Using (6) and (8) in (2) yields the optimal allocation of workers across sectors.
Integrating over the interval [0,1] gives:
A (i)e(e-1)
( (i) — ^ 1 Ш 1 ’
(10)
' A (j)βl<-11 dj
Note that more productive sectors attract more workers (as long as e > 1) because
the value of marginal productivity of labor has to be equalized. Profits generated
by the sale of machine i are a fraction β (1 — β) of the value of sectoral output:
π (i) — β (1 — β) p (i)1^β A (i) I (i).
(11)
The evolution of technology combines Ricardian elements with endogenous tech-
nical change. The productivity index A(i) in each sector is the product of two com-
ponents, an exogenously given productivity parameter, φ (i), and the level of current
technology in use in sector i, a (i):
A (i) — a (i) φ (i).
While φ (i) is fixed and determined by purely exogenous factors, such as the specific
environment of a country, a (i) can be increased by technical progress. For simplicity,
the model assumes that all the countries in the North share the same productivity
schedule φ — (φ (i)). Innovation is directed and sector specific. To simplify, without
loss of generality, innovation is modelled as incremental:14 in the R&D sector, μ
units of the numeraire can increase the productivity of machine i by ∂a (i). Once
an innovation is discovered, the innovator is granted a perpetual monopoly over its
use. The patent is then sold to the producer of machine i. Free-entry in the R&D
sector drives the price of any innovation down to its marginal cost μ. The monopolist
decides how much innovation to buy by equating the marginal value of the quality
improvement, the present discounted value of the infinite stream of profits generated
by the innovation, to its cost. Along the balanced growth path, where ∂π (i) ∕∂a (i)
14This description of innovation is equivalent to the expanding variety approach of Romer (1990).
See Gancia and Zilibotti (2003) for more details on growth through expanding variety of interme-
diates and how to rewrite the present model in that context.
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