Economic growth and economic development 396

Introduction to Modern Economic Growth
which requires that the discounted market value of the capital stock in the very far
future is equal to 0. This “market value” version of the transversality condition is
sometimes more convenient to work with.
We can derive further results on ³
the consumption
´ behavior of households. In
Rt
particular, notice that the term exp − 0 r (s) ds is a present-value factor that

converts a unit of income at time t to a unit of income at time 0. In the special
case where r (s) = r, this factor would be exactly equal to exp (−rt). But more
generally, we can define an average interest rate between dates 0 and t as
Z
1 t
r (s) ds.
(8.17)
r¯ (t) =
t 0

In that case, we can express the conversion factor between dates 0 and t as
exp (−¯
r (t) t) ,
and the transversality condition can be written as
r (t) − n) t) a (t)] = 0.
lim [exp (− (¯

(8.18)

t→∞

Now recalling that the solution to the differential equation
y˙ (t) = b (t) y (t)
is
y (t) = y (0) exp

µZ

t

0

we can integrate (8.14), to obtain
c (t) = c (0) exp

àZ

0

t

ả
b (s) ds ,

ả
r (s)
ds
u (c (s))

as the consumption function. Once we determine c (0), the initial level of consumption, the path of consumption can be exactly solved out. In the special case where
εu (c (s)) is constant, for example, εu (c (s)) = θ, this equation simplifies to

àà
ả ả
r (t)
c (t) = c (0) exp
t ,
θ
and moreover, the lifetime budget constraint simplifies to
Z ∞
Z ∞
c (t) exp (− (¯
r (t) − n) t) dt = a (0) +
w (t) exp (− (¯
r (t) − n) t) dt,
0

0

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