Introduction to Modern Economic Growth budget constraint of the form: (8.9) ¶ c (t) L(t) exp r (s) ds dt + A (T ) t àZ T ả àZ Z T = w (t) L (t) exp r (s) ds dt + A (0) exp Z µZ T T t 0 T ¶ r (s) ds , for some arbitrary T > 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 ) ≥ 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 Z t ả (r (s) − n) ds ≥ (8.10) lim a (t) exp − t→∞ 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: µ Z t ¶ (r (s) − n) ds = (8.11) lim a (t) exp − t→∞ 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 377