Many markets ban monetary transfers. Rather than exogenously imposing this constraint, we introduce discrimination-freeness as a desideratum based on egalitarian objectives. Discrimination-freeness requires that an agent's object assignment is independent of his wealth. We show that money cannot be used to Pareto-improve ordinal and money-free assignments without violating discrimination-freeness. Furthermore, if a discrimination-free assignment of objects and money is implementable then the respective object assignment is also implementable without money. Once money can be used outside a market designer's control, further restrictions than only money-freeness might be required to address discrimination concerns.