As discussed previously here, one good reason for prolonging the duration of unemployment insurance insurance coverage in a deep recession like the last one is that it would be unfair to expect from the unemployed workers to find as easily a job as in normal circumstances. Or, in other words, even if they apply the same work effort, they have a smaller chance of finding a job given the labor market tension and should be allowed to be protected for a longer time.
Alan Krueger and Andreas Mueller study what happens to search effort when there is a smaller change of finding a job. Specifically, they interviewed several thousand unemployed weekly during the last recession about their reservation wage and their search effort. They find that the reservation wage is essentially constant, expect for older workers with sufficient cash reserves, but workers are willing to go below that reservation wage for part-time work. The time devoted to search, however, dips quickly over the unemployment spell. That should not be surprising given how little time people spend looking for a job (from a study by the same authors). What is more interesting is how all this compares to a normal recession. Unfortunately, Krueger and Mueller offer no discussion in this regard.