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The point is there are a lot of people in this position.

Even if there are, that's not going to lower the unemployment rate significantly; it's more likely to raise it. Say there are 100 million people in the labor force, of which 4 million don't have jobs. The unemployment rate is 4% (4/100). Now 10 million previously retired people suddenly realize that they need to go back to work, and they all get hired instantly. The unemployment rate falls to 4/110=3.6%, not a huge change. And in reality all of those people won't get jobs immediately, and while they're looking they're counted as unemployed. If 1 million out of the 10 million former retirees haven't found jobs yet, the unemployment rate rises to 5/110=4.5%.




What is the mechanism to identify and count people who were voluntarily retired but now wish they could find a job but can’t?


If they're actively looking for work, then they're unemployed. If they're not looking for work because they assume there are no opportunities, they're "discouraged workers": https://www.investopedia.com/terms/d/discouraged_worker.asp. Discouraged workers aren't counted in the primary unemployment rate, but are counted in U4 and U6, which are also historically low; see https://fred.stlouisfed.org/series/U4RATE and https://fred.stlouisfed.org/series/U6RATE.




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