Various markets ban or heavily restrict monetary transfers. This is often motivated by moral concerns. However, it appears to be disputable whether the observed restrictions on transfers are the appropriate market design answer to these concerns. Instead of exogenously restricting transfers on a matching market, we introduce a desideratum based on fairness objectives and study its market design implications. The desideratum we concentrate on is discriminationfreeness, i.e. one’s access to certain resources is independent of one’s wealth endowment. A key assumption in our model is that preferences are not quasilinear but wealth has an impact on the willingness to pay. We show that matchings without transfers based on ordinal object rankings are at the efficient frontier of discrimination-free social choice functions. Implementable social choice functions are discrimination-free if and only if an agent’s object assignment only depends on ordinal object rankings and her money assignment is constant. If money can be used outside the market designer’s control even externality-freeness is needed: an agent’s object assignment has to be independent of other agents’ types. We discuss several applications in the context of discrimination-freeness including compensation for kidney donors.