这个作业是分析商业周期模型对经济体的支出冲击
Homework Assignment #1
ECON6032
1.基本实际商业周期模型中的政府支出冲击
(50分)考虑一个经济体,该经济体具有相同的具有无限偏好的无限生命的代理商
由
00
X∞
t = 0
β
Ť
</ s> </ s> </ s> </ s> </ s>
1个
1 −σ
C
1−σ
t −
ν
1 + ϕ
ñ
1 + ϕ
Ť
</ s> </ s> </ s> </ s> </ s>
0 <β<1
哪里Ct
是消耗量,Nt是工作小时数,ϕ> 0,σ> 0和ν> 0。每个
消费者在一系列预算约束下寻求最大化效用:
Ct + QtBt = WtNt + Bt-1 + Dt-Tt
对于所有t和Wt代表实际工资的州,Bt代表1的存量
定期债券(净供给为零),Qt表示债券的实际价格Dt
是公司股利,Tt是一次性税。代表公司有权使用
以下生产技术:
Yt = AtN
1−α
Ť
分别输出Yt和At和外部技术冲击。而且,
在≡登录
遵循固定的AR(1)过程:
在=ρaat-1+ε
一种
Ť
其中ρa∈(0,1)和ε
一种
Ť
是一次i.i.d震惊。
最后,假设政府希望购买外来消费品流{Gt}
∞
t = 0,在每个时期它通过收集来为购买Gt筹集资金
代表家庭的一次性税收。每个时期的说法不同
产出是由私营部门生产的,政府购买了该产出的外生金额Gt,其余部分由代表人消费
家庭。另外,ˆgt≡log Gt-log G遵循平稳的AR(1)过程:
gˆt = ρggˆt−1 +ε
G
Ť
其中ρg∈(0,1)和ε
G
Ť
是一次i.i.d震惊。
1个
(a)(8分)获得家庭和企业问题的一阶条件。
(b)(7分)写下政府预算约束。
(c)(10分)列出所有市场清算条件。
(d)(10分)定义该经济体的市场均衡。
(e)(15分)对所有平衡条件进行对数线性化,并在课堂上进行恢复
一个由三个平衡条件组成的子系统,其函数为
,cˆt + 1,nˆt
,
,aˆt和ˆgt
只要。建议:在对数线性化中,您可能需要使用以下符号χ≡
G
ÿ
其中G和Y是非随机稳态下的Gt和Yt的值
状态。
2. The Planner Problem in a two Sector Economy
(50 points) Consider an economy where there are two goods, Good V and Good Z,
agents are all identical and infinitely-lived. Agent preferences are given by
E0
X∞
t=0
β
t
[U(CV,t, CZ,t)]
with CV,t and CZ,t being the consumption of Good V and Good Z respectively. U(CV,t, CZ,t)
is well behaved (i.e., UCV ,t > 0, UCZ ,t > 0, UCV CV ,t < 0, UCV CZ ,t ≥ 0 and UCZ CZ ,t < 0)
Goods V and Z are produced with a decreasing return to scale technology:
YV t = AV tF(NV,t)
YZt = AZtF(NZ,t)
where NV,t and NZ,t are the quantity of labour input (hours worked) and AV,t AZ,t
the productivity shocks in Sector V and Z. Moreover F(NV,t) and F(NV,t) are well
behaved (i.e., FNV ,t > 0, FNV NV ,t < 0, FNZ ,t > 0 and FNZNZ ,t < 0). The total amount
of time available that households allocate to labour is equal to H. For the rest the
model is as the basic real business cycle model discussed in class.
(a) (10 points) Write down and explain intuitively the constraints to the planner
problem.
(b) (7 points) Write down the planner problem. State the variables with respect to
which the social planner maximises.
(c) (7 points) Find the first-order conditions to the planner problem.
(d) (8 points) Express the marginal rate of transformation of Good Z into Good V in
terms of the marginal product of labour in Sector V and Z.
(e) (8 points) Interpret the conditions found on point (c).
2
(f) (10 points) Suppose the planner wants to decentralise the Pareto efficient allocation as a competitive equilibrium (i.e., he wants to find the prices that can
support the planner allocation as a competitive equilibrium) and sets
PZ,t =
UCZ ,t
UCV ,t
wV,t = AV,tFNV ,t
wZ,t = PZ,tAZ,tFNZ ,t
where PZ,t, wV,t and wZ,t represent the price in Sector V and wages in Sector
V and Z respectively expressed in terms of Good V units (i.e., Good V is the
numeraire). Provide an economic interpretation of the conditions above.
(g) Extra question: Which condition must the relative wage wV,t
wZ,t
satisfy to ensure
that the Pareto efficient allocation can be decentralised as a competitive equilibrium? Is this condition likely to hold in the market equilibrium? Explain
intuitively why or why not.