The Bureau of Labor Statistics, which produces the official unemployment rate, tries to follow discouraged workers by defining them as those who say that they are not looking for work is because of poor economic conditions. The BLS includes these people in an alternative unemployment rate called U-4. In December, U-4 was 9.1 percent, meaning that discouraged workers accounted for 0.6 percent of the labor force that includes the officially discouraged.
If this rate of discouragement seems a bit low given that we are entering the fifth year since the start of the Great Recession, you are right. This is becauses people who have given up looking for work but have not looked for work in over a year are not counted as being officially discouraged. The BLS doesn't give a name to this group who are too discouraged to be a discouraged workers. Perhaps the BLS needs a new category, such as "despondent workers". At any rate, regardless of how long they have been discouraged, the weak economy is responsible for these people not being in the labor market, so an accurate measure of the weakness of the labor market needs to account for them.
I have just finished a paper that describes a way to do this by using pre-recession labor-force participation rates as an estimate of what labor-force participation would have been if economic conditions were normal. After adjusting for demographic changes since 2007, my estimate of the true unemployment rate for the U.S. in December 2011 is 10.2 percent. The figure below compares the official unemployment with my measure of true unemployment.
Notice how the description of the recovery differs when looking at the true unemployment rate rather than the official rate. According to the true unemployment rate, there was effectively no improvement in unemployment between October 2009 and June 2011. The most sustained recovery was in the last half of 2011 as the true unemployment rate fell from 10.9 percent to 10.1 percent.
I should note that the idea of using labor-force participation rates to estimate the number of discouraged workers did not originate with me. I have seen a couple of people, as reported in their blogs, doing something very similar. But for the life of me, I can't remember who they were. They know who they are, however, and they deserve to be acknowledged.
