We propose a novel framework that integrates the "task approach" for a more precise production modeling into the search-and-matching model with low- and high-skilled workers, and wage setting by labor unions. We establish the relationship between task reallocation and changes in wage pressure, and examine how skill- biased technical change (SBTC) affects the task composition, wages of both skill groups, and unemployment. In contrast to the canonical model with a fixed task allocation, low-skilled workers may be harmed in terms of either lower wages or higher unemployment depending on the relative task-related productivity profile of both worker types. We calibrate the model to the US and German data for the periods 1995-2005 and 2010-2017. The simulated effects of SBTC on low-skilled unemployment are largely consistent with observed developments. For example, US low-skilled unemployment increases due to SBTC in the earlier period and decreases after 2010.