Art on debt

8/3/2019 Art on debt

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Index

1. Introduction 2

2. A simple model

2.1. Dynamics 7

2.2. Main features and economic policy 9

3. A model with growth and uncertainty 11

3.1. Dynamics 13

3.2. Economic policy 14

3.3. The long run 20

4. What happened in 2007 21

5. Some empiric results 23

6. Conclusions 24


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1. Introduction

Since 2007 many economists support the use of public spending to

minimize the effects of the crisis. Others argue that it is high time we reduced

public spending to stimulate private investment. The former say that multiplierseffects of public spending can sustain economic activity during recessions,

change expectations, influence positively investment and, in the end, drive the

economy back to previous levels of activity. Some of them even argue that

increasing public spending may even reduce public debt or, at least, its ratio to

GDP since public spending increases GDP and this favors public revenues

growth. Taken to its limit, according to this reasoning, heavily indebted countries

are in this situation because they have spent too little. Among neokeynesians

economists we can cite Paul Krugman or Joseph Stiglitz.

The latter, liberal economists1, argue that increased levels of public

spending and deficit raise the risk premium on interest rates crowding out

investment and deepening the crisis. Among them, the Bundesbank, most

European Central Banks, European Ministries of Finance and, notably, the

German one. They ask for cuts in public spending even at times when it is

almost the sole source of demand, when firms and families have almost ceased

consuming and investing. Their alternatives are the so called structural reforms

and, among them, labor reform, which, in the end, consist almost exclusively in

lowering the cost of labor and, among these costs, firing costs and wages.

However, when sales have almost disappeared, stocks are mounting and firms

1Other liberal economists used to argue that deficits were irrelevant since under the

Ricardian equivalence, private savings would rise by the full amount of the deficit anticipatingthe moment when tax payers will have to pay back the extra debt plus its interests. However,they seem to have disappeared in this crisis from the public debate.


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are losing money, it is far from clear that firms will add to the loses hiring more

labor just because wages or interest rates are lowered.

Despite that some economists on both sides see this controversy in

terms of black and white, many of us believe that sometimes public spending

does more good than harm and sometimes a little bit of austerity is just

unavoidable. If a single, short and deep negative shock hits the economy, public

spending can be a powerful tool but, if the recession lasts more than a few

years, then public spending just cannot be sustained. The central point is,

therefore, how much information do we have about the shock and, if put in

place, when to draw back fiscal stimulus, at what pace and what should be the

roll of monetary policy.

Besides, some economists consider taxes, public spending, deficit anddebt from a different view, more like a game theory problem where

governments act strategically in the political cycle to favor their chances of

being reelected. According to them, as elections get closer public deficit and

debt increase. To avoid this type of behavior some fiscal rules or discipline

should be imposed. They are the last example of a long tradition. We may cite

the following types of rules:

Those focused on the roll of public expenditure in the

management of aggregated demand like the full employment

equilibrated budget or the cyclically equilibrated budget.

Those focused on the sustainability of public finances like the

capability of servicing the debt or paying the interest on debt or the

long term sustainable deficit.


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Both aspects have to be considered. We will not, however, state any

general rule on how deficits and budgets should be fixed. We will just explore

what will happen to GDP and to debt if a given combination of public

expenditure and taxes is chosen at any time. To do that we will have to do

some mathematics related with spending, debt and GDP.


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2.A simple model

The model is quite simple since its equations, except for the last one, are

almost accounting identities, and this last one can be as general as we wish.

Variables should be read as follows: B is the stock of public debt, r is the

interest rate, G is public spending, T is the average tax rate, y is GDP, C is

consumption, I is private investment, XN are net exports and and are

shocks.

= + (1)

= ((1 )) + (,) + + + (2)

= , (3)

Equation (1) says that the increase in public debt in time t equals the

interest paid on the existing debt stock plus the primary (or non financial)

budget deficit. This deficit equals public expenditures minus revenues that we

will call taxes. In some sense, its like the old LM equation that said that

money supply should equal the demand for liquidity but, in this case, written in

terms of debt and interest rates and with no behavior implicit in the equation.

Equation (2) is the usual GDP equation as seen from the demand side

considering also a shock. It is important to notice in this equation that net

exports can play a very important roll sine an increase in net exports can raise

GDP and, therefore, lower interest rates. We can add complexity to this

equation saying that imports depend on income.


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Equation (3) says that interest rates depend on the ratio of public debt to

GDP2 and other variables that we can consider shocks such as the reputation of

the country, its current account deficit, monetary policy, risk aversion level and

so on. After solving equation (1) the model is:

=

+

(

)

()du (1)

= ((1 )) + (,) + + + (2)

= , (3)

We should look closely at equation (1). This equation takes the effects of

shocks, policies or whatever happens in the model in time t to future periods. In

some sense, this equation resumes the consequences of our acts and makesthe model consistent in time. It is also important in the sense that if we divide

both sides by yt we get what can be called the equation of sustainability of

public debt. If GDP is growing almost all deficits are sustainable as long as its

growth rate is greater than the interest rate. If this is the case, interest rates will

go down and we will enter something like a virtuous circle. On the other hand, if

GDP growth rate is smaller than interest rates we have a vicious circle. GDP

growth and interest rates are, therefore, the main variables driving the dynamics

2Some authors find that the debt service ratio to public revenues works better, that is

rB/Ty instead of B/y. To be honest, the evidence is mixed even on whether there is a fiscaleffect on interest rates. Barth et al (1991) surveyed 42 studies of which 17 found apredominately significant, positive effect of deficits on interest rates; 6 found mixed effects;and 19 found predominately insignificant or negative effects. They conclude that Since theavailable evidence on the effects of deficits is mixed, one cannot say with complete confidencethat budget deficits raise interest ratesBut, equally important, one cannot say that they do nothave these effects. Other reviewers of the literature have reached similar conclusions.

Elmendorf and Mankiw (1999) note that Our view is that this literature...is not very informative.Bernheim (1989) writes that it is easy to cite a large number of studies that support anyconceivable position.


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of the economy and there is little public spending can do to cancel their effects

except for a short period of time.

2.1. DynamicsWe will study the dynamics of this model using a simple graphic. In the

vertical axis we have the interest rate and in the horizontal axis we have GDP.

The line = 0 is just the combination of interest rates and GDP that keepsdebt constant, that is, all the

points where there is a primary

surplus just to service the debt. At

the right of this line debt will

decrease and at the left it willincrease. The line = 0 is thelocus of combinations of interest

rate and GDP that leave the latter

unchanged. Above this line

interest rate is too high reducing

investment and diminishing GDP,

below this line the opposite. Both

lines divide our graphic in four

areas. In area I it is clear that

interest rates go up and GDP decreases, the economy enters a vicious circle

where action has to be taken. On the other hand, in area III the economy is in a

virtuous circle.

Area III

= 0

y = 0

< 0 > 0

< 0

> 0

r

Area I

Area IIArea IV


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In areas II and IV things are a little bit more complicated, since there is a

line where

=

and interest rates remain unchanged. This line is given by

equation:

=

=() (4)

Our graphic, therefore, can be redrawn adding a third line, the dotted

line. This new line, the line, = 0, will allow us to redefine our four areasaccording to the dynamics of the system. Above this line interest rate will

increase and below it will decrease.

Lets imagine, now, that

from an initial equilibrium point

there is a shock in the financial

markets that drives interest rates

up. Servicing the debt becomes

more expensive and debt starts

to grow. At the same time

investment falls and GDP starts

to fall, were in area I. A sort of

vicious circle drives interest rates

and debt up and GDP down

since the system is unstable.

Things can get even worse if, in

an attempt to stimulate

aggregate demand and GDP growth, the government increases public

spending. In both cases the system is explosive once it enters area I and theonly thing to do is to use monetary policy to bring down interest rates. In area III

= 0

= 0Area III

= 0

Area IV

Area II

Area I

y

r


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the opposite takes place, if a shock lowers interest rates, GDP will start growing

and interest rates falling in a kind of virtuous circle.

2.2. Main features and economic policyA couple of things should be noticed. First of all, while GDP is basically

stable, interest rate is unstable. The only path that goes back to the equilibrium

point is a horizontal straight line at the equilibrium interest rate. This is so

because we have only the demand side of GDP. There is no technology nor

capital accumulation driven growth. If GDP is static and there is no growth,

deficits tend to grow in an explosive manner if they are not quickly cancelled by

cuts in public spending since in the future there will be more debt servicing. This

will drive interest rates up and deficit will mount. Consequently, debt is more or

less constant in time except for some variations that average zero in the long

run.

Second, we do not have a model to determine the interest rate, we don't

have a LM line nor a money market. Our model determines the risk premium

component of interest rates or, at least, one of the factors that may influence

this premium when, as economist say, all things remain equal. It is very

arguable that risk premia respond to the proportion of debt to GDP and even

more that it does so and not to other things that change with GDP like trade

surplus since imports depend on domestic demand. Anyway, if we want our

model simple enough, some hard to swallow assumptions have to be made.

Next, we will remove some of these hypotheses.

Accordingly, our main suggestion to policy makers living in a world with

no economic growth at all would be don't mess around with the risk premium

since it is quite unstable. Second, if it happens that you are in a middle of a


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financial storm, don't worry, no matter what, times will be harder in the future

than they are right now. But, if we want to do some exercises with the Maths,

find multipliers and so on and give some advice, the main questions would be:

what level of public debt and deficit are sustainable3 and for how long. In this

simple case, debt is constant and deficit is zero, both in the long run.

3

We will say that public debt is sustainable if it grows slower than GDP, that is, if itsproportion to GDP remains constant or decreases in the long run. A deficit path that keepspublic debt sustainable we will say it is a sustainable deficit path.


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3.A model with growth and uncertainty

We will introduce now economic growth. The important thing here is not

where this growth comes from, what really matters is just that the economy

grows. Our model, therefore, becomes:

= + (1)

= (2)

= ((1 )) + (,) + + (3)

= , (4)

After solving equations (1) and (2) this model with deterministic growth

becomes:

= + ( )()du (1)

= (2')

= ((1 )) + (,) + + (3)

= , (4)

From equation 1 we know that any primary deficit is sustainable as long

as it grows at a slower pace than the sum of the rate of growth of GDP minus

the interest rate. In this last crisis, GDP growth became negative or slightly

above zero and interest rates grew since the risk premia went up and monetary

policy didn't compensate. If GDP growth is going to remain low in SouthernEurope and interest rates go up as Germany recovers, Southern European


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countries are going to experience higher risk premia and the situation become

explosive. However, GDP growth is not constant in time; there are shocks that

affect GDP growth and expectations become important. Investors in public debt

will care about whether the government is going to pay back its bonds or not,

that is, to default. One important thing for them is how robust GDP growth and

therefore public revenues are going to be in the near future. We will assumethat interest rates depend on the expected proportion of debt to GDP from time t

onwards, something that depends on past values but also on other information.

We will have to change once again our model and it will become:

= + (1)

= where gt = g(Gt/yt, XNt/yt, Tt, rt, t) (2)

= , (3)

After solving equations (1) and (2) this model with stochastic growth

becomes:

= + ( )()du (1)

= , ,,,

(2')

= , (3)

We will assume there is a lower bound to interest rates, that is, 0 r but

also an upper bound, since in this crisis we have seen that interest rates don't

have to reach the infinite before you have severe restrictions to the amount of

money you can borrow. Therefore, 0 r r. On the other hand, the demand

side of the GDP doesn't affect its total, only its composition, how GDP is


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Area III

= 0

g = 0

< 0 > 0

< 0

> 0

r

Area I

Area IIArea IV

distributed among consumption, public expenditures and so on. To end up with,

if the model is going to reflect real economies we have to introduce some

complexities. We should take into account a lower bound to public spending, G,

and an upper bound to the resources the government can get from the

economy, 0 < T-, < 1, a variable that governments can discretionary change. We

can also take into account a limit to how much governments can borrow in

financial markets and the alternative, not paying in due time governments

suppliers. Investors won't probably wait till debt becomes infinite to stop lending.

There is a ceiling to the debt to GDP ratio as there was one with interest rates

and above it investors presume the country will default, they won't buy bonds,

the government won't be able to renew its borrowing and it will default. Let's call

that ratio d.

3.1. DynamicsIn this case we will

substitute in our first graphic y

by g. In equation (1') we find

that the line that keeps the

interest rate unchanged has apositive slope since it is the line

where the equation +

=

holds. The line where thereal side of the economy is in

equilibrium has a negative

slope since it depends on the

effect of interest rates on


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private investment, that is, =

< 0.

3.2. Economic policyNow, let's tell a few stories. If the government increases public spending,

the year it does so GDP will grow at a faster rate than it would otherwise have,

unemployment will go down, taxes will grow but less than public spending and

deficit will increase. Public debt will also increase and, assuming it grows more

than the economy, Interest rates will also go up. GDP will grow more the closer

the economy is and the greater the sensibility of private investment to GDP

growth and, also, if it is quite rigid to interest rates. Public debt will grow

because of two factors: because interest rates are higher and because we have

a bigger primary deficit. Now, if we don't maintain the fiscal stimulus, GDP will

return to previous levels, that is, will grow at a rate below g, but public debt will

continue to grow and also the interest rate if it is bigger than the rate of growth

of the economy. According to equation (1'), public debt is sustainable in the long

run only if interest rates plus the rate of growth of the primary deficit are less

than the rate of growth of the economy. Therefore, in the long run, primary

deficit has to grow less than the economy, its proportion to GDP decrease in

time and its effect on the economy disappear. In conclusion, fiscal stimuli just

cannot be maintained forever since public debt becomes explosive. Therefore,

we have to use them wisely. What is the measure of our wisdom? It depends

on:

The magnitude of the multiplier effect of G on GDP. It depends on

the crowding out effect or how much private investment is

displaced by G and how close is the economy.


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How much deficit G creates since part of the increase in GDP

goes to the government as taxes. The bigger T the lesser the

multiplier effect of G but also the lesser the deficit generated.

The sensibility of interest rates to the ratio between debt and GDP.

The effect of an increase in expenditure on debt depends, on its

turn, on how effective it is to boost the economy and public

revenues.

As we said above, there is a ceiling to debt to GDP ratio and to

interest rates. The closer the economy is to that point, the lesser

the margin for an expansive fiscal policy its government has. Also,

the bigger the interest rates the faster the economy will approach

d with a given primary deficit path.

Let us now see some examples of all of this. To see how it works let's

assume a baseline scenario where the economy and public expenditures grow

at the same rate (we know this is not

sustainable). The economy will

eventually reach a debt to GDP ratio

that makes investors think the

government won't pay back its bonds.In this case, we have primary

surpluses every year but they are not

enough to avoid the explosive

evolution of public debt since the high

value of interest rates makes

servicing the debt an ever increasing

cost. All key variables evolve as

shown in graphic 1. The interest rate is the variable that shows a more

Graphic 1


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explosive behavior4. In graphic 1 all variables are normalized to 100 to see their

proportional changes. In graphics 2, 3 and 4 we will give the original values.

4

Except for the deficit since it was normalized. We gave 100, like to all other variables,its first value that was zero. Therefore, its first change would be infinite in proportional terms,something that disappears with normalization.


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Now we will see what happens to this baseline scenario if our

government increases government spending by 10% of GDP at time 0 (graphic

5) and compare it with what happened in our first case. First of all, it only takes

8 years for the debt to collapse. GDP and debt grow faster than in the previous

case but debt grows faster than GDP so the ratio increases faster when

Graphic 3 Graphic 4

Graphic 2


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compared with our first case. before the collapse the economy may have been

at full employment, but we have to compare that with the cost of default which

means reducing deficit to zero quite abruptly, graphic 6. In this model, if private

investment is not affected by interest rates and only by GDP, after the collapse

of financial markets GDP will fall but it will always be above, even though

slightly, of what it would otherwise have been without the stimulus of 10% ofGDP in public spenditure (graphic 8).

Graphic 5 Graphic 6


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Graphic 7 Graphic 8

What now if our economy starts from sustainable figures, the rate of

growth of the primary deficit plus the interest rate is less or equal than the rate

of growth of the economy. Now the economy grows at 2%, interest rate is 1%,

public expenditure grows at 1% and initial debt is 38% of GDP. At time 1 GDP

suffers a negative shock of, let's say, 5% of GDP, which causes a fall in GDP,

after the automatic stabilizers come to play, of -3.6%, but at a cost of a public

deficit of almost 7% of GDP. Debt enters an explosive path and the economy

will default, more or less, in time 15 even though we have primary surpluses

from year 13 on. The problem here is that fiscal consolidation comes too late

and in an automatic way. If the multiplier of public spenditures were 1.5 instead

of 0.8 (as supposed initially in the exercise) public debt would have not followed

an explosive path. Assuming this is the correct value, even an initial increase of

public spenditures of around 20% of GDP would have caused only a modest

increase in interest rates in the first period. Therefore, knowing the correct value

of the public spenditure multiplier is of the first importance when it comes to

designing the most suitable economic policy in the face of a negative shock.


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Also, if the rate of growth of the economy is 3% instead of 2%, even with

a low multiplier of 0.8 and a big fiscal stimulus of around 10% of GDP or more

will be sustainable from the point of view of the evolution of the debt even

though with an initial deficit of 12% of GDP in time 1.

3.3. The long runWhile deficits can have favorable effects on economic performance in the

short run, they have unfavorable effects in the long term because of potentially

differing economic situations over different horizons. In the long run, the typical

assumption is that the economy is at full employment and the only way to raise

economic growth is to expand the economy's capacity to produce. By reducing

national savings, deficits hinder that ability. Over shorter horizons, when the

economy is well below full employment, the deficit can boost aggregate demand

and increase the use of existing capacity, thus reducing unemployment.

Also, deficits may increase the aggregate supply. Spending on public

infrastructure increases current deficit but its total net effect depends on

whether its return is bigger or not than the cost of borrowing like in any

investment project, public or private.

.


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4. What happened in 2007

The economy was in area III till 2007 and from then on it entered area I.

Using this framework, lets consider a shock on financial markets that drove

interest rates up, that is, that we had a positive value for the shock t and our

debt curve moves up. First, investment, consumption and GDP fell and

unemployment rose given that monetary policy didnt hold down interest rates.

Since this was not the case,

automatic stabilizers came into play,

public deficit increased and public

debt started accumulating till we the

economy reaches a new

equilibrium. Then, governments

decided to fight back and increase

discretionarily public spending. GDP

fall was mitigated but deficit grew

and public debt accumulated faster

but eventually the economy would

have reached a new equilibrium at

point B. End of the story? We willhave to see, since B is not a stable

point interest raise will keep on rising and governments will have to cut public

spending in a desperate effort to curb deficit and interest rates down. These

cuts will cause further GDP falls and unemployment but the closer we get to r

the bigger the cuts in spending. If r reacts slowly we will have more time and

cuts will be lesser and more spaced in time to cause less harm but if r is very

sensitive to Debt over GDP, drastic action will have to be taken. There is littledoubt so far that if the ECB had acted otherwise and lowered interest rates

A

B

g

r

r


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things would have been different but, is there anything that national

governments deprived of the use of monetary policy can do?

The key point here is whether the effect of public spending on income is

strong enough so that the ratio of public debt to GDP goes up or down. We

have assumed so far the most likely, that it goes up. But, if interest rates are not

very sensitive to the debt to GDP ratio, they move slowly and the economy

recovers, that is, g increases, then the economy may return to the stable

equilibrium point by itself or with little help from a restrictive fiscal policy.


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5. Some empiric results


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6. Conclusions

At the beginning of a crisis, governments need to know if it is going to

last or not. In the first case, they should be cautious when it comes to increase

spending since deficits may grow fast and accumulate causing debt to grow. Ifrisk aversion increases and there is a restrictive monetary policy they may

approach more than wanted dangerous waters where the interest rate starts

walking an explosive path leading to default. On the other hand, if we have just

a one period shock, governments can use public spending to compensate the

shock avoiding unemployment and social pain. However, financial crisis

resulting from a excessive expansion of credit all around the world will likely last

for long periods.

Expansionary monetary policy can help a lot by lowering interest rates

and increasing nominal growth, that is, with inflation reducing the debt to GDP

ratio.

Art on debt

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