Economic growth and economic development 664

Introduction to Modern Economic Growth
in the expanding varieties model but it is important in the model of competitive innovations.
Exercise 14.11. Consider a version of the model of competitive innovations in
which innovations reduce costs instead of increasing quality. In particular, suppose
that the aggregate production function is given by
¸
∙Z 1
1
1−β
x(ν, t) dν Lβ ,
Y (t) =
1−β 0

and the marginal cost of producing machine variety ν at time t is given by MC (ν, t).
Every innovation reduces this cost by a factor λ.
(1) Define an equilibrium in this economy.
(2) Specify a form of the innovation possibilities frontier that is consistent with
balanced economic growth.
(3) Derive the BGP growth rate of the economy and show that there are no
transitional dynamics.
(4) Compare the BGP growth rate to the Pareto optimal growth rate of the
economy. Can there be excessive innovations?
(5) What are the similarities and differences between this model and the baseline model presented in Section 14.1.
Exercise 14.12. Consider the model in Section 14.2, with R&D performed by
workers. Suppose instead that the aggregate production function for the final good
is given by

∙Z 1
¸
1
1−β

q(ν, t)x(ν, t | q) dν Lβ .
Y (t) =
1−β 0
(1) Show that in this case, there will only be R&D for the machine with the
highest q(ν, t).
(2) How would you modify the model so that the equilibrium has balanced
R&D across sectors?

Exercise 14.13. Consider the model of competitive innovations in Section 14.1,
with one difference: conditional on success an innovation generates a random improvement of λ over thehprevious technology,
i where the distribution function of λ is
−(1−β)/β ¯
,λ .
H (λ) and has support (1 − β)
(1) Define an equilibrium in this economy.
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