We did this exercise and the previous two in the meeting format of this lecture. Now let us go back to the model and do some kind of manipulations of the government valuation equation so of the equilibrium condition which in some views of macroeconomic models may be a
budget constraint and other views may be an equilibrium condition.
You see what I have done here is basically that I start out with the equation which you all know, the real value of outstanding debt must be equal to the expectation of current and future discounted real surpluses.
Important here is that the sum starts at i equal to zero. So this is the current surplus ST as the first component of this sum, and this is of course not discounted r to the power of zero is one, but all future surpluses are then discounted.
So we can also write this such that the current surplus which is not an expected surplus because we have already information conditional on period T available. We can also therefore write this as ST plus then the same sum starting at i equal to one.
So let's take this part of the equation here, the right hand side here and the left hand side here and just shift this by one period.
So we would have BT divided by PT plus one if we replace any T here by T plus one. So T plus one minus one gives us BT and here PT becomes PT plus one is then the expectation conditional on period T plus one information
of a sum starting again at i equal to zero. Well, and then this term here with the T's replaced by T plus ones.
Now we can move this also to a sum which starts with i equal to one because obviously where we start at adding things to T is actually one plus i.
So if i here is zero, we could equally well say at rather than starting at i plus one, we just let's i start at one and then we have a simpler expression which is ST plus i divided by r to the power of i multiplied by one r because obviously now we have one factor of r.
We have one factor of r too much here when i starts at one, therefore we have to correct this and multiply by one r factor here.
So what we have here is basically r times this thing here but with a different expectations operator namely ET plus one rather than ET.
Now recall that the law of iterated expectation lets us know that the expectation conditional on period T information of some random variable XT plus tau is the same thing as the expectation with respect to period T of the expectation with respect to period T plus one of XT plus tau.
So we can actually rewrite equation 56 in the following form that BT minus one over PT is equal to ST plus and now rather than using this term here i use the law of iterated expectations and replace this thing here which is as you see identical to this thing here
by the expectation of BT over PT plus one so that i get one over r expectation conditional period T of BT over PT plus one
which is then as you see a simple difference equation in the real value of outstanding debt.
Here it is period T's outstanding debt and here it is period T plus one's outstanding debt or better. Here it is period T minus one's nominal debt evaluated in period E or computed as the real debt of period T and here the same thing for period E plus one.
This we can also write in terms of the nominal interest rate because we see of course he is the real rate and then he is the expectation of the inverse price level of the next period so by just
multiplying in and dividing through by PT we get BT over PT here and PT over PT plus one here so the PT factor here cancels and then we have inverse the expectation of inverse inflation in this equation here.
Well, and this is the real interest rate factor so inverse real interest rate factor times the expectation of the real of the
of the inverse inflation factor gives us of course nominal interest rates so we arrive at ST plus one over the nominal interest rate factors factor times the real value of current issued debt in period T.
Now the sector one over one plus IT is just our QT so we are now at the equation which we discussed already in the meeting format 57 becomes now equation 58 which simply says that BT minus one over PT is equal to current surplus plus the real value of revenue from bond sales in period T.
An equation which we had already derived from the budget equation there.
So the intuition for this equation here is that the real value of bonds coming due in period T so this thing here will be financed either by Texas or by debt finance and this result obtains because we know we cannot finance it by money printing since there is no demand for money.
At the end of the period.
Now what he can do next is that we equate the right hand side of equation 58 here with the right hand side of equation 56.
This thing here, then the ST term drops out on both sides of the equation, and we get the equation that the real value of revenue from the sale of nominal bonds must always be equal to the expectation of future discounted surpluses.
So not the current discounted futures surplus anymore, the sum here starts at I equal to one, so it is just the expectation of future surpluses discounted by the real interest rate factor, the current surplus is missing.
Now, let us stay in the fiscal theory of world and suppose that the government has fixed all the surpluses.
But it wants to sell more debt BT at the end of period T. So this relates to the last exercise that we have just discussed in the meeting.
From equation 59 we now see that QT BT over PT must be constant.
We see that if the government has fixed all these surpluses here.
If this is a constant, then obviously QT BT over PT must also be constant.
Now how does this relate to our assumption that the government tries to increase nominal bond sales BT.
So, we assume the government wants to sell more debt BT at the end of period T but we know this thing here must be constant.
Obviously what we discussed in the exercise is that either QT must change or PT must change or both of them must change.
But what we have not yet discussed in the exercises that we also have to have equation 55 equation 55 was the equation on unexpected inflation, which I showed to you.
Again, is it. Yeah. Right.
Unexpected inflation is determined by this expression here. So essentially since PT minus one is given and BT minus one is given and PT minus one is given here.
We can say that the price level is determined by the value of all the surpluses which the government by assumption has set.
So, including surplus ST.
So, PT is actually determined by equation 55.
So therefore, in this analysis here, this analysis here, we have that PT cannot change anymore.
So, we now know that QT times BT must be constant by virtue of equation 59.
So, QT BT being constant is precisely the condition which we discussed in the meeting that the demand elasticity for bonds must be equal to one.
So here we have the result which I have already alluded to when I commented on the exercise in the meeting format.
Sorry for this.
I should turn off my phone. Just a minute, please.
Sorry for that. So, this I had already alluded to in the meeting format of the session the negative format QT BT is constant is necessarily constant.
If all the surpluses have been fixed, including surplus of period ST. So if the fiscal policy is active in the sense that the government has fixed all current and future surpluses, then the demand elasticity for bonds must be constant.
And this means that the government is simply not able to raise more revenue by selling bonds. So they may intend to have higher revenues there, because they have taken the decision to sell more debt, but it's actually not possible to get more revenue
because the interest rates would adjust accordingly.
So we conclude that the government bond demand curve, the demand for gone bonds by consumers demand for government bonds by consumers as a unit elasticity throughout for a given series of surpluses.
And this means that the nominal interest rate factor one plus it moves just one for one with bond sales BT.
Okay, so this I think complements exactly what we've done in the meeting format.
Now changes in the nominal interest rate it are obviously changes in expected inflation it plus one, because we know that the nominal interest factor one plus it is just the real interest rate factor divided by the inverse of expected future inflation.
So since the real interest rate factor is constant in our model it's just one over beta, we see that changes in the nominal interest rate directly relate into changes of expected inflation.
And that implies that by bond sales, the government has perfect control over the normal interest rate and over expected inflation.
Now this may sound very peculiar to you that is that I say a government has perfect control over the normal interest rate and over expected inflation, because you may think the control of interest rates is typically something which is exercised by central bank.
And you may say, not by government.
But remember the term government is actually used ambiguously.
Government comprises both the central bank and the fiscal authority is actually both the monetary authority and the fiscal authority.
Here we would say that using bond sales, the government in its function as monetary authority has perfect control over the nominal interest rate and over expected inflation so that's the way a central bank controls the interest rates.
The normal interest rate by either selling or buying bonds or other more short term papers and thereby tries at least to control expected inflation.
So, view the government here as the monetary authority, the monetary authority can control the nominal interest rate and expected inflation without interfering with fiscal variables, since we have assumed that surpluses are constant.
And revenue from bond sales is also constant. So there's no interference with the fiscal side of the government.
And this then is exactly what I already mentioned here we describe or by this behavior we describe a central bank and its monetary policy.
The central bank would typically try not to interfere with government surpluses and perhaps doesn't even have at least a direct instrument to interfere with government surpluses because these are decided upon by the government or by the parliament.
And they would not interfere with revenue from bond sales or they should not interfere with revenues from bond sales so their own open market open market operations should not change bond prices.
And that is what is the typical view of a central bank.
The ECB, by the way, is possibly doing just that, that they buy bonds currently at such a scale that they do affect market prices.
They claim that they do not, but it is at least debatable whether this is true. I think it is not true at all. The low interest rates that we observe on European bond markets are certainly, to my view, due to the huge purchases of government bonds
from the side of the European central bank. So the European central bank has depressed interest rates, normal interest rates, and thereby has had an impact on
the revenue from bond sales for the fiscal part of the government, for the fiscal authority, and thereby the separation between the monetary authority and the fiscal authority in a state that doesn't work anymore.
There is no complete separation anymore, but rather the central bank interferes with decisions of the fiscal authority.
Okay, so the important takeaway from this slide here is that when we fix the fiscal variables, both the surpluses and the revenue from selling debt,
then we describe the behavior of the central bank, and then we speak of monetary policy.
And when the central bank sells bonds, then it reduces the quantity of money and the nominal interest rates rise, obviously. When it buys bonds like the central bank, the ECB is doing this right now, then it increases the quantity of money, of high-powered money, as you know, and the nominal interest rates go down.
Suppose the central bank sells bonds, as I have put it here on this slide, then the price of the new bonds falls because there's more supply of bonds.
So since more bonds are sold at a lower price, each bond, however, has the same face value and will be redeemed at this face value one period later, namely at the beginning of period E plus one.
From this, it follows that people now receive more money in period T plus one, and they can't spend this money in goods, and this of course then drives up inflation in period T plus one.
So by this mechanism, by providing more money to people via bond sales, via reduced prices of bonds, and the same principle, the same face value of bonds, by this mechanism, the central bank causes inflation, and therefore we can say that the central bank actually controls inflation.
So the takeaway from this is that bond sale, holding constant the level of surpluses, is called a monetary policy.
But this is important that surpluses are being held constant. We call a bond sale a monetary policy if and only if surpluses are held constant. Okay, so that is what you should remember.
Now suppose that the government sells more bonds at the end of period T and changes its plans for future surpluses. So in this case, we do not hold future surpluses constant.
And since there are many ways to have future surpluses, not constants, we will just focus on one particular way, which in some way is a polar case to the case we have just considered with constant future surpluses.
Namely, we now consider the case where future surpluses are increased exactly by such an amount that the price of bonds, QT, and therefore also nominal interest rate, IT, stays constant.
So when nominal interest rate stays constant, then, as we know, there is no change in expected inflation by virtue of this equation here. So if IT is constant, then expected inflation is also constant.
So let us assume that surpluses are changed from period T plus one onwards in exactly such a way that the intertemporal price, IT, is unchanged, is left constant. So this is in some way a polar case.
Let us also suppose that the change in surplus is such that PT stays constant. So we do not only not change the intertemporal price, we also do not change the intertemporal price, namely PT.
Then by virtue of equation 58, the increase in bond sales implies that the current surplus ST decreases. And this is the other case which I had already discussed in the meeting, which we had before the break when we discussed the third exercise, that it is possible that an increase in BT
implies that the current surplus ST decreases. So it may be not possible to hold ST constant if we increase BT and want to hold constant QT and PT, because obviously when QT is constant and PT is constant, then the only other variable which may react to the increase in BT is ST.
So in that case, the government just changes the timing of surpluses and replaces tax revenue in T by debt and by higher taxes, of course, from T plus one onwards because surpluses are basically taxes.
This is actually exactly what Adrian already alluded to in his answer when he said somehow the surpluses must rise either now or in the future. That's exactly the case here that in this particular instance, the
taxes, which we have in period T, are reduced. So surplus ST decreases, but this must be made up for by increases of surpluses in later periods, because otherwise the government equilibrium conditional valuation equation does not hold.
So such a policy we would call a pure fiscal policy. It would be a policy which does not affect monetary variables. Interest rate stays constant and inflation stays constant, but we do change fiscal variables. So we would say here we see the government in its function as a fiscal authority.
In this setting now, as you may already have noticed, the fiscal policy of the government is able to raise more revenue by selling more debt in period T.
So it is also polar in that respect that in the monetary policy framework that we analyzed before, we had the property that QTBT was constant. So there was the constant unit elasticity of bond demand.
But here when government surpluses change or when the structure of government surpluses change, then the government is indeed able to raise more revenue by selling more debt in period T.
This additional revenue is then just a direct measure of how much the government has persuaded markets that it will raise future surpluses.
Because the future surpluses are needed to pay off the additional debt and markets will only sign up to the bonds issued by the government. Now, if they are confident that the government is in a position to service this debt in terms of interest and principal in later periods,
for the servicing of the debt, the government needs future revenues, which can only be future surpluses.
So the extent by which the government is able to raise more revenue is exactly the extent by which the government can convince capital markets that it is able to raise higher surpluses in the future.
If the investors are skeptical of such promises, then interest rates will rise, obviously, right? So investors will then, would they like to have some type of risk premium?
So they would only be willing to hold the bonds to buy the bonds if interest rates are higher than they currently are.
So interest rates would rise and bond prices would fall. And since bond prices then fall, there is less additional revenue than intended. Actually, it may be that bond prices fall so strongly that there is no additional revenue at all.
If we want bond prices to stay constant, this implies then that the sequence of surpluses, which we have current surplus and future surpluses, must be rearranged in such a way that its present value is unchanged.
So we can always have changes in our financial planning, in our multi annual financial framework. Most governments have medium term financial frameworks in which they have certain plans on what amount of surplus they would like to achieve at what point in time.
And it may be that the medium term financial forecast is constructed in such a way or is changed in such a way that they say, well, we would like to have less surplus in periods, for some reason, for instance, there's a pandemic, which we
need to fight or whose consequences we need to fight, but then we have to achieve higher surpluses in future periods. Therefore, we have to change our plans for the future for the surpluses, which will be generated in the medium term.
The problem with medium term planning and practical policy is, of course, that is always medium term and not long run planning. So what governments can always do is that they say, well, we do lower the surplus now, so we have some type of expansionary fiscal policy.
We lower the taxes, we increase the expenditure, so surplus is going down right now, but we do not provide for an increase in surpluses over the course of the medium term planning horizon.
Let's say it's seven years like it is in the European Union, for instance. So the government may announce, yes, of course, we will have higher surpluses, but we do not really say when that will be the case.
Or we just say that will be the case sometime after the seven year planning horizon, so that this is postponed to such an extent that it is unclear how credible the increase in future surpluses will be to bond investors.
Obviously, the government will learn how credible this is when it goes to the capital markets and issues bonds because the reply of the investors will then be in terms of the interest rate, which they would like to see for them to hold the bonds.
So it is not necessarily a clever strategy to say that future surpluses will rise, but only at a point in time which is very distant in the future, or perhaps not even specified at all, just a general declaration of course we will pay back the debt later by means of tax revenues.
If it is, however, within the planning horizon of the government and here in our model, the planning horizon of the government is infinite, so we always have the present value of all future surpluses.
Then it is clear that we must have some rearrangement of the structure of SS here in such a way that future surpluses properly discounted make up exactly for the decrease in surplus in period t which we currently may have due to say COVID-19.
So, the wrap up of this discussion is that bond sale funded by increasing future surpluses is called a fiscal policy.
So in some sense, we have now defined fiscal policy and we have defined monetary policy. In both cases, the policy is actually bond sale.
But if the bond sale is done with constant future surpluses, then this is a monetary policy. However, if the surplus is done with increasing future surpluses, then this thing is called a fiscal policy.
Now, obviously from this distinction, we see directly that a fiscal policy is financed by future taxes, so by changes in future surpluses.
And a monetary policy is financed by future inflation, because we held surpluses constant, so we have an inflation tax, rather than fiscal tax.
Good. Are there any questions in particular relating to this definition of fiscal policy and monetary policy in the corporate model?
Okay. Don't see anything. So, if a government now wants to keep inflation low, we learn from this analysis that it has to finance debt by future surpluses.
And here perhaps on the slide I should better have written if a fiscal authority wants to keep inflation low, then it has to finance debt by future surpluses.
Because here we do not speak of the government as the union of monetary and fiscal authority. Here we just take government as signifying the fiscal authority.
So, if the fiscal authority wants to have low inflation, if the fiscal authority wants to ensure that the central bank does not have to step in and generate inflation in order to save the government from default.
If this is true that the government wants to keep inflation low or the fiscal authority wants to keep inflation low, then it has to finance debt by future surpluses.
Obviously, in the Eurozone, it is the case that all governments have committed to price stability, so to low inflation. So it should be actually the policy of all governments to keep inflation low, and therefore we may try to have an empirical check of whether surpluses
are changed in such a way that reductions in surpluses now are matched by increases in surpluses later.
So, we have as an implication of the theory that the correlation between surpluses should be negative when we correlate current surpluses and present value of future surpluses.
Now we do not know exactly what the plans of the government for surpluses are, and therefore we do not have a present value of future surpluses as observable data.
We would need all the intentions of the government all the way up to the very distant future. Obviously, even the government would not know what kind of surpluses it aimed for in 20 or 30 years from now, not to speak of surpluses even farther in the distant future.
But we could take the real surpluses, the realized surpluses in the past, as an approximation to the intended surpluses, assuming that the government has mostly followed up on its planned surpluses in some way.
So, we could check for the type of correlation between the current surplus and future surpluses, or between current surpluses and lagged surpluses because the current surplus was the future surplus back in time.
So back in time, when some surplus was made, there was an expectation or a plan by the government that a certain type of surplus would be realized in the future, and the future is now so we may also look back, or actually that's the only thing we can do since we don't have data on future surpluses.
We can only look back in time and here the correlation should also be negative if government has financed its debt by future surpluses. So if a government lift up to the target of keeping inflation low.
Conversely, of course, when surpluses display long lasting positive auto correlation, the government is probably funding debt by inflation, rather than by fiscal taxes.
Now let's look at these implications by looking at data from the Eurozone.
What we see here is a pattern for Belgium and France, which in some sense is similar for the two countries to understand what I have
given here to you as an output. This column is the auto correlation of surpluses of realized surpluses for the government of Belgium, actually for all levels of government, including regional and municipality governments.
So, this is the auto correlation and you see the auto correlations which are computed for a time series of government surpluses in Belgium are such that they are positive for at least six years, and then they turn into negative territory.
You have a graphical representation of these numbers right here on the left.
So, this year indicates the size of the auto correlation to the right is positive to the left is negative. So this year is the value of 0.765.
And this is the value of 0.617, and so forth. And you see that the auto correlations decline cross zero between year six and year seven.
And then there is negative auto correlation here.
Now that's hard to reconcile with the model that we have because the way I framed it, we would actually expect that either there's positive auto correlation or there's negative auto correlation but here we have both.
So it is helpful to also look at the significance of the auto correlations.
And what we see here for Belgium is the two sigma confidence interval as a dotted line.
So all the auto correlations which exceed this two sigma line are significant auto correlations.
Therefore, that Belgium has a positive first order auto correlation for government surpluses with a time lag of one year, and with a time lag of two years.
From years three, four, five, and six, the estimates are not significantly different from zero anymore. But given that there was auto correlation here significant so.
And here it is quite likely that these things here are also positive auto correlations, because auto correlations which typically if they exist for one leg.
Decay over time, it's such a more or less geometric order. So, quite positively, we may say that Belgium has positive auto correlation for its surpluses.
However, then the estimates turn negative, and there is one estimate here which is just barely significant, just barely outside the two sigma.
That's harder to interpret. Most of these things here are insignificant, but not always very far off the significance spent and one is actually just barely significant.
So, one could interpret this in such a way that one would say where Belgium is a country which tends to go into debt with considerable considerable amounts, and not necessarily paying off the step quickly or planning to pay off the step quickly.
Because we have the strong positive auto correlation here, much as if Belgium would try to pay off the debt just by inflation tax.
But in the long run, it seems that there is some type of correction also fiscally for Belgium, not completely clear how strong it is because the significance is lala, but it is too strong to just discard it and say there's certainly no correction.
So we can also not go that far. So it may be that Belgium in its policy combines both ways of accommodating for increased debt.
In the short run, it just raises debt by considerable amount and hopes that parts of the burden induced by this debt is borne by inflationary processes.
Hopes for a number of years and then tries to also counter the level of debt by fiscal policies.
And as you see for France, the situation is similar, even though the positive correlations are after year one, not as pronounced as they were in Belgium.
Not so clear what to make out of these estimates here. It looks a little bit like these are really insignificant, like as if there is no reaction anymore, no significant influence anymore of the debt making in period one on the surplus debt making period three, four, five and six.
But here's one positive auto correlation, which again would suggest that France also hopes that some of its debt is being borne or some of the debt load is being borne by private consumers who suffer from inflation and only much later they may be very similar to the Belgium experience.
Some try to stabilize debt also by means of fiscal policy variables.
Here's the same thing for Finland.
Much the same thing as for Belgium for the short run period, so for the first six years essentially, two of the auto correlations clearly positive and significant.
And then negative auto correlations starting in year eight, all of them insignificant though.
Some of them not far off the 2% error band.
So approximately the 5% confidence level, but not to say that one of these is really significant so again we are left with some kind of unclear impression what actually happens.
Looking just at the picture, I would actually say it seems that Finland also counts on some of the debt being taken away by inflation, but if this is not sufficient.
They also tried to fight this by means of fiscal variables, but there Finland takes its time like Belgium has taken its time like France has taken its time.
Some surprise to me was Greece, where the first auto correlation is significant, but it is actually less significant than it was for Finland and France, Belgium.
So I would have expected that Greece is particularly a particular typical example of a country which finances its debt, not by means of fiscal revenues by by taxes, but rather hopes that can get around the debt by some other mean for instance by means of inflation.
And indeed, the first auto correlation is still positive, but it is not as significant as it was before. And then the second auto correlation is already much smaller.
And there's no positive auto correlation whatsoever in data between period three and seven.
So, apparently, Greek does Greece does not really try to finance its debt by inflationary processes over a very long horizon, rather, again, there is some evidence of fiscal policy in Greece, and in order to accommodate a bit.
And then there's Belgium, France again.
Here come Spain and Italy. Well, not much is there to say about Spain this looks very similar actually to France.
Two significant auto correlations which are positive, quite a few which are negative and setting in earlier than in the case of Belgium and France already in period five here.
So, probably again, first the speculation that inflation may eat up some of the debt, but later on, then, Spain seems to be seems to be quite reasonable and try to use future purposes in order to in order to fight the debt.
France to me was also Italy, because there's hardly anything to see in Italy. So, most auto correlations are just insignificant.
Well this one here is still insignificant, even though it is the greatest of the negative auto correlations. There's hardly any evidence of positive auto correlations either, because auto correlations here for instance, with a leg of two years are insignificant
three years four years five years, so very very small values here, only for a year of one year we have a positive auto correlation, and it is just barely a significant.
In any case, as compared to Spain for instance one would perhaps say that Italy seems to have less effort in eventually
financing the debt by means of surpluses. So, this year is also rather weak, right. This thing here is compared for instance to the experience of Spain or other countries with pronounced negative correlations.
See Finland here, see Greece there, see Belgium here, and see France here.
So in comparison to these developments, Italy looks rather meager here.
And this may support the claim that Italy is perhaps not absolutely resolute in servicing its debt by fiscal means.
Germany and, interestingly, Germany doesn't look much different from Italy. See there is weak first auto correlation, it is positive.
It is not very strongly. So, so perhaps in Germany there is not a big tradition of inflation financing of debts, of deficits, so surpluses may play a greater role, but actually we don't see this great role of surpluses.
Surpluses here. Everything down here is completely insignificant. So there is very little reason to suspect that Germany tries to use surpluses in the future in order to make up for decreases in surpluses in the presence.
The only rarely significant auto correlation is the first auto correlation, which would actually suggest that even in Germany there is some type of inflation financing of debts.
And then comes Austria where everything is insignificant, actually. No positive first auto correlation, so no evidence at all of inflation financing of debt, rather negative auto correlations already very early, even though all of them insignificant.
But still, a pattern, which we haven't seen so often, some evidence of this perhaps also in the German data but much weaker than in Austria, perhaps the clearest indication that a country may also think of increasing surpluses in the future when the current surplus goes down.
But not strong evidence again because it is not significant at 5%.
And here is the same thing for Ireland and Malta. Not much to say here. Ireland also has a positive auto correlation which is significant. Everything else insignificant with slight evidence of fiscal financing of deficits in later periods,
setting in already at period four, but all of that insignificant and Malta apparently has no inclination whatsoever to do fiscal financing of deficits. The only thing that we can see here is inflation financing of deficits.
Okay, at this point I would like to conclude today's lecture.
This is the last lecture before Christmas so I wish you all the best for the Christmas break. Happy holidays to all of you come good and healthy and to the new year.
And we see each or hear each other again in the first week of January. I think it is January 7 when the next lecture on the fiscal theory takes place.
Until then, you have time to do other things or work on stuff we have already covered. In any case, as I say, have good holidays and so long.