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