And just go back a little bit into the slides which we covered already in the last lecture in December, just to remind you what we defined there. Basically, we just pushed between monetary policy and fiscal policy by trying to give these terms some type of definite precise meaning.
And the wrap up for monetary policy was that we will speak of monetary policy in our model when a bond sale is conducted is conducted or let's say purchase, of course, so some type of
bond transactions between the private sector and the government sector. So bond sales or purchases take place holding surpluses constant. So this is the defining
circumstance of monetary policy that we say if surpluses primary surpluses of the fiscal authority are kept constant or more precisely the present value of primary surpluses of the fiscal authority are held constant. If this is the case, then we call a bond sale monetary policy.
And we distinguish a monetary policy from a fiscal policy by defining that fiscal policy is a bond sale if this bond sale company nice and increase or decrease some purchase sale of bonds company nice increase or decrease of future surpluses.
So we call this a fiscal policy if we have a bond sale, which is funded by increasing the future surpluses so that the bonds which are sold to the public are actually covered in the sense of a solvency condition by future surpluses of fiscal authority.
So the defining condition to distinguish between monetary policy and fiscal policy is whether future surpluses are kept constant. In this case, monetary policy that we sell the bonds or whether future surpluses are changed. So in this case for bond sale are being increased.
And we were looking at a number of data from the Eurozone to apply this concept and also look at the auto correlation structure of surpluses there.
I will not go back into this, but rather I would like to move on to come to the theoretical implications of different types of policies.
For this purpose, we still have two or three slides to cover in this set of lectures before we move on to slides number four and these two or three slides which I still have here in slides number three, make the point that obviously it is not so clear whether a monetary policy really should be defined in terms of a bond sale.
Because if we think of the central bank, we could also think of the board of the central bank deciding not on a level or amount of bond sales, but rather decide on fixing the nominal interest rate.
So this point I would still like to make to complement what I have said on monetary policy or even fiscal policy in the last lecture.
Think of it here in terms of a monetary policy where surpluses are fixed and the central bank decides on an interest rate target.
So a very specific case, a very special case actually of an interest rate rule where we have a target in the sense that we just peg the interest rate, nominal interest rate at some level I bar.
So that would be a possibility for the central bank to conduct its monetary policy by saying that bond sales will always be done in such a way as to ensure that the nominal interest rate which results in the market will be at the peg level I bar.
So in this case it would mean given surpluses, given a certain price level, given nominal surpluses basically here, given nominal surpluses, given debt from the previous period which cannot be changed anymore, new bonds would have to be sold or perhaps purchased so as to guarantee that the nominal level of the interest rate is equal to I bar.
And in this case, obviously there is one to one correspondence between bond sales and the level of the interest rate but you may actually read in the news about a central bank which conducts its monetary policy in terms of the peg nominal interest rate rather than reading that the
governing council of the central bank has decided to purchase or to sell bonds in a certain quantity.
So equation 62 here makes clear that these are actually basically the same things at least as long as we have an interest rate rule which comes about in the form of a fixed target for the interest rate.
So, regardless of what else happens in the economy, the central bank is stubborn in realizing nominal interest bank I bar.
Few central banks have done this very few do this today if any, I'm not sure if any central bank exists which banks currently the interest rate, but they have been experiments with pegging the interest rates and actually they were not very successful.
They are also theoretical arguments is kind of behavior that we will not deal much with this possibility of an interest rate bank but rather we will move on already in this lecture to a more flexible behavior by the central bank in which we postulate a certain interest rate rule.
So we say that the government is not necessarily committed to realizing a certain level I bar for the nominal interest rate, but rather that the government follows a rule which says nominal interest rate is adjusted in response to inflation or perhaps even in response to output or in response to some type of exogenous shock.
We will call this rule Taylor rule and you're probably acquainted with Taylor rules from your lectures on dynamic macroeconomic theory.
So this is probably nothing new, but you should just recognize that obviously pegging the interest rate at a fixed level I bar is just a special case of a rule is a special case of a rule which says that the rule.
Well, usually depends on say inflation and output gap. In this case, the dependency is zero so coefficients are zero, and therefore there's just a fixed nominal level of the interest rate, which the central bank realizes.
And this then would, as I have said, be just sort of the other side of a policy which is framed in terms of selling bonds by means of equation 62 here.
Now, I will come back to equation 62 in a couple of minutes when we are in the next set of slides. Before moving there, I just would like to give you an exercise, which is rather easy actually show that one percentage point increase in the nominal interest rate requires approximately
1% increase in bond sales. So basically show that there is an elasticity of one, which then implies that even in an economy which does not have frictions.
So, an economy without money demand or without frictions motivating money demand. The variations in bond sales required to control the interest rate are quite small.
Right. I mean that there may have been the concern actually that you would say in a frictionless economy elasticities become really great.
So, rather tiny movements in prices may perhaps cause big movements in quantities because there are no frictions.
So that in order to control the nominal interest rate, the central bank would need to have great bond sales or great bond purchases, but equation 62 shows you that this is not the case that actually the elasticity is just one resulting from this equation.
And this you shall in this exercise, find out on your own.
The second exercise, which you may want to think about, which is perhaps more thought provoking than insight giving.
Can the central bank given fixed surpluses fix the real interest rate. Right. So far we have spoken about the nominal interest rate but obviously you can also ask the question,
can the central bank fix the real rate rather than the nominal rate and may also contemplate about whether it makes sense for the central bank to fix the real rate rather than the nominal rate because most economists would say it is the real rate which matters in economic decisions and not so much the nominal rate.
Okay, coming back now to the definitions of monetary and fiscal policy which I have introduced just a couple of minutes ago.
We see from equation 55 which I display here again for your convenience that only fiscal policy can generate unexpected.
Right. Recall equation 55 which we discussed last lecture and in equation 55, we had on the right hand side of the equation, the unexpected change in surpluses and basically the present value of surpluses.
So in the discounted sum of all current and future surpluses current because I start at zero here so ST is included and then we sum all the surpluses which will
which we will have both in the current period and period T and in our future periods. And this notation here was telling us that we compare the expected value conditional on period T information with the expected value conditional on period T minus one observation
so that everything drops out which was already known in period T minus one because this information is also contained in the information set of the expectation operator with respect to period T information.
And what is left is just the new information the surprise information which was coming in in periods T so that these are actually the changes in surpluses which were not expected in period T minus one.
So that's why I call this the unexpected changes in surpluses.
And obviously, equation 55 tells us that unexpected inflation occurs because of unexpected changes in surpluses, because this term here on the left hand side of the equation is predetermined.
This is debt, real debt in period T minus one. So, nothing can change about this term anymore in period T. And this term here is just the inverse of unexpected inflation, as you know.
Therefore, equation 55 tells me that it is only fiscal policy which generates unexpected inflation.
Conversely, taking equation 54, which I also display here again for you convenience shifted forward in time by one period.
We see that monetary policy can generate only expected inflation, right, because if we shift equation 54 forward one time, then we get basically the same type of equation as an equation 55 here.
But now we have the monetary policy instrument here.
This is the real level of bonds to be sold in period T. BT over PT is now an instrument of the central bank of the central bank can sell these bonds.
It is a period T term which we have here so it is open to being manipulated by the central bank.
I think that monetary policy is characterized by constancy of this term here constancy of all the surpluses as I had just defined monetary policy to be that we know that this expectation here on the right side, right hand side is just some constant
which then means that by changing bond sales in period T, what has to address is the expectation of PT over PT plus one. And this year is of course expected inverse inflation I should actually write.
I don't know why I wrote inverse here and didn't write it here, but obviously this year is also inverse inflation and the expectation of inverse inflation, which means that monetary policy changes expected inflation, whereas fiscal policy
changes unexpected inflation.
Now, these are the two basic types of policies that can be enacted by the two authorities which together comprise the government so the monetary authority of the fiscal authority and we have now a clear concept of how to distinguish monetary policy from fiscal policy,
which however is somewhat cloudy when we look at actual policy, because in the real world, very clearly, politicians typically mix fiscal and monetary policy.
So you will rarely see that there are along with bond sales, not also changes in expected surpluses or even unexpected changes in surpluses, so that it is not so easy to say what is actually monetary policies since surpluses basically tend to change any time.
And on the other hand, you would not like to come to the conclusion that monetary policy just doesn't exist and everything is always just fiscal policy because surpluses change.
So, in empirical work it is quite difficult to distinguish between monetary policy and fiscal policy, even if we have these kind of theoretical distinguishing characteristics of the two policies, and you should be well aware of the fact that here we make our
life easier in theory that is in practice.
That was all I wanted to add to my lecture last year, last lecture in December. Supposing that there are no questions and I didn't see anything coming in, I would now move to the next set of slides, which is slides four.
And I should note that when I prepared for today's lectures I made some changes to the slides because I found that some issues I wanted to explain were not so well explained on the slides.
So the slides you have downloaded from STEAM for slides four have minor discrepancies as compared to the slides, which I will prepare, which I will explain now.
I have actually not yet uploaded the new set of slides but I will do this after this lecture.
So tomorrow let's say, give me some time, tomorrow you may also download this version of the slides or you may take notes in your version of the slides where my presentation will be a little different.
It's not a dramatically different thing but it is more than just correcting typos.
All right.
And now we talk about monetary policy in the fiscal theory of the price level model. And we do this in order to understand well what the role of shocks, monetary policy shocks and fiscal policy shocks in the sense of shocks who expect the surpluses is in a fiscal theory of the price level model.
Because what we want to do next and what we will actually do already in this lecture is that we compare the role of these shocks in the fiscal theory price level of in the FTP L model with the role of these shocks in the new Keynesian model and actually
we will arrive at a very systematic comparison between the new Keynesian model, almost the standard new Keynesian models, simply titled it a bit in terms of output but otherwise the standard new Keynesian model and the fiscal theory of the price level.
And I think we will derive some quite surprising results when we do this comparison. But in order to do this, it is important that we understand the role of the shocks in the fiscal theory of the price level model and therefore here we look at monetary policy.
Now as I have just said, a fiscal policy determines unexpected information. And this was the equation which used for that. Here it is written again in terms of the equation being shifted forward in time by one period but otherwise it's precisely the same equation that we have seen already on the previous slides.
Now let's manipulate this equation a little bit because what I want to do is actually that I would like boil down the FTP L model now to a very, very simple set of equations.
Since I'm a little worried that these type of equations here may be a little difficult to understand when we go into studying the dynamics of the model since there are so many terms here like all those S's being summoned over the complete time horizon and these kind of things.
So I would like to simplify the model and go first some very basic analysis.
In order to do this, let us first define small bt as the real value of outstanding debt in the beginning of period.
So, bt minus one is, as you know, the nominal value of outstanding debt, which retires in period t, so which must be paid back by the government in period t, and the government has to pay this back then of course in terms of prices, p t so it makes sense
to look at bt as the ratio of bt minus one over p t as the value which is being redeemed by the government.
Okay, so this is small bt just the definition of real value of outstanding debt.
Using this definition we can transform equation 60 to this form here, where you must be aware that here now I have p t plus one, rather than bt because what I did was that I divided through by bt plus one in this sum here.
Okay. And it is important to understand that here I divide by bt plus one for every term in the sum. So, st plus one plus i is also just divided by bt plus one when i is positive.
So there's no i here and there's no i missing here. All future surpluses are divided by the same quantity, namely by bt plus one.
So that actually what we have here is the ratio of surplus over current debt, debt of period t plus one.
And by this operation basically by introducing this small bt plus one here as bt divided by p t plus one.
By this operation I get the p t plus one in here and the small bt plus one here which is certainly not so difficult to understand.
Now why is this helpful? We leave the right hand side as it is and just look at the left hand side where we see that the ratio p t plus one over p t is just the inflation factor.
One plus pi t plus one and then here comes e t plus one minus e t unchanged. Well, and p t over p t plus one is one over the inflation factor as you know.
So here we have two expressions which almost cancel and I will show you in a minute that parts of it actually do cancel.
Just these expectation operators here prevent currently that they cancel completely because we have the expectation here in period t of one plus p pi t plus one, which is not the same thing as one plus pi t plus one without expectation operators so that this term of course does not cancel.
But what is true is that the expectation conditional on t plus one information of one over one plus pi t plus one is just one over one plus pi t plus one because given information of period t plus one we know what the inflation rate in period t plus one is.
Therefore, we do not need the expectation operator here anymore. We know exactly that this here is the expectation and using this our equation simply simplifies to one minus inflation factor times conditional expectation of the inverse inflation factor is equal here to the surprises in surplus relative to existing real debt.
Now, still the equation is not in a very nice and easy to the test form. So the next thing we do is that we take Taylor expansion actually of this expression here.
Because usually inflation rates are numbers which are close to zero. Oh, we have 2% 3% 5% inflation but usually we do not deal with great inflation rate so typically do not deal with hyperinflation like we used to have them in the 1980s Latin America on the 1920s in Germany, but rather moderate degrees of inflation where we can safely say that the inflation rates are close to zero.
So that if we take a Taylor expansion of this term here, or actually just of this term the inverse inflation factor at pi equal to zero, we get that one over one plus pi t plus one is approximately equal to one minus pi t plus one.
And when we now replace the inverse inflation factor here by this Taylor approximation of it, then obviously our equation becomes easier because then we have that your left hand side of the equation which had this form here
Can be written as or is approximately equal to one minus, and then the product of the inflation factor times one minus the expected inflation of years, plus one.
Plus one expectation in period t, which if I multiply out this term here, results in the one canceling against the product of the ones here, we are being left with minus pi t plus one plus the expectation of pi t plus one plus a term which is pi t plus one times the expectation of pi t plus one.
Now, we said already that pi t plus one is a small number close to zero. And when I multiply a small number close to zero with another small number close to zero, it becomes smaller and closer to zero.
So we approximate again and say this thing here is approximately equal so that we are left with minus pi t plus one plus expectation of pi t plus one, which then results in a 60 as well or in the approximate form we can then write 60 as pi t plus one minus the expectation of pi t plus one.
So that we here have unexpected inflation pi t plus one is equal to unexpected changes in this surplus ratio here.
And I will in the future, not speak of a surplus ratio anymore but just again speak of surpluses but you please bear in mind that this is surpluses relative to the current level of real data.
So real surpluses divided by the real value of that. This is here our surplus term. And all of this I just call epsilon t plus one s.
So the unexpected part of the surpluses is like a shock, which realizes in period t plus one. Right. So what we have here, this whole term, this whole term here is actually just something which happens in period t plus one.
Because whatever has already materialized in period t is being subtracted from what we have in period t plus one. So here is just the surprise which happens to this sum here.
In period t plus one. And this surprise I just call epsilon t plus one. And the superscript s is for surplus not for surprise. Right.
This is a surprise is expressed by more or less the epsilon because epsilon is the standard symbol for shock for something which comes as a surprise. But this is surplus shock, which I have here.
And I write it as minus epsilon because there was a negative sign here so that this thing taken positive is actually equal to epsilon, which means a positive epsilon is tantamount to an increase in primary surpluses either today or in the future.
An increase in the present value of primary surpluses.
Okay, and this is what I've written here epsilon t plus one s is an innovation is a surprise and innovation to the present value of future surpluses.
Be aware that is now future surpluses. So I start at zero, and we start here with s t plus one plus I. So it is future surpluses or period t plus one surpluses, which we speak here which corresponds to the fact that we speak about unexpected inflation in period t plus one.
So that unexpected inflation is positive when future surpluses are lower than expected. So there's an inverse relationship apparently between unexpected inflation and future surpluses.
Now does this make sense from the point of view of the fiscal theory as we so far got to know this theory. Yes, indeed, it doesn't make sense to speak of it in this way or have this inverse relationship, because think of what what happens.
Suppose that we have a change in the present value of surpluses, in the sense that there are fewer surpluses in the future. So we have some type of expansionary fiscal policies taxes are being lowered or expenditures are being increased so that surpluses are being decreased.
If this is the case, then we can cover less debt with the present value of surpluses because the present value of surpluses is lower.
And this would mean that the real value of that has to go down, but the real value of that can go down only if the price level increases. So if we have inflation.
So this means if we have negative shock and epsilon if epsilon is negative, then we have a positive development in inflation and vice versa, of course, of course, also if we increase the present value of future surpluses.
So if we can expect higher primary surpluses in the future, then inflation should actually go down or there should be less pressure on prices prices should not rise as fast as before, because surpluses are sufficient to cover real debt.
So this is actually the first this equation 69 is actually this first equation of which I said that we try to simplify the model to write the model in a really easy form.
Because, forget about this part of the equation here which still has this ugly sum in it and the difference of the expectations operators.
And we have a very easy equation here which just says expected inflation by t plus one minus expectation of by t plus one is equal minus shock minus a fiscal shock.
Right. And that's all. So this is the first of the really easy equations which we now will analyze in the course of the analysis of the FTP model.
The other easy equation is the Fisher equation which you know already from my previous presentations.
We know that the real interest rate factor capital RT is the nominal interest factor one plus it times the expectation of inverse inflation.
And so this is the Fisher equation, which we have here, and we will also take this linear form so I linearize this equation here, assuming again that both inflation and the interest rates the real interest rate at the nominal interest rate are close to zero, so that we can approximate
the real rate of interest are, which is actually constant as we know in our particularly simply simple model, we can approximate the real rate of interest as nominal interest of interest minus expectation of future expectation of future inflation.
And that is probably the standard way by which you know the Fisher equation in the first place so in your undergraduate macro economics lectures probably one has presented the Fisher equation to you always in the sense as saying the real interest rate is nominal interest rate minus expected inflation,
which is actually not the exact Fisher equation, the exact Fisher equation is this nonlinear equation here, but it is for small inflation and interest rates, the linear approximation to the Fisher equation and that is what we usually deal with.
The second easy equation, which we have we have equation 69 on the previous slide, and now equation 70, which are just two very easy linear relations.
And what happens to the rest of the model. Now let us assume for simplicity that the central bank has a nominal interest rate target it. So just the setting that explained at the beginning of this lecture.
If the interest rate is pegged at some level, I bar.
In this case, we have seen equation 62 I have commented on this a couple of minutes ago.
And we know if the interest rate is pegged if the nominal interest rate is pegged as at I bar, then this implies that the central bank has to have a certain level of bond so they either have to sell bonds or they have to purchase bonds at a certain level.
So we play much of a role here what the central bank is doing, we can just put this equation aside and say okay equation 62 of course it does exist but it determines nothing else but the amount of bonds which are being purchased or sold by the central bank.
Otherwise, it is of no great value to us currently so we can basically forget about equation 62. If the center bank has a normal interest rate target and focus on equation 70 and equation 69 on the previous slide.
So this equation here which says an unexpected inflation is equal to the negative of the fiscal shock.
And in this sense, we have boiled down the fiscal theory model of monetary on the sea to two equations, namely two equations 69 and two equations 70 so to these two very simple equations.
So we may now for the first time make an analysis in the fiscal theory model by just studying these two linear equations, which are of course simplified by means of failure expansions, where we take note of equation 71, which is actually
an example that inflation in period t plus one can be decomposed into expected inflation and unexpected inflation right expected inflation would be the expected value in periods t of it plus one.
And then the difference between actual party plus one and expected party plus one is unexpected inflation.
So this I have here as equation 71. So using now equation 69 and 70 I obtain this simple equation here inflation pi t plus one is equal to the nominal interest rate minus the real rate minus a fiscal shock.
And that's sort of the news we have here.
You are used to knowing from the Fisher equation that the expectation of party plus one is equal to normal rate minus real rate.
Here, we do not have the expectation of party plus one on the left hand side of the equation but we have actual inflation on the left hand side of the equation and perhaps we are more interested in actual inflation than expected inflation.
However, this may be we also have an equation for expected inflation as you know equation and 69 but here we now now have an in an equation for actual inflation.
And we know now that the difference between expected inflation which is it minus are an actual inflation is just well what is unexpected inflation and that is as we have seen the negative of the fiscal shock.
So that's actually fairly easy right after all the technical effort that we have put into the model you may be surprised to see that we bought it down to something so easy and so trivial and be assured it will become a little more complicated
But it is nevertheless I think helpful to understand the model in its most stripped down almost naked form of just two linear equations in the way in which I have put them here because these equations are so easy to understand.
So, this, by the way, is the slide which I have changed most when I re work the slides this morning.
So here you see something which you do not see that form on the previous slide even though the general content is the same and I meant to say the same thing but I think it is now better presented.
Let us study now the effects on the normal interest rate and on inflation of three different types of shocks that I would consider.
One shock, I call a well basically monetary shock that shall be permanent shock which I denote by new and this shall be a shock to the nominal interest rate it in let's say period one.
Suppose that we look at the model in deviations from the steady state and suppose that we are in the steady state of the model prior to period one, at least now when we look at new.
So we may happen what actually changes if the central bank decides to change the interest rate target by an amount of new.
So by setting it equal to new for t greater than or equal to one.
Whereas previously it had been equal to zero which did not mean zero interest rate but it means zero deviation from the steady state so some state state level of the interest rate and there the interest rate was being kept kept cuts and now the center bank changes this interest rate target increases the interest rate
target by the amount of new. And we may ask, what will actually happen with inflation. If the interest rate goes up.
And why is this a relevant question that should actually be clear but perhaps in an old mess of formula which I have presented in this lecture it is not that clear to you.
You may know that central banks typically increase the nominal interest rate when they think that inflation is too high.
And typically they lower the nominal interest rate when they want inflation to increase so when they think we are dangerously close deflation, then typically central banks try to lower the interest rates and thereby stimulate money demand and by higher money holdings
they hope that inflation will pick up and there will be higher inflation.
So, typically, we would think with this kind of macro economic knowledge most of you will have assembled in your undergraduate studies that when the central bank changes the interest target here in the sense that there is a positive new so short run nominal interest rates are being being raised.
Then, typically, we would expect from more or less newspaper reader knowledge that this depresses inflation so inflation should go down.
And we will look at the model here and see that this is not true.
In our model inflation will actually go up when the central bank raises the short run nominal interest rate.
And that's quite important to understand it is an insight which is actually very clear when you look at the structure of the model.
And we will see that is not special to the fiscal theory of the price level model, but that the same insight actually will also develop in the standard you can see it.
Contrary to what most new Keynesian theorists will have told you that central banks raised the interest rate and then inflation goes down and they have showed him consequences and all these kind of things.
What we will learn shortly, is that even in the non Keynesian model, nominal interest rates and inflation rates are positively correlated and not negatively correlated like, like you would probably expect they should be.
So, that puts some question marks as to whether central banks actually have the right type of policy, and I will also come to the empirical part of this problem here by showing you data for the European Central Bank.
You know that the European Central Bank has for years now lowered the interest rate, the nominal interest rate, lowered so strongly that actually key interest rates like interest rate on the deposit facility of the central bank are negative.
So, unprecedentedly low interest rates in the hope that inflation would pick up that inflation would rise to the desired level of close to but lower than 2% but actually that has not happened.
Right. So, where the models property which we, as we expose it here as I exposed here for for you this positive correlation between nominal interest rate and inflation may be a surprise to you because you thought it is perhaps just the other way around.
It may not be an unpleasant surprise in perspective of trying to understand what's happening in the real world, because perhaps in the real world we see precisely that that inflation and interest rates are positively correlated rather than negatively correlated.
So, not counter movements of inflation and interest rates but rather co movements of inflation and interest rates in the real world, and the members of the central board may sit there and scratch their heads and think why is this happening and you
may come up to them and tell them well just look at your models, just look at the new Keynesian model which you have in your research department and developed up to such that nobody really understands what's going on in this model anymore.
If you boil it down to the key equations of the model, as we have just done for the theory interpretation of the model and we will shortly do it also for the Keynesian model.
If you boil it down to the key equations of the model then it is very plain to see that actually inflation and interest rates are positively correlated and not negatively correlated as central bankers seem to think they are.
Clearly, this could also be used as a criticism of the model. So if you adhere to the view that in reality interest rates and inflation rates are negatively correlated then you may say well obviously we are working with the wrong model because it has the wrong type of prediction.
If you want to draw this conclusion, it's fine with me, you may do that and you may say well then we forget about these models and look for different models.
But then we also have to look for different models, then we also have to construct different models because the standard model all economists work with the macroeconomist and the Keynesian model
would also go to the waste paper basket, not just the fiscal theory model as I will show.
Okay, so that's why I look here now in the very basic view at a permanent shock to the nominal interest rate. We think of this shock as being an positive shock so the nominal interest rate rises obviously.
Over lower than before, just the opposite would happen so no surprise there. The central bank changes the interest rate target and we will look at what happens with inflation.
Well, it's not so difficult to see right what happens with inflation just look at this equation here.
Pi t plus one is equal to the nominal interest rate minus the real rate minus the fiscal shock. So if we have a monetary shock only.
So, the central bank raises the nominal interest rate. Suppose, no surprise in fiscal shock so this shock here zero real rate fixed, because this is something of something which is determined technologically as the real return to capital.
The only thing which happens is that the central bank raises its interest rate raises its pegged target for the nominal interest rate that our model tells us pi t plus one goes up also right so we would have a one for one change
in the nominal interest rate one percentage point plus for the normal interest rate would lead to a one percentage point increase in the inflation rate. Sorry, I think I just said
interest rate but of course here on the left hand side is the inflation rate. So the two are clearly positively correlated.
So this is the one shock, we will start. Now the other shocks are two variants of fiscal shocks. First, the way which is perhaps the most natural to look at it.
We look at an unexpected increase in surplus of period one.
And here I have to be careful with what I mean by unexpected and that's actually why I made changes to this slide here. You know that this epsilon is already the surprise, which happens.
Right. So this is already the unexpected development in the present value of surpluses.
What I mean here is that the epsilon itself comes along unexpectedly. Right, so we have no information prior to period one about any surplus changing.
And suddenly, in equation one, in period one, we are given new information which tells us unexpectedly that surpluses have changed in period one.
So epsilon s in period one is greater than zero.
Let's suppose that this increase or differently, the epsilon is, as you know, the sum of all surpluses from period one up to the distant future.
But suppose the news which we receive is actually that the surplus of period one increases.
When we receive this news that the surplus of period one increases, we receive this news in period one.
So exactly in the minute in which the surplus of period one increases, we are informed about that fact. So this comes completely unexpected for us.
We assume that surpluses in period two and later are not changed at all. So there's no effect of the higher surplus in period one on surpluses in succeeding periods to make things easy.
Right, so all other surpluses stay constant.
Then we have epsilon one s greater than zero.
So this is a trivial to you and you may think why does it discuss this so long. I do this because here comes a slightly different fiscal shock, which I call an expected increase in period one surplus announced in period minus two.
Three periods in advance, essentially. So in period minus two, we already learn about a surplus at a later period, in a later period, namely in period one, being higher than we have so far expect this surplus to be.
Which means that we are given three periods in which we can already expect that the surplus in period one is greater than so far we have thought it to be.
Again, I assume that all the other surpluses are not effective.
And here, actually, I should improve my wording one more time because here I'm writing not if affecting unexpected surpluses in period minus one and later, which is not quite true because period one is later than period minus one.
So obviously in period one the surplus has changed but in period minus one period zero and then period two, three, four, five and so forth.
Surpluses are not changed. This is what I mean here would probably make another change.
Anyway, what does this mean?
Here perhaps you understand why I bothered a little bit about the wording I had on the previous slide, which is still having your documents as I suppose about the meaning of the word expected
because the word expected comes up here in two different meanings actually. When we are informed in period one, about a highest, sorry, when we are informed in period minus two, about a higher surplus in period one, then we can expect already starting with period two, with period minus two,
we can already expect the surplus in period one to be higher and in that sense it is an expected increase in period one surplus.
Nevertheless, the news we received in period minus two came completely unexpected to us.
They were completely unexpected to us. Nobody had any clue that there would be information in period minus two about a higher surplus three periods later.
Right. So in that sense, it is something unexpected which happened in period minus two.
And the unexpected developments about surpluses we always capture in the epsilon shock. So what we here actually have is an unexpected development in period minus two, which is why this event here expected increase in period one surplus announced in period minus two
is the same thing as a positive epsilon s in period minus two.
Right. So not in period one as here is epsilon s positive, but in period minus two is epsilon s positive. In period minus two it was unexpected that the value of future surpluses rises.
So there was news, it was something unexpected and this is why it is part of the epsilon.
But from period minus two onwards, we know about that. So we expect the surplus in period one to be greater than it used to be.
Of them used to expected.
So we have three different types of shocks at which we will now look in graph and one monetary shock new and two versions of fiscal shock.
One, which has sort of instantaneous increase in surpluses and one which has an increase in surpluses in the later periods period and this I capture by epsilon one s and epsilon negative two s both of which I assume to be positive.
When I study the effect of these three shocks here I will always assume that when I studied the effect of a monetary shock, then nothing happens on the fiscal side so then all the epsilon shocks are just zero.
And when I study the effect of a fiscal shock, this one, this fiscal shock here, then I assume that there's no monetary shock and no competing fiscal shock so I just study one shock at a time.
And here is what happens.
Perhaps it's a little difficult to read this graph.
But I'll explain it to you and I hope that you familiarize with this type of analysis or this type of graph because diagrams like these will come up really frequently but every once in a while now in the lecture to visually illustrate what happens in the model.
Now, what do we have here. We have time here on the x axis.
So there is time from minus three, minus two, minus one, period zero, period one, and then two, three, four, five.
And here on the y axis I just give percentage changes of the variables at which I look. Basically I look at the nominal interest rate and I look at inflation.
Okay, so start with the blue line here. This is the development of the nominal interest rate, given a monetary shock, which is in this graph labeled an interest rate shock.
So how does the nominal interest rate behave when we have a monetary shock new in period one. Well, that's very easy, of course.
All variables always start at zeros, because we just look at deviations from previous levels. So basically from the steady state.
So we know the interest rate did not deviate from the steady state deviation was zero until period zero, nothing happened, but in period one there is a shock of size new equal to one.
And then obviously the interest rate is higher, permanently higher by the size of the shock, which in this case is new equal to one.
So that's very clear because what happens here is just that the central bank decides to peg the interest rate permanently for our purposes at a higher level, higher by the amount of new.
So this is why we have this type of I, this type of reaction of the nominal interest rate to a monetary policy shock.
Okay.
So, basically,
let's see. Yeah, basically just this just says well if we increase I then I will be higher.
Now what happens to inflation in this case. Here we have to look at equation 72 and I spoke about that already when I pointed to the positive correlation between inflation and interest rates.
Inflation rises when the nominal interest rate rises. So suppose new is equal to one.
So it goes up by one unit by one percentage point, then obviously inflation also goes up by one percentage point, right, this is what the equation 72 tells us.
However, inflation is doing this only with a leg of one period. So, the interest rate goes up in period t inflation will go up only one period later period t plus one, which is why the impulse response function of pi.
In response to an interest rate shock so in response to be equal to one looks pretty much like the impulse response function for the normal interest rate, just taking off one period later.
Right. Otherwise, it's exactly the same, but here is this lag in the reaction of I to the shock in I so inflation is latecomer in this model because there is a lack of one period.
And that's it already for the analysis of the monetary policy shocks. Now we know how the normal interest rate and how inflation rate is affected by a change in the central bank policy by a higher value of the pegged nominal interest rate.
That's all currently because the model is so, so simple.
Next, let's look at the fiscal shock. So let's look at this development here there's an unexpected increase in surplus of it one and we learn about this unexpected increase in the surplus of period one and period one.
So the same period. So epsilon one s is greater than zero. So look at our model, what does this tell us for inflation pi t plus one here depends on epsilon t plus one s.
So there is no lag when we have impurity plus one it change in epsilon, then inflation adjusts immediately, same inflation adjust so you can write the same equation here for p t, obviously, right.
So this is an instantaneous reaction of inflation when epsilon changes.
So if we have a higher surplus or in periods t plus one, say, in our diagram here t plus one is period one so set the t equal to zero.
If this is one unit higher than obviously inflation will be one unit lower in period one.
How about period to inflation. Well period to inflation is not affected at all, because epsilon and periods to zero. Again, right, epsilon t plus two is not changed, we have already digested the news about the higher surplus in period one.
There's no change in surpluses of period two, three, four, five or something like this. So inflation in period two will not be affected at all.
Inflation in period two will be at the same level as before. So we'll be back at the level of zero, which explains why the impulse response function for inflation looks like this.
In period one, the news comes in about the surplus of period one being greater than previously thought, greater by one percentage point.
Inflation is negatively correlated to surpluses, to surplus news, right. So inflation goes down when surpluses go up.
It goes down by one percent point if surpluses go up by one percentage point and in the next period in period t inflation is already back at its initial level at zero. So everything is constant then.
So this is why we have this type of reaction of inflation to a unexpected rise in surpluses of period one.
Now finally, what happens when we have an expected increase in period one surplus being announced in period minus two? Well, obviously, same thing.
The news was now coming in in period minus two.
In periods minus two, we learn that the present value of future surpluses is greater, not because the surplus in period minus two was already greater, but only because we know that in period one, three years later, three periods later, there will be a higher surplus.
Nevertheless, the news, the unexpected thing, happens in period minus two. So the shock, as I have explained, is a minus two shock, which means that inflation in period minus two goes down.
So the epsilon is positive, inflation will go down by one percentage point.
Nothing else happens to inflation in any other period, because there are no news anymore. In particular, in period one, nothing happens to inflation because nothing unexpected happens in period one anymore.
We have digested the information of the surplus being greater in period one than previously thought. We have digested this information already in period minus two.
So this is why, for the expected fiscal shock, inflation behaves like this, goes down in period minus two, because expected fiscal shock is positive, goes back to normal in period minus one, and nothing happens since.
Nothing happens in any later period, in particular in period one, inflation does not react to higher surpluses, because the information is already in the world since period minus two.
Okay, so these are the two impulse responses for two different versions of a fiscal shock that we have for inflation.
Now you may ask, and actually I would like to ask this to you if we had a usual lecture where we could have interacted, but since the question is rather trivial, perhaps I don't ask you now.
Usually I would ask you, what about the interest rate? How does the interest rate behave when we have a fiscal shock, either the expected fiscal shock or this year, which is sort of the unexpected fiscal shock.
Actually the wording expect fiscal shock which Cochrane has in this book here is not so good. The shock is also unexpected in period minus two.
It's an unexpected shock there, but from period minus two onwards, we can expect the increase in surpluses in period one and there doesn't play a role anymore, as I have laid out.
All right. Now, why is here no impulse response function, which shows us how the nominal interest rate reacts when we have these type of fiscal shocks?
Perhaps I should actually ask you, does anyone know the answer, then please wave to me.
Does anyone know the answer why there is no impulse response function here for the eyes?
I don't think too complicated the answer is trivial.
No guess? Well, very easy, of course, the interest rate is pegged, right? We said the interest rate is pegged. The only way to change the interest rate is if the central bank decides to change
the peg, raise it or lower it by some amount new, that has not happened. We have here just considered the fiscal shocks and no monetary shocks which involve the new.
So obviously the nominal interest rate stays completely constant under pegged interest rate when we have the fiscal shocks either this version of the fiscal shock or that version of this fiscal shock and therefore it doesn't make any sense to draw the line for IT shocks in the presence of fiscal shocks.
It's just zero, everything's zero.
Okay, good. So that is basically what we can learn from this very primitive, very simple version of the money.
And while you may say it is trivial and perhaps not interesting, I would actually contradict this assessment because it leads us to one of those key properties which I will spend more time on later.
That inflation and the interest rate and nominal interest rates are counterintuitively when you think of it in terms of central bank typical monetary policy statements counterintuitively, they are positively correlated.
Okay.
The other property which is not so nice actually is that inflation only reacts with one period delay to an interest rate shock.
Now, well, of course, you may say there is some time lag until interest rates are channeled into inflation rate so perhaps it's not that unreasonable.
But it is of course a little mechanic here just comes from assumptions we've made when we set up the model so it is not really well explained by economic thinking that we have this.
Note that even if the interest rate shock were not say k periods ahead inflation would not move until after the shock is realized so there is no forward looking behavior here in inflation, in the same way as we had it for inflation with looking at the
and the surplus developments in our model such as we have it here inflation just reacts to the interest rate of the previous period.
And even if we know in period t plus one that interest rates would be higher in period.
Plus two t plus three plus four or whatever. No change would take place so prices would not react to that certainly a weakness of the model that inflation doesn't react, at least so far.
In this very simple version of the model inflation does not react to news about future interest rates, it's just reacting to use amount future fiscal developments.
Consequently, in this model expected monetary policy, which would be the case if we know how the interest rate develops in the future as the same effect as unexpected monetary policy there's actually no useful distinction between expected monetary policy unexpected monetary policy
in this very simple setup of the model. And then you may say well this is perhaps not so very interesting and not highly realistic.
Yeah, and in particular, you don't have the property that higher interest rates do not lower inflation that sense you may say it is not realistic because most central bankers think that this is actually the case.
Fiscal shocks, on the other hand, and there's more interesting lead to a one period disinflation as soon as the news becomes known. So it is not important according to this model when actually the fiscal authority is able to achieve higher surpluses.
It is important when markets learn about highest surpluses being likely or even can safely be assumed in the future. So whenever financial markets are convinced that in the future primary surpluses will be higher than they have currently expected.
And then there will be this inflation according to this model here. So inflation is forward looking with respect to fiscal policy, but it was not forward looking with respect to interest rate policy.
So inflation obviously adjusts to the news which come in about future developments and saying that means inflation adjusts before the economic tradition actually observes the change in fundamentals right inflation comes about just by reading say
newspaper announcements or things which come to the capital market by means of Twitter or tickers or whatever so some type of news about future developments.
We do not yet see the future developments actually in collected data, we may only see this years later that surpluses have increased.
But markets may already have taken this development into account because they received the news and found the news credible.
This is quite challenging actually for any kind of econometric analysis, because econometric analysis, we are used to explaining observed data by observed data.
But this model here would tell us we cannot explain inflation, which we observe in period t by data which we have observed in period t or prior to period t.
So, any type of econometric regression which would try to explain inflation by regressing inflation say on current and past interest rates would be misspecified.
Actually, oh, now I said interest rates, I should say surpluses so regressing inflation on current surpluses or past surpluses would not make sense.
If this model were a good explanation of what happens in reality, because inflation is dependent on future surpluses and on news about future surpluses.
So not only can we cannot even say that we regress inflation on future surpluses. Let's say for instance we have a certain time span of data from 1950 to 2020.
And then we can of course say we regress the inflation rates of the 1980s, the 1990s on the inflation on the surpluses which later have materialized in the first decade or the second decade of the
21st century, because now we know what those surpluses are. But that again would not necessarily make sense because we don't really know whether these surpluses were already anticipated whether they had been used about the surpluses which came in later.
Now we know they came in that way but we need actually to know whether there was news about these surpluses already in the 1980s or 1990s.
So that's quite challenging for economic analysis.
The actual timing of when the surplus is increased or decreased is completely irrelevant in this model.
Relevant is when the news comes in about the development of the present value of surpluses. So that's quite a complicated structure to think in and quite a difficult framework to draw economic conclusions from.
But I think it may well be the case that this is actually what happens in reality that people that economic agents anticipate future developments. It's quite rational to do that and as soon as they have some means of anticipation, as soon as they have some news,
they will have to bother about these news. They will have to digest and process these news and make something out of them. And we don't really know when people receive what kind of news, how credible these news are and
by how much they change people's behavior. So all of that makes it very difficult to do vertical analysis.
Good. So that was the analysis of which I have now spent quite some time for the very simple boiled down version of the model.
The section of the lecture, which I currently talk about is actually overwritten monetary shocks in the fiscal theory or monetary policy and the fiscal theory of the price level.
I have now talked not only about monetary policy, but also about fiscal shocks to sort of sharpen your understanding of what happens in terms of fiscal policy and in terms of monetary policy, but now I would like to focus more on what topic of this
section actually is namely monetary policy by modeling monetary policy in a less trivial way than just taking the interest rates.
As I have already said, today, hardly any central bank fixes nominal interest rates.
Since the 1970s, actually, we know that this is not a good monetary policy. So what was quite fashionable, at least until we ran into the financial crisis, not clear whether it is still an accurate description of monetary policy was to model the behavior of a central bank by some kind of
state dependent rule, which is often called Taylor rule, and such a rule basically expresses the idea that the interest rate decisions by the governing council of the central bank respond to certain conditions in the economy, in the economy.
We know that all central banks have big departments research departments which look at the conditions in the economy and then make suggestions to the board of the central bank or its governing council as to how adjust interest rates given conditions in the economy.
So it's very reasonable to have a rule which is not just fixing interest rates but rather responding to conditions in the real economy.
So, Taylor's famous approach, which he also supported by empirical work was to say that the central bank adjust the nominal interest rate when certain targets are not achieved.
Certain economic targets are not achieved in the macro economy in particular target for inflation and the part target for output, and we will hear
focus our attention just on a target for inflation or on some kind of responsiveness of the central bank policy to inflation by means of a rule which I now call it Taylor rule as expressed in equation 73.
So this rule would tell me the central bank, it just the level of the normal interest rates, given first some type of real interest rate are plus some response to the nominal inflation rate, where phi is just some parameter,
which allows the central bank to have responses to inflation, which are either more or less than one for one. So five or equal to one, then it would be equal to our plus pi t and this would mean that the central bank raises the normal interest rate whenever inflation
raises the normal interest rate by one percentage point whenever inflation rises by one percentage point, but that need not be the case.
So it is quite possible that the central bank raises the normal interest rate more strongly than just one percentage point and in this case five would be greater than one or in principle it would also be possible that the central bank raises the interest rate by less than one percentage
point or less than one for one. So in this case, five would be smaller than one.
So, the nominal interest rate would be equal to the real interest rate or this is what the central bank desires be equal to the real interest rate plus phi times the inflation rate plus new t, which is some disturbance which occurs, for instance, some development from
various capital markets. Let's say the European Central Bank as the authority which follows this Taylor rule here, notes that in the far east, Hong Kong, Asian capital markets are in the far west, so in the Americas, interest rates go up.
Right. And then they may think, well, this is some type of shock which occurs to the central market that other central banks raise their short run nominal interest rates.
So we also have to take this into account and possibly we will raise our short run normal interest rate also and adjust to that.
So, think of new t just as some shock, and we will take this shock to be a shock which is autocorrelated over time, which is why we model it as a AR one process on auto regressive first order auto regressive process new t let's say is row times new t minus one plus some real surprise
correlation epsilon t i with a coefficient of autocorrelation row being smaller than one absolute value.
So we have to adjust to allow for the possibility that the that the autocorrelation, which we observe in real world data for, for instance, about developments in foreign interest rates that we can depict this in our model.
We have to allow for autocorrelation here and take the epsilon t is then as the real monetary shocks.
So what do we have to do, we just insert the Taylor rule in equation 72. Recall equation 72 was this equation here. So pi t plus one was equal to normal interest rate minus real rate minus epsilon s plus one.
So now for the nominal interest rate we introduce, we insert the Taylor rule here, then you see already we have a minus R and there was a plus R in the Taylor rule so these two terms cancel.
Right here's this plus R which we have in here. So when we insert 73 and 72, then we get pi t plus one is equal to five times it plus new t minus epsilon t plus one s.
Essentially here we see now with some type of responsiveness in central bank interest rate policy, we would have an autoregression for the inflation rates pi t plus one depends with autoregressive parameter phi on previous inflation.
But phi may actually be greater than one. So can be an explosive autoregressive process. And then there is some shock which is added to that as we usually do with an autoregressive process. However, this shock consists of two components, namely a monetary shock which by itself is already assumed to be
an autoregressive process of first order and some fiscal shock of which we have not made any assumption on how it evolves over time.
Okay. And this year would be basically the equation describing the development of inflation when the center bank is responsive to microeconomic conditions in the economy.
Now suppose that all variables are initially at zero. So same assumption as before we are in the steady state and we have a monetary policy shock of epsilon t i equal to one in period one.
So things are a little more complicated than the previous impulse response analysis that we have done because in equation 75 we don't see our epsilon t i.
Epsilon t i was in fact in equation 74 and we see here that 1% increase in epsilon t i would raise new t by 1%.
Right? Rho times new t minus one, rho times new t minus one would be unaffected because the shock happens in period t, so anything which has happened previously is not changed.
So the first thing that happens is actually that by means of epsilon t i being increased by 1 percentage point, well the new t increases.
Right? So what happens when epsilon t i is equal to one, then we simultaneously that new t increases by one and therefore we see that pi t plus one would in the next period increase by one.
Okay.
This is what I write here. Inflation will react only with one period lag again to the monetary policy shock epsilon t i. So if we have this shock in period one, then we see that inflation in period one is still equal at zero, but inflation in period two, so that's the one period lag, will be at one because new one is equal to one.
What about inflation in period three? In the previous analysis we had inflation falling back to zero. This is not the case here anymore because what we would have now is that pi three would be equal to phi plus rho.
Why phi plus rho? Well, in period two, look at this equation here, we would have that pi two is equal to phi times pi one.
Sorry, excuse me. I should say it with three, of course, right? So pi three I want to compute. Pi three would be equal to phi times pi two.
And pi two was equal to one. So the effect would be phi as far as this term here is concerned.
Plus, of course, nu three. Well, what is nu three? Nu three is rho times nu two, and nu two is affected by the shock in period one, by the epsilon i in period one by means of this equation here.
So we would have nu three is equal to rho times nu two, and nu two was equal to rho times nu one plus epsilon two.
So we would actually have pi three equal to phi plus rho here because via the rho effect we would have the increase by one of the of the monetary policy shock.
Okay, so now we see that the reaction of inflation to a monetary policy shock is not as trivial anymore as in the previous analysis in the previous variant of the model because we see that inflation increases in period one, in period two, excuse me, in period two.
And it may actually increase further in period three because then is phi plus rho and phi plus rho may well be greater than one.
Even phi may already be greater than one and if rho is positive, then phi plus rho is certainly greater than one. So inflation may actually pick up.
And we could continue our analysis here to see where does inflation actually go and I won't do this by hand or more or less orally and by discussing it here, but we can simply compute it.
We do this after the break. Okay, so this shows us that in this type of model here, the reaction of inflation to monetary policy is more difficult, perhaps more realistic.
At least certainly more interesting than in the previous variant of the model, although in principle we do the same thing as we have done with fixed interest rate as the central banks, as the assumption on the central bank behavior.
So just briefly, before I leave you for a break, the interest rate shock and the interest rate of course will react as a tendency.
So when we have the epsilon t i equal one, then nu t will immediately react to that and the normal interest rate it will also immediately react to that because it is determined by the Taylor rule as being
these terms here plus nu t and nu t reacts immediately instantaneously to the change in epsilon.
Also note that again inflation rises when interest rates go up. Right. So we have interest rate going up here, and we have inflation going up here, so we still have the positive correlation so that does not change.
Okay, let's discuss the graphs after the break we make a 10 minute break, and I reopen the lecture then five minutes.
Don't remind me to restart the recording then, which I don't know.
Okay, so I have discussed already formally more or less what happens when we have a monetary policy shock of size one. And here you see this now for selected parameter values.
So in this case I set the persistence of the monetary policy shock new equal to zero point seven so this row parameter, which came up in the equation, new t is equal to row times new t minus one plus epsilon t i is equal to zero point seven, and the
parameter for the Taylor rule is zero point eight. And in this case, we get the behavior depicted in this diagram where perhaps, or where, please, you just ignore this pink curve here which I haven't discussed it this comes later.
But the other three curves are relevant for us. So first, what you see here is that the monetary policy shock new response to the innovation epsilon t i being equal to one in period one, and you see this here.
This time zero here is time one.
So you see that in time one there is a shock epsilon t i of size one, which moves new t up to the level one. And then by means of row being smaller than one in particular being equal to zero point seven.
And this effect dies out in succeeding periods because there is no new innovation in the epsilon i variable there was just this one shock in period one and all other epsilon t i shocks are equal to zero.
But the shock still has effects in succeeding periods, since new t depends on new t minus one by means of this row parameter here.
So the shock dies out slowly. And after say 10 periods, hardly anything is left over. So that's the development of new.
So for i, we have the red line here. And we know that in the first period, the epsilon t i shock and just pushes up the nominal interest rate by precisely this amount of one.
And then later periods this being pushed up by the parameters row and five as we know, so it comes to a level of 1.5 later on, which is just the sum of row and five as we have seen on i think the previous slide in the formula.
Where the second period effect was row plus five so 0.7 plus 0.8 is 1.5, which is exactly this thing here.
And then it goes up further but reverts after some time and begins to fall. So there is a prolonged effect on the effect on the nominal interest rate, which has not completely died out even after 15 periods, as you can see it here, but obviously the effect will go down to zero eventually.
So, the nominal interest rate increases due to a positive exogenous monetary policy shock for instance bird market developments or something like this increases for a number of periods.
Here in this case there are four periods approximately that we have an increase in the nominal interest rate, and then it reverts and falls again.
And exactly like in the simpler version with an interest rate, inflation reacts with a one period leg. So the increase in inflation comes only in period two, and then we have the same development further increases up to the height of approximately
1.7 or something like this, but with a one period leg so it's essentially the same impulse response for pi as it is for i just shifted in time by one period.
So, as you know, in the initial situation all variables were at zero, and we had now shocked the new variable or we had shocked epsilon ti which shifted the new variable so we had a monetary policy shock.
Now let's suppose that we have a fiscal policy shock that epsilon t s is equal to one in period one, and all other all other shocks are zero.
Sorry, this was my phone forgot to turn off.
The fiscal policy shocks are zero. So when we have epsilon t s equal to one positive shock in terms of surpluses. This means that we have a fiscal tightening either Texas are increased or government expenditures are reduced.
This is why the surplus is increased. So what happens with a positive epsilon talk in period, one of size one.
So, from equation 72 we see that we will have deflation. So, pi the inflation rate will now react instantaneously and with opposite time. So pi one is equal to minus one.
And from equation 73 we see that the normal interest rate will decrease so I one will actually be minus five. Let's go to equation 72 and 73 to verify this.
This is equation 72 here. So from equation 72 we see that when epsilon is minus is equal to one, then pi t plus one will be equal to minus one, obviously, and from equation 73.
So, from the data rule, we see that when pi changes, then the nominal interest rate will also change by the amount of five times the change in pi.
So this is how the same thing translates over to the nominal interest rates when we have a fiscal policy shock.
So, by affecting inflation, we provoke a response of the central bank, which adjusts interest rates.
Now when you go back to this equation here, then you now see that the change in the interest rate will give rise to a change in inflation in the next period.
So we do not have this Bing Bang behavior of inflation anymore that we had in the fixed interest rate setting so where the central bank tagged the interest rate, but now the inflation rate will also respond smoothly to a shock in the fiscal policy
so to shock in surpluses, because there is an initial reaction which comes directly from the epsilon s shock, and then there is an indirect reaction which comes from changes in the interest rates which have been provoked by previous changes in the inflation rate.
So, this is where we are here. And I think it is clear how the first period reactions come about.
Then when we look at equation 75, the inflationary effect will be seen to be smaller in the following periods, because pi two can then be computed as minus five two and pi three can be computed as minus five squared and pi four can be computed as minus five to the power of three.
So, fourth, and for the interest rate, we have the same thing, just that this goes a little faster I two is equal to minus five squared and I three is equal to minus five to the power of three and so forth, which we can derive from equation 73.
Let's have a look again at this, these equations here. This was equation 75, which specified the inflation rate to be now an autoregressive process. So once we set something in motion by means of the shock epsilon t plus one s here,
the previous inflation rate will cause another change of the current inflation rate. And this means that once we have an initial change in pi t due to the shocks and epsilon s, the succeeding inflation rates will also be changed.
But these changes will die out if the phi is smaller than one, which in our parameterization is the case and equation 73 I had already discussed that's the Taylor rule, so nothing new about that.
These two equations 75 and 73 yield these type of impulse responses to a unit impulse in the surplus shock.
Now note that the responses to fiscal shock are independent of the value of row. So the persistence of the monetary shock doesn't play any role here because we don't have a monetary shock central right so this doesn't come into play.
So it just depends on phi, whereas when we have the monetary shock, the persistence parameter row played a role and the phi played a role. So that is different in the sort of impulse responses we can read from this model, depending on whether we have a monetary shock or we have a fiscal shock which you plug into the model.
Okay, here are now the responses.
Ignore this little blue line here. This has nothing to tell you. Important is the blue line which goes down here and of course the red line, you see that both variables now react in period one.
Simultaneously, with the shock in surpluses taking place, and both reactions are negative. So again, interest rate and and inflation rate are positively correlated contrary to what people typically believe, even though inflation reacts to inflation.
In the time series representation we see that there's a clear positive relationship between interest rates and inflation. So what I already commented on in the first part of the lecture, the current the dominating believe that increases in interest rates decrease inflation rate are not really supported
by the time series properties of the data generated by this particular model.
Yeah, inflation as I said will react instantaneously to unexpected fiscal policy shocks and the fiscal shock will of course generate unexpected inflation, as we have seen already.
So, the first reaction of inflation is unexpected everything else can then be expected, because it depends on the interest rate rules, thereby on predictable monetary policy decisions.
This type of shock which I have discussed here is the unexpected shock where we can think of
the surface of the same period increasing. As a small exercise not too hard to solve, consider please how inflation and the normal rate would react to fiscal tightening announced two periods before before it takes effect.
Given what I have said already about this mechanisms of this model, it should not be difficult for you to solve this exercise.
Good. And are there any questions up to here, because I will now proceed to solve the model formally.
Please wave to me if there are any questions raise your hand.
I don't see any questions. Okay, good. Now, and solving the FTP L model, formerly, what we do again is that we forget about equation 62, say that this basically just amounts of bonds being purchased given certain interest rate decisions by the central bank.
What we can put this equation aside is still valid, even though we do not have an interest rate peg anymore, but an interest rate rule.
The interest rate rule also just says that certain interest rates are set by the central bank and whatever the central bank sets implies a certain amount of bonds to be sold to be purchased so this is what is being determined by equation 62.
Otherwise, we do not need to consider the equation so it's just a sort of supplementary equation to the model, but has nothing to do with the core dynamics of the model.
Okay, so, in this said, the model consists of three equations. One equation, which defines unexpected inflation. We have already seen this equation it's equation 69 so nothing new.
In one equation which defines the Fisher relationship in its linear approximation. So the real rate is equal to the normal right minus expected nation, also nothing new. And then finally, the Taylor.
So that's basically it for the fiscal theory model, and nothing of that is completely unconventional. Right.
Actually I will show you that this type of model can also be interpreted as a new Keynesian model we do this a little later, but all the sort of complicated dynamics are now more or less outside of the model or we have simplified the notation
taking linear approximations and approximations and these type of things to such an extent that the model is actually reduced to just these three type of very easy and partially standard equations.
The only perhaps not so standard equation is equation 69 which says that unexpected inflation is due to surprise changes in surfaces.
And what can we do now where we can substitute a 70 into 73 and thereby eliminate the difference between the nominal interest rate at the real rate.
Because both in 70 and in 73 we have this difference. Right.
Just move this area over to the other side you have it minus r here, move the r here over to the other side you also have it minus r there.
So you can equate the two i minus r terms, and you're left with the expectation of future inflation is equal to five times current inflation, plus the monetary policy shock here the new variable, not the innovation, not the epsilon ti, but the shock which shocks the system.
So, equation 70 and 73 can be condensed in this way. And
then we still have the expectation in here and you may think of now we have to solve an expectation of difference equation and that is kind of messy, usually, but things become even easier, because the expected inflation rate is also included in equation 69.
So we can solve equation 69 for the expected inflation rate and then substitute out the expectations term here against actual inflation rates, which would give us equation 76 so that we have it plus one is equal to five times pi t plus the monetary policy shock minus the fiscal policy shock.
So very easy formula actually which captures completely the content of these three equations here.
Out of three equations we've made one equation which is now equation 76 and this equation is of simple form it doesn't involve expectation terms anymore.
It is a simple autoregression actually pi t plus one being dependent on its own past with parameter phi and two shocks with which shock this relationship.
Now there is a stability condition for the such a difference equation, as you know, and this stability condition in order to be able to solve its backward looking would be that phi is smaller than one in absolute value.
We will just impose this condition here and then use successive substitutions of lagged versions of the same equation 76 here in order to solve the model that we have done already previously it's not so very difficult.
So basically, this is the same equation as 76. And what we now do is that for pi t here, we insert the same equation just with the previous time indices.
So we get this equal to phi squared times pi t minus one plus phi times new t minus one minus epsilon t s plus, well, this term here the shocks, the same thing as what was what was added there.
So now, pi t plus one depends on pi t minus one again we can substitute out it minus one.
And if we do this over and over again, we arrive at a representation of pi t plus one which consists of two infinite sums, one is finite sums, some, some, the monetary policy shocks new, and the other sums the fiscal policy shocks, epsilon t plus one.
So these two shocks are being summed both by the way with the same coefficients phi to the power of tau in both sums and well the difference between these two sums determines pi t plus one.
So this has no economic interpretation so far. It is just technical stuff, which I have done here.
We're not quite done with solving the model because we know the new is just the monetary policy shock which depends again on this innovation epsilon t i and the relationship is an autoregressive process of first order which I gave to you in equation 74.
So you know that such an AR one process can be solved in its moving average form as new t being equal to an infinite sum over rho to the power of j times epsilon t minus j to the power of epsilon t minus j i.
So the interest rate shock here or monetary policy innovation and distinguish if I formulate properly between epsilon being the innovation and new being the shock right to make clear that new in some sense is still endogenous and the truly exogenous development
is this innovation epsilon t minus j even though new doesn't depend on anything else but something which is completely exogenous in that sense new t is of course also exogenous.
But given this relationship here between new t and lagged epsilon innovations, we can substitute out the new here and insert in place of the new t minus tau, this infinite sum here, and then our expression becomes more complicated.
And you may think, well, why do we actually do this? Well, it is easier to analyze the dynamics of the model if we have shocks which are white noise so which are innovations which are not autocorrelated anymore
than when we have an autocorrelated variable in here like this new. This is why I substitute this as and actually this expression here is more complicated much more complicated than it need be, we can simplify.
But it is a little messy to simplify this expression since the expression itself is already messy and you may not know the techniques of doing that so I will show this to you in detail now rather than sort of moving to interactive mode
because I think you may not be so familiar with how to manipulate such sums, but since such double sums occur over and over again in macroeconomics when you solve dynamic models, it is probably useful that you know how to handle these sums and
in this case in particular reduce this double sum here double infinite sum to a single sum which makes the impression easier to handle. So we'll show you this now in detail how to do it.
As I said, it's not economics what I teach you here, it's just sort of basic mathematical tools of how to manipulate such an equation.
Yes, and this you see here. So the only thing I want to show you on the slide here is that a double infinite sum of this form here can actually be written as a single infinite sum of that form here.
And this is not obvious. So I will explain to you in detail how you come from the double sum to the single sum here.
As you see, this thing looks rather complicated with different indices which we have here in particular actually four indices, we have the T, we have the tau, we have the j, no sorry just three indices, t, tau and j which interact here.
The T is actually just a time index which doesn't change. We just hold the T here constant. The two indices which really do change, which run over a certain range from zero to infinity, r, tau, and j.
And the first thing I do is that I just exchange these two sums here which I can do in a certain conditions which I assume as given or actually they are given assuming certain certain sizes of phi and of rho here.
So I can just revert the order of the two summation signs here, which I have done here this first equation and nothing else changes.
You see the phi tau rho to the power of j epsilon i t minus tau minus j. That's exactly the same thing as I had here, just that the order of the summation signs is reverted because I want to preserve the tau as an index down here so the time index.
And I do not want to preserve the j, I want to eliminate actually the j index. Okay.
So this sum here I can write in that form.
And why this is so I will explain on the next slide. So currently you just please believe me that this double sum here is equal to that double sum here which is not obvious, because in this first double sum, we had
two sums which were going up to infinity as you see as the upper summation limit here, whereas here I just have one sum being left, which goes to infinity and the other just goes to tau, so it goes to the summation index of the first sum.
And then as you see here the powers here have also changed from tau as a power here to tau minus j, the road to the power of j is unchanged and here the time index epsilon t minus tau minus j has been changed to t minus tau.
So this is different and I don't think that any of you sees it immediately that these sums are equal, but I approve it to you.
Actually I give you an intuition of why they are equal and that's fairly compelling I would say, so it should be come clear to you why this equality sign here holds after you've seen the next page but currently you just believe it.
So here on with a finite upper limit here, things are easier. Right. So first thing what we can do is that we can say well, are there any elements in this these terms here, which do not depend on the j.
In fact, there is the epsilon team minus tau does not depend on the j. So we can actually move this out of this inner sum and just write it as a factor in front of this inner sum, which I have done here.
Also phi to the power of tau minus j depends on j but I can take this apart and write this as phi to the power of tau times phi to the power of minus j and the phi to the power of tau I can then put out of the inner sum so that in the next line I have phi to the power of tau epsilon t minus tau i
and then the inner sum which has the road to the power of j divided by phi to the power of j because here I had phi to the power of minus j and obviously a negative exponent means that we divide by the phi here with a positive exponent.
So that's easy to see that this is the same thing.
But now observe that this year is a simple geometric sum.
This runs from j equal to zero to an upper limit of tau over some coefficient of which we do not know whether it is greater or less than one.
So it is quite well known that the value of this sum here is one minus phi minus row over phi to the power of tau plus one. So the exponent here is by one greater than the upper limit of the sum here.
It's always like this divided by one minus row over phi.
You should have learned something like this in your undergraduate mathematics lectures. If you don't see it right away, try to prove the property that this sum here is equal to this sum here or perhaps I will actually show it to you in the next slide.
Not quite recall it. And anyway, it should not be too difficult to prove it by just saying, well, this sum here is some s and then you multiply by row over phi both sides of the equation and subtract the first equation from the second.
And then you find immediately this result. So that's not difficult.
Nothing has changed with respect to the first sum or the factors that I have figured out of the inner sum.
But now I have a phi to the power of tau here and I have a phi to the power of tau plus one here actually as something by which we divide.
There's a file also here. If I just put all the files in this expression here, I can rearrange this to get phi to the power of tau plus one minus row to the power of tau plus one divided by phi minus row and this is being multiplied by this
sum over the epsilon t minus tau here. So these are coefficients which also depend on tau here and there. So depend on the index, sorry, on the index of the sum. Therefore, the
enumerator of this ratio I cannot pull out of the sum. I could actually pull out the denominator here if I like.
Anyway, given that you believe me that this equality here is correct, you should be able to understand this rearranging of terms here. And now I'm going to show to you that this equality here is actually correct.
And I do this in a rather general way because I prove to you or give you an intuition, but actually it's basically a proof that this here is an identity so that this holds true for any elements a, i, j, which I would like to sum over the first index.
Let's say the row index i going from zero to infinity and the other index, the second index or let's say the column index j going also from zero to infinity.
So if I want to calculate such a sum here, which you may encounter sometimes in economic models, then you may find it useful to know that this is equal to this sum here.
Where the first sum again goes from zero to infinity and the second sum just goes from zero to j.
We have changed the column index here from j to j minus i.
And at first set, you may think this is not really possible why this should be the case, because here for every i, I sum infinitely many a, i, j's, whereas here for every j, I just sum finitely many j's.
And so it seems that this must be fewer j's, which I sum here, but you'll see that this is not.
Okay, look at the first sum and try to understand graphically what is happening here.
For this purpose, we first write down all the a, i, j's as an infinite matrix, that's basically. So we just order them so that the i is the row index and the j is the column index.
Then I have a zero zero here, one zero here, two zero here, three zero here, and so forth. And along the first row it would be, so going over the columns, there would be a zero one here, a zero two here, a zero three here.
So going all the way up to infinity here, and then all other rows, and also of course here in all columns, we move all the way up to infinity.
So we can rearrange our data in that way, we can rearrange our a coefficient or write down our a coefficients in this way here.
Now look at what the first sum, the left hand side of the identity is doing. This is the usual way of writing this as two infinite sums.
i runs from zero to infinity, j runs from zero to infinity over a, i, j. What is this doing? It is actually summing the elements here row wise, because how would we think of that?
Let's say we would think of first we have i equal to zero, and for i equal to zero, which would correspond to the first row here because i is the row index, right?
For i equal to zero, we sum then all the j's from zero to infinity. So all the column elements from zero, one, two, three, all the way to infinity.
So we sum this row here, and then we move on and i becomes one, and by the same token we sum this row here.
And then we move on and i becomes two, and we sum this row here, and therefore letting i go from zero to infinity, we will eventually sum all the a, i, j's in this matrix.
Now look at the right hand side of the identity. So at the double sum where the second sum doesn't have a limit of infinity anymore,
but it just goes from zero to j, and the column index here is changed from j to j minus i.
What you just need to understand is that this also sums all the a coefficients, but it doesn't do this row wise anymore, but it does this across the off diagonals of the matrix.
So first this off diagonal, and then this off diagonal, and then this off diagonal, and then this off diagonal, and if i move on like this, summing all the off diagonal elements, then i will of course eventually also sum all the a, i, j elements which i have arranged like this.
So the result will be obviously the same.
So the only thing you need to understand is that this expression here actually sums the elements in such a way that they are summed across the off diagonal elements.
And we'll just go into the form here and look at this. So starting with j equal to zero, we would sum what?
i would go from zero to j, but j is also zero, so i would just assume one value, namely zero. So at first we would just sum a zero, and then, well, zero minus zero is also zero, so we would just sum a zero, zero.
So we just sum this element here.
Now for j equal to one, we would sum here i equal to zero, and i equal to one because j would be equal to one, right, at the upper limit.
So i would run from zero to one, which would mean we would sum a, i, and i can take on either value zero or value one, and j minus i would be either one minus zero,
which is zero, or it would be zero minus zero. Am I saying this correctly? No, I think I'm not saying that correctly, so j is one, j is one, so it's always greater than i, and either we would have i equal to one, then the difference here would be zero, or i would be equal to one, then the difference here,
sorry, again j is one, i would be either one or zero, so the difference here is either one or zero, okay, so that's the previous way to say it.
Obviously, when i is equal to one, then here we have a zero, so we would have a one zero, which is being summed, and if i is equal to zero, then this thing is equal to one, so we would have
a zero one being summed, so these two elements are then summed.
It is a little hard to explain, the easiest way for you would actually to write it down, now for j equal to two, and you would see that you sum precisely these three elements here, and then for j equal to three, we would sum these four elements here,
and then by the token that I explained, we would sum all of the elements in this square organization of the aij data, so that the result would be exactly the same as the result we had here, just the order of summation is different.
So, this proves this identity here, and this identity here is exactly the equality sign, which I have made use of here, okay.
Now in the fourth equality here, I have used that this here is geometric sum, and has this value, and I told you already, I did have it on the slides, which, but just by the result, I didn't prove it to you here, and you should try to prove it at home,
that the sum from zero to n over some numbers c to the power of i is equal to one minus c to the power of n plus one divided by one minus c for any c in the range of real numbers, so we do not require the c to be smaller than one absolute value as you know it from
the geometric sum with infinite upper limit, right, you know the sum you probably from say things in multiplier analysis and these kind of things.
If n goes to infinity and c is smaller than one absolute value, and obviously c to the power of n plus one will go to zero, and then you have the Keynesian multiplier one over one minus c so you know these type of things they occur very often in economic models,
that this relationship is being used, and this just the finite equivalent of that which actually holds for any c.
So, what we do here is that we sum all the coefficients which are relevant for a particular surplus shock, namely the surplus shock.
No I'm sorry not the surplus shock here it is the interest rate shock, the interest rate shock in period t minus tau.
So wherever this influence comes from when it can be traced back to the innovation epsilon t minus tau i so to the innovation to the monetary shock in period t minus tau.
And there are various coefficients which relate to the same innovation here, so to the same time index here, and these are precisely the off diagonal elements which we have some, namely those elements where i plus j is the same thing as tau.
Since tau denotes the time lag here we are looking for those elements where i plus j sum up to tau so where the sum of i plus j is equal to tau.
And this is exactly what I have written down here.
So it starts in period zero so when tau is equal to zero by the effect phi to the power of zero row to the power of zero so both would be equal to one.
So we would just be left with epsilon t i so just the epsilon shock of say size one and in when time moves on, we would see that with the leg of one period between the innovation epsilon.
So epsilon happening in period t minus one the effect on current dependent variables would be made up of phi to the power of one row to the power of zero, which is of course one plus row to the power of zero.
So phi to the power of zero is again one so that we would have here phi plus row and we had already computed this band earlier on that the effect with one period lag would be just equal to phi plus row, this was the 1.5 which we saw in the in the diagram.
And these expressions here in front of the epsilon t shocks build up so they become more and more complicated. So that's the second degree effect here.
The defining characteristic always being that the powers to which we raise the variables here add up in each product to the time lag.
So here the power is zero and zero.
Here the powers add up to one one plus zero is one.
And here the powers add up to one zero plus one is one. So this is exactly this time like here.
Here the powers add up to two two plus zero is what is to one plus one is to zero plus two is to so there are three possibilities. We have them all here they add up to two.
That's the time back.
And by this token, it becomes more and more complicated.
You can express it more compactly in writing the double sum that way. And we have seen that this double sum is exactly the sum, or is exactly equal to the sum with both upper limits being equal to infinity and changing the indices here.
Okay, so that was the type of technique that I wanted to meet you in this later part of today's lecture. Time is actually off.
There are exercises which I leave to you to solve at home. So prove this finite version of the geometric some property prove this identity here which is not the same identity as the identity I have proven to you.
Above. So again, a double infinite some being transformed, sorry, being transformed to one infinite some and one finite some but now indices are different. So this is why it's not the same as above and verify this thing here.
Everything just mathematical technique. But after you've done this, I'm sure you will have understood how to manipulate this model here and to how to bring it in a form in which we can exploit it and interpret it and come back to economic
inference, which I will discuss next time on this slide here. So, not so much for today. And may I ask you if there is need or desire from your side to go to an interactive meeting now if yes, then even though time is over, of course, I'm very willing to do that.
If you want to raise questions or comment on anything or whatever you may want to discuss and then please wave to me now.
Nobody.
Once again, please use the waving function if you want to go into interactive mode and meet me in the under the meeting link.
Which apparently nobody wants to do. So then I say bye bye until next week. Thank you very much for your attention. And we will continue at this point in the next lecture.