The results previously obtained on the finiteness property in vertically differentiated markets, with cost functions having increasing or decreasing returns, are extended to a much larger class of cost functions with local properties in a vicinity of the zero output. Moreover, existence results are provided to prove that the demonstrated properties have cases of application.
Lahmandi-Ayed, R. (2004). Finiteness property in vertically differentiated markets: a note on locally increasing and decreasing returns. Economic Theory, 23(2), 371-382. https://doi.org/10.1007/s00199-003-0370-7 (Original work published 2004)