We devise a Directed Technical Change (DTC) multisector Schumpeterian growth model in which both wage inequality and wage polarization are analysed. To that end, we introduced tasks in the model, some of which can be automated -replaced by robots or machines -, thus combining the DTC and task-based growth literature in an unified framework. This model produces positive relationships both (i) between the relative supply of high-skilled workers and the skill premium and (ii) between automation and wage polarization. Moreover, within the model, we analyse Lobbying as an activity that can affect the wage distribution and integrate it in the strategic interations between firms. We find that it can reduce the effects of automation on wage polarization, and through this channel possibly affecting the wage distribution without affecting the skill premium.