Consumption Externality

Assume, there are two roommates A & B, who have preference over money and smoke. It has been assumed that both of them prefer money but A likes to smoke while B like clean air. As a consequence,
consumption externality occurs due to the action of A.
In an Edgeworth box, we can determine point of endowment based on assignment& property rights. If property right is given to person A then endowment bundle will be WA and in that case they will have to exchange in order to move to the final allocation XA which is market equilibrium as well as Pareto efficient. Likewise, if property rights are assigned to B, then person B can sell his right in exchange for money and in that scenario, final allocation would be XB and endowment would be WB which implies if property rights are properly defined and through exchange we can attain efficient allocation. However, problem arises when property rights are not well defined which is when market failure arises.