Ec 502 Problem Set 5

1.资金需求：

t
X∞
s = 0
β
s
（log（Ct + s）+γlog（Mt + s / Pt + s））

Bt + Mt =（1 +它）Bt-1 + Mt-1 + Wt − PtCt

（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

（d）现在假设名义货币余额Mt以恒定速率增长
Mt + 1 =

1克

</ s> </ s> </ s>

Ct + 1 =

1克
C
</ s> </ s> </ s>
Ct

PtCt =ΦMt

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

，债券持有Bt

X∞
s = 0
β
s

Mt +秒

</ s> </ s> </ s>

ñ
1 +φ
t +秒
1 +φ
＃！

PtCt + Bt + Mt =（1 +它）Bt-1 + Mt-1 + WtNt +Πt

，Wt

Et-1Πt（i）= Et-1 [Pt（i）Yt（i）-WtNt（i）]

Yt（i）=

t

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
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