Natalie Angier has an excellent article on back-of-the-envelope calculations. This is one of the main skills that we try to teach in Frontiers. I think her article describes the process very nicely!
This skill is very empowering once you’ve gotten some practice with it. In some ways, it gives a certain illusion of control that probably isn’t warranted, but on the other hand, it can come in very handy and I would consider it as a sort of extension of knowing our world’s “geography”. Or maybe, it’s more like having some basic skills in navigation.
Now, one might ask, why can’t I just look facts up online or using some reference and that’s that. Well, you can, but isn’t it a good idea to have some way to double-check that what you are reading about makes sense?
For example, recently there has been lots of news about the economy. We’ve been reading that the so called GDP of the US is something like 14 trillion dollars. Can we try to understand how a number like this comes about? Let’s do a back-of-the-envelope calculation.
The first step is to figure out what is meant by GDP–what are the components that go into it. Well, it is the economic output of the entire US economy in one year, measured in dollars (over a year). How do these dollars get split up?
The most obvious pieces are the amount of money that the citizenry makes (i.e. the total amount of the annual salaries of everyone who works). Then there’s the amount that the government spends. Finally, we buy things from other countries, but we also sell things to other countries, so net exports.
How the heck to we estimate those things from scratch? Well, the key is not to get scared. One just needs to get a little creative.
There are about 300 million people in the US. Let’s say 200 million work. Suppose that the average salary is about 50,000 dollars a year. This comes out to about 10 trillion dollars.
Actually, this 10 trillion dollars component can be refined a bit. We know that the government taxes us, so people don’t get to keep their full salaries. Let’s say about a quarter of the money goes to taxes. That’s 2.5 trillion in taxes to the government. We also know that the government spends more than it makes in taxes! Over the last several years, we’ve run up budget deficits of 500 billion dollars or so. So this means that the government plus citizenry piece is about 11 trillion dollars.
Net exports gets a bit trickier. First, let’s ask how much income people don’t spend (i.e. how much do we save?). It’s a well known fact that for a while now, people in the US weren’t saving very much at all. That means that all the income was spent on stuff. Let’s break this problem up a bit: what is the average percentage of income spent on food? Perhaps 10%? Most of this is food that gets made here. Next, what percent on services and such things (school, etcetera), perhaps 30%? I don’t know–let’s go with that. In fact, let’s round up and say that between food and services, 50% of the income is locked up here in the US. Now let’s split the difference and say 25% of the other income is spent on stuff from here, while 25 % is spent on stuff from abroad.
That would mean that we spend about 3750 billion on imports. In fact, let’s just round this up to 400 billion dollars for simplicity.
We certainly do not export more than something on the order of 100’s of billions of dollars. This means that the net exports component of GDP will be a decimal place compared to what we’ve already estimated. So in fact, I will argue that for the purpose of back-of-the-envelope, we neglect it.
This means that our BOE calculation tells us that GDP is roughly 11 trillion–not bad, certainly in what I would consider the ballpark.
We can do more with this number. We’ve been reading reports of terrible job losses–it seems about 600,000 jobs a month. At this rate, we will lose about 7 million jobs over the whole year. In fact, let’s be “pessimistic” and put the figure up at 10 million jobs.
Unemployment insurance generally pays a fraction of people’s usual salaries. Let’s assume it’s about half. This means that personal income falls by about 250 billion dollars. Out of the 11 trillion dollars we estimated before, this amounts to a 2% drop in the GDP.
Let’s check these estimates: if you go to the Bureau of Labor Statistics, you can look up the different components of GDP that they gather up. They find that people’s private consumption during 2008 was about 10 trillion dollars, while people’s private investment was about 2 trillion, giving 12 trillion dollars for people’s overall personal income. We estimated 10 trillion before taxes and about 7.5 trillion after (so, clearly somewhat off, but not catastrophically). They found that the government spent about 2.8 trillion dollars, we estimated 3 trillion dollars (not bad!). Net exports were apparently about -600 billion or so. We said that we’d neglect this since it would alter things fractionally and we didn’t care about the fraction.
BOE’s are a really useful tool for cross-checking facts and more importantly, giving us a sense of place in the world, in a manner analogous to knowing where different important places in the world are relative to one-another, and relative to ourselves.