Good morning, everybody, and welcome to the lecture on the fiscal theory of the price
level.
In this lecture, I would like to move early to an interactive part of the lecture because
we would want to discuss the input responses of the shocks in the fiscal theory model.
And I think it is useful if we really do this in some type of discussion.
So with your participation, which is why I have turned the recording on already now just
to review what I have done last lecture and to give a little bit more of formal presentation.
Or I will discontinue the video and ask you to move to the interactive format as complicated
as that is, unfortunately, the case.
So then we have a chance to discuss the shape of the impulse response functions in interactive
mode.
This is perhaps somewhat unfortunate for those which do not attend the lecture in time, but
rather watch the videos because I'm not allowed to record the interactive meeting since then
your voices will be heard and then there are data protection issues and these kind of things.
But nevertheless, I think it is useful to do it.
And so I would like to appeal to those who do not attend the lecture now, but rather
listen to the videos later, then you try to discuss on your own with yourself or with
perhaps fellow students what the interpretation of the impulse responses is.
I may summarize a little bit the discussion when we move back then to the lecture format
of this lecture to give you a sense of what we have been discussing in the interactive
session. But as I say, I think this discussion is actually useful.
So after this introductory remark, I just say that I will not ask for any questions now
regarding content of last lecture because we'll move to the interactive format shortly.
And then you may have the opportunity and you will have the opportunity to ask any question
you may want to ask, but rather I will now start with repeating what I've done last
time, just to remind you where we are, which is actually the fiscal theory model in its
most primitive form summarized here by three equations, which, as you may recall, were
derived by taking a linear approximation to a model which actually is nonlinear and the
three equations we will work with quite a bit in this lecture and in the following lecture.
Because they capture the main content of the fiscal theory and it is also useful to
compare them with the standard new Keynesian model, which I will do later on. So solving
the fiscal theory model as we have started doing last lecture and as we will continue
now is only the first step in this kind of analysis because not much later I will solve
the new Keynesian model or a model which is very similar to this fiscal theory model
but has a new Keynesian interpretation to work out the differences and the similarities
between the two models. Currently we're still dealing with a solution of the fiscal theory
model and a solution for us means that we will look at the impulse responses generated
by this model in response to two types of shocks which we have in the fiscal theory model, namely
a surplus shock, so some unexpected development in surpluses in this model, budgetary primary
surpluses, real surpluses, and an interest rate shock which is currently not visible in these
three equations here, which is some shock epsilon ti, but as you may recall the epsilon ti shock
is a shock which basically essentially drives this shock new here, did I call it new or yeah I think
I did, or did I call it v, I don't quite recall, so anyway this shock here was an autoregressive
process of first order which was driven by an interest rate shock epsilon ti, so the shock
epsilon ti is of course exogenous and since it's the only thing driving this new here the news also
an exogenous shock just that it is autocorrelated and we've been dealing with the new computing the
impulse response of the model with respect to epsilon ti shocks in the last lecture.
The three equations have simple names so there is an unexpected inflation equation here which
basically is an approximation to the government debt valuation equation or government budget
equation whatever you would like to call it, so the equation which tells us that
the present value of surpluses must be equal to the real value of current debt,
which is sort of the cornerstone of the fiscal theory because it implies that when surpluses
change somehow then the real value of current debt must change and since the nominal value of current
debt is given the only thing which can change is the price level so the price level has to adjust
to changes in surpluses and this is why we get the linear approximation this very simple equation
here where surprise changes in inflation occur when we have shocks and surpluses.
So that's a key equation of the fiscal theory model we will see that formally the same equation
may be included in new Keynesian models but with a different number. Then there is the Fisher
equation of which we do not need to say much it is well known the real interest rate and the
nominal interest rate are related to each other by expected inflation rates so the real interest
rate is the nominal interest rate minus expected inflation very simple relationship and we have
the Taylor rule a policy rule which is being followed by the central bank or we hypothesize
that the central bank has such a policy rule which reveals how the central bank adjusts
nominal interest rates if there are developments either in inflation or somewhere else in the
economy something which we call a nominal interest rate chart and you know brand new.
Now what we've done last lecture and what I'm just quickly repeating is that we solve this model so
we can use those two equations here to substitute out the i t minus r term then we derive a simple
expectation difference equation in pi of this form here we can actually replace the expected
inflation term by the equation 69 so that we have a simple difference equation without expectation
terms as in from 76 and then imposing a stability a condition a stability condition
phi being smaller than one in absolute value we can use successive substitutions to solve this
equation here and please bear in mind that we have imposed the stability condition here in order to
solve this equation backwards because we will when we deal with the new Keynesian model look at the
opposite way of solving a model solving it forwards and assuming that phi is greater than one in
absolute value. So we've done this and then we had a number of technical things I won't go through
we could write pi as a function of exogenous shocks here as a function of the news and the
surplus shocks that the news as I say are a function of the innovations in interest rates
of capital markets epsilon t i this is why we solve the ar1 representation so replacing the new
in this formula here with this thing there gives us this ugly double sum here and we have dealt
with how to simplify this double sum and arrived eventually at the end of last lecture at this type
of representation are better um where do I have it um sorry um
we we have the double sum here pi t is uh this double sum minus the sum over the epsilon both
sums go to infinity and oh yes exactly um so uh and I showed you that the double sum can be
simplified by a single infinite sum of this form here that was basically at the end of last lecture
I won't go through the slides again again anyway uh we can use this result to derive equation
80 which tells us that pi t is one over the difference between phi and rho times a sum which
also relates to differences between transformations of phi and rho both of them being taken to the
power of tau plus one in this expression here times the monetary policy shock minus then an
infinite sum of the surplus shocks so this representation here as you see is a moving
average representation of pi t pi now depends only on exogenous shocks actually on the innovations
so on the uncorrelated innovations of the pure shocks epsilon i and epsilon s so that's the
fiscal shock here and that's the monetary policy shock and therefore we can use this representation
80 uh in order to derive the impulse responses of pi two impulses by either a monetary policy shock
or a fiscal policy shock now you may recall that in impulse response analysis what we do
is basically that we say well what happens if we have just single shock um and all the other
shocks being equal to zero what would then be the effect on pi t and it may be either a contemporaneous
single shock so say shock epsilon t uh which hits pi t so we can see what is uh the simultaneity
effect uh the simultaneous effect of epsilon t i increasing or taking on a value different from
zero on pi t or we can take lag shocks of late shocks obviously because we have terms minus one
minus tau here the same thing we can do with the epsilon t minus tau so then essentially
the impulse response is nothing else but the partial derivative of pi t with respect to
one of these shocks here by the way and we also can derive uh an analogous an analogous
representation for the nominal interest rate and on the next slide i will actually show you the
steps on how we derive this representation here here i just give it to you for the sake of
completeness so that i have both formula on the same uh slide and you see that here also i have
a moving average representation of the nominal interest rate it because the nominal interest
rate just depends on constant r which is our model of course real interest rate or the real interest
interest rate is constant in our model and then we have sums here which are very similar actually
to the sums we have in the moving average representation of the pi's here and we have
also a an infinite sum depending on the fiscal shocks epsilon t s so we look at this formula
a little later currently we first focus on this formula here and now just have a look at
the mathematical structure of this formula here the first question you may ask is well how does a
monetary shock epsilon t i impact the inflation period t so impact on pi t so here we would look
at the simultaneity we would have we would ask the question suppose we have a shock monetary
policy shock epsilon t i what is the partial derivative of this right hand side here with
respect to epsilon t i well and and that's actually not difficult to see but the result may surprise
you because well obviously this thing here is independent of epsilon t i and here we just have
to ask where where do we have an epsilon t i now watch out the index tau here starts at zero and
then increases it goes through all the numbers all the way up to infinity so tau is at best zero so
that's being subtracted here and here we have a minus one so suddenly we see that there is no
epsilon t i in this formula the earliest shock we can study is actually epsilon i in period t minus
one this would be the case when tau is zero and we can of course study epsilon i t minus two this
would be the case when tau is one right so when tau increases then the lag here increases and we
study the partial derivatives of pi with respect to shocks which are more distant in in the past
but there's no epsilon t i here which is the first remarkable result we have if in the fiscal theory
model there is a an innovation on say world capital markets say rise in the nominal interest
rate of a different country for instance then inflation does not react in the same period the
partial derivative is actually zero right the partial derivative of pi t with respect to epsilon
t i is zero from this formula here um as soon as we start looking at partial derivatives with
respect to epsilon i lagged so epsilon i t minus one for instance or epsilon i t minus two then
there's always some value of tau which makes this term here either epsilon i t minus one or epsilon
i t minus two or other lags and obviously then whatever we find here in the parentheses is the
partial derivative um divided by five minus zero of course so then we would have a partial derivative
which may be non-zero but for epsilon t i we have a zero partial derivative the situation is different
for the surplus shock as you see here because we may assume that we have a surplus shock epsilon t
s and we would have non-zero effect on pi t so this simultaneous effect is non-zero here
because if tau is equal to zero then we have epsilon t s here so that's uh the same period
type of shock and obviously the partial derivative of pi with respect to epsilon t s
is just phi to the power of tau or tau is equal to zero so the partial derivative is just one
since phi to the power of tau is equal to one so that that effect is definitely non-zero
all right so this would be the way of our computation of the partial
derivatives and therefore the way we construct the impulse response function
the same way would apply of course to this formula here in d1 which we can derive using
a number of shocks steps which you see here i go through this briefly so these are the steps to
derive 81 what do we do where we take the Taylor rule right it is equal to r plus phi times pi t
plus nu t we know what pi t is what the moving average representation of pi t is we had it in
the equation 80 so we just plug in this representation this moving average representation
for pi t and multiply by phi so this is this phi is this phi here and the rest is just the
formula for pi t and then we get this equation here and now we can rearrange terms a little bit
and i have sometimes indicated with red with changes taking place so the change from this
line here to that line here basically takes place in this part here where i have said well if i have
tau plus one here and tau plus one here and here basically subtract tau plus one from t
because i subtract one it has subtract tau so i subtract tau plus one i can equally will then
starting tau with zero let tau start with one and decrease the exponents here and there and
the index here by one unit so i would then have phi to the power of tau and rho to the power of tau
and epsilon t minus tau here and let the tau start running at one so that would be exactly the same
thing the other terms here are unchanged okay so next step going to this line here is just
that i multiply out this term here or separate terms here so i would have one term relating
to phi tau which is this term here and i would have one term relating to the rho to the power of tau
which would be this term here so both are multiplied by this coefficient right which
is this coefficient here but i have i have just separated terms here and the other change i have
done is that i separated this sum which starts here at tau equal to zero you see here i let it
start at tau equal to one so i also have to provide for tau equal to zero well what is this
term here if tau is equal to zero rho to the power of zero is one and epsilon i t minus zero
is just epsilon i t so this is this thing here epsilon i t right this is this is the first term
of the sum here for tau equal to zero so i have separated this sum in this way here
well why do i do this you see i move the epsilon t i here and then go to the front of the equation
which is of course doesn't make much of a difference and then you see this sum here
starts with tau equal to one is the same thing as this sum here which also now starts with tau
equal to one but it has of course a coefficient here of phi over phi minus rho and this has a
coefficient of one so basically i have to subtract one here and this then yields rho over phi minus
rho right so this coefficient here changes i have the same sum here by this rearrangement i have
put together this sum and that sum here nothing else has changed when i go over to this line here
now again we have sums going from tau equal to one and going up infinity here and there
and we have the epsilon i shock here and the epsilon i shock there so we can
take these two sums put these two sums together which i do to just here where as you see
this sum here where there's phi to the power of tau this is being multiplied by one more phi
so the index is actually increased to tau plus one and here the this rho is multiplied by this
rho so the index is also increased to rho to the power of tau plus one so that's what happens
here and this term is unchanged and well the epsilon ti here i can again put into this term
by changing the the index here so that eventually i arrive at this representation
and this is precisely what i have given to you in equation 81
now when we compare equation 80 and equation 81 you see already that the expressions are rather
similar actually it is the case that inflation has almost the same impulse response to a monetary
shock as the nominal rate as this impulse response except that inflation has a one period lag
right so the r is just a constant so this doesn't change the response
the impulse response but otherwise the the only difference actually is that here we have epsilon
i t minus tau and here it is epsilon i t minus one minus tau so inflation reacts exactly the
same way as the nominal rate but with a period of with a lag of one period okay so that's the
only difference in terms of the response to monetary policy shocks and in terms to fiscal
policy shocks you see that here the exponent is tau and here it is tau plus one so the response
of the nominal interest rate is equal to the response of the inflation rate multiplied by
phi i one phi okay so the impulse responses to shots for expected surpluses are different
only by a constant factor phi all right so that basically gives us the following results
we saw already the response of pi t with respect to epsilon t i so to a shock in the same period
is zero and the response of pi t plus one plus tau with respect to epsilon t i is equal to phi
to the tau plus one minus rho to the tau plus one or tau being greater than or equal to zero
well divided by phi minus zero of course and if i do the same thing for i then i get
basically the same expression here just that pi has as we have just discussed this one period
lag so it is pi t plus one plus tau here whereas it is i t plus tau here so the response of i
which has the same coefficient as the response of pi happens one period earlier
now the first two of these equations can actually be put together in this form here i can also write
this means this is the partial of pi t plus tau with respect to epsilon t i being of this form
here because in the case of tau being equal to zero we would have one minus one which is just
zero which captures this inflation here and we can do the same thing for the surplus shock
epsilon t s in both cases i assume and that is important for later purposes that the size of
the shock is just one okay so the reaction then of pi t plus tau to a epsilon t s shock would be
minus phi to the tau here for any value of a tau and the reaction of i t plus tau to an epsilon
t s shock would be minus phi to the tau plus one for all values of tau and this gives me
these type of impulse response functions and those i would like to discuss with you
in the interactive session so i will stop the recording here