Soda tax incidence and design under monopoly

We consider an unhealthy good, such as a sugar-sweetened beverage, the health damages of which are misperceived by consumers. The sugar content is endogenous. We first study the solution under "pseudo" perfect competition. In that case a simple Pigouvian tax levied per unit of output but proportional to the sugar content is sufficient to achieve a first best solution. Then we consider a monopoly. Market power affects both output and sugar content, possibly in opposite directions, and these effects have to be balanced against Pigouvian considerations. We show that, nevertheless, a tax per unit of output achieves an efficient solution, but it must be an affine function of the sugar content; taxing "grams of sugar" is no longer sufficient. Interestingly, both the total tax as well as its sugar component can be positive as well as negative.