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ECMT2160: Computational Assignment
创建日期：2024-06-05 14:53:18
所属分类：数学mathematics
标签： ECMT2160
Computational Assignment

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ECMT2160: Computational Assignment

‘is assessment task requires you to use Matlab to perform some calcula-
tions based on a simple economic model. You should prepare your submission
as a Matlab Live Script €le (i.e., a .mlx €le). Submit your answers through the
Canvas course website. Your submission should include a mixture of wri‹en re-
sponses forma‹ed as text, blocks of Matlab code, andMatlab output, including
graphs. You should submit two versions of your answers: the original .mlx €le,
and a version exported to .html.
You may work on this assessment individually, or in pairs. If you work in
pairs, it is important that you clearly indicate the student ID number of your
partner in your submission. Your submission should not be identical to your
partner’s submission.
Answer all questions. ‘e assignment is worth a total of 25 points towards
your €nal assessment. Points will be deducted for poor presentation, including:
excessive typos, poor wri‹en expression, poor organization, etcetera.
In the questions you will be presented with an economic model which you
have not been given sucient information to fully understand, and equations
you have not seen before. ‘is is intentional, and a common situation when
providing quantitative support to research teams. ‘ere is enough information
provided to carry out the required computations.
1
‘is assignment is based on a simpli€ed version of an economic model de-
veloped in 2010 by an economist working for the US Federal Reserve Board. You
can €nd the original FRB discussion paper here if you are interested, but reading
it will be dicult and is unlikely to help you solve the questions below.
Imagine you are a research assistant working for the US Federal Reserve
Board in 2010. Your boss, Dr. Bernanke, wants you to calculate some numbers
based on an economic model he €nds interesting. ‘is model has a number of
parameters. You should begin by entering these intoMatlab.
β = 0.96 (annual discount rate of agents)
γ = 0.12 (fraction of agents who own a business)
α = 0.36 (capital share parameter in Cobb-Douglas production function)
δ = 0.08 (annual depreciation rate of capital)
υ = 0.975 (annual probability that an agent does not retire)
pi1 = 0.003 (annual probability that a worker starts a business)
pi2 = 0.02 (annual probability that a business closes)
τL = 0.3 (average labor income tax rate)
τK = 0.4 (average capital income tax rate).
Also enter the following 2 × 1 vector p and 2 × 2 matrix Π intoMatlab:
p =
[
1 − γ
γ
]
, Π =
[
1 − pi1 pi1
pi2 1 − pi2
]
.
Finally, enter the following simple 2 × 1 vectors intoMatlab:
e1 =
[
1
0
]
, e2 =
[
0
1
]
.
To answer the following questions youwill need to know how to de€ne functions
inMatlab, and how to use the command fsolve inMatlab to solve systems
of nonlinear equations. You can read about how to do this in a separate document
I have made available in Canvas, titled Using fsolve.mlx.
2
1. (2 points) Create a function inMatlabwhich takes a positive real number
w as an input, and returns the following 2 × 1 vector h(w):
h(w) =
[ −(1 − τK)δ
(1 − τK)
(
α
( 1−α
w
) 1−α
α − δ
)] .
Report your calculation of the vector h(1.2).
2. (2 points) Create a function inMatlabwhich takes two positive real num-
bers R andw as inputs, and returns the following 2 × 2 matrix A(R,w):
A(R,w) = βΠ
[
R R
1 − (1 − τK)δ 1 + (1 − τK)
(
α
( 1−α
w
) 1−α
α − δ
)] .
Here, the symbol refers to the entrywisemultiplication ofmatrices, which
we implement inMatlab using the “.∗” operation. It does notmean ordinary
matrix multiplication, implemented inMatlab using the “∗” operation.
Report your calculation of the matrix A(1.04, 1.2).
3. (2 points) Dr. Bernanke tells you that the variablesR andw in your function
A(R,w) refer to the interest rate (plus one) and the average wage in the
economy. For instance, when R = 1.04, this corresponds to an interest rate
of 4%. ‘e average wagew is measured in multiples of $50,000, so ifw = 1
then the average (annual) wage is $50,000.
Dr. Bernanke also tells you that, in his model, the aggregate excess supply
of money is the following function of R andw :
f (R,w) = βRw(1 − υ)(1 − τL)
υ(R − υ) p
>(I2 −A(R,w))−1e1 − (1 − τL)w
R − υ .
Here, the notation I2 refers to the 2 × 2 identity matrix.
Create a function inMatlabwhich takes two positive real numbers R and
w as inputs, and returns f (R,w). Report your calculation of f (1.04, 1.2).
4. (2 points) Dr. Bernanke tells you that, in his model, the aggregate excess
demand for labour is the following function of R andw :
д(R,w) = βRw(1 − υ)(1 − τL)
υ(R − υ)
(
1 − α
w
)1/α
p>(I2 −A(R,w))−1e2 − 1.
Create a function inMatlabwhich takes two positive real numbers R and
w as inputs, and returns д(R,w). Report your calculation of д(1.04, 1.2).
3
5. (7 points) Dr. Bernanke tells you that his model is said to be in equilibrium
when the interest rate and wage are such that the aggregate excess sup-
ply of money and the aggregate excess demand for labour are both equal
to zero. In other words, in equilibrium, R and w must solve the pair of
nonlinear equations
f (R,w) = 0 (1)
д(R,w) = 0. (2)
Use the fsolve command inMatlab to €nd the values of R andw solv-
ing this pair of nonlinear equations. You may assume that there is exactly
one value of R and one value of w such that both equations are satis€ed.
Use R = 1.01 andw = 1 as starting values for fsolve.
6. (2 points) Let Req andweq be the unique values of R andw solving the pair
of nonlinear equations (1) and (2). Dr. Bernanke tells you that, in his model,
equilibrium aggregate tax revenue is given by
Teq = τLweq +
βReqweq(1 − υ)τK(1 − τL)
(Req − υ)(1 − τK) p
>(I2 −A(Req,weq))−1
((Πh(weq)) e2) .
Calculate the equilibrium aggregate tax revenue inMatlab.
7. (6 points) Dr. Bernanke tells you that he would like you to use his model to
explore the economic e‚ects of increasing the average capital income tax
rate. Speci€cally, he is interested in cases where τK is somewhere between
0.4 and 0.6, and wants to knowwhat the e‚ect would be on the equilibrium
interest rate and wage, and on equilibrium aggregate tax revenue.
Using a “for loop” in Matlab, recalculate Req, weq and Teq for every value
of τK in a €ne grid extending from 0.4 to 0.6. (Say, about 20 evenly spaced
points between 0.4 and 0.6.) Create three plots in Matlab showing how
Req,weq andTeq vary with τK. Set the options forfsolve so as to stop this
command producing a large amount of unnecessary output during your
loop.
8. (2 points) In your answer to the previous question, what was the general
e‚ect of raising the capital income tax upon the equilibriumwage? Do you
have any economic intuition which may help to explain this e‚ect?
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