WU The Inflation Rate & Growth Rate of Real GDPare Constants Essay

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Macroeconomics B: Assignment 2 Similan Rujiwattanapong Waseda University January 21, 2022 • Your work must be submitted on the course’s Moodle page by Friday 28th January 2022. • Every group member must in principle submit the same copy of their joint work on Moodle. • The first page should contain the names and student numbers of all group members. • The main texts must be typed and not be handwritten (with the exception of diagrams). • The answers should be no more than 20 pages. They should be in one single PDF file. • To receive good marks, you must complete all exercises and answer all questions (read the questions carefully). Furthermore, your arguments must be complete and precise. For mathematical questions, you must demonstrate how you obtain your results in each step. Be careful to label every curve and axis, and use the notations as given in the questions. 1 Question 1 Consider an economy where the nominal interest rate (i), the inflation rate (π) and the growth rate of real GDP (g) are constant unless otherwise stated. (1.1) Interpret the following equation and argue why it must hold: ∆Dt+1 = Dt+1 − Dt = Gt − Tt + rDt where Dt is the real government debt at time t, Gt is the real government expenditure at time t, Tt is the real tax revenue at time t and r is the real interest rate. Subsequently, show that the law of motion for the debt-to-GDP ratio (dt ≡ ∆dt+1 = Dt Yt ) is Gt − Tt + (r − g ) d t Yt (1.2) Assume that there is a primary budget deficit and that the real interest rate is strictly smaller than the growth rate of real GDP. Illustrate the debt(-to-GDP ratio) dynamics in the (dt , ∆dt+1 )-space. In addition, calculate and interpret the debt-to-GDP ratio in the steady state/equilibrium. (1.3) Now consider a scenario where the government increases its expenditure even further from its position in (1.2) to attract/please the voters. Also assume the debt-to-GDP ratio is initially large and above the steady-state debt-to-GDP ratio. This action from the government raises doubts about its ability to repay its debt and, as a result, the interest rate on the government bonds increases to the point where it is higher than the growth rate of real GDP. Illustrate and explain the debt(-to-GDP ratio) dynamics in the (dt , ∆dt+1 )-space. What happens to the growth rate of the debt-to-GDP ratio? 1 (1.4) (This question continues from (1.3)) In response to this interest rate hike, the government tries to stimulate the economy but they only manage to bring the growth rate of real GDP to be equal to the real interest rate. Illustrate and explain what happens to the debt-to-GDP ratio in the (dt , ∆dt+1 )-space. Lastly, can the government stabilise the debt-to-GDP ratio in this scenario? Explain. 2 Question 2 Consider a small open economy with a fixed exchange rate regime. It can be summarised by the following system of equations: = C (Y − T, Y e − T e , i − π e , A) + I (Y e , i − π e , K ) + G + NX (ε, Y, Y ∗ ) Y = V (i ) ee 1 + i ∗ = (1 + i ) e T = = T + τY Y M P Assume the exchange rate target (e⊗ ) is credible, i.e. ee = e⊗ = e, and that the economy is initially in a short-run equilibrium where prices are fixed. T can be interpreted as a lump-sum tax and τ as an income tax rate. (2.1) Interpret the four equations and illustrate the model graphically. Is it possible for the central bank to have full control over the domestic nominal interest rate in this setup? Explain carefully. (2.2) Analyse graphically, verbally and mathematically the effects of a devaluation of the exchange rate. Provide intuitions behind your answer. Lastly, how does the value of τ affect your answer? You can assume that the exchange rate target is always credible in this question (2.2). (2.3) A currency devaluation comes with a risk of changes in the expected future exchange rate. Analyse graphically, verbally and mathematically the effects of an expected devaluation of a currency in the future. Provide intuitions behind your answer. You can answer this question independently from the situation in (2.2). Hint: You need to consider how the UIP condition (along with the variables inside this condition) may be affected in this scenario. Particularly, it will be useful to recall which variable(s) are fixed. (2.4) Compute mathematically the effect of an exogenous increase in government expenditure on real GDP and compare its magnitude to that from a closed economy setup (assuming the same tax system as in this question). Does the behaviour of the central bank in the closed economy setup matter for your answer? Explain. 2
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Explanation & Answer

Please view explanation and answer below.

1

Macroeconomics B: Assignment 2
1. Question 1
Consider an economy where the nominal interest rate (𝑖), the inflation rate (𝜋) and the
growth rate of real GDP (𝑔) are constants unless otherwise stated.
1.1 Interpret the following equation and argue why it must hold:
∆𝐷𝑡+1 = 𝐷𝑡+1 − 𝐷𝑡 = 𝐺𝑡 − 𝑇𝑡 + 𝑟𝐷𝑡
Where 𝐷𝑡 represents the real government debt at time 𝑡, 𝐺𝑡 is the real government
expenditure at time 𝑡, 𝑇𝑡 is the real tax revenue at time 𝑡 and 𝑟 is the real interest rate.
Subsequently, show that the law of motion for the debt-to-GDP ratio (𝑑𝑡 ≡
∆𝑑𝑡+1 =

𝐷𝑡
𝑌𝑡

) is

𝐺𝑡 − 𝑇𝑡
+ (𝑟 − 𝑔)𝑑𝑡
𝑌𝑡

Solution
Taking a macroeconomic point of view, it is sensical to acknowledge the fact that an
escalation in the government expenditure such that 𝑮∆𝒕 > 𝟎 will eventually result into
the stated government deficit cutting in with an equivalent proportion such that
∆(𝑻 − 𝑮) = −𝑮 < 𝟎
Another indistinguishable case of a similar economic impact is a cut in tax represented
as ∆𝑻 < 𝟎 which would follow into a splendid rise in government deficit by a proportion
that can be illustrated as
∆(𝑻 − 𝑮) = ∆𝑻 < 𝟎
A real time instant for this exposition is the United States’ Trump tax cuts. The
ramification of this tax cut is an evident upsurge of the United States public debt.
However, the Trump tax has stood as an enormous benefactor of sharp unemployment
declination as far as the U.S economic history is concerned following predictions by the
Keynesian prediction model.
Taking 𝑫𝒕 to denote the real government debt at time 𝒕, 𝑮𝒕 representing the real
government expenditure at time 𝒕, while 𝑻𝒕 represents the real tax revenue at
time 𝒕 and 𝒓 is the real interest rate, then the law of motion of government debt is
illustrated as
∆𝑫𝒕+𝟏 = 𝑫𝒕+𝟏 − 𝑫𝒕 = 𝑮𝒕 − 𝑻𝒕 + 𝒓𝑫𝒕
From the above equation, the term denoted by
𝑮𝒕 − 𝑻𝒕

2

denotes the proportion of the government primary deficit experienced within the given
period of time 𝒕. The law holds with relation to the instance given above. The change in
the government debt, denoted as ∆𝑫𝒕+𝟏 , gives the resultant government deficit and is
corresponding to the sum of the government primary deficit denoted as (𝑮𝒕 − 𝑻𝒕 ) and
the government interest payments as expressed by the relation
∆𝑫𝒕+𝟏 = 𝑫𝒕+𝟏 − 𝑫𝒕 = 𝒓𝑫𝒕 + (𝑮𝒕 − 𝑻𝒕 )
Rearranging the equations
𝑫𝒕 = 𝑫𝒕+𝟏 − 𝒓𝑫𝒕 − (𝑮𝒕 − 𝑻𝒕 )
From the above provided equation, the evolution of the law of motion for the debt –toGDP ratio (𝒅𝒕 ≡

𝑫𝒕
𝒀𝒕

) is given by substituting the above expression into the debt term as
𝑫𝒕 𝑫𝒕+𝟏 − 𝒓𝑫𝒕 − (𝑮𝒕 − 𝑻𝒕 )
=
𝒀𝒕
𝒀𝒕

The above expression can be decomposed by separation of terms into the expression

𝑫𝒕 𝑫𝒕+𝟏 − 𝒓𝑫𝒕 (𝑮𝒕 − 𝑻𝒕 )
=

𝒀𝒕
𝒀𝒕
𝒀𝒕
Denoting the debt to GDP ratio by the term 𝒅𝒕 the expression reduces to
𝒅𝒕 =
=

𝑫𝒕
𝒀𝒕

𝑫𝒕+𝟏 − 𝒓𝑫𝒕 (𝑮𝒕 − 𝑻𝒕 )

𝒀𝒕
𝒀𝒕

Rearranging the above equation
𝑫𝒕+𝟏 − 𝒓𝑫𝒕 (𝑮𝒕 − 𝑻𝒕 )
=
+ 𝒅𝒕
𝒀𝒕
𝒀𝒕
Assuming that the GDP grows at a rate 𝒈 then we have the expression
𝒀𝒕
=𝟏−𝒈
𝒀𝒕+𝟏
Then for a change in the debt over time 𝒕, ∆𝒅𝒕+𝟏
∆𝒅𝒕+𝟏
𝟏
𝑫𝒕+𝟏 − 𝒓𝑫𝒕
𝟏 (𝑮𝒕 − 𝑻𝒕 )
)
=(
=
.
+ 𝒅𝒕
𝒓−𝒈
𝒓−𝒈
𝒀𝒕
𝒓−𝒈
𝒀𝒕
Hence

3

∆𝒅𝒕+𝟏
𝟏 (𝑮𝒕 − 𝑻𝒕 )
=
.
+ 𝒅𝒕
𝒓−𝒈 𝒓−𝒈
𝒀𝒕
Multiplying through by (𝒓 − 𝒈) the expression above reduces to the law of motion for
the debt –to-GDP ratio (𝒅𝒕 ≡

𝑫𝒕
𝒀𝒕

) given by

∆𝒅𝒕+𝟏 =

𝑮𝒕 − 𝑻𝒕
+ (𝒓 − 𝒈)𝒅𝒕
𝒀𝒕

1.2 Assume that there is a primary budget deficit and that the real interest is strictly smaller
than the growth rate of real GDP. Illustrate the debt (-to-GDP ratio) dynamics in the
(𝑑𝑡 , ∆𝑑𝑡+1 ) −space. In addition, calculate and interpret the debt-to-GDP ratio in the
steady state/equilibrium.
Solution
Taking an assumption that a primary budget deficit denoted by (𝑮𝒕 − 𝑻𝒕 ) exists and
that the real interest is strictly smaller than the growth rate of real GDP, such that 𝒓 <
𝒈, the illustration of the debt (-to-GDP ratio) dynamics in the (𝒅𝒕 , ∆𝒅𝒕+𝟏 ) −space can be
determined from the law of motion for the debt –to-GDP ratio (𝒅𝒕 ≡
∆𝒅𝒕+𝟏 =

𝑫𝒕
𝒀𝒕

) given by

𝑮𝒕 − 𝑻𝒕
+ (𝒓 − 𝒈)𝒅𝒕 … … . 𝒆𝒒 𝟏
𝒀𝒕

The given circumstance that the real interest is strictly smaller than the growth rate of
real GDP, such that
𝒓 𝒈 , the debt to GDP ratio goes to infinity. That is, when the real interest is
greater than the growth rate of real GDP the debt to GDP ratio goes to infinity.

6

Case 2
b. If 𝒓 < 𝒈 , the debt to GDP ratio goes to zero. That is, when the real interest is
smaller than the growth rate of real GDP the debt to GDP ratio goes to zero.
This situation in the economic terms is termed as dynamic inefficiency.
Case 3
c. If 𝒓 = 𝒈 , the debt to GDP ratio stays constant. That is, when the real interest is
equal to the growth rate of real GDP the debt to GDP ratio stays constant as
illustrated above.
1.3 Now consider a scenario where the government increases its expenditure even further
from its position in (1.2) to attract/please the voters. Assume the debt-to-GDP ratio is
initially large and above the steady state debt-to-GDP ratio. This action from the
government raises doubts about its ability to repay its debt and, as a result, the interest
rate on the government bonds increases to the point where it is higher than the growth rate
of real GDP. Illustrate and explain the debt (-to-GDP ratio) dynamics in the
(𝑑𝑡 , ∆𝑑𝑡+1 ) −space. What happens to the growth rate of the debt-to-GDP ratio?
Solution
There are times when the government increases its expenditure with the intention of
attracting or pleasing voters. The assumption here is that the debt-to-GDP ratio is
initially large and above the steady state debt-to-GDP ratio. This action from the
government is prone to raising doubts about its ability to repay its debt and, as a result,
the interest rate on the government bonds increases to the point where it is higher than
the growth rate of real GDP.
For this case where the real interest takes a higher value compared to the real growth
rate of the GDP, the government economy has to have a primary surplus. This should
be such that the government should have an income of greater value that its expenses
and its obligations of interest. This is from the view that the government might own a
surplus on the offset of meetings its expenses and still attain an interest obligation
towards the value of borrowed money.
When the government increases its expenditure and the GDP goes splendidly high,
investors or voters are attracted. The voters similarly demand appreciable value of
investment returns taking the form of investment interests with regards to the increase
in the interest rates. The increase in GDP is a key conducive breeding ground for higher
chances of the government’s economy suffering an inflationary trend which can be
economic depressing.
As a mechanism of controlling the booming increase in GDP, the government tends to
counter the excessive investment through declaration of hiked interest rates on the
government bonds to curb investment and put a control on the possible inflation rise.
The resultant debt to GDP ratio is represented as

7

𝟏 + 𝒓 𝒕−𝒕𝟎
)
𝒅𝒕 = (
𝒅𝒕𝟎
𝟏+𝒈

{𝒕 ≥ 𝒕𝟎 }

If 𝒓 > 𝒈 , the debt to GDP ratio goes to infinity. That is, when the real interest is greater
than the growth rate of real GDP the debt to GDP ratio goes to infinity.
In this case, the GDP set to decrease, hence lowering the debt-to-GDP ratio growth rate
as related by the law of motion for the debt –to-GDP ratio (𝒅𝒕 ≡
∆𝒅𝒕+𝟏 =

𝑫𝒕
𝒀𝒕

) given by

𝑮𝒕 − 𝑻𝒕
+ (𝒓 − 𝒈)𝒅𝒕
𝒀𝒕

When the interest rate on the government bonds increases to the point where it is higher
than the growth rate of real GDP, disproportionately larger magnitude shares of the
government’s national income are being acquired by the savers. The government’s
producers, speculators and borrowers undergo effective taxation in effort to subsidize
the owners and savers of the monetary value assets.
1.4 This question continues from 1.3. In response to this interest rate hike, the government
tries to stimulate the economy but they only manage to bring the growth rate of real GDP
to be equal to the real interest rate. Illustrate ad explain what happens to the debt-to-GDP
ratio in the (𝑑𝑡 , ∆𝑑𝑡+1 ) −space. Can the government stabilise the debt-to-GDP ratio in
this scenario? Explain.
Solution
As a mechanism of controlling the booming increase in GDP, the government tends to
counter the excessive investment through declaration of hiked interest rates on the
government bonds to curb investment and put a control on the possible inflation rise.
The government puts in effort to stimulate the economy. There are cases where it only
manages to bring the growth rate of real GDP to be equal to the real interest rate,
illustrated as follows. The assumption is that future primary surpluses are taken to
amount to zero after time 𝒕 ≥ 𝟎. This is illustrated as
𝑮𝒕 = 𝑻𝒕

{𝒕 ≥ 𝟎 }

Hence for a debt-to-GDP ratio in the steady state/equilibrium, the law of motion for the
debt –to-GDP ratio (𝒅𝒕 ≡
𝒅𝒕 =

𝑫𝒕
𝒀𝒕

) reduces as shown below

𝑫𝒕+𝟏 − 𝒓𝑫𝒕 (𝑮𝒕 − 𝑻𝒕 ) 𝑫𝒕+𝟏 − 𝒓𝑫𝒕 (𝑮𝒕 − 𝑻𝒕 ) 𝟎

=

=
=𝟎
𝒀𝒕
𝒀𝒕
𝒀𝒕
𝒀𝒕
𝒀𝒕

Now
𝒅𝒕 =

(𝑮𝒕 − 𝑻𝒕 )
𝑫𝒕+𝟏 − 𝒓𝑫𝒕
𝒔𝒊𝒏𝒄𝒆
=𝟎
𝒀𝒕
𝒀𝒕

𝒇𝒓𝒐𝒎 𝑮𝒕 = 𝑻𝒕

8

For simplicity of interpretation his can be rewritten as
𝒅𝒕 =

𝟏+𝒓
𝒅
𝟏 + 𝒈 𝒕−𝟏

The resultant debt to GDP ratio is represented as
𝟏 + 𝒓 𝒕−𝒕𝟎
)
𝒅𝒕 = (
𝒅𝒕𝟎
𝟏+𝒈

{𝒕 ≥ 𝒕𝟎 }

The interpretation of the debt-to-GDP ratio in the steady state/equilibrium follows that
if the growth rate of real GDP becomes equal to the real interest rate, 𝒓 = 𝒈 , the debt
to GDP ratio stays constant. That is, when the real interest is equal to the growth rate of
real GDP the debt to GDP ratio stays constant as illustrated above.

2

Question 2

Consider a small open economy with a fixed exchange rate regime. It can be summarised by
the following system of equations:
𝑌 = 𝐶(𝑌 − 𝑇, 𝑌 𝑒 − 𝑇 𝑒 , 𝑖 − 𝜋 𝑒 , 𝐴) + 𝐼(𝑌 𝑒 , 𝑖 − 𝜋 𝑒 , 𝐾) + 𝐺 + 𝑁𝑋(𝜀, 𝑌, 𝑌 ∗ )
𝑀
𝑌
=
𝑃 𝑉(𝑖)
𝑒𝑒
1 + 𝑖 = (1 + 𝑖)
𝑒


𝑇 = 𝑇̅ + 𝜏𝑌
Assume the exchange rate target (𝑒 ⊕ ) is credible, i.e. 𝑒 𝑒 = 𝑒 ⊕ = 𝑒, and that the economy is
initially in a short-run equilibrium where prices are fixed. 𝑇̅ can be interpreted as a lump-sum
tax and 𝜏 as an income tax rate.
2.1 Interpret the four equations and illustrate the model graphically. Is it possible for the
central bank to have full control over the domestic nominal interest rate in this set-up?
Explain carefully.
Solution
Equation 1 is given as:

9

𝒀 = 𝑪(𝒀 − 𝑻, 𝒀𝒆 − 𝑻𝒆 , 𝒊 − 𝝅𝒆 , 𝑨) + 𝑰(𝒀𝒆 , 𝒊 − 𝝅𝒆 , 𝑲) + 𝑮 + 𝑵𝑿(𝜺, 𝒀, 𝒀∗ )
The first equation shown above is a representation of a summarised simple national
income determination Keynesian process. The equation is a description of the goods
marks equilibrium. The variables 𝒀 denotes the national income, 𝑪 denotes the private
consumption expenditure as a function of disposable income. The term 𝑻 denotes the
taxes while 𝑰 denotes the private investment as a function of interest rate. The
term 𝑮 denotes the total government expenditures. The term 𝑵𝑿 denotes the net exports
and current account balance of the international balance of payments as a function of
foreign exchange rate, national income and foreign income. The term 𝜺 denotes the
foreign exchange rate which is the number of units of foreign currency per unit of the
national income. An escalation in this variable is equivalent to an appreciation of the
national currency in regards to the foreign currency. The term 𝒀∗ denotes foreign
income. The terms 𝑮 and 𝑻 variables are dependent on the fiscal policy.
The income decomposition follows components of expenditure into private
consumption, private investment, government expenditures that are given exogenously,
and the foreign current account balance. Private consumption can be termed a positive
function of income that is disposable i.e. the difference between income and taxes. A
negative function of interest rate, private investment is included. Foreign current
account balance stands as of negative function of the level of exchange rate and the
national income. It serves as a positive function of the foreign income though.
Equation 2 is given as follows:
𝑴
𝒀
=
𝑷 𝑽(𝒊)
It describes the money market equilibrium, a typical money demand function of the
Keynesian model. There is a positive relation between money demand and income while
the demand for money is negatively related to the rate of interest. The term 𝑴 , a
significant exogenous variable, denotes the monetary policy variable. The term 𝑷 is
price.
Where the interest rate is provided, the intersection of the curves given by equation 1
and 2 will determine the national income level at which the money market and the
goods market reach equilibrium. The exchange rate is similarly determined by the
equilibrium of these two markets for the variable exchange rate system.
Equation 3 is given as shown below:
𝟏 + 𝒊∗ = (𝟏 + 𝒊)

𝒆𝒆
𝒆

The equation relates the exchange rate target dynamics.
Equation 4 is given as shown below:

10

̅ + 𝝉𝒀
𝑻=𝑻
̅ and the product of income tax rate
This equation relates taxes 𝑻, to lump-sum tax, 𝑻
𝝉 and the national income denoted as 𝒀.
The IS curve provided by the first equation is as shown below

The graph depicts the equilibrium in the goods market (𝑰𝑺), the money market (𝑳𝑴)
and the balance of payments (𝑩𝑷).
𝒀 is the total output or total economy income. When the quantity of goods and services
demanded attain equity with the output of goods and services, the equilibrium occurs.
This is illustrated by the intersection point of the goods market (𝑰𝑺), the money
market (𝑳𝑴) curves shown above.
In this setup, the central bank has no full control over the domestic nominal interest
rate. The domestic nominal interest rate which is the interest rate before the accounting
of in a small open economy with a fixed exchange rate regime is not fully regarded as
the central bank often targets the general exchange rate level.
2.2 Analyse graphically, verbally and mathematically the effects of a devaluation of the
exchange rate. Provide intuitions behind your answer. Lastly, how does the value of
𝜏 affect your answer? You can assume that the exchange rate target is always credible in
this question 2.2
2.3 Currency devaluation comes with a risk of changes in the expected future exchange rate.
Analyse graphically, verbally and mathematically the effects of an expected devaluation
of a currency in the future. Provide intuitions behind your answer. You can answer this
question independently from the situation in 2.2. Hint: You need to consider how the UIP

11

condition (along with the variables inside this condition) may be affected in this scenario.
Particularly, it will be useful to recall which variable(s) are fixed.
Solution for 2.2 and 2.3
A small open economy adopting a fixed exchange rate regime is not an economic
assurance that the exchange rate will always be constant. Parity or official exchange
rate changes are frequent. The official exchange rate devaluation follows a tariff-like
operation such that it shifts off the foreign onto domestic output the world of demand
for goods and services.
The devaluation of the exchange rate defines a deliberate effort of the official exchange
rate downward adjustment such that the currency’s value decreases. This results in the
domestic currency being cheaper in relation to other currencies. Devaluation
discourages importation such that it brings about a situation where the foreign products
are relatively more expensive for the country’s domestic consumers. This in turn
encourages exportation such that it makes the country’s exports relatively cheaper for
other foreigners.
The overall result is a push to decrease importation and escalate the exportation in the
given country, a situation where current account deficit may be reduced.
Rewriting the first and second equations:
𝒂 + 𝜹 + 𝜱𝑩𝑻 + 𝑫𝑺𝑩
𝝁
𝒎∗
𝝈
𝒀=

+ ∗

𝒔+𝒎
𝒓(𝒔 + 𝒎) 𝒀 (𝒔 + 𝒎) 𝑸(𝒔 + 𝒎)
𝟏 𝑴
𝜺
𝒓∗ = − ( + 𝜱𝑴 ) − 𝝉 + ( ) 𝒀
𝜽 𝑷
𝜽
Here, 𝒓∗ denotes the domestic real interest rate that is determined by world market
conditions. 𝑴 defines the nominal money stock, 𝜱𝑩𝑻 is the exogenous shocks to the
trade balance. 𝒎∗ and 𝒎 denote the foreign and domestic marginal properties to import
while 𝝉 represents the expected rate of domestic inflation. 𝑫𝑺𝑩 is the debt service
balance while 𝑸 is the real e...


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