Sustainable debt and oil fund
Halvor Mehlum Nov 2012
1
Sustainable government debt
What is the maximum level of debt that a government can handle? What level is sustainable?
Sustainable debt can be defined in a number of ways. One definition is that it is a level of debt such that
the debt does not grow faster than GNI. Hence it should be possible to hold the DEBT/GNI ratio constant over
time. That implies that if the growth rate of GNI is 3% a year the growth rate of DEBT should be no more than
3% a year.
The two main factors determining the evolution of debt, ∆B are 1) down payment H and 2) interest on the
existing debt r ∗ B (where r is the rate of interest). Abstracting from other factors it follows that
∆B=rB − H
Assume that the expected growth rate of GNI is g. From the definition above, B is sustainable if the growth
∆B is equal to g ∗ B. Assume that the government surplus realistically can be a fraction a times the GNI (Y ).
Hence, a requirement for sustainable debt (B̄) is that
g ∗ B̄ = rB̄ − aY
It follows that sustainable the debt ratio S is
S=
B̄
a
=
Y
r−g
What determines S? Assume that a = 0.01 (that the government surplus is expected to be positive and equal to
1% of GNI) and assume that r = 0.07 (7%) then
S=
0.01
0.07 − g
If the growth rate is zero (g = 0) then S = 14%(and D̄ = 0.14 ∗ Y ), if g = 2% then S = 20%, if g = 5% then
S = 50%, if g = 6% then S = 100%. (If the growth rate is large, g > 7%, then a can be negative. That implies
that if the growth rate of GNI is higher than the interest rate it is possible to have a small government deficit
at the same time as the debt is sustainable. The important condition is that debt does not grow at a faster rate
than the GNI).
Anyway, the main message is that the answer to the question ”What level of debt is sustainable?” depends
critically on the assumptions regarding r, a and g. If one for example is too optimistic with regards to the growth
rate g and expect g = 6%, a debt ratio of 100% may appear sustainable. Crisis debt accumulation based on that
premise will prove to be dangerous if g turn out to be 2% instead (in which case only a debt ratio of 20% is
sustainable).
As a result of the financial crisis realistic values of r may go up (for a country like Greece) while realistic
values of a and g drops. Before the crisis the realistic scenario for starting to control the debt, was perhaps:
g = 5%, a = 2% and r = 6%. Then S = 200%. After the crisis it might be the case that a = 1%, r = 8%,
and g = 2%, then S = 17%. If a goes to zero then obviously S = 0. The problem with austerity measures is
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that increasing a might at the same time lower g. Increasing a most typically involve increasing taxes and firing
government employees. This depresses demand and the growth of GNI is hurt. BUT increasing a could also be
seen as a signal of low taxes and stable economy in the future, it could therefore stimulate investments, which
would be good for growth.
1.1
Greek example
”Table 1. Greece: Debt Sustainability Baseline, 2009–2030” in ”Greece: Preliminary Debt Sustainability Analysis
February 15, 2012” shows several examples of these calculations. On the previous pages we defined
S=
a
r−g
In Table 1 IMF asks what primary surplus a would make the existing debt sustainable. The last number in
the table is 1.5 % with the heading ”Debt-stabilizing primary balance” In footnote 10 they write ”Assumes that
key variables (real GDP growth, real interest rate, and other identified debt-creating flows) remain at the level
of the last projection year. ” What they in effect do is to solve the equation above with respect to a and insert
r and g form the 2030 column. They get
a = S(r − g) = 100% · (3.0% − 1.4%) = 1.6% ≈ 1.5%
They also ask the question: what is the ”automatic debt dynamics”? ( see the 2011 column) What they then
do is to write the debt rate (sustainable or not) as
D=
B
Y
(1)
Because they focus on ”automatic debt dynamics” they set a = 0. It then approximately follows that
∆D = (r − g)D = rD − gD = 3.0% · 164% − (−6.1%) · 164% ≈ 4.7% + ·9.2%
1.2
(2)
Sovereign default
In the article Is there the will to save the eurozone? from Financial Times December 7 2010, Martin Wolf writes
What has been happening is familiar to experts on emerging countries: this is a ”sudden stop”.
Before 2007, credit was available on easy terms to fund asset price bubbles, construction and consumption, private and public. Then, suddenly, markets shifted towards sobriety: funding dried up,
house prices collapsed, construction crashed, governments guaranteed the debts of raddled financial
systems, economies slumped and fiscal deficits exploded.[...]
Why might one be this pessimistic? The salient characteristic of lending to sovereigns is the absence
of collateral. Thus, the safety of the creditors depends on their ability to sell debt to others at
reasonable prices. If this confidence disappears, liquidity dries up and sovereigns are driven into
default. What, then, determines confidence? The short answer is: sustainability. That itself depends
on the relationship between prospective economic growth and the real rate of interest. The lower
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Figure 1: Vicious spiral between confidence and sustainability
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B
r
A
r = r0 + e(a)
a=
B
Y (r
− g)
•
•
g
a∗
a
the growth and the higher the interest rate, the bigger the primary fiscal surplus (before interest
payments) needs to be - and so the greater the political costs of achieving it. The bigger these costs,
the less confident will investors be and the higher the interest rates will become. This, then, creates
a vicious spiral.
The argument can be formalized as follows: The primary fiscal surplus, a, needed in order for actual debt B
to be sustainable is
a=
B
(r − g)
Y
(3)
The credit market punishes a high a with high interest rates the bigger the primary fiscal surplus (before interest
payments) needs to be - and so the greater the political costs of achieving it. The bigger these costs, the less
confident will investors be and the higher the interest rates will become. Therefore
r = r0 + e(a), where e(a) = 0 when a < a∗ , and e0 (a) > 0 when a ≥ a∗
(4)
Here r0 is the risk free interest rate while e is the risk premium, which increases with a as long as a is larger
than a critical level a∗ .
The possibility of a vicious circle is shown in Figure 1. Here A is the stable equilibrium. B is an unstable
tipping point. When a is lower than B then the interest rate is moderate and a can be lowered without making
the debt unsustainable. This is possible all the way until A. If a is above B, however, the interest rate is so high
that a has to be raised. But as a rises r increases even more. This, then, creates a vicious spiral.
Such spirals are a core problem in the current crisis in the euro-area. It has been suggested that this may be
solved by ECB guaranteeing a maximum interest rate for countries such as Spain. (Remember that guaranteeing
maximum interest rate is tantamount to guaranteeing a minimum price). If ECB does that for a r = r∗ which is
less than the r in B. Figure 1 will change. Figure 2 illustrates such a guarantee. It illustrates that the critical
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Figure 2: ECB preventing vicious spiral by guaranteeing maximum interest rate
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r
A
r = r0 + e(a)
a=
B
Y (r
− g)
r = r∗
•
g
a∗
a
point B goes away and only the good equilibrium A remains. Note that in A r is well below r∗ and ECB does
not have to intervene. The promise of intervention is enough.
Now, it might be that ECB is too optimistic with regards to where a∗ is located. If a∗ is to the left of A
then after a guarantee of r∗ , there will only be one equilibrium C (which is stable, due to relative slopes). In
this equilibrium r = r∗ . Note that the private investors are not willing to buy Spanish bonds at that low rate of
interest. Hence ECB will have to buy all Spanish bonds in circulation. ECB’s promise will prove to be expensive
for ECBs owners (e.g. Germany) both in terms of cash and in terms of risk. Figure 3 illustrates that case
2
Sustainable national debt
What is the maximum level of debt that a nation (as opposed to the national government only) can handle?
What level of national debt is sustainable?
Now the relevant surplus is the current export surplus. From the definition above, B is sustainable if the
growth ∆B is equal to g ∗ B. Assume that the export surplus realistically can be a fraction a times the GNI (Y ).
The logic is as above. Just replace fiscal surplus with export surplus.
S=
B̄
a
=
Y
r−g
where a now is export surplus in proportion to GNI, r is the interest rate on the foreign debt, while g is
growth rate of GNI. As a result of the financial crisis realistic values of r may go up (for a country like Greece)
while realistic values of a and g drops. In order to improve on a a country would like to weaken it’s currency. But
when most countries wants to weaken their currencies (even the major surplus country china), you get a currency
war. The Euro countries cannot weaken their currencies vis-a-vis the others. The only way to improve a is by
lowering costs (wages) at home and/or depressing demand at home. Both can be achieved by firing government
employees. But GNI down implies g down.
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Figure 3: ECB preventing vicious spiral by guaranteeing maximum interest rate and have to buy all Spanish
bonds.
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r = r0 + e(a)
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C
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•
a=
B
Y (r
− g)
r = r∗
r
g
a∗
3
a
Oil fund
What is the sustainable use of a oil fund? What spending rule is sustainable? Now the relevant deficits are
both the current account deficit and the fiscal deficit. From the definition above, B is sustainable if the growth
∆B is equal to g ∗ B. Assume that we want to determine the sustainable deficit a (now a negative number), in
proportion to GNI for a given oil fund B (now a negative number as a fund is negative debt). As before
a
B
=
Y
r−g
Assume that we in some years time reach
B
Y
= −2, i.e. the oil fund is 2 times GNI . What level of a assures
that the fraction can remain at this level? The answer is
a = −2(r − g)
If r = 4% and g = 3% then a = −2%. Hence, we can run a deficit of 2% (fiscal and export) of GNI. As we have
assumed that the fund is two times GNI, we can only use 1% of the fund.
If r = 4% and g = 4%, however, then a = 0%. In that case the GNI grows so fast that we must leave the oil
fund alone in order for it to grow at the same speed.
The “handlingsregel” implies that four percent of the fund is used. This rule is only sustainable in the sense
that if r = 4% the fund itself stays constant. The size of the fund relative to GNI will most probably decline
under this rule. Note that when the deficit is 4% of the fund while the fund is two times GNI, then a = −8. Such
a high deficit is only consistent with a constant
B
Y
(the above meaning of the word sustainable) if r − g = 4%.
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