underinvest in their contractual relation, reducing their joint profits.5
_ . ∙ , - ∙
Let expressions s = ds∕dt and m = dm∕dt represent the flows of new final assembler and
intermediate supplier entrants respectively. The number of new upstream-downstream matches at
time t is determined by the following constant returns to scale matching function:
∙ ∙ ∙ ∙ , ∙
mιn(s, m). If we define r ≡ m∕s, the matching probability of a final assembler entrant and an
intermediate supplier entrant can then be rewritten as η (r) ≡ / (s, m) ∕s and η (r) ∕r respectively.
The blueprints that correspond to unmatched entrants are instantaneously destroyed.6
After a successful match, intermediate suppliers produce their relation-specific inputs. Then each
matched pair bargains on the division of its joint surplus, given by the prospective revenues from the
sales of the corresponding variety. Since neither party has an outside option, they will eventually
agree on a share that makes both better off than if they had not met. We denote the bargaining
weight of the intermediate input producer by ω.
2.4 Timing
In each period t the following sequence of actions take place. Independent labs engage in R&D to
innovate new patents corresponding to vertically integrated firms, upstream specialized intermediate
producers and downstream specialized assemblers. In the production sector firms choose their mode
of entry by purchasing the respective blueprints. Firms who have purchased specialized blueprints
search for partners to form an upstream-downstream chain. Their effort could end in a successful
or an unsuccessful match. Each matched intermediate producer manufactures the input needed by
its partner, while unmatched entrants exit and their patents are destroyed. Once input production
is completed, the outsourcing pair bargain over the share of total revenues from final sales that goes
to each partner and inputs are handed over to assemblers. Final assembly then takes place and the
5This approach is similar to the transaction-cost approach adopted by Grossman and Helpman (2002, 2003). Marin
and Verdier (2003) as well as Antras (2003) take on a different approach in line with the property rights theory of
Grossman and Hart (1986) and Hart and Moore (1990), which states that agreements among stakeholders within a
vertically integrated firm are also incomplete.
6This assumption is made for analytical convenience. The survival of unmatched blueprints would increase the
number of statuses through which blueprints can transit without adding much insight.