这个作业是计算不同生活时期储蓄、借贷收益率等

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
在阶段2中。家庭在第二阶段赚取工资(1 + r) W,可以储蓄或借贷
时期2的收益率为(1 + r),其中r为净利率。家庭
在第一阶段没有获得工资。家庭在第一阶段选择消费
2 (c, c)和储蓄S最大化
U (C1) + chulu (C2)
受其预算限制。假设效用为常数相对风险规避:
U (C) = C
1−γ
1−γ
有一个0。
(a)写出第一期的预算约束和第一期的预算约束
两个时期。将它们合并成一个终生预算约束。
(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
作为r,报文,报文和W的函数。
(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
对r增加的响应?用经济直觉来解释(提示:你应该这样做
提到收入效应和替代效应)。你的规模有多大
对代换的跨期弹性的回答?
(f)当利率1 + r上升时,C1会上升还是下降?你的回答如何?
取决于σ?用经济直觉来解释。用你的答案来解释原因
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
储蓄回报是不确定的。具体来说,储蓄的总回报率(1 + r)是a
随机变量R˜
R˜=
(
(1−x) (1 + r)概率是1/2
(1 + x) (1 + r)的概率是1/2
为了简单起见,假设t = 1/ (1 + r)。
(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
不确定性x Ch和Cl(你不需要解Ch和Cl)
1
(c) x的增加(即回报率的不确定性增加)如何
储蓄)影响储蓄率?储蓄利率是如何反应的
增加的不确定性取决于相对风险厌恶系数。解释
凭直觉。
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
intuitively. Relate your answer to the permanent income hypothesis.
(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
your answer to empirical evidence we discussed in class.
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