In intermediate microeconomic textbooks the reciprocal nature of externalities is presented using numerical examples of costs and benefits. This treatment of the Coase theorem obscures the fact that externality costs and benefits are best understood as being on a continuum where costs vary with the degree of intensity of the externality. When these cost-and-benefit functions are portrayed as continuous, it is possible to analyze optimal levels of the externality. Total costs and benefits can be observed so that efficiency can be evaluated under conditions of zero transactions costs, positive transactions costs, and auction bidding for property rights. This approach also reinforces several other general principles of microeconomic theory.