Economic growth and economic development 524

Introduction to Modern Economic Growth
To do this, let us substitute for c(t) from equation (11.11) into equation (11.8),
which yields
(11.13)

à
ả
k (t) = (A − δ − n)k (t) − c(0) exp 1 (A − δ − ρ)t ,
θ

which is a first-order, non-autonomous linear differential equation in k (t). This type
of equation can be solved easily. In particular recall that if
z˙ (t) = az (t) + b (t) ,
then, the solution is
z (t) = z0 exp (at) + exp (at)

Z

t

exp (−as) b(s)ds,

0

for some constant z0 chosen to satisfy the boundary conditions. Therefore, equation
(11.13) solves for:
(11.14)n
Ô1 Ê
ÂÔo
Ă 1
Ê

1
1
,
c(0) exp (A − ρ)t
k(t) = κ exp((A − δ − n) t) + (A − δ)(θ − 1)θ + ρθ − n

where κ is a constant to be determined. Assumption (11.12) ensures that
(A − δ)(θ − 1)θ−1 + ρθ−1 − n > 0.

From (11.14), it may look like capital is not growing at a constant rate, since
it is the sum of two components growing at different rates. However, this is where
the transversality condition becomes useful. Let us substitute from (11.14) into the
transversality condition, (11.10), which yields
Ê
Ô1
Ă Ă
 Â
c(0) exp A δ)(θ − 1)θ−1 + ρθ−1 − n t ] = 0.
lim [κ+ (A − δ)(θ − 1)θ−1 + ρθ−1 − n

t→∞

Since (A − δ)(θ − 1)θ−1 + ρθ−1 − n > 0, the second term in this expression converges

to zero as t → ∞. But the first term is a constant. Thus the transversality condition
can only be satisfied if κ = 0. Therefore we have from (11.14) that:
Ô1 Ê
ÂÔ
Ă
Ê

(11.15)k(t) = (A − δ)(θ − 1)θ−1 + ρθ−1 − n
c(0) exp θ−1 (A − δ − ρ)t
¡
¢
= k (0) exp θ−1 (A − δ − ρ)t ,

where the second line immediately follows from the fact that the boundary condition
has to hold for capital at t = 0. This equation naturally implies that capital and

output grow at the same rate as consumption.
510