one unit of the intermediate component produced by their partner for each unit of the final good.
Accordingly, outsourcing leads to productivity gains that stem from specialization in production.
Labs invent new blueprints at a marginal cost that depends on their types. R&D faces a learning
curve, a larger number of a certain type of blueprints successfully introduced in the past makes
researchers more productive in inventing that type of blueprint. For specialized blueprints, what
matters is not only the number of invented patents, but also the number f of those that have
actually been matched and used in production. In particular, as in Grossman and Helpman (1991),
we consider a linear learning curve such that the marginal costs of innovation are kυ∕v, km∕f, and
ks∕f (with kυ, km and ks all positive) depending on the type of the blueprints.4 Given this functional
form, some initial stocks of implemented blueprints is needed to have finite costs of innovation at
all times. We call them Vq > O and fɔ > O for vertically integrated and specialized blueprints
respectively. We finally assume that ks + km ≤ kυ to capture the idea that the R&D cost of more
complex vertically integrated blueprints is higher.
2.3 Matching and Bargaining
Outsourcing also faces additional costs that result from search frictions and incomplete contracts.
After buying a patent, specialized entrants of each type must bear a search cost of finding a suitable
partner in a matching process that may not always end in success. Matched intermediate suppliers
also suffer hold-up problems as they each produce a relation-specific input. This input has no value
outside the relation and its quality is too costly to observe by courts. Thus, the final assembler
can refuse payment after the input has been produced. This gives rise to a hold-up problem in so
far as, the variety-specific input having no alternative use at the bargaining stage, its production
cost is sunk. The transaction costs involved in ex-post bargaining may then cause both parties to
4The assumed shape of the learning curve serves analytical solvability and the comparison with Grossman and
Helpman (1991). In equilibrium it yields a ‘size effect’, meaning that larger countries grow faster. As this prediction
runs against the empirical evidence, the size effect could be removed by assuming that the intensity of the learning
spillover is lower, i.e. kυ∕r^, km∕f^, and ks∕f^ with O < ξ < 1 (Jones, 1995).