We investigate how trade unions influence the process of technological change at the workplace level. Using matched employer-employee data, comprising all Norwegian workplaces and working individuals in the period 2000-2014, we exploit exogeneous changes in the tax rules for union members to identify how changes in unionization rates affect the structural composition of occupations within workplaces. Making a distinction between routine and non-routine workers, based on their estimated probabilities of being replaced by automation technologies, we show how labor unions contribute to raising the relative wage of routine workers over non-routine workers. As routine workers on average have lower earnings than non-routine workers, unions thereby contribute to compress wages at the workplace level. The direct implication of this policy is shown to reduce the relative demand for routine workers over non-routine workers in unionized establishments. However, our results also suggest that unions influence the relative demand for routine workers, conditional on relative wages. Our findings thus give some support to bargaining theories where unions force firms off their demand curves.