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

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

Ct + QtBt = WtNt + Bt-1 + Dt-Tt

Yt = AtN
1−α
Ť

Ť

Ť

t = 0，在每个时期它通过收集来为购买Gt筹集资金

gˆt = ρggˆt−1 +ε
G
Ť

G
Ť

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

，cˆt + 1，nˆt

，aˆt和ˆgt

G
ÿ

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.