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EC201 Intermediate Macroeconomics
EC201 Intermediate Macroeconomics

Lecture 31-32: Government Debt and Budget
Deficit
Lecture Outline:
- Sustainability of government debt;
- Some unpleasant monetarist arithmetic;
- Ricardian equivalence;
Essential reading:
Mankiw: Ch
...
Political debates about the future course of
fiscal policy and the need to keep government debt under control and the
sustainability of public finances is probably the most widely discussed topic in
economics these days
...
To get the extra resources needed to finance
this extra expenditure the government must borrow money (normally by issuing
government bonds that are sold to investors)
...

Therefore the government debt is a stock variable while the budget deficit/surplus is
the flow that affects the stock
...
For example in UK in 2009 public
expenditure was 1
...
7 billion of dollars
...
3 billion of dollars
...
123
trillion of dollars and so the deficit was 11
...
This will then increase the UK
government debt in 2010
...


1

If we have a budget surplus ( G < T ) then next period government debt will decrease
...
The biggest parts of public spending in all developed
countries are represented by Pensions, Health, Welfare expenditure (unemployment
benefits, family support, etc
...
), Education and Interest payments on outstanding
debt
...
In those models the public expenditure was spending on goods and
services that can create extra income
...
The same applies for interest expenditure on outstanding debt
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13

Pensions

7
...
40

Education

5
...
1

Defence

2
...
ukspending
...
uk

The next figure shows the evolution of the government debt as a percentage of GDP
in UK
...
Notice that as public debt is measured in
current prices we divide it by the nominal GDP in each year to get the debt-GDP
ratio
...
Because of the credit crunch the
government debt has increased and it is now well above the 50% threshold
...

Sustainability of Government Debt
In the following table we report the government debt (as a percentage of GDP) in
various countries in 2009
...
6

Japan

192
...
8

Greece

113
...
6

Germany

73
...
2

Ireland

64
...
5

Spain

53
...
9

Source: CIA World Factbook

Why Ireland and Greece had a public debt crisis while Italy and Japan did not (yet)
3

even if they have bigger debt as percentage of GDP?
The problem is not only the size of the debt but it is the sustainability of that debt
...
The same applies to government debt
...
In that case governments may not be able to borrow money
in the market because investors think it is too risky to lend money to those them and
they will ask for very high interest rates to cover themselves from the risk of a
possible default (when the government decides not to pay its debt, like Argentina did
recently on a part of its external debt)
...
If a government cannot borrow it will face a debt
crisis (exactly like any other borrower)
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56%
...
29%
...

The fact that the spread increased so massively reflects two things:
a) Greek debt was considered very risky
...

Therefore investors asked for a high return on those bonds to be willing to
lend money to Greece;
b) The interest payments by the Greek government will increase and this will
increase even further the size of the debt, everything else constant;
To avoid the spiral above tight and painful fiscal policies (decrease in Government
Expenditure and/or increase in Taxes) should be put in place at certain point
...


4

The Government Budget Constraint
Define with Bt the level of government debt at the end of time t
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Seigniorage is not a very important source of revenues for government
in developed countries nowadays however it may be a relevant source of revenues in
less developed countries
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All variables are nominal
...
So the budget deficit
at time t is:
Def t ≡ ∆Bt = Bt − Bt −1

2)

If ∆Bt < 0 we have a budget surplus at time t
...

What equation 1) implies? Suppose that we start at time 0 and we decrease taxes by 1
keeping constant government expenditure for that period only (meaning at time 1
primary deficit will be zero)
...
In period 1 equation 1)
says:

B1 = (1 + i )1 + G1 − T1 − S1
If the debt is fully repaid at time 1 so that B1 = 0 we have:
0 = (1 + i )1 + G1 − T1 − S1
⇒ T1 − G1 + S1 = (1 + i )
The government must run a primary surplus or must monetize the debt to raise
seigniorage revenues
...
Then the government must run a primary surplus equal to
the interest payments on outstanding debt
...
Then the primary surplus required is going to be:
Tt − Gt = (1 + i ) t

5

Although taxes have been reduced only in period 0, the debt increases continuously
from year 0 onwards, at a rate equal to the interest rate
...

This example tells us that if government spending remains unchanged, a reduction of taxes
today must be offset by an increase in future taxes
...


The Evolution of the Debt-GDP Ratio
In analysing government debt we normally look at the debt-GDP ratio
...

First define the following:

Yt
= (1 + π )(1 + g )
Yt −1

4)

this is the ratio of nominal GDP at time t and at time t-1 and it depends on two
components since nominal GDP is real GDP multiplied by the general price level
...

See the Appendix of this lecture note for the derivation of 4)
...

Divide equation 1) by Yt :

Bt
B
D S
= (1 + i ) t −1 + t − t
Yt
Yt
Yt Yt

5)

Bt
B Y
D S
= (1 + i ) t −1 t −1 + t − t
Yt
Yt −1 Yt
Yt Yt

6)

Re-write equation 5) as:

define with b =

B
D
the debt-GDP ratio and with d =
the primary surplus-GDP
Y
Y

ratio and with s =

S
the seigniorage-GDP ratio
...
Now notice
(1 + π )(1 + g )

that i − π = r is the real interest rate
...


6

bt = (1 + r − g )bt −1 + d t − st
From that equation we can write the following:
bt − bt −1 = (r − g )bt −1 + d t − st
And by using the notation: ∆b = bt − bt −1 we have:
∆bt = (r − g )bt −1 + d t − s t

8)

Equation 8) describes the evolution of the debt-GDP ratio over time
...

The Steady State Level of Debt-GDP Ratio
The long run (steady state) equilibrium level of the debt-GDP ratio is defined as the
situation where the debt-GDP ratio is constant, meaning bt = bt −1 = b *
...
From equation 8) this means (since in steady state all variables are constant
we get rid of the time subscript):
(r − g )b * + d − s = 0
From which:

b* =

d −s
g−r

9)

This is the equilibrium level towards which the debt-GDP ratio should tend to in the
long run
...
If growth rate of real GDP

is higher than the real interest rate, the steady state level of debt-GDP ratio is
positive if the government is running a primary deficit bigger than the seigniorage
revenues
...


We can see the implications of equation 8) in a graph
...

Firs assume that: g > r and d − s > 0
...
The line intersects the horizontal axis at the steady state
level given by 9) (where ∆b = 0 )
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Suppose
we start with b1 < b *
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More interestingly suppose we start with a high level of
debt-GDP ratio b 2 > b *
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The debt-GDP ratio is stable when g > r
...
Nevertheless even if the debt-GDP
ratio is very high it will converge to the steady-state
...
Independently on what the initial level
of the debt-GDP is, it will converge to the steady state level
...
The economy will converge towards it
...
Then equation 8) implies a positively sloped
line with a negative intercept
...
In this case the steady
state is unstable
...
Suppose we
start with a debt-GDP ratio b1 < b *
...
In order to have that the government must
run a primary surplus d < 0 or at least have a deficit smaller than seignirioge
revenues
...
Then if we do
not do anything the debt-GDP ratio will explode over time
...


∆b

b*1

b2

b*2

b

1

(d-s)

(d-s)2

By doing that we shifts the line down and we move the steady-state, for example from

9

*

b *1 to b2
...

Result: if r > g then the government can stop the debt-GDP ratio to explode only by
running primary surpluses or running deficits less than seigniorage revenues
...


Conditions for Debt Sustainability
What do we mean by debt sustainability? There are various criteria that can be used to
define sustainability but we are going to focus on only two
...
Moreover a sustainability
condition makes sense only if r > g , otherwise the debt-GDP ratio is always stable
and so sustainable
...
Then a debt-GDP
ratio is sustainable if b * ≤ b
...

b) No Ponzi3 game condition: this condition says that a country cannot finance
all its outstanding debt by issuing other debt (meaning by borrowing)
...

In general it is easier to implement condition a) than b)
...
Moreover it does not say anything about the level of debt that a country
can run while condition a) clearly does
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A negative number for a primary balance
means a primary deficit (a positive number means a primary surplus)
...
He ended up in jail
...
A recent example of a Ponzi game scheme is the one made by
Bernard Madoff that was the largest financial fraud committed by a single person in history
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2

2
...
6

5

Ireland

-9
...
3

Spain

-6
...
6

UK

-9
...
2

Italy

Now we have framework to understand why Greece and Ireland had a debt problem
and why Italy did not (yet)
...

For example in Greece in 2009 we had the following: the growth rate of real GDP was
-2%, the inflation rate was 1
...
4% and primary deficit was
13
...
The interest spending as a percentage of GDP was
5%
...
Therefore if we divide ibt −1 by
bt −1 we get the unit cost of debt i
...
05
= 4
...
It was
1
...
This fact associated with a primary deficit made the
Greek debt explosive
...
Assume seigniorage is zero
...
134

(1 + 0
...
136 = 1
...
012)(1 − 0
...
This is consistent with the forecasts made by the IMF in May 2010 as
shown by the table below
...


11

Some Unpleasant Monetarist Arithmetic
When r > g and government debt is unstable there is an interesting link arising
between monetary and fiscal policy
...

Remember the steady state level of the debt-GDP ratio:
G −T − S 


Y


*
b =
g−r
If r > g and

10)

G −T − S
> 0 meaning the government is running a primary deficit then
Y

the debt-GDP ratio is explosive
...
In this case, monetary and fiscal policy cannot be chosen
independently of one another
...

Suppose also that there is a limit to the amount of government bonds the public can
hold (for example the public is willing to hold up to 50% of their wealth in
government bonds)
...

Moreover define the seigniorage revenues as: s =

∆M
that is the real growth rate of
P

money supply
...
If it is known that the
government wants to maintain a constant debt to GDP ratio at some point in the future
this may imply that money growth will have to be higher in the future to help pay off
the debt
...
If agents anticipate
this the result may be higher inflation now
...
This will
increase the cost opportunity of holding money ( i = r + π e )
...
So we have a paradoxical result: a restrictive monetary policy now may lead
to higher inflation now
...
Even if the result here may appear odd it points out a very
simple thing: governments with high debts and deficit may have an incentive to print

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money
...

Fortunately there is little evidence that the link between fiscal and monetary policy is
important
...


The Traditional View of Government Deficit and Ricardian Equivalence
Deficits become are a problem when they result in rapid accumulation of debt and
when the economy is experiencing low growth rates of real GDP
...
Deficits (and
surpluses) can help to redistribute the burden of taxation over time
...

We distinguish between the traditional view or Keynesian view of government deficit
and the Ricardian view
...

Traditional View
In the short run an increase in government deficit made by an increase of government
expenditure (in goods and services) or made by a decrease a taxes will increase real
output and a decrease in unemployment
...

Short Run: ↑ G or ↓ T → Y ↑ and u ↓ , r ↑ and I ↓
...
In an open economy
we also have that nominal exchange rate will increase e ↑ (if floating) and
the trade balance NX ↓ will decrease
...
Prices can be

permanently higher at the long run equilibrium but inflation will be zero
...
In an open
economy we have also that the real exchange rate will increase and trade
balance will decrease
...

Therefore the traditional says that a government deficit increases both aggregate
demand and interest rate in the short-run
...
4
The Rcardian view says that a government deficit should not have any effect on
aggregate demand and so on the interest rate
...
The effect on aggregate demand should be the same
...
There were two options: a temporary increase in
taxes now or borrowing money issuing government debt
...
Suppose that the government keeps government expenditure
constant and it decreases taxes and so it creates a deficit
...
Now the public
has an asset that pays them an interest rate
...

However suppose that the public realises that this increase in the deficit means that at
certain point in the future taxes must increase to pay off this deficit
...
Therefore total saving in the economy will not change
...
Therefore aggregate demand will not change and the
interest rate will not change
...

A tax cut today is equivalent to a tax increase tomorrow
...
The
holders of the government bonds view it as an asset
...


4

Robert J
...


82, 1095-1117
...
There is a sort of
“fiscal illusion” going on
...

Because they perceive an increase in the present value of their taxes that just
offsets deficit-financed expenditures, the stock of government bonds is not
part of their net wealth
...
According to Ricardian equivalence, a debtfinanced tax cut has no effect on consumption, national saving, the real
interest rate, investment, trade balance or real GDP, even in the short run
...

The idea of Ricardian Equivalence has not been considered for more than a century
till it was resurrected by Robert Barro that linked the original idea of Ricardo with the
idea of rational expectations
...


A Simple Model of Ricardian Equivalence
Consider an economy that lasts for only two periods
...

We have two agents in the economy: a government and a representative consumer
...

In period 1) the government budget constraint is:

G1 = T1 + B1

11)

government expenditure can be financed by taxes (T) or by issuing debt (B)
...

Combine equations 11) and 12) together noticing that 11) can be written as:

15

B1 = G1 − T1 while 12) B1 =

T2 − G2

...
It is the lifetime (in
our case only two periods) constraint faced by the government
...
The government intertemporal budget
constraint shows how changes in fiscal policy today are linked to changes in fiscal
policy in the future
...

In particular equation 13) says that if T1 decreases, for given levels of government
expenditure, then T2 must increase
...
Notice that if S1 > 0 our consumer is a lender while if S1 < 0 our
consumer is a borrower
...

A Decrease in Current Taxes
Now suppose that the government decreases taxes in period 1 by ∆T
...
Therefore
(1 + r )

T2
∆T2
must not change
...
Therefore: ∆T2 = −(1 − r )∆T1
...

By putting those facts into equation 16) we have:

16

C1 +

C2
[Y − (T2 + (1 + r )∆T1 )]
= [Y1 − (T1 − ∆T1 )] + 2
1+ r
1+ r

17)

Notice that a decrease in taxes in period 1 increases disposable income in period 1 and
an increase in taxes in period 2 decreases disposable income in period 2
...
What happens is the following: the decrease in taxes in period 1 results
in higher saving by the same amount so that C1 does not change
...
Those bonds are bought by the consumers that will increase their
saving
...

A deficit created by a tax cut and financed by borrowing does not alter the
consumption decisions of the individuals and so it does not stimulate aggregate
demand nor the interest rate (since total saving = public saving + private saving does
not change)
...
In different words, debt is neutral with respect to consumption
...

Suppose that a tax cut is associated with a decrease in government expenditure by the
same amount then there will be real effects and consumption will increase
...
A debt financed tax cut does not
...

They must know the intertemporal budget constraint of the government
...
So they
are able to calculate exactly the present values of future tax increases
...
A tax cut today made by a
government may mean an increase in taxes in a very long future made by the
same government or by a different government
...
Suppose instead that the consumer dies at end of period
1
...
Robert Barro assumed that consumers have
altruistic behaviour and they care about their children (intergenerational
altruism)
...
Suppose a tax cut
today
...

c) Consumers should not be liquidity constrained
...
So if there is a
tax increase today that implies a tax cut tomorrow consumers can borrow
today to maintain the same level of consumption as before the tax increase
...

d) Taxes are lump-sum
...
If taxes are distortionary then a tax cut or increase will always
have effects on consumption behaviour
...
Indeed we can have:
a1) Myopia: Not all consumers think so far ahead in the future some see the tax cut as
a windfall and so they will feel wealthier after the tax cut;
b1) Future Generations: If consumers expect that the burden of repaying a tax cut

18

will fall on future generations (maybe not even born yet), then a tax cut now
makes them feel better off, so they increase consumption;
c1) Borrowing Constraints: in reality consumers can have constraints in borrowing
and in general a government can borrow at a lower rate than consumers;
d1) In reality most of the taxes are not lump-sum;

Empirical Evidence on Ricardian Equivalence
There is no consensus in the empirical literature about Ricardian equivalence
...
The same should
be true with inflation, trade balance etc
...
In general the evidence results are mixed
...

However by looking at a particular example of debt financed tax cut we see that:
•

in 1982, in US, Reagan tax cuts increased deficit
...
All those results are consistent with the traditional view or Keynesian
view
...
Consumers may have expected the debt to be repaid
with future spending cuts instead of future tax hikes
...

Because the data is subject to different interpretations and because empirical evidence
is still mixed, both views of government deficits survive
...
Where Pt is the general price level
(the GDP deflator) and Qt is the real GDP
...
Then we have: t − 1 = π ⇒ t = 1 + π
Qt −1
Pt −1
Pt −1
and

Qt
= 1+ g
...

Note: when g and π

are small then the following approximation holds:

(1 + π )(1 + g ) ≈ π + g
...

From equation 4) we can see that one way to write the growth rate of nominal GDP is:

Yt
= (1 + π )(1 + g )
Yt −1
⇒

Yt
− 1 + 1 = (1 + π )(1 + g )
Yt −1

⇒

Yt
− 1 = (1 + π )(1 + g ) − 1
Yt −1

⇒

Yt − Yt −1
= (1 + π )(1 + g ) − 1
Yt −1

A numerical example: suppose Pt = 20 , Pt −1 = 10 , Qt = 5 and Qt −1 = 1
...


Pt Qt
100
=
= 10
...
Moreover the growth rate of nominal GDP is:

Pt Qt − Pt −1Qt −1 100 − 10
=
= 9
...

Pt −1Qt −1
10

20



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