这个作业是计算货币的利率和消费的增长率

Ec 502 Problem Set 5

1.资金需求:
考虑一个购买名义债券Bt和实际货币余额Mt / Pt的家庭
最大化:
t
X∞
s = 0
β
s
(log(Ct + s)+γlog(Mt + s / Pt + s))
受名义预算约束:
Bt + Mt =(1 +它)Bt-1 + Mt-1 + Wt − PtCt
哪里铂
是价格水平,家庭收入工资Wt
一项劳动
无弹性地供应。
(a)推导持有债券的家庭最优条件?
(b)推导持有货币的家庭最优条件?
(c)比较这两个条件。货币需求作为名义货币会发生什么
利率为零?什么时候消费增加?解释经济学
换句话说,在决定如何选择家庭时要侧重于权衡
持有大量实际货币余额。
(d)现在假设名义货币余额Mt以恒定速率增长
Mt + 1 =

1克
中号
</ s> </ s> </ s>
公吨
消费以恒定速度增长:
Ct + 1 =

1克
C
</ s> </ s> </ s>
Ct
并且货币的速度是恒定的:
PtCt =ΦMt
假设经济处于长期均衡状态。什么决定名义
利率?实际利率?实际利率如何随
货币和消费的增长率?直观地解释。你可以
假设g
M> gC。 (提示:使用Fisher方程。)
2.新的凯恩斯模型:考虑以下一般均衡模型。住户
选择劳动力供给Nt
,债券持有Bt
以及要解决的货币余额
最大Et
X∞
s = 0
β
s

对数(Ct + s)+γ对数
Mt +秒
铂+
</ s> </ s> </ s>

ñ
1 +φ
t +秒
1 +φ
#!
受名义预算约束
PtCt + Bt + Mt =(1 +它)Bt-1 + Mt-1 + WtNt +Πt
在哪里
以名义利率持有名义债券
,Wt
是名义上的
工资,铂
是价格指数和Ct
是消费。 Πt是名义利润(股息)
家庭从平等拥有的公司那里得到的收益。家庭请客
优化时给出的这些红利。
经济中存在单位数量的公司。我为每个公司选择的价格Pt(i)
提前期以最大化其预期利润
Et-1Πt(i)= Et-1 [Pt(i)Yt(i)-WtNt(i)]
受需求曲线向下倾斜的影响:
Yt(i)=
铂(i)


t
在哪里
是总需求,Yt(i)是公司i的商品需求,并且ε>1。公司使用
仅需人工的CRS生产技术
Yt(i)= AtNt(i)
(a) Show that households choose their bond holdings Bt
, their labor supply Nt
, and
their real balances Mt/Pt to satisfy:
1
PtCt
= βEt (1 + it+1)
1
Pt+1Ct+1
Wt
Pt
1
Ct
= N
φ
t
1
PtCt
= βEt
1
Pt+1Ct+1
+ γ
1
Mt/Pt
1
Pt
Interpret these conditions in one sentence each. If a derivation is identical to
problem 2, you may simply state your result.
(b) Consider the firm’s problem.
i. Show that each firm’s marginal revenue is
MRt =
ε − 1
ε
Pt (i)
Is marginal revenue above or below price? Explain why marginal revenue is
not equal to price, as is the case in competitive markets.
ii. At the optimum, a firm maximizing profits:
Et−1Πt (i) = Et−1 [Pt (i) Yt (i) − WtNt (i)]
sets
MRt = Et−1 {MCt}
Show that each firm in the economy thus chooses its price Pt (i) to satisfy:
Pt (i) = Et−1 {µMCt}
where µ = ε/ (ε − 1). Interpret µ and explain why it depends on ε.
2
iii. Show that the nominal marginal cost is:
MCt = Wt/At
and so:
Pt (i) = Et−1 {µWt/At}
(c) Assume that the growth rate of technology satisfies
log At − log At−1 = v
A
t
where v
A
t
is a mean zero iid shock to the growth rate of technology. The monetary
authority sets the growth rate of money to satisfy
log Mt − log Mt−1 = v
M
t
where v
M
t
is a mean zero iid shock to the growth rate of money.
In equilibrium, labor, goods, output, and bond markets clear:
Nt (i) = Nt
Pt (i) = Pt
Yt (i) = Yt
Ct = Yt
Bt = 0
Show that in equilibrium these conditions imply
Mt = ΦPtYt
for some constant velocity of money Φ. Taking logs and differencing for periods
t and the expectation of period t at time t − 1, what does this expression imply
about monetary neutrality in this model (I am looking for an expression for
log Yt − Et−1 log Yt
in terms of v
M
t
)? Explain in a sentence why money is neutral
or non-neutral in this model.
(d) Show that in in equilibrium the quantity of labor satisfies
N
1+φ
t =
1
µ
Wt/At
Et−1 (Wt/At)
Use this condition to argue that
Et−1Nt = N

for some constant level of labor N∗ which we interpret as the natural rate of
employment. What is N∗
? How does it vary with the markup µ? Provide some
intuition about how this relates employment if prices were flexible.
(e) Derive expressions for the growth rate of output and inflation in terms of current
and past shocks to money growth and technology. How does output growth and
inflation today and tomorrow respond to a money shock v
M
t
and a technology
shock v
A
t
? To answer, draw the impulse response functions with and without
price stickiness, and explain the economics of your impulse response functions in
a few sentences.


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