I'm definitely using this one in class when I talk about negative externalities and Coase (Warning: a little NSFW language):
Commercial barbecue cookers are not exempt from causing a nuisance odor. If a sufficient number of complaints, representing different households, are reported and an Inspector witnesses the problem, they can issue a Warning Letter.
N' then they get a stern talkin' to.
From a Coasian perspective the answer seems pretty straight forward. The County has established the property right in favor of the neighbor (no smoke allowed). A Pareto-improving solution exists. If the value of the BBQ to the owner is high enough, he should be willing to pay the neighbor to tolerate the nuisance of the odor and smoke. If the offered payment is high enough to cover the cost of the odor and smoke to the neighbor, then the neighbor should be willing to accept the offer. If the offered payment is not high enough to cover the cost to the neighbor then the BBQ'er is SOL.
Whether a deal is struck or not is largely irrelevant, the point is by assigning the property right (in this case to the neighbor) the county has created an opportunity for the market to solve the problem. Without the property right assignment, this becomes a case of a neighborly dispute with no mechanism for a solution: "I have the right to BBQ." "No, I have a right to smell fresh air." The assignment of property rights creates the mechanism for bargaining to begin. The final piece is the price that needs to be bargained.
So, I offer the following as a starting point for negotiations: The BBQ'er should offer the neighbor a plate of BBQ any time he barbecues. If he wants to sweeten the offer, he might offer all the fixin's to go with it (I'm sure he makes a mean slaw). If that's not enough, a glass of momma's sweet tea to go along might be the key.