Economic growth and economic development 595

Introduction to Modern Economic Growth
with the Pareto optimal allocations, we set up the problem of the social planner
and derive the optimal growth rate. Notice that the social planner will also use the
same quantity of all types of machines in production, but because of the absence of a
markup, this quantity will be different than in the equilibrium allocation. The social
planner will also take into account the effect of an increase in the variety of inputs
on the overall productivity in the economy, which monopolists did not because they
did not appropriate the full surplus from inventions.
More explicitly, given N (t), the social planner will choose
#
"Z
Z N(t)
N(t)
1
1−β
β
x(ν, t) dν L −
ψx(ν, t)dν,
max
[x(ν,t)]v∈[0,N (t)] ,L 1 − β
0
0
which only differs from the equilibrium profit maximization problem, (13.5), because
the marginal cost of machine creation, ψ, is used as the cost of machines rather than
the monopoly price, and the cost of labor is not subtracted. Recalling that ψ ≡ 1−β,
the solution to this program involves

xS (ν, t) =

L

(1 − β)1/β
so that the social planner’s output level will be

,

(1 − β)−(1−β)/β S
N (t) L
Y (t) =
1−β
S

= (1 − β)−1/β N S (t) L,

where superscripts “S” are used to emphasize that the level of technology and thus
the level of output will differ between the social planner’s allocation and the equilibrium allocation. The aggregate resource constraint is still given by (13.3). Let us
define net output, which subtracts the cost of machines from total output, as
Y˜ S (t) ≡ Y S (t) − X S (t) .
This is relevant, since it is net output that will be distributed between R&D expenditure and consumption. We obtain
Y˜ S (t) = (1 − β)−1/β N S (t) L −
−1/β

= (1 − β)

Z

N S (t)

ψxS (ν, t) dν

0

N (t) L − (1 − β)−(1−β)/β N S (t) L
S

= (1 − β)−1/β βN S (t) L.
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