There was a bit of interesting news that came out Friday–the nations unemployment rate actually declined, from 9.5% to 9.4%. This is true despite the fact that there was a net loss of jobs of 247,000 (see the NY Times article). How could this happen?

Well, the unemployment rate is calculated by taking the number unemployed and dividing by the labor force: Unemployment Rate= Number Unemployed / Labor Force.

The numerator in the equation, Number Unemployed, is defined as the number of people not employed minus anyone who hasn’t looked for a job in the last 4 weeks. The denominator of the equation, Labor Force, is defined as the Number Unemployed plus the number of people currently working (either full or part-time).

Thus, if people give up (and giving up is defined as not looking for the last 4 weeks), they are no longer counted in either the numerator or denominator of the equation. And that is exactly what happened between June and July of this year. According to the BLS (bureau of labor statistics), 637,000 people left the labor force between June and July. Thus, even though the number of people employed fell (by a seasonally adjusted 155,000), the unemployment rate also fell, because the number of people looking for work fell also (267,000). The net result was a drop in unemployment even though fewer people were working and more people lost jobs than found jobs.

A note about the math. At first blush, you may wonder whether it matters, since the people not looking are removed both from the numerator (Number Unemployed) and denominator (Labor Force). But mathematically, it does matter. Suppose we have a ratio 2/10, which equals 20%. Subtract 1 from the numerator and 1 from the denominator and you have 1/9, which equals 11.1%. Thus we subtracted the same number from the numerator and denominator but we did not end up with the same 20%. Instead we ended up with far less (11.1%).

The general rule is that the ratio falls when subtracting the same number from the numerator and denominator as long as the ratio is less than 1. So, 2/10>1/9 but 20/10

I would guess that the labor force drop-offs would be far higher during deeper recessions where many despair of getting work or decide to take a break from their search, and this guess is borne out by recent information on the BLS site, which cites the increase in discouraged workers this last year: “Among the marginally attached, there were 796,000 discouraged workers in July, up by 335,000 over the past 12 months. (The data are not seasonally adjusted.) Discouraged workers are persons not currently looking for work because they believe no jobs are available for them.”

This NY Times chart of unemployment uses a more reasonable definition and shows unemployment far higher than the official 9.4% rate. It includes all those who have looked for a job in the past year as well as part-time workers who want full-time work as part of the unemployed, and the unemployment rate is between 10 and 20%, depending on the state.