ECON2040 PROBLEM SET 2

Md = P·L（Y，i），

L（Y，i）在Y中增加而在i中减少。

L（Y，i）= Yαi

，（1）

L（Y，i）的函数形式。
1.证明处于平衡状态
ln（m）= ln（α）-βln（i），

2.接下来，您将使用澳大利亚数据估算货币需求关系。你会

•访问并下载以下时间序列（使用搜索

–系列ID：MANMM101AUQ189S。这对应于M1的时间序列

–系列ID：AUSGDPNQDSMEI。这对应于标称的时间序列

–系列ID：IR3TBB01AUQ156N。这对应于3个月的银行票据收益率
（即短期利率的时间序列）在澳大利亚每季度一次。
•将所有数据合并到一个Excel文件中，并将样本限制在期间内
1968年第一季度至2019年第四季度。
3.绘制散点图，纵轴为ln（m），横轴为ln（i）

4.使用第3部分中的采样周期，估计α和β。为此，请计算
Excel中第3部分中散点图的线性趋势线。 β的估计是否与
L（Y，i）的性质在课堂上学习过吗？为什么或者为什么不？
5.根据第4部分的估计，Md对GDP（Y）的1％变化或对

6.现在绘制一个散点图，垂直轴上为ln（m），水平轴上为ln（i）

0

U(C,l,C
0
) = log(C)+αlog(l)+βlog(C
0
)
subject to her lifetime budget constraint
C +
C
0
1+r
= w(h −l)(1−τ)+π+
π
0 −T
0
1+r
,
where α > 0, β ∈ (0, 1), h is the total number of hours available for productive work, w is the
hourly wage, r is the real interest rate, π and π
0
are dividends in the first and second period,
τ is a proportional tax levied in the first period, and T
0
is a lump-sum tax levied in the second
period.
1. Show that the household chooses consumption and leisure according to the optimality
condition:
α
C
l
= w(1−τ). (2)
2. Suppose that output is produced by a representative firm. In the first period, the representative firm produces output Y by operating the Cobb-Douglas technology:
F(K,N) = zKθ
(N
d
)
1−θ
,
where θ ∈ (0, 1), z is total factor productivity, K is the capital stock, and N
d
is labour demanded in the first period.
The firm chooses labour demand based on the optimality condition FN (K,N
d
) = w. Show
that this optimality condition boils down to
w = (1−θ)
Y
Nd
. (3)
3. Show that equilibrium employment, N

, satisfies
N
∗ =
(1−θ)h
1−θ +
C
Y
α
1−τ
. (4)
4. Next, you will use equation (4) to quantify the impact of the income tax τ on the level of
employment predicted by the model. Follow these steps:
• Let θ = 0.3224, α = 1.54, and let h = 100.1 This last value should be interpreted as the
total number of productive hours per week.
• Set C/Y = 0.7, which roughly matches the contribution of consumption to GDP in
Australia.
Using these values, plot the relationship between N

and τ in Excel. Use a grid of values
of τ between [0, 0.8] with a step size of 0.05.
5. Now set τ = 23.6%, which corresponds to a measure of the average income tax rate in
Australia for a single person with no children (according to OECD data) for 2019.
(a) What is the level of employment (N

) predicted by the model for Australia? Note:
Given our model calibration, you should interpret N

as the number of hours worked
by a representative individual on a weekly basis.
(b) What is the empirical counterpart of N

? That is, how many hours did a “typical” Australian work per week in 2019? To answer this question, browse the OECD
website and look for a meaningful counterpart to N

. Provide a brief description of the statistic that you use. How
would you judge the ability of the RBC model to match the number of hours worked
in Australia?
Problem 3: Macro Effects of a Lockdown [25 marks]
In this problem you will analyse the macro effects of a lockdown through the lens of the RBC
model studied in class.
Suppose that to contain a pandemic crisis, the government shuts down a number of sectors in
the economy. The sectors affected by the shutdown are those that require close contact with
customers, like restaurants, sports and recreation, and hospitality.
This policy has two key implications. First, it reduces the supply of labour by the households,
as workers within affected sectors cannot work as much as much as they would desire. Second,
the lockdown reduces aggregate demand. This is mainly because households in affected sectors experience a reduction in labour income, so they consume less. Assume throughout this
problem that the corresponding drop in aggregate demand is larger than any potential drop in
aggregate supply.
1. Consider the closed economy RBC model with money studied in class. Assuming that
prices are flexible, analyse the effect of the lockdown on the equilibrium values of employment, output, the real wage, the real interest rate, and the aggregate price level. Use
the equilibrium diagrams for the current labour market, the current goods market, and
the money market to answer this question.
2. Now assume that the price level is fixed, and that the central bank follows an interest rate
target. Analyse the short run equilibrium after the lockdown assuming that the central
bank does not change its interest rate target. Use the diagrams for the current labour
market, the current goods market, and the money market to answer this question. Show
that the lockdown results in a negative output gap.
3. Following up on the short-run equilibrium resulting from the lockdown, assume that to
close the output gap the central bank changes its interest rate target. Illustrate the effects
of this policy using the diagrams for the current labour market, the current goods market,
and the money market. How does the composition of output change as a result of the
monetary intervention?
Note: To answer this question, you should start from the short run equilibrium you characterised in part 2, i.e., the short run equilibrium that results after the lockdown is implemented.
4. As discussed in class, the zero lower bound poses a challenge to conventional monetary
policy interventions like the one you analysed in part 3. In class we focused on the Australian case, and looked at different forms of unconventional monetary policy interventions implemented by the RBA to provide further stimulus to the economy.