The Deceptive Unemployment Rate

So we hear a lot about the unemployment rate. It is one of the most widely used macroeconomic measurements. We all know that high unemployment rates are bad, and low unemployment rates are good. Simple enough, right? What exactly is the unemployment rate, though? It a percentage that represents the number of unemployed people divided by the number of people in the labor force, or \[Unemployment \; rate=\frac{number \; unemployed}{number \; in \; labor \; force}\] times 100, technically, if you want to express it as a percentage.

Now it's important to understand precisely the definitions of these terms. The labor force is defined as anyone who is able and willing to work. that last part is important. A person has to not only be of an appropriate age and be physically/mentally able to work. He or she must also be willing to work! (This will factor in later) Those that are considered "unemployed" must be a subset of the labor force. One is considered unemployed if he or she is willing an able to work, but is unable to find a job.

Now, with this in mind, what ways is it possible for the unemployment rate to be decreased? (that is what we want to happen, right?) Well, the most obvious way for this to happen is if a larger percentage of the labor force becomes employed. In terms of our equation, that would mean that, the size of the labor force held constant, people would simply leave the numerator, thereby decreasing the value of the expression as a whole. This is probably what we usually think about when we think of a decrease in the unemployment rate.

There is actually a second way to decrease the unemployment rate, though. Let's look really carefully at our definition of what the labor force is. Remember, those in the labor force must be both able and willing to work. Let's consider someone who has been searching for a job for a long time, and has given up hope. this is what we call a "discouraged worker". It is a person who is perfectly able to work, but has stopped actively searching for a job. This person will not be recognized as part of the labor force, and consequently, will not be recognized as an unemployed person. It is true that he or she is not employed, but to be unemployed one must be in the labor force. If this person becomes a discouraged worker and leaves the labor force, he or she will be removed from both the numerator and the denominator of our original equation. It turns out that this will also reduce the value of our expression. This means that if people just stop looking for work (meaning that no one new has started working) the unemployment rate will still decrease. Crazy, huh?

Now this is all well and good, but I actually didn't know that reducing a numerator and denominator by the same amount would always reduce the value of the a given (regular) fraction. I know this is a long-known mathematical fact, but I had never come across it, so I decided to prove it to myself. Let's try to do it.

Consider a fraction \(\frac{x}{y}\) which represents the unemployment rate. To stay consistent with our earlier discussion, let's say that \(x\) represents the number of people unemployed and \(y\) represents the number of people in the labor force. Let us suppose that a certain number of people (let's say \(a\) people) become discouraged after not finding a job for a long time and decide to stop looking, and thereby leave the labor force. This means that the unemployment rate is now \[\frac{x-a}{y-a}\] What we have to do is figure out which is bigger, and which is smaller. Since we don't yet know, I will use a question mark in between the two sides until we can figure it out. Like this: \[\frac{x}{y}?\frac{x-a}{y-a}\] Well if we take this expression, and multiply the denominators on both sides of the expression we will get \[x(y-a)?y(x-a)\] Multiplying through, we get \[xy-xa?xy-ya\] We can then subtract \(xy\) from both sides of the equation and get \[-xa?-ya\] Then divide by \(-a\) to get \[x?y\] So what does this tell us? Well, we said that \(x\) represented the number of people unemployed, and \(y\) represented the number of people in the labor force, and we said earlier that the unemployed is a subset of the labor force, meaning that the number of people unemployed must be less than or equal to the number of people in the labor force. I think we can safely assume that this will be strictly less than, so this means that \[x < y\] Now all we have to do is get back to the original expression. No problem. Just do the opposite of everything we just did in reverse order. So we'll start by multiplying both sides by \(-a\). Don't forget to switch the direction of the inequality, since we are multiplying by a negative. \[-xa>-ya\] Next, add \(xy\) to both sides: \[xy-xa>xy-ya\] and factor out an \(x\) on the left side and a \(y\) on the right side: \[x(y-a)>y(x-a)\] Finally, divide by \(y\) and by \((y-a)\) to give our final expression: \[\frac{x}{y}>\frac{x-a}{y-a}\] And there you have it. If you decrease a fraction by the same amount in the numerator and the denominator, the original fraction will be greater than the new fraction (assuming that the numerator is smaller than the denominator).

So what is the moral of the story? Be careful when someone tells you with great joy that the unemployment rate has gone down. Such a decrease does not necessarily imply that more people are working. It might just as well mean that some people have given up and have stopped looking for work entirely--not exactly a reason for joy.

Now, once again, I know this is nothing new. I just didn't know it for sure myself until I sat down and proved it. I hope you enjoyed it as much as I did.

Have a magical day!