Introduction to Modern Economic Growth will grow in response to a gap between this rate of return and the discount rate, which is related to the elasticity of marginal utility of consump
Trang 1Introduction to Modern Economic Growth will grow in response to a gap between this rate of return and the discount rate, which is related to the elasticity of marginal utility of consumption, εu(c (t)) Notice that εu(c (t)) is not only the elasticity of marginal utility, but even more importantly, it is the inverse of the intertemporal elasticity of substitution, which plays a crucial role in most macro models The intertemporal elasticity of substitu-tion regulates the willingness of individuals to substitute consumpsubstitu-tion (or labor or any other attribute that yields utility) over time The elasticity between the dates
t and s > t is defined as
σu(t, s) =− d log (c (s) /c (t))
d log (u0(c (s)) /u0(c (t))).
As s↓ t, we have
(8.16) σu(t, s)→ σu(t) =− u
0(c (t))
u00(c (t)) c (t) =
1
εu(c (t)). This is not surprising, since the concavity of the utility function u (·)–or equiva-lently, the elasticity of marginal utility–determines how willing individuals are to substitute consumption over time
Next, integrating (8.12), we have
µ (t) = µ (0) exp
µ
−
Z t 0
(r (s)− ρ) ds
¶
= u0(c (0)) exp
µ
−
Z t 0
(r (s)− ρ) ds
¶ , where the second line uses the first optimality condition of the current-value Hamil-tonian at time t = 0 Now substituting into the transversality condition, we have
lim
t→∞
∙
exp (− (ρ − n) t) a (t) u0(c (0)) exp
µ
−
Z t 0
(r (s)− ρ) ds
¶¸
= 0,
lim
t→∞
∙
a (t) exp
µ
−
Z t 0
(r (s)− n) ds
¶¸
= 0,
which implies that the strict no-Ponzi condition, (8.11) has to hold Also, for fu-ture reference, notes that, since a (t) = k (t), the transversality condition is also equivalent to
lim
t→∞
∙ exp
µ
−
Z t 0
(r (s)− n) ds
¶
k (t)
¸
= 0, 381