your measurements, but I think I can adapt it to apply to giving out numbers like 7 billion without a sense of scale.