Economic growth and economic development 590

Introduction to Modern Economic Growth
machine production to ψ ≡ (1 − β), so that
χ(ν, t) = χ = 1 for all ν and t.
Profit-maximization also implies that each monopolist rents out the same quantity
of machines in every period, equal to
(13.10)

x (ν, t) = L for all ν and t.

This gives monopoly profits as:
(13.11)

π (ν, t) = (χ(ν, t) − ψ) x (ν, t)
= βL

for all ν and t.

The important implication of this equation is that each monopolist sells exactly the
same amount of machines, charges the same price and makes the same amount of
profits at all time points. This particular feature simplifies the analysis of endogenous technological change models with expanding variety.
Now substituting (13.6) and the machine prices into (13.2),
1
N (t) L.
(13.12)
Y (t) =
1−β
This is the major equation of the expanding product or input variety models. It
shows that even though the aggregate production function exhibits constant returns
to scale from the viewpoint of final good firms (which take N (t) as given), there
are increasing returns to scale for the entire economy; (13.12) makes it clear that
an increase in the variety of machines, N (t), raises the productivity of labor and

that when N (t) increases at a constant rate, so will output per capita.
The labor demand of the final good sector follows from the first-order condition
of maximizing (13.5) with respect to L and implies the equilibrium condition
β
N (t) .
(13.13)
w (t) =
1−β
Finally, free entry into research implies that at all points in time we must have
(13.14) ηV (ν, t) ≤ 1, Z (ν, t) ≥ 0 and (ηV (ν, t) − 1) Z (ν, t) = 0, for all ν and t,
where V (ν, t) is given by (13.7). Recall that one unit of final good spend on R&D
leads to the invention of η units of new inputs, each making profits given by (13.7).
This free entry condition is written in the complementary slackness form, since
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