Outsourcing, Complementary Innovations and Growth

By (7), (12), (17) and (18), the condition can be rewritten as

(V               ∙^s , ^{n             a} L⅛ P _i, ,

- + J + k_s— + k_m — + υλA[-)   + -A (αω) ^l~“ .

3.3 Organization

In any instant t there is never simultaneous invention of both vertically integrated and specialized
blueprints. This would be the case if all equalities in (22) held at the same time. This is generally
impossible. To see this, proceed in two steps. First consider that new outsourcing agreements are
signed only if there is new creation of both intermediate supplier and final assembler blueprints,
which requires

^fπ^e_m = f⅛
k_m      k_s ∙

Using (14) and (15), this yields a fixed ratio of intermediate suppliers over final assemblers

k_s (1 — a) —                                           , .

r = r ≡ — ʌ----^ω-.                              (24)

k_m 1 — —

This implies that the two types of specialized blueprints have to be invented in fixed proportions.

Turning to the second step, a case with only vertically integrated firms reflects Grossman and
Helpman (1991), as the model has no transitory dynamics and jumps instantaneously to its balanced
growth path.⁷ Simple inspection reveals that, by analogy, the same property applies when only
specialized firms or all types of firms are simultaneously active. Along the balanced growth path
all variables either grow at the same rate or do not grow at all. Therefore, for both vertical and
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specialized blueprints to be generated at the same time, f/f = v/v = g must hold. Under this
constraint, υ = -_Qe^3t and - = foe^gt always hold. Then, substituting J_υ = k_υ/- and Jj = kj/— for
j = m,s into (20) and (21) gives

- v₀e^3t    π^tjf0e^3t .

—-,--- = —-,---^, j = m,s

kυ          kj

which implies

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