Outsourcing, Complementary Innovations and Growth



C (t):


■ n(t)


p(i,t)a/(1—a)di


(1 α) j α


(3)


Households then allocate their expenditures across all varieties, which yields the instantaneous
demand function

c(i,t) = A(t)p(i,t)1/(1 a) г [0,n(t)]

(4)


for each variety. In (4) p(i, t) is the price of variety i and

A(t) = , AB(t)1—г                                       (5)

v 7    P(t)a/(1—a)                                               v

is aggregate demand. Throughout the rest of the paper, we leave the time dependence of variables
implicit when this does not generate confusion.

2.2 Innovation and Production

There are two factors of production in the economy. Labor is inelastically supplied by households.
Each household supplies one unit of labor; we can hence use a single index
L to refer to the number
of households as well as the total endowment of labor. Labor is chosen as numeraire. The other
factor is knowledge capital in the form of blueprints, the creation of which leads to the production
of differentiated varieties. As in Grossman and Helpman (1991), while the length of patents on the
blueprints is infinite, they depreciate at a constant rate
δ.

There are two sectors, production and innovation (R&D). Perfectly competitive labs invent differ-
ent types of blueprints depending on the corresponding production processes. Vertically integrated
processes need a single blueprint. Fragmented processes require two blueprints (‘innovation net-
work’): one for the intermediate component and one for the final product. Firms enter by buying
patents from the R&D labs. A firm can thus choose the type of patent and enter as a vertically
integrated firm, an intermediate supplier or a final assembler. The number of each of these types of
blueprints available at time t will be referred to as
v, m, and s respectively. The marginal cost of
production for vertically integrated firms is λ
1 units of labor, whereas specialized intermediate
producers only require 1 unit of labor per unit of input. Specialized final assemblers in turn need



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