ECU33011 Assignment 2020/21
Michael Wycherley
Department of Economics, Trinity College, Dublin
Current Version: September 28, 2020
Note: All questions (but not necessarily all subparts) are equally weighted.
1 Consumption and savings
Households live for two periods and choose consumption (Ct) and savings (St) to maximise
U =
X2
t=1
t1E1 futg where ut =
C11=
t
1 1=
where and are positive constants. The budget constraint is
Ct + St + Tt = Yt + (1 + rt) St1
S2 0
where Tt are lump sum taxes, Yt is household labour income and rt is the interest earnt at time t on
savings from the previous period.
1.1 Solve for consumption in the current period.
1.2 Explain how current consumption responds to changes in interest
rates and to changes in the expected present discounted value of
lifetime income.
1.3 Does the empirical evidence support this consumption function?
1
A simple real business cycle model
Output in time t is given by
Yt = AtK
t L1
t = Ct + It + Gt
where Ct is household consumption, It is total investment, Gt is government consumption, At is total
factor productivity, Lt is total hours worked and Kt is the capital stock, which evolves according to
Kt+1 = It + (1 )Kt
Firm decision-making
De
ning the real wage and rental price of capital as Wt and Rt; perfectly competitive
rms will
hire labour and capital from households such that the rental price equals the marginal product.
Rt = AtK 1
t L1
t
Wt = (1 )AtK
t L
t
Household decision-making
There is an in
nitely lived representative household that chooses consumption, savings and hours
worked to maximise
U =
1X
t=0
tut where ut =
C11=
t
1 1=
L
1+ 1
‘
t
1 + 1
‘
where ; ‘ and are positive constants. The budget constraint is
Ct + Kt+1 + Tt = WtLt + (1 + Rt )Kt
where is the depreciation rate1 and Tt is the lump sum tax rate.
The optimal choices of consumption and hours worked are given by
C
1
t+1 = C
1
t (1 + Rt+1 )
L
1
‘
t =
Wt
C
1
t
Simulating the RBC model
The model has nine variables (output, private consumption, investment, government consump-
1This implies that the rate of return on capital purchased at time t is 1 + Rt+1 . This is because one unit of
capital in period t has a value in period t+1 of 1 + Rt+1 . This is composed of a rental income in period t+1 of
Rt+1 plus the value of the remaining 1 units of capital. Therefore Rt+1 is the rate of interest in this model.
2
tion, capital, labour, productivity, wages & the interest rate) and nine equations:2
Yt = AtK
t L1
t
Yt = Ct + It + Gt
Kt+1 = It + (1 )Kt
Rt = AtK 1
t L1
t
Wt = (1 )AtK
t L
t
1
C
1
t
= Et
(
1 + Rt+1
C
1
t+1
)
L
1
‘
t =
Wt
C
1
t
logAt+1 = A logAt + “A;t+1
logGt+1 = + G logGt + “G;t+1:
The last two equations are the law of motion for productivity and government consumption respec-
tively. “A;t and “G;t are shocks to productivity and government consumption at time t and A and
G represent the persistence of the shocks. The steady state value of government consumption is
related to and is set to 20% of output. The shocks are normally distributed random variables with
mean zero and standard deviation A and G.
Solving the model with < 1 is complicated as capital does not fully depreciate so savings depends
on both income and capital. Therefore the model can only be solved numerically by assigning values
to the parameters, linearizing the model and using a computer algorithm to solve it3.
We choose what are considered reasonable parameter values for quarterly data, drawn from the
available evidence (some microeconomic and some macroeconomic4).
‘ A G
0:36 0:025 0:99 0:5 3 2 0:95 0:95
De
ning lower case variables as the percentage deviation of the variable from steady state5, these
2To pin down the lump sum tax rate an additional equation is actually needed, namely the intertemporal government
budget constraint. We abstract away from this here by assuming a balanced government budget requirement Tt = Gt
in all periods. Ricardian equivalence applies here so the timing of taxes does not a¤ect the results.
3Which computer algorithm to use and how it works is not a part of the course and will not be examined. Get in
touch with me if youre interested, but this is normally postgraduate level.
4 comes from labours share of income, from the idea that machinery wears out at the rate of 10% per year,
from evidence for discount rates of about 4-7% per year, from the evidence that people spend about a third of their
available time working, and ‘ from the macroeconomic evidence on the responsiveness of labour supply to wages.
5So negative numbers do not necessarily mean that the variable is negative, just that the value is below normal.
3
parameters imply that (to a
rst order approximation)
at+1
gt+1
=
0:95 0
0 0:95
at
gt
+
“A;t+1
“G;t+1
(1)
2
666666664
yt
ct
lt
rt
wt
it
kt+1
3
777777775
=
2
666666664
0:2685
0:9658
0:1429
0:7315
0:4114
1:0002
0:9500
3
777777775
kt +
2
666666664
1:5593 0:0635
0:4968 0:1706
0:8739 0:0992
1:5593 0:0635
0:6854 0:0357
5:0282 0:1707
0:1257 0:0043
3
777777775
at
gt
(2)
2 Understanding the impact of shocks
2.1 Explain how and why the economy responds to a productivity shock
over the following 100 periods (equivalent to 25 years).
Assume the economy starts in steady state (k1 = 0 and a0 = g0 = 0) and is hit by a one time
productivity shock of 1% (“A;1 = 0:01, “A;j = 0 for all j 6= 1, while “G;j = 0 for all j). Explain why
the capital stock, output, etc respond the way they do.
This involves
rst using 1 to get the value of productivity in each period and then using this, 2
and the initial capital stock to get capital in each subsequent period. Finally, now that you have
productivity and capital in every period you can use 2 to get the other variables in every period.
Then you just have to plot them and explain the behaviour.
However, there is a spreadsheet on my website that does this for you. So if you open the spread-
sheet go to the Impulse responses tab, make sure the correct shocks have been entered in columns B
and C, then the spreadsheet will show you the response of the model economy to the shock and you
just have to use the model to explain the behaviour of the variables.
3 The model and the data
In the previous question you were asked to explain how the model worked. In this question you
will be asked to explain how well the model matches the real world. There are a number of ways of
doing this, we are going to focus on whether the model can generate similar behaviour to what we
observe in the data, not whether the underlying assumptions are strictly accurate (no assumptions
will always be accurate).
3.1 How well does the RBC model match the stylized facts?
In the real world productivity does not grow smoothly at the trend rate, and is not hit by one shock
every 25 years. Therefore, we want to see whether the model can match the stylized facts we observe
in the real world given a plausible set of shocks. We are interested in the e¤ect of productivity
4
shocks and we are going to abstract away from other shocks (such as government spending), so
assume A = 0:0045 and G = 0:
In this question we use the calibrated RBC model (equations 1 and 2) to see whether this version
of the RBC model matches the stylized business cycle facts from the lecture and the textbook. You
can ignore the ones related to imports and exports and nominal variables, as this version of the model
is a closed economy without monetary e¤ects.
One way to do this is to use excels random number generator (in the analysis toolkit6) to draw
a long sequence of shocks (perhaps 11,000 periods) from the normal distribution with mean zero
and standard deviation (A) of 0.00457. Use these values for “A;t and equation 1 to generate a time
series for at. Then use this time series for at and equation 2 to generate time series for the other
variables. Assume the economy starts in steady state (k1 = 0 and a0 = 0) and discard the
rst
1000 observations in the generated time series so that the starting values might be thought of as
reasonable. Use the generated time series for output etc to calculate variances and correlations and
compare these with the stylized facts below8.
Im sure youll be please to see that much of this has been done for you on the Monte Carlo tab in
the spreadsheet. Dont worry if the numbers on this tab seem to change every time you do anything,
new random shocks are being drawn.
Complete this table
The mo del matches the stylized fact
strongly weakly not at all
1 Investment is much more volatile than GDP
3 Employment (here hours worked) is less volatile than GDP
4a Private consumption is strongly p ositively correlated with GDP
4b Investment is strongly p ositively correlated with GDP
5a Employment (here hours worked) is pro cyclical and
5b more strongly correlated with GDP than real wages and output p er hour worked
5c Output p er hour worked is pro cyclical
5d Real wages are weakly correlated with GDP
7 Employment (here hours worked) is a lagging variable
8 GDP is p ersistent and consumption p ersistence is ab out the same
9 Employment tends to b e more p ersistent than GDP
10a Lab our input varies in a strong pro cyclical manner, explaining most of the output gap
10b TFP also varies pro cyclically
6Alternatively entering =NORMINV(RAND(),0,0.0032) into 10,000 cells will generate 10,000 random numbers
from a normal distribution 0 and standard deviation 0.0032
7Given the other parameters, = 0:0028 implies a standard error of yt of about 0.0176, which corresponds to
empirical
ndings for the US economy.
8Thre is no need for you to use the HP
lter. The HP
lter is used on real world data to separate the data into the
trend and the cycle. The stylized facts are obtained from the resulting estimate of the real world business cycle. You
are being asked to compare how well the model is able to match the stylized facts (i.e. the real world), as represented
in the table.
5
3.2 How well does this version of the RBC model appear to describe
the economy and how might it be improved?9
Remember, all models are simpli
cations of the real world and so are “wrong” or “overly simplistic”.
You are being asked about how this simple RBC model matches the data and which of the underlying
assumptions might be altered in order to match the data better. You are not being asked to identify
assumptions you disagree with at a micro level, unless modifying those assumptions also enables the
model to match the data better. Its also worth noting that more complicated versions of this basic
model are generally used for actual analysis.
9Dont go overboard, this should follow from what youve done. No more than a couple of pages at most.
6
What Students Are Saying About Us
.......... Customer ID: 12*** | Rating: ⭐⭐⭐⭐⭐"Honestly, I was afraid to send my paper to you, but you proved you are a trustworthy service. My essay was done in less than a day, and I received a brilliant piece. I didn’t even believe it was my essay at first 🙂 Great job, thank you!"
.......... Customer ID: 11***| Rating: ⭐⭐⭐⭐⭐
"This company is the best there is. They saved me so many times, I cannot even keep count. Now I recommend it to all my friends, and none of them have complained about it. The writers here are excellent."
"Order a custom Paper on Similar Assignment at essayfount.com! No Plagiarism! Enjoy 20% Discount!"
