Price vs. Quantity Competition in a Vertically Related Market Revisited

In a recent paper, Alipranti et al. (2014, Price vs. quantity competition in a vertically related market, Economics Letters, 124: 122-126) show that in a vertically related market Cournot competition yields higher social welfare compared to Bertrand competition if the upstream firm subsidises the quantity setting downstream firm's production via negative wholesale input prices. However, the assumption of negative input prices is not economically viable as it would encourage the downstream firms to buy an unbounded amount of inputs knowing that the upstream firm would pay the downstream firms for each unit of input they purchase. We show that the welfare ranking may be reversed once we introduce a non-negativity constraint on the input price