Strategic bilateral exchange of a bad

A private bad is a commodity that causes its owner disutility. We study the bilateral exchange of a bad for a good that provides utility. Considering the exchange price to be fixed, we determine the first-best choice of a single agent and study its properties, then investigate the equilibrium strategies of the two-player game. We characterize the normalized equilibrium à la Rosen (1965), and also provide a Pareto optimal allocation scheme that is beneficial and fair for the two players.