Ec 502 Problem Set 3

1. 假设一个家庭生活在两个时期，在时期1和时期C2消费C1

2 (c, c)和储蓄S最大化
U (C1) + chulu (C2)

U (C) = C
1−γ
1−γ

(a)写出第一期的预算约束和第一期的预算约束

(b)推导家庭储蓄的最优条件。
(c)定义替代的跨期弹性= 1/解出C1和C2

(d)描述W的增加对第一阶段消费C1和第二阶段消费C2的影响。为这个结果提供经济直觉。
(e)计算C1和C2相对于利率的比值弹性
1 + r(弹性定义为d log(C2/C1))
d日志(1 + r)
)。c / c是增加还是减少

(f)当利率1 + r上升时，C1会上升还是下降?你的回答如何?

C1对利率r完全不敏感，log效用U (C) = log (C)?
2. 用U (C) = C重新考虑上述问题
(1−a)，但现在假定为

R˜=
(
(1−x) (1 + r)概率是1/2
(1 + x) (1 + r)的概率是1/2

(a)再次推导家庭储蓄的最优条件。
(b)推导出储蓄率S/W作为程度的函数的表达式

1
(c) x的增加(即回报率的不确定性增加)如何

3. Consider an infinitely-lived household who makes consumption and savings decisions
to maximize
X∞
s=0
β
sU (Ct+s)
subject to:
St = (1 + r) St−1 + (1 − τt) Yt − Ct
where τt
is the percentage tax on wage income in period t. There is no uncertainty so
all future wages and tax rates are known to the household. Assume β =
1
1+r
.
As shown in class, in this framework the Euler equation holds:
U
0
(Ct) = β (1 + r) U
0
(Ct+1)
so with β =
1
1+r Ct = Ct+1 and we have perfect consumption smoothing. Plugging into
the budget constraint, we can write period zero consumption as a function of current
financial assets (1 + r) S0 and human wealth:
C0 =
r
1 + r

(1 + r) St +
X∞
s=0

1
1 + r
s
(1 − τt) Yt+s
#
.
You do NOT need to derive these results.
(a) Consider a temporary increase in the labor income tax. In particular, suppose
τ0 increases from τ to τ
0
for the current period but τt = τ for all future periods.
What effect would such a tax increase have on current consumption? Explain
intuitively.
(b) How would your answer differ if the tax rate increase was permanent? Explain
(c) How would your answer to part b differ if the consumer faced a credit constraint
that limits borrowing of the form:
St ≥ S¯ ∀t
and in period zero the consumer was indebted to the extent that S0 = S¯? Relate