To ensure the achievement of the climate and energy policy targets and simultaneously guarantee affordability and inclusiveness, a thorough consideration of institutional settings is necessary, such as the structures of housing provision (SHP), and taking into account low-income households. This paper deals with the following research questions: (1) Is a CO₂ tax sufficient to achieve decarbonisation in the Austrian housing sector, particularly considering different SHP and low-income households? and (2) What is the impact of regulatory policy instruments like a renovation obligation in combination with a CO₂ tax, in particular on low-income households? In the first step, key institutional structures affecting housing in Austria are identified through a literature review, regulation analysis, and interviews with housing-related organizations. The second step expands the Invert/EE-Lab building stock model by integrating SHPs, household income levels and age groups. We use the model to compare scenarios with different stringencies of renovation obligation. The results show that CO₂ taxes alone do not provide sufficient incentives for building owners to renovate their properties, leading to a burden on tenants, particularly low-income households. Mandatory, well-prepared, and long-term renovation targets can help compensate for these negative effects and unlock the full potential of CO₂ taxes for decarbonisation.