Let's say a real estate agent is laid off and, at some point, needs to start working elsewhere in the economy.
One scenario is that the former agent searches for twelve months and finds a job in the health care sector. The economy loses twelve months' worth of output from that individual. (These numbers are chosen for illustrative simplicity and they are not estimates of actual variables.)
A second scenario is that the former agent is reemployed immediately, improving the energy efficiency of school buildings, and he is paid by stimulus funds.
Two years later, that stimulus money ends. The former real estate agent then searches for twelve months and finds a job in the health care sector.
The net effect is to sub in two years of insulating work for two more years working in the medical sector, or wherever that individual ends up in the later years of his career.
Both scenarios involve the cost of twelve months unemployment and the associated foregone outputs.
If you measure the progress of the stimulus early on, it will appear to yield higher employment and gdp growth prospects. Those benefits are to some extent an illusion because you are not picking up the possibility that labor market search is postponed but not avoided.
A plausible intermediate scenario is that an economic downturn is a mix of real and weak AD factors. So, after the fiscal stimulus, and after the insulation work is over, the former real estate agent can reemploy himself in six months rather than twelve. By this point in time AD is higher (perhaps) and labor market adjustment is easier. Still, the short-run measure of stimulus benefits will be about twice as high as the actual net benefit, all long-run adjustments considered. It will look, in the short run, as if twelve months unemployment have been avoided when in fact the savings are a net of only six months unemployment avoided (toss in discount adjustments plus consider the costs of taxation and debt).
Many people argue that "we're not yet out of the water" or that we are seeing a "jobless recovery." I agree on both counts. I would stress that those arguments do not unambiguously favor the case for more stimulus. On one hand, troubled times may suggest that we can't let the economic recalculation happen all at once. On the other hand, if the labor market is still sluggish when the stimulus ends, we are just postponing search and unemployment, not much reducing it.