By-product lobbying with rival public goods

A by-product firm uses the profits from the sale of a private good to finance provision of a public good. If the public good exhibits any degree of rivalry, an increase in population will lead to a reduction in the provision of the public good, when the number of by-product firms is constant. An increase in the number of by-product firms raises provision of the public good, if population is constant. When population and the number of by-product firms are increased in the same proportion, the effect on provision of the public good depends upon the degree of rivalry exhibited by the public good.