Strategic incentives when supplying to rivals with an application to vertical firm structure

We consider a vertically integrated input monopolist supplying to a differentiated downstream rival. With linear input pricing, at the margin the firm unambiguously wants the rival to expand-unlike standard oligopoly with no supply relationship-for either Cournot or Bertrand competition. With a two-part tariff for the input, the same result holds if downstream choices are strategic complements, but is reversed for Cournot with strategic substitutes. We analyze vertical delegation as one mechanism for inducing expansion or contraction by the rival/customer.