Economic growth and economic development 392

Introduction to Modern Economic Growth
using current borrowings. The consequence of this scheme would be that the asset holding of the individual would tend to −∞ as time goes by, clearly violating

feasibility at the economy level.

To understand where this form³ of the no-Ponzi-game
condition comes from,
´
RT
multiply both sides of (8.9) by exp 0 r (s) ds to obtain
à Z t
ả
à Z T
¶
Z T
c (t) L(t) exp −
r (s) ds dt + exp
r (s) ds A (T )
0
0
0
à Z t
ả
Z T
=
w (t) L (t) exp −
r (s) ds dt + A (0) ,
0

0

then divide everything by L (0) and note that L(t) grows at the rate n, to obtain
à Z t
ả
à Z T
¶
Z T
c (t) exp −
(r (s) − n) ds dt + exp −
(r (s) − n) ds a (T )
0
0
0
à Z t
ả
Z T
=
w (t) exp
(r (s) n) ds dt + a (0) .
0

0

Now take the limit as T → ∞ and use the no-Ponzi-game condition (8.11) to obtain
à Z t
ả
à Z t
ả
Z
Z
c (t) exp

(r (s) − n) ds dt = a (0)+
w (t) exp −
(r (s) − n) ds dt,
0

0

0

0

which requires the discounted sum of expenditures to be equal to initial income plus

the discounted sum of labor income. Therefore this equation is a direct extension
of (8.9) to infinite horizon. This derivation makes it clear that the no-Ponzi-game
condition (8.11) essentially ensures that the individual’s lifetime budget constraint
holds in infinite horizon.
8.2. Characterization of Equilibrium
8.2.1. Definition of Equilibrium. We are now in a position to define an
equilibrium in this dynamic economy. We will provide two definitions, the first is
somewhat more formal, while the second definition will be more useful in characterizing the equilibrium below.
Definition 8.1. A competitive equilibrium of the Ramsey economy consists of
paths of consumption, capital stock, wage rates and rental rates of capital,
[C (t) , K (t) , w (t) , R (t)]∞
t=0 , such that the representative household maximizes its
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