Economic growth and economic development 391

Introduction to Modern Economic Growth
budget constraint of the form:
(8.9)

¶
c (t) L(t) exp
r (s) ds dt + A (T )
0
t
àZ T
ả
àZ
Z T
=
w (t) L (t) exp
r (s) ds dt + A (0) exp
Z

àZ

T

T

t

0

0

T

ả
r (s) ds ,

for some arbitrary T > 0. This constraint states that the household’s asset position
at time T is given by his total income plus initial assets minus expenditures, all
carried forward to date T units. Differentiating this expression with respect to T
and dividing L(t) gives (8.7) (see Exercise 8.2).
Now imagine that (8.9) applies to a finite-horizon economy ending at date T . In
this case, it becomes clear that the flow budget constraint (8.7) by itself does not
guarantee that A (T ) ≥ 0. Therefore, in the finite-horizon, we would simply impose

this lifetime budget constraint as a boundary condition.

In the infinite-horizon case, we need a similar boundary condition. This is generally referred to as the no-Ponzi-game condition, and takes the form
à Z t
ả
(r (s) n) ds 0.
(8.10)
lim a (t) exp −
t→∞

0

This condition is stated as an inequality, to ensure that the individual does not
asymptotically tend to a negative wealth. Exercise 8.3 shows why this no-Ponzigame condition is necessary. Furthermore, the transversality condition ensures that
the individual would never want to have positive wealth asymptotically, so the noPonzi-game condition can be alternatively stated as:
à Z t
ả
(r (s) n) ds = 0.

(8.11)
lim a (t) exp −
t→∞

0

In what follows we will use (8.10), and then derive (8.11) using the transversality
condition explicitly.
The name no-Ponzi-game condition comes from the chain-letter or pyramid
schemes, which are sometimes called Ponzi games, where an individual can continuously borrow from a competitive financial market (or more often, from unsuspecting
souls that become part of the chain-letter scheme) and pay his or her previous debts
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