Outsourcing, Complementary Innovations and Growth

"■ = (1 - ω)p_sy_s,

(10)

and the remaining share ω goes to the intermediate supplier. The latter must decide in the previous

stage how much input x_m to produce anticipating this share, which incurs a cost of x_m units of

labor. Therefore, it maximizes

ʌ m = ωpsys — X_m,

(11)

which implies an intermediate and final output equal to

1

X_m = y_s = A (αω)

(12)

with associated final price

1

p_s = —.
aω

(13)

Using these results in (10) and (11), and recalling that specialized intermediate and final entrants

face probabilities η (r) and η (r) /r of being matched, their expected profits are respectively:

Q
^πS = η(r) (1 — ω) A (αω) ^{1 q}

(14)

and

π^e_m = (1 — a) η^ ^ ωA (aω) ^{1 q} .
r

(15)

Substituting (8) and (13) into (3) and (5) allows us to write aggregate demand as

(16)

where υ is the number of vertically integrated firms and f is the number of matched pairs of spe-

cialized producers that are active at time t.

3.2 Innovât ion

In the entry stage, the output from the R&D labs determines the laws of motion of υ and f. For
vertically integrated firms, we have

v = E⅛ — _δυ                              (17)

k_v

11