Outsourcing, Complementary Innovations and Growth

from alternative uses. Differently, stronger time preference (larger p) has a negative impact on
the rate of innovation but a positive one on expenditures since it biases intertemporal decisions
towards consumption and away from saving. Finally, higher costs of innovation (larger k_υ) increases
expenditures and slows innovation whereas a larger economy (larger L) supports proportionately
larger expenditures accompanied by a faster rate of innovation.

4.2 Outsourcing

When condition (26) holds, no labor is allocated to vertical innovation (L¹_υ = 0), so no vertically
integrated blueprints are ever created: v = 0 and asymptotically v = 0.⁹ Along a balanced growth
•                                                                                                                                  ∙

path, we have /// = g/ and E = 0. This allows us to write the long run full employment condition
(23) and the Euler condition (2) as:

^k_s + ^k_mr

L = ---—7— (q/ + δ) + aωE

η (r)

and

η(^r)(¹ - ^ω) ^e

⁰ =-----k--з/ ^- p ^{- δ}∙

^kS

Given the definition of r in (24), these can be solved together to yield

k_s 1 — ωα                 L

^e/ = ^{l + p            ,}g/ = rj^{(r) (1 - ω}) - ^{pωa - δ},                 ⁽²8⁾

^j           η (r) 1 - ω ^j                   k_s

which depend on the matching probability of assembler entrants η (r). Hence, there are two cases. If
there are fewer assemblers than intermediate entrants (r > 1), then the former are surely matched,
so η (r) = 1. Accordingly, (28) becomes:

^e(³ = ^{l +} p^ks —-----^{, 3}S = ^{(1 - ω})  --p^{ωa - δ}∙                  ⁽²⁹⁾

1 - ω               k_s

If there are more assembler than intermediate entrants (r < 1), then the latter are surely matched,
so η (r) /r = 1. This allows us to write (28) as:

sC -r         1 — ωα s ∕ι ʌ L                                ._on,

^Em = ^{L + pk}m (1 _{- a}) _ω ^{, Q}m = ^{(1 - a) ω}J.--^{pωa - δ}∙                (³⁰)

⁹The initial stock of vertically integrated blueprints depreciates through time and asymptotically disappears since
it is not refilled. See Naghavi and Ottaviano (2008a) for details.

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