Outsourcing, Complementary Innovations and Growth

where v = dv∕dt, L¹_v is labor employed in inventing new blueprints for vertically integrated produc-
tion, v ∕kυ is its productivity, and J is the rate of depreciation. For specialized pairs we have

I        '                   ^m ' ^fLs _n^   ^fLm

j = η (r) s — ∂f with r = —, s = ——, m = —-- (io)

S       ^ks        ^km

•

where f = df∕dt, L¹_s and L¹_m are labor employed in inventing new final assembler and intermediate
supplier blueprints, and f ∕k_s and f∕k_m are their respective productivities.

Learning implies that the values of blueprints are not constant. As innovation cumulates, it
becomes increasingly cheaper to create new patents. Being priced at marginal cost, their values fall
through time. Specifically, if we call Jj the asset value of a patent, patents are priced at marginal
cost due to perfect competition in R&D requiring J_υ = k_υ∕υ, J_m = k_m∕f and J_s = k_s∕f. This

implies

Labs pay their researchers by borrowing at the interest rate R while knowing that the result-
ing patents will generate instantaneous dividends equal to the subsequent expected profits of the
corresponding firms. Arbitrage in the capital market then implies

J υ
^r = JT
υ

and

•

where v∕v and f ∕f represent the rate at which new blueprints are innovated in the case of vertical
integration and outsourcing respectively. These results give

vJυ   v   fj^e_s   f   fj^f_m   f

---=---=---

^kυ   ^{v    k}s    ^{f    k}m    ^f

which pins down the interest rate in the Euler equation (2).

Finally, the aggregate resource constraint (or full employment condition) closes the characteriza-
tion of the instantaneous equilibrium. Since labor is used in innovation and in intermediate produc-
tion by both vertically integrated and specialized producers, we have L = L¹_υ+L¹_s +L¹_m +vλx_υ +fx_m.

i2

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