Answers to Assignment 3

ECON 222
Macroeconomic Theory I
Winter 2014
Assignment 3
Due: Drop Box 2nd Floor Dunning Hall by March 14, 2014 at noon
No late submissions will be accepted
No group submissions will be accepted
No “Photocopy” answers will be accepted
Remarks: Write clearly and concisely. Devote some time to give the graphs, plots and tables a format
easy to understand.
The assignment is on 100 points. Questions 1 and 3 are worth 30 points each and Question 2
is worth 40 points
Question 1: Solow-Swan model (30 Points)
Assume that Canada has the following production function:
Y = AK α N 1−α
where Y is real output, K is real stock of capital, N is the number of workers and α is the elasticity of output
with respect to capital and it is equal to 0.3. Also assume that savings are proportional to real output i.e.
S = sY .
Make the (false) assumptions that Canada is a closed economy and there’s no government.
Part A
i) Derive algebraically an expression for the capital-labour ratio in the steady state, k ss and the consumption
per capita in the steady state, css .
Production function in per capita term: y = Ak α
We know that the steady state is such that Investment equals depreciating capital + equip new workers and
that investment equals savings (closed economy): I = sY = (n + d)K
In per capita term we get:
α
sy = sAk = (n + d)k ⇒ k
1−α
sA
=
⇔ k ss =
n+d
sA
n+d
1/(1−α)
Consumption = Output - Savings:
css = (1 − s)y = (1 − s)Ak α = (1 − s)A
sA
n+d
α/(1−α)
ii) Derive algebraically an expression for the golden rule capital-labour ratio, k G and the consumption per
capita at the golden capital-ratio, cG
Golden rule is such that MPK = n+d:
∂y
= αAk α−1 = n + d ⇒ k G =
∂k
αA
n+d
1/(1−α)
And the consumption is:
G
G
G α
G
c = y − (n + d)k = A(k ) − (n + d)k = A
αA
n+d
α/(1−α)
− (n + d)
αA
n+d
1/(1−α)
Part B
Use the data from the table below to answer the following questions:
Year
2010
2011
2012
Real Capital (Billion $)
1815
1858
1929
Real GDP (Billion $)
1593
1634
1662
Labour Force (Billion)
0.01704
0.01731
0.01751
Total Factor Productivity
2900
2923
2916
iii) Use the capital accumulation formula (i.e. It = Kt+1 − (1 − d)Kt ) to find It . Use the year 2012
for t + 1 and 2011 for t. Assume that d, the depreciation rate is 10%.
It = I2011 = 256.8
iv) In a closed economy, you know that St = It . Now assume that total saving is proportional to real
output (i.e. St = sYt ). Find the saving rate. Note that we will assume that s is constant over time.
s = 0.157 (15.7%)
v) Use the number of workers in 2010 and 2011 to find the growth rate of the employed workers, n. As
in iv) assume that this rate is constant over time.
n = 0.0158 (1.58%)
vi) Using your results of part i), find the steady state capital-labour ratio, k ss and the consumption per
capita in the steady state, css . Where is k 2011 compared to k ss ? What are the implications?
k ss = 138, 172.1
css = 85, 842.0
k 2011 = 107, 336.8
It means that the Canadian economy is still in the transition phase i.e. The economy is growing at a
rate greater than the steady state growth rates.
vii) Using your results of part ii), find the golden rule capital-labour ratio, k G and the consumption per
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capita at the Golden rule, cG .
k G = 347, 962.3
cG = 94, 056.0
ix) What would be the necessary saving rate such that the Canadian steady state is at the Golden rule?
We want to find s such that k ss = k G . From i) and ii) we can see that this is the case when s = α = 0.3
Question 2: Solow-Swan model with Government (40 Points)
There’s an island on which lives a dictator. Assume the production function per capita follows a CobbDouglas:
yt = Aktα
where α = 0.3 and A = 10. In addition assume that the growth rate of the working force, n, is 2%, the
depreciation rate, d, is 10% and the constant saving rate is 25%.
Part A
Assume that the dictator taxes his population at a rate proportional to output and uses these government
earnings to bribe supporters in other countries. In other words, the government spending is unproductive.
Note that Gt = τ Yt and that disposable income is now (1-τ )Yt . Where τ = 20%
i) Derive algebraically an expression for the capital-labour ratio in the steady state, k ss and the consumption
per capita in the steady state, css .
We still have that Investment = Savings. However, savings are now proportional to disposable income:
1/(1−α)
)sA
.
S = s(1 − τ )Y . Therefore in the steady state we have: s(1 − τ )Ak α = (n + d)k ⇒ k ss = (1−τ
n+d
css = y −i−g = y −s(1−τ )y −τ y = (1−τ )(1−s)y = (1−τ )(1−s)A(k ss )α = (1−τ )(1−s)A
(1−τ )sA
n+d
α/(1−α)
Using the numerical values we get:
css = 20.04
k ss = 55.65
Part B
Seeing that their taxes are used for bribes, the workers started revolting and a civil war ensued. 25%
of the workers died during this war.
ii) Assuming that initially the economy was at its steady state and no capital was destroyed during the
war, what happen to k, y and c right after the war?
K1 = capital before war and K2 = capital after war. N1 labour before war, N2 labour after war.
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We know that
Therefore
K2
N2
K1
N1
=
= 55.65. We also know that K2 = K1 and N2 = 0.75N1 .
K1
0.75N1
=
55.65
0.75
= 74.2
y2 = A(k2 )α = 10 ∗ (74.2)0.3 = 36.40
c2 = (1 − s)(1 − τ )A(k2 )α = 0.8 ∗ 0.75 ∗ 10 ∗ (74.2)0.3 = 21.84
Consumption and the capital-labour ratio increased right after the war.
iii) Draw a graph of this economy. Clearly indicate where is the steady state and where are the capital-labour
ratio and the saving per capita after the war. Secondly, show where would be the steady state if there was
no tax (τ = 0).
Part C
Assume that Part B never happened. Seeing that the population is unhappy, the government now decides to use its earnings to build roads, bridges, a port, etc... In other words, the government spending is
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now productive. The economy’s per capita production function becomes:
yt = Aktα gtβ
where gt is the per capital government spending. α = 0.3 and β = 0.1. Note that the government runs a
balanced budget.
iii) Derive algebraically an expression for the capital-labour ratio in the steady state, k ss and the consumption per capita in the steady state, css .
First we know that G = τ Y therefore g = τ y. We can rewrite the production function as:
yt = Aktα (τ y)β ⇔ yt = Ak α τ β
As in part A. we have: s(1 − τ )y = s(1 − τ ) Ak α τ β
Therefore k ss =
1/1−β
s(1−τ )(Aτ β )
n+d
1/(1−β)
1/(1−β)
= (n + d)k
(1−β)/(1−α−β)
And css = (1 − τ )(1 − s)y = (1 − τ )(1 − s)(A(k ss )α τ β )1/(1−β)
Numerical values:
k ss = 76.37
css = 27.49
Question 3: Money and Prices (30 Points)
Assume that the real money demand is:
Yα
Md
= β
P
i
where Y is real output, P is the price level and i is the nominal interest rate. α = 0.5 and β = 0.3. The
money supply (M s ) is controlled by the central bank.
i) Derive the elasticity of real money demand with respect to (a) real output, (b) the nominal interest
rate.
d
Elasticity of real output =
∂ MP
∂Y
Y
Md
P
=
αY α−1
Y
iβ
M d /P
= α. Same way for the nominal interest rate = −β
ii) Assume that the asset market is in equilibrium and that the nominal interet rate is fixed. Use the
formula for the growth rate of prices in term of money growth and real income growth to answer the following questions:
The formula is:
∆P
∆M
∆Y
=
− ηY
P
M
Y
(a) Assume that the real money demand is held constant, what will happen to prices if the money supply
is doubled? What is the name of this concept? If the economy were to take 10 years to adjust to the increase
in the money supply, what would be the annual rate of inflation?
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If
∆Y
Y
= 0 then
∆P
P
=
∆M
M .
So the prices are doubled.
The inflation rate during these 10 years is 100% however the annual inflation rate is be: (1 + π) = (2)1/10 ⇔
π = 0.0718 (or 7.18%).
(b) Now assume that the money supply is doubled and the real output also doubles. Find the inflation rate. (Hint: Use the elasticity you found in i)
∆P
= 100% − 0.5 ∗ 100% = 50%
P
iii) Assume that Y = 100, the real rate of interest is 2.5% and the price level is 1.0. What would be the
expected inflation rate and the nominal rate of interest if the real money supply were 24.56?
Start with the money demand equation and the definition of the nominal rate of interest, which implies:
M
Yα
= β ⇒ i = (π + r) =
P
i
Yα
M/P
1/β
=
1000.5
24.56
1/0.3
≈ 0.05 ⇒ π ≈ 0.025
iv) Given the above money demand function, write an analytical expression for the velocity of money. Does
this expression differ from what you would get using the quantity theory of money? If so in what way would
it differ? Based on what you found in iii) what is the value of velocity?
Start with the definition of velocity as:
V =
PY
PY
= α β = Y 1−α iβ
M
Y /i
The above differs from the quantity theory in two important respects: First, velocity is not constant; it
varies positively with both income and the interest rate over the cycle. Second, it is sensitive to the interest
rate, which does not appear as an argument in the quantity theory of money demand. Given Y=100 and
i=r=π = 0.05, the velocity is 4.0609.
v) The central bank has an inflation target of 2% and real potential growth is 1.5%.
(a) What must the Bank do to achieve this target over the long run? Briefly explain the reasoning
behind the banks actions.
Based on the money demand function, inflation would be:
∆P
∆M
∆Y
=
− ηY
P
M
Y
which implies that the growth in the money supply would have to be:
∆M
∆P
∆Y
=
+ ηY
= 2.0% + 0.5 ∗ 1.5% = 2.75%
M
P
Y
The Bank has to supply enough money to meet transactions demand (Y) as well its inflation target of 2%.
(b) Because of innovations in the banking system, individuals can now hold smaller money balances
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to do their transactions. In particular, the elasticity of money demand with respect to real income falls by
50%. How will this change affect central bank actions?
The improvement in efficiency would lower to 0.25. The given the inflation target of 2% then the new
money growth target would be:
∆M
∆P
∆Y
=
+ ηY
= 2.0% + 0.25 ∗ 1.5% = 2.375%
M
P
Y
Individuals would no longer need to hold as much money as before.
(c) Suppose that at the same time that there were efficiency improvements in the banking system,
the government introduced new policies that raised potential growth from 1.5 to 2.0%. What should the
bank do to maintain its inflation target?
Given a higher potential growth rate as well as the efficiency gains, the new money supply target becomes:
∆M
∆P
∆Y
=
+ ηY
= 2.0% + 0.25 ∗ 2.0% = 2.5%
M
P
Y
The bank would have to increase the growth rate if money to both accommodate the new, higher potential
growth rate and at the same time maintain its inflation target.
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