Economic growth and economic development 681

Introduction to Modern Economic Growth
Similar to the demands for machines in Chapter 13, these are iso-elastic, so the maximization of the net present discounted value of profits implies that each monopolist
should set a constant markup over marginal cost and thus a price of
χL (ν, t) = χZ (ν, t) = 1 for all ν and t.
Substituting these prices into (15.13) and (15.14), we obtain
xL (ν, t) = pL (t)1/β L for all ν and all t,
and
xH (ν, t) = pH (t)1/β H

for all ν and all t.

Since these quantities do not depend on the identity of the machine, only on the
sector that is being served, profits are also independent of the machine type. In
particular, we have
(15.15)

π L (t) = βpL (t)1/β L and π H (t) = βpH (t)1/β H.

This implies that the net present discounted values of monopolists only depend on
which sector they are supplying and can be denoted by VL (t) and VH (t).
Next, combining these with (15.5) and (15.6), we obtain derived production
functions for the supply of the intermediate goods of the two types:
(15.16)

YL (t) =

1−β
1
pL (t) β NL (t) L
1−β

and
(15.17)

YH (t) =

1−β
1
pH (t) β NH (t) H.
1−β

These derived production functions are similar to (13.12) in Chapter 13, except for
the presence of the price terms.
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