A private bad is a commodity that causes its owner disutility. This article studies the bilateral exchange of a bad for a good that provides utility. Considering the price of exchange to be fixed, we first characterize the first-best choice of a single agent and study its properties, then investigate the equilibrium strategies of the two-player game in cooperative and non-cooperative settings. The non-cooperative solution is characterized by the normalized equilibrium à la Rosen (1965). In the equilibrium, the agents are assigned an exogenous r-weight, and their weighted dissatisfactions resulting from the exchange are equalized. We show the relationships between the first-best, cooperative and non-cooperative solutions. We then study market-based policy instruments such as taxes or subsidies that aim to direct agents to the cooperative outcome.