WEBVTT - autoGenerated
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Okay, and we are in section two of the lecture, Lieper's model, the first explicit model of
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the fiscal theory of the price level, which I have begun to introduce to you in the last
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lecture. In particular, you remember that we had a rather standard consumer optimization
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problem which is at the core of this model where the consumer maximizes the intertemporal
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utility function which characterizes his preferences. Momentary utility, so period T utility, depends
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on real consumption CT and on real money MT or real balances MT. The latter part is
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a little unconventional or not completely unconventional. I mean, there are money and
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the utility functions in the literature, not actually a few of them, but it is not the
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standard approach to use money into the utility function as I have pointed out. Last time
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it is in some way a shortcut to something similar to a cash in advance constraint.
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Lieper is using the shortcut here, so he assumes that the availability of real money also contributes
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to the felicity of the consumer to some extent. And the consumer then maximizes this intertemporal
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utility function subject to his budget constraint and the budget constraint is equation two,
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which I have also already explained last time, just to repeat and reintroduce you to
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the model if you haven't dealt with it last week. And then you see that the standard budget
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equation which basically here on the right hand side you have the resources available
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to the consumer, which is some fixed real income Y and then there is real money balances
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he's carrying over from the last period to this period. So nominal money MT minus one,
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which he has accumulated last period has to be divided by today's price level, PT by today's
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price of consumption PT to tell you in today's consumption units how much this money is worth.
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So how many consumption units can be purchased from the stock of money. And then there is
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the stock of government bonds accumulated last period. This is multiplied by the nominal
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interest rate factor so little contrary to intuition. Lieper uses the symbol R for the
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nominal interest rate rather than the real rate as today's practice and it is the interest
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rate factor so it is one plus the nominal interest rate which we have here. In this
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case the nominal interest rate of periods T minus one and of course because these were
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bonds issued in period T minus one. So we would have the proceeds from selling or redeeming
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the bonds in period T plus paying out interest due on the bonds and this nominal amount of
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RT minus one times BT minus one has to be again divided by price of consumption in period
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T to make it real and to tell us how many units of consumption we could in principle
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purchase from the bond holdings of the previous period. So these are the resources income
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plus basically the wealth the consumer holds and carries over to period T and this can
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then be spent either on consumption which is here in real form or we can again accumulate
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money balances to be carried over to the next period or we can again accumulate bond holdings
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for the next period to be carried over to the next period both of them in real terms
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and then the customer the consumer also has to pay real taxes tau T to the amount. Obviously
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by multiplying equation two by the price of consumption PT you can write the whole budget
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constraint also in nominal terms but for us it is more convenient to have it in real
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terms so this is why I have written it already in this form here and we will actually often
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make use of shortcut notation for real quantities like for instance capital MT divided by PT
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is equal to MT which is real balances which we see already here we will also write it
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down in the budget constraint of the customer the consumer. Now I told you already that
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government consumption shall be constant and that therefore the resource constraint of
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this economy is very easy CT plus GT is equal to YT so all goods which are produced Y have
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to be divided into either private consumption or public consumption and public consumption
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for simplicity is constant which then implies that the private consumption level is also constant.
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Nevertheless that's just sort of what happens then in equilibrium that a private consumption
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is constant the consumer maximizes his intertemporal utility also with respect to CT so he determines
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his demand for consumption CT and it has to be the case that his demand is eventually equal to
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constant consumption C which means that prices in particular the interest rate but also the price
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of consumption in this model have to adjust in such a way that his private consumption demand CT is
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equal to C in equilibrium. Now then I told you that after substituting out the Lagrangian multipliers
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the first order conditions for this models or two of the first order conditions of this model
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are actually equations four and five which I show you here again equation four relates the real
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quantity of money to the level of consumption and to the ratio of RT divided by the nominal
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interest rate so that is something like a money demand function which we have here and this thing
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here relates the nominal interest rate or better the nominal interest rate factor here it's the
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inverse of this normal interest rate factor to beta and beta was a preference parameter which
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tells you how strongly over time instantaneous utility is being discounted because instantaneous
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utility as you see here is weighted by the factor of beta, beta to the power of t so beta is smaller
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than one which we assume then utility which is far in the future will have a rather small weight
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because number beta smaller than one raised to the power of t becomes rather small over time so
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future consumption has less weight than consumption closer to the present time if beta is smaller than
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one and here we have this parameter beta which is on the right hand side of equation five and then
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we have the expectation of PT over PT plus one which actually is the inverse of the expected
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inflation factor and period t plus one now just one question do you have any idea how this equation
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here is interpreted or how we usually would interpret this equation or what name it typically
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bears so then please wave to me what is the usual interpretation of an equation like this
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there's somebody coming in the chat this is a different question which I come to later sorry
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first for my question here how do you how would you call this equation here you know the name of
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this equation I suppose since undergraduate times first natural lecture any suggestion okay so
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that's basically the Fisher equation you remember the Fisher equation essentially tells you that
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the nominal interest rate is equal to the real interest rate plus expected inflation and that's
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essentially what we have here because here we have the nominal interest rate factor and this beta
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is essentially the real interest rate of this model even though we don't write it down as an
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interest rate but this beta we can interpret as 1 over 1 plus r so it's the inverse of the real
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interest rate factor because typically in the equilibrium of such models we would find that 1
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over 1 plus r is equal to beta at least in mean there may be fluctuations around but beta basically
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governs the real interest rate and well this here then is expected inflation or the inverse
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the expectation of the inverse inflation factor so this equation here basically relates the
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nominal interest rate to the real interest rate and inflation and therefore it is essentially the
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Fisher equation now let me come back to the question in the chat the question was the first
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order condition the denominator is t minus 1 or r minus 1 it is r t minus 1 right i don't quite
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understand why you pose the question but it is exactly as i have written it here it is the
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interest nominal interest rate factor in period t minus 1 yeah i can't really tell you much more
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because i don't really understand what the point of the question is unless you write something to
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me now perhaps something still unclear you can come back to me and oh there's a question answer
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just because the one was too big i just wanted to contract the key oh okay no no the one here
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you know i understand the question the one is not part of this index here right it is really meant
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to mean we subtract one from r t because r t is essentially one plus the nominal interest rate and
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so you now subtract another one so basically you divide here by nominal interest rate you divide
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the nominal interest rate factor one plus i by i if you call the normal interest rate i
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okay so these are the two first order conditions you derive from the derivatives you take off the
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Lagrange function another first order condition is of course the restriction equation two and the
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fourth first order condition is actually the transversality condition about which we'll talk
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a little later we'll derive it a little later i will briefly state it now but i derived the
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transversality condition only a little later good now um i gave that already to you as an exercise
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derive four and five um since nobody required interaction in this uh lecture uh i couldn't
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check whether you could actually derive four and five but be aware that it is not completely
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trivial to derive a four and five for the following reason um we have um this problem here
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and you may solve it in a standard way and then it's actually easy to derive the two
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first order conditions four and five but you may also derive it and let's make some sense here
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under the restriction that money balances must be positive and bond holdings must be
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positive or let's say non-negative both of them are negative so um if you ignore those two
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restrictions obviously in principle you could provide for negative money holdings
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which however doesn't really make sense because you have uh money holdings here the logarithm
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obviously logarithm of a negative number would be a complex number which is not
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useful here um but also for the bond holdings i mean in principle you could not only think about
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the possibility that the consumer saves in terms of bonds but he could also go to debt in terms of
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bonds and this is something which we will not analyze in this model actually we will not allow
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the consumer to take up debt from the government to be invented vis-a-vis the government but rather
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the government will be indebted vis-a-vis the consumer so the bt must always be a positive
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number and that means that this maximization problem actually has two inequality constraints
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which i haven't written down here one of which would be empty is greater than or equal to zero
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and the other would be bt is greater than or equal to zero and in this case you don't have
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to solve like a vouch problem but a kun takka problem and then you can still derive these type
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of conditions here but it is a little bit more of mathematical work and if you are not so familiar
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with kun takka anymore then perhaps you would like to check and see whether you can confirm
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these two equations here okay i also mentioned already that the government also has something
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which i call the government budget equation i hesitated a little bit to call it a constraint
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because this is actually a big issue in the fiscal theory of the price level of whether this
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equation here is a constraint or whether it is some other type of equation some people call it
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a government debt valuation equation or something like that and that's a controversial issue in the
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whole theory we'll come back to that later but currently you don't need to bother about it because
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equation six has already pointed out last week is redundant since it is implied by equations two
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and three and then in order to complete the description of the model and its necessary
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conditions we have the transversality condition which gives us some limit condition on the
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variables and this condition says and i don't derive it now do this later that the product of
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beta so the individual discount factor to the power of capital t multiplied by real debt
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holdings b some sort of final period is capital t that the limit of this expression for t going
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towards infinity is equal to zero which is a property which basically rules out explosive
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equilibrium solutions but it also is a necessary condition so you can derive it as a necessary
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condition from the consumer's optimization problem symbol bt i have already used on the
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previous slide i define it just here bt is just normal bond holdings divided by the price level
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of course and the inflation factor pi t is equal or is defined to be equal to p t divided by p t
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minus one note that this is not the inflation rate but it is one plus the inflation rate which
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i define here so that's what i call the inflation factor using these two notations and then
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collecting everything we have about the model the model can actually be summarized in four equations
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which we have all mentioned already but just for completeness completeness i rewrite them
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here again now in real quantities so the bond holdings are real quantities and the money holdings
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on real quantities only so small letters so we have the consumer's budget constraint here equation
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two we have the resource constraint here equation three and then we have the two necessary
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conditions which we derive by taking the derivatives the partial derivatives of the
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equations four and five dm so these are four independent equations and as i mentioned the
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fifth equation the budget equation of the government is redundant it is by valras law
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implied by these four equations here so we need not to consider it they're just four equations
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in order to analyze a model it is always useful to compare the number of equations
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in the model with the number of endogenous variables and so let's now count the number
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of endogenous variables we have consumption here money here bonds here and taxes here these are
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already four endogenous variables then we have the inflation rate which is number five
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money we had already why is exogenous be given and then we have the interest rate factor of
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interest factor nominal interest are here which is the sixth endogenous variable
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bond holdings we had already inflation we had and here's nothing new there was just consumption
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which we had already money again interest factor again interest factor again inflation again
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so just counting the endogenous variables in the equation two gives us all of them then six
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so we have six endogenous equations here and we have just four
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excuse me six endogenous variables here but we have just four equations so this means the model
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so far is not fully determined or the variables in this model are not fully determined obviously
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we could solve this issue by saying that certain variables are just exogenous for instance we could
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say the government just exogenously sets this tau t so then tau t would not be an endogenous
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variable anymore we are just left with five endogenous variables and then we could say the
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central bank sets the nominal interest rate and thereby the nominal interest factor and therefore
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this one this variable here would also be exogenous and then we have only four endogenous variables
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and then we would have the good hope that this model here determines all the remaining
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endogenous variables because we would have four endogenous variables and four equations actually
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that's just a rule of thumb it is not always the case that a model with four equations determines
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four variables there are also cases which this is not possible and later in the lecture i'll show
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you one of these cases but mostly it's a pretty good rule of thumb actually to count endogenous
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variables and and the number of equations here however we do not want to take the cheap
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resort of just saying well that we fix two variables exogenously in this case taxes and
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the nominal interest rate factor but rather we want to apply some more realism to this model and
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we'll say the government is also not completely unresponsive to conditions in the economy when
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determining how much taxes the consumers have to pay and the central bank is not completely
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unresponsive in its interest rate decisions because it looks for instance at inflation
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and may think that different levels of interest rates are appropriate for different levels of
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inflation which is observed in the economy so what we do rather than fixing two variables tau
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and r as exogenous we close the model by adding two more equations which we call two policy rules
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because they tell us how in terms of policy the government and the central bank set
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the taxes and the nominal interest rate so if i'm careful i will always distinguish now between
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this part of the government which is the fiscal authority and the part of the government because
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which is the central bank because actually both of them are parts of the government both of them
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are parts of the executive branch in a country the fiscal authority let's say the ministry of
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finance and the central bank are both parts of the executive branch of a country both are parts of
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in some sense of the government even though in many countries the central bank has some or even
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quite a bit of independence from what we in conventional language called the government
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but it is certainly sometimes useful to be careful about language here and if i am so i won't promise
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that i always manage to then the i would speak of the fiscal authority as the authority responsible
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for taxes and the central bank as the authority responsible for the nominal interest rate and
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monetary policy in general and will think that the government is actually both can mean both of
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this year sometimes when i'm perhaps not so careful i may accidentally speak of the government when
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i actually mean the fiscal authority so that may also happen and then you have to understand
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from the context of whether the government actually means the fiscal authority or means the
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government the broader sense anyway for the fiscal authority we'll say that taxes are set
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in response to the level of debt essentially so i assume a linear policy rule here taxes are equal
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to some constant gamma mud and then taxes respond to the level of debt in the previous period
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because real bonds which we have here are of course the same as real debt of the government
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and a useful way of thinking about this policy rule would be to say that we think that a
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responsible government may perhaps rise raise taxes when debt levels are high because the
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government may be concerned about the high debt levels and about its solvency about its
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possibility to service its debt so when that becomes too high governments may want to increase
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taxes in order to reduce debt levels so it is certainly useful to think of this gamma parameter
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here as a positive parameter and then taxes are typically not so easily described by just a linear
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function or an affine function of gamma naught plus gamma times bt minus one so we also add some
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shock some random shock which has no particular structure which just explains that periods perhaps
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taxes are not exactly as given by this policy rule here and a similar policy rule we define
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for the normal interest rate number interest rate factor we say that the central bank sets
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the nominal interest rate as some constant alpha naught plus some response to the current level
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of inflation in the economy so alpha times pi t so the managing board of the central bank
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observes that inflation is high in the economy i want to do something against inflation so
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the easy thing to do would then be to raise a normal interest rate so it is useful to think
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of alpha as a positive coefficient again increases in inflation leads to increases
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in the nominal interest rate would be the the interpretation of equation 11 here
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and then we again add an exogenous shock which we call theta because interest rates do not always
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follow in a very strict sense such a linear relationship with the inflation right so there's
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another shock coming in here which you may think of as perhaps some type of world capital market
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shock so for instance if the european central bank has this type of reaction function here
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i doubt that it does but whatever suppose it has it then it may still be the case that it
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realizes other central banks on the world capital market have certain interest decisions so for
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instance the the fed raises or lowers interest rates in the united states of america and the
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bank of japan is doing something and the bank of email is doing something and the bank of
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uh swiss national bank is doing something um so all these things impact of course via
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their influence on bird capital markets also on the interest decision which the european central
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bank may take and we would think that these are then exogenous shocks because the euro zone or
00:27:06.000 --> 00:27:11.000
the european central bank has no very little influence on what other central banks are doing
00:27:11.000 --> 00:27:17.000
on what kind of conditions are that they are creating on birth capital markets where you
00:27:17.000 --> 00:27:24.000
can think of this as some kind of exogenous nominal interest rate shock however we assume
00:27:24.000 --> 00:27:31.000
that both psi and theta are shocks with the expected value of zero so anything which is
00:27:31.000 --> 00:27:37.000
systematic is in the constants gamma naught and alpha naught and here we have then the responses
00:27:37.000 --> 00:27:46.000
to debt levels and inflation and now joined with equations 10 and 11 we have six equations
00:27:46.000 --> 00:27:53.000
for six variables so it seems to be the case that we can solve the model because now it may be well
00:27:53.000 --> 00:28:03.000
defined and it actually is as we will see later now to solve the model is not really trivial and
00:28:03.000 --> 00:28:11.000
this is why we now have to go a little bit into the realities of solving such a model it is not
00:28:11.000 --> 00:28:18.000
trivial because we have dynamic equations here you see for instance equation two is a difference
00:28:18.000 --> 00:28:25.000
equation where we relate the level of debt in period t to the level of debt in period t minus
00:28:25.000 --> 00:28:34.000
one so we will have to solve a difference equation here which is connected to the other
00:28:34.000 --> 00:28:41.000
equations here for instance via the nominal interest rate factor r t minus one in this case
00:28:42.000 --> 00:28:50.000
that as you know is then well related to equation four and equation five and you see in equation
00:28:50.000 --> 00:28:57.000
five again we have then the expectation of pi t plus one or the inverse of pi t plus one which
00:28:58.000 --> 00:29:04.000
relates to the pi t which we have here and there in equation two so basically we have here a system
00:29:04.000 --> 00:29:11.000
of equations where the variables are interrelated and the variables are actually interrelated in
00:29:11.000 --> 00:29:17.000
non-linear way because we have various non-linear transformations in here so it's not so easy to
00:29:17.000 --> 00:29:24.000
solve the system directly and therefore what we will have to do in order to come to at least an
00:29:24.000 --> 00:29:32.000
approximate solution of this model is that we linearize the model there is no known solution
00:29:32.000 --> 00:29:38.000
as far as i know to this model as a non-linear model we do what all the people typically do with
00:29:38.000 --> 00:29:45.000
non-linear model we take a linear approximation to it using first degree Taylor approximations
00:29:46.000 --> 00:29:52.000
these Taylor approximations are typically done around these the model's steady state
00:29:52.000 --> 00:29:59.000
and this is why as a first exercise here i give you the exercise to compute the model's steady
00:29:59.000 --> 00:30:09.000
state assuming that real debt b is constant in the steady state and of course that we use for
00:30:09.000 --> 00:30:14.000
the steady state the expectations of the exertion with shocks so both of these expectations are
00:30:14.000 --> 00:30:25.000
equal to zero which simplifies the policy roots now i'm not sure if everybody knows exactly what
00:30:25.000 --> 00:30:33.000
a steady state is even though this should have been covered in your first year's lectures at
00:30:33.000 --> 00:30:39.000
the very latest but let me just repeat a steady state is not a state of the economy in which
00:30:39.000 --> 00:30:47.000
everything is constant while a state in which everything is constant is also a steady state
00:30:47.000 --> 00:30:55.000
if that can be so far a reasonable solution while a situation in which everything is
00:30:55.000 --> 00:31:02.000
constant is typically a steady state the definition of a steady state is more general a steady state
00:31:02.000 --> 00:31:10.000
is defined to be some state of the economy in which all variables have a constant growth rate
00:31:12.000 --> 00:31:18.000
and if this constant growth rate happens to be zero then of course the variables are all at the
00:31:18.000 --> 00:31:23.000
level zero at a constant level excuse me so if the growth rate is zero then all the variables in the
00:31:23.000 --> 00:31:29.000
steady state are constant which is some type of special case of steady state but it is also
00:31:29.000 --> 00:31:36.000
possible that you have a steady state where all variables grow at constant growth rates with some
00:31:36.000 --> 00:31:44.000
of these growth rates or all of these growth rates being non-zero so this is why i give you here
00:31:45.000 --> 00:31:52.000
the hint that you should compute the model steady state or possible steady state of this model
00:31:52.000 --> 00:31:59.000
under the assumption that b is constant so the growth rate of debt is zero obviously this need
00:31:59.000 --> 00:32:04.000
not be the case perhaps there are steady states where the growth rate of debt is constant so
00:32:06.000 --> 00:32:13.000
debt grows over time at a constant level and perhaps income also grows at a constant level
00:32:13.000 --> 00:32:17.000
actually not in this model because income y is fixed but the principle can be
00:32:17.000 --> 00:32:23.000
so as i say it is not usually legitimate to assume that the steady state is characterized
00:32:23.000 --> 00:32:32.000
by non-growing variables but in this particular case here i tell you well suppose b is non-growing
00:32:32.000 --> 00:32:38.000
and then so for the model to spend steady state and you'll find probably that the steady state
00:32:38.000 --> 00:32:46.000
is then a steady state where everything is constant taking this steady state which you derive in exercise
00:32:46.000 --> 00:32:53.000
one go to exercise two and think about whether the price level is determined in the steady state
00:32:53.000 --> 00:33:01.000
so just look at the solution you get for p assuming p is constant you get an equation for
00:33:02.000 --> 00:33:08.000
p in the steady state for the price level in the steady state and you may actually also
00:33:08.000 --> 00:33:12.000
think about whether you get an equation for pi in the steady state
00:33:12.000 --> 00:33:18.000
this exercise i give to you because the determination of the price level is one of the
00:33:18.000 --> 00:33:25.000
key questions in the fiscal theory of the price level which is already implied by its name that
00:33:25.000 --> 00:33:31.000
perhaps in some settings price levels are not so easily determined or just determined by fiscal
00:33:31.000 --> 00:33:39.000
decisions and not by monetary decisions exercise three here relates to exercise one on this
00:33:39.000 --> 00:33:46.000
slide the question is is it reasonable to assume that real debt b is constant in the steady state
00:33:46.000 --> 00:33:53.000
or could there be a steady state in this model where real debt b grows at a non-constant rate
00:33:53.000 --> 00:34:00.000
either positive rate or negative rate so as you can you derive a steady state in which b is not
00:34:00.000 --> 00:34:08.000
constant this exercise is a little bit more difficult than the other two so there are more
00:34:08.000 --> 00:34:13.000
computations involved in finding the solution to this exercise i think it's nevertheless quite
00:34:13.000 --> 00:34:21.000
helpful to just consider the possibility of a steady state which has non-growing and non-constant
00:34:21.000 --> 00:34:31.000
variables in it so in this case a growing or shrinking real debt and finally um exercise
00:34:31.000 --> 00:34:37.000
four which is similar to exercise three can you derive a steady state in which real balances
00:34:37.000 --> 00:34:44.000
m are not constant so that would also be a legitimate question similar to bonds which
00:34:44.000 --> 00:34:51.000
is one part of the assets the consumer can hold money is also an asset the consumer can hold
00:34:51.000 --> 00:34:56.000
and when we consider the possibility that bonds need not be constant in the steady state then we
00:34:56.000 --> 00:35:01.000
may also want to consider the possibility that money need not be constant in the steady state
00:35:01.000 --> 00:35:06.000
so you can also look at that question and you'll find this question is actually easier to answer
00:35:06.000 --> 00:35:12.000
than questions three or exercise three um so perhaps you want to consider both of them
00:35:13.000 --> 00:35:18.000
and if you prepare solutions for that i'm happy to discuss your solutions next time
00:35:18.000 --> 00:35:23.000
in the interactive session but if you don't do anything about it then i won't solve those
00:35:23.000 --> 00:35:29.000
equations for you and i will just carry on with the lecture rather than going into the interactive
00:35:29.000 --> 00:35:38.000
part now as i already said we have to do some technicalities now and they are important because
00:35:38.000 --> 00:35:46.000
we will make use of these techniques in particular linearizing at the steady state of a model
00:35:46.000 --> 00:35:54.000
over and over again in the lecture i suppose that you have heard or learned these techniques
00:35:54.000 --> 00:36:01.000
already in the first year's masters lectures on how to solve dynamic macroeconomic models but in
00:36:01.000 --> 00:36:09.000
case you have not i will just go through this theory here briefly but completely or at least as
00:36:09.000 --> 00:36:15.000
much as you as we need it in this lecture and then you again get exercises you can get many many
00:36:15.000 --> 00:36:22.000
exercises today where you will be asked to verify certain results of the linearization which i am
00:36:22.000 --> 00:36:30.000
going to present for you so first on the principle linearizing at the steady state why at the steady
00:36:30.000 --> 00:36:37.000
state well if we have tater approximations we always develop a kind of a function around a certain
00:36:38.000 --> 00:36:46.000
point and this point which we take as sort of the origin of a approximation of a tater approximation
00:36:47.000 --> 00:36:52.000
is most reasonably the steady state because we can interpret the steady state as the longer
00:36:52.000 --> 00:36:59.000
equilibrium of an economy so the economy should be somehow close to the steady state and therefore
00:36:59.000 --> 00:37:07.000
it makes sense to linearize a system of equations at the steady state of a model first in general
00:37:07.000 --> 00:37:13.000
just to repeat what a Taylor expansion was suppose that we have f as some one-dimensional
00:37:13.000 --> 00:37:21.000
non-linear function of a variable xt and let us denote by x so without time index the steady
00:37:21.000 --> 00:37:30.000
state value of this variable xt then a first degree Taylor expansion is given by the following
00:37:31.000 --> 00:37:40.000
approximate equation f of xt so the non-linear value f which the function takes which the non-linear
00:37:40.000 --> 00:37:46.000
function f takes at point xt is approximately equal to the value at the steady state f of x
00:37:47.000 --> 00:37:54.000
plus the difference between xt and the steady state value xt minus x times the first derivative
00:37:54.000 --> 00:38:01.000
of the function at the point where the steady state is so f prime of x always be where that
00:38:02.000 --> 00:38:09.000
xt pops up in the equation over here and all the other expressions here relate to the steady state
00:38:09.000 --> 00:38:18.000
value of x so to x f of x here x there and x there now that's easy and you know that's
00:38:18.000 --> 00:38:25.000
just the first order approximation here you can principle add second order approximations to that
00:38:25.000 --> 00:38:30.000
thereby usually increase the quality of the approximation we will do this because then we
00:38:30.000 --> 00:38:34.000
wouldn't have a linear system anymore but rather with a second order term we would get a quadratic
00:38:35.000 --> 00:38:40.000
system or higher over terms which again lead us into the sphere of non-linear models and we
00:38:40.000 --> 00:38:46.000
just want to make a linear model out of a non-linear model. Now what happens if we have a
00:38:46.000 --> 00:38:51.000
function which is not a function of one variable but of several variables? Let's look at the case
00:38:51.000 --> 00:38:58.000
of two-dimensional functions and functions with more than two variables are just analogous then
00:38:58.000 --> 00:39:04.000
suppose that g is a two-dimensional non-linear function so has two functional arguments
00:39:05.000 --> 00:39:17.000
then we know that g of xt and yt is by the Taylor expansion approximately equal to g evaluated at
00:39:17.000 --> 00:39:24.000
the steady state quantities of x and y so g of x and y is the steady state value plus now two terms
00:39:25.000 --> 00:39:32.000
namely the deviation of xt from its steady state value which I denote by x tilde t
00:39:33.000 --> 00:39:42.000
so x tilde t is equal to xt minus x deviation of xt from its steady state value so this deviation
00:39:42.000 --> 00:39:50.000
which we had already here this was also x tilde right x tilde t times now the derivative again
00:39:50.000 --> 00:40:01.000
with respect to x so times g index x at the point of the steady state at x and y so that's
00:40:01.000 --> 00:40:08.000
the derivative the partial derivative of g with respect to x taking at the steady state
00:40:08.000 --> 00:40:18.000
namely at the point x y and then the same thing with respect to y so plus the deviation of yt
00:40:18.000 --> 00:40:24.000
from its steady state value so y tilde t would be analogous to this equation here up to this
00:40:24.000 --> 00:40:32.000
definition here would be yt minus y so that's the deviation of yt from its steady state value
00:40:32.000 --> 00:40:40.000
times the partial derivative of g with respect to y at the steady state okay I wrote this again
00:40:40.000 --> 00:40:48.000
here g x of x y is equal to the partial of g with respect to x at the point x and y in your slides
00:40:48.000 --> 00:40:56.000
there was a misprint as I detected this morning I wrote g index y is equal to del g del x which
00:40:56.000 --> 00:41:03.000
is of course nonsense no sorry I wrote g index x is equal to del g del y which is of course nonsense
00:41:03.000 --> 00:41:09.000
might be x so I corrected this in the slides here but you probably have still the old version on
00:41:09.000 --> 00:41:18.000
your computers now let's suppose that in our model we have a nonlinear equation which we can
00:41:18.000 --> 00:41:27.000
write as f of x t so just the one-dimensional nonlinear function is equal to g of x t and y t so
00:41:27.000 --> 00:41:36.000
a nonlinear function with two arguments this would be a nonlinear equation here and we want to
00:41:36.000 --> 00:41:43.000
simplify this equation by taking its linear approximation so how would we do this we would
00:41:43.000 --> 00:41:50.000
set out by saying well in the steady state obviously we would have f of x is equal to g
00:41:50.000 --> 00:41:59.000
of x and y so this equation here evaluated at the steady state gives us f of x equal to g of x and
00:41:59.000 --> 00:42:07.000
y but we know already how f is approximated linearly by a Taylor expansion and we know how
00:42:07.000 --> 00:42:15.000
g is approximated linearly by a Taylor equation and we know that the first term in the Taylor
00:42:15.000 --> 00:42:20.000
expansion here and the first term in the Taylor expansion there for just the steady state values
00:42:20.000 --> 00:42:27.000
f of x and g of x y right so we had here the Taylor expansion which starts with the steady
00:42:27.000 --> 00:42:33.000
state value f of x and here we had the steady state value which the Taylor expansion which
00:42:33.000 --> 00:42:42.000
starts with the steady state value g of x y so now we know that f of x is equal to g of x y
00:42:42.000 --> 00:42:49.000
so when setting f of x t equal to g of x t y t or if this is the equation then we know in the
00:42:49.000 --> 00:42:55.000
Taylor approximation the f of x and the g of x y just cancel on both sides because they are equal
00:42:55.000 --> 00:43:02.000
and therefore the linear approximation to the non-linear equation just consists in the first
00:43:02.000 --> 00:43:11.000
order approximation terms which from the formula on the last slide are x t tilde times f prime of x
00:43:13.000 --> 00:43:20.000
is approximately equal to x t tilde times the partial of g with respect to x at the steady
00:43:20.000 --> 00:43:28.000
state plus y tilde times the partial of g with respect to y at the steady state and now note
00:43:28.000 --> 00:43:35.000
that these expressions here are steady state expressions so these are just some constants
00:43:35.000 --> 00:43:42.000
we can compute them if we have calibrated the model and usually it's not so difficult to
00:43:42.000 --> 00:43:49.000
calculate the steady state so we can exactly derive what kind of constants these are here
00:43:50.000 --> 00:43:56.000
there and there in any case for our purposes currently all we need to know is that these are
00:43:56.000 --> 00:44:04.000
constants and the variables are x tilde and x tilde again and y tilde so in this particular
00:44:04.000 --> 00:44:10.000
case we can write the linearized equation with just one coefficient because since we have the
00:44:10.000 --> 00:44:18.000
x tilde here and there we can just bring this part over to the other side and then so for x tilde t
00:44:18.000 --> 00:44:25.000
so we will get x tilde t is equal to some constant namely this ratio here times y tilde t
00:44:26.000 --> 00:44:32.000
so x tilde t will be a linear function of y tilde t and there will be some coefficient here which
00:44:32.000 --> 00:44:38.000
is constant and which is actually the partial of g with respect to y divided by f prime of x minus
00:44:38.000 --> 00:44:45.000
partial of g with respect to x provided of course that this derivative here at the steady state
00:44:45.000 --> 00:44:50.000
minus this partial at the steady state is different from zero otherwise we would not be
00:44:50.000 --> 00:44:56.000
able to solve it that way but we can check this this condition here and mostly it is the case
00:44:56.000 --> 00:45:01.000
that is different from from zero so then we can write x tilde t as a simple linear function of
00:45:01.000 --> 00:45:10.000
y tilde t and that is of course a much easier function than f of x t is equal to g of x d y t
00:45:11.000 --> 00:45:13.000
the equality of two non-linear functions
00:45:17.000 --> 00:45:23.000
now doing that is just math it doesn't add anything to our economic understanding
00:45:24.000 --> 00:45:30.000
so i will not go through the math here assuming that actually you have learned these techniques
00:45:30.000 --> 00:45:38.000
already one year ago in here first masters lectures on dynamic macro i leave it to you
00:45:38.000 --> 00:45:46.000
as an exercise to verify that whatever i have shown you here as linear approximations to the
00:45:46.000 --> 00:45:56.000
model is correct so what do i do i don't take a linear approximation to all the six equations
00:45:56.000 --> 00:46:05.000
we have here but i use some of those equations to substitute out certain variables for instance
00:46:05.000 --> 00:46:14.000
i use the policy rule of the central bank equation 11 to substitute out the the interest
00:46:15.000 --> 00:46:25.000
rate factor r t remember r t was equal to alpha naught plus alpha pi t plus theta t and then i
00:46:25.000 --> 00:46:32.000
get on the left hand side of this equation just an equation which relates to pi t and on the right
00:46:32.000 --> 00:46:39.000
hand side of this fissure equation remember my equation five i had pi t plus one so essentially
00:46:39.000 --> 00:46:49.000
now i have a a an equation a non-linear equation which depends on one variable here namely namely
00:46:49.000 --> 00:46:56.000
on pi t plus one like my f and it depends on two arguments here namely namely on pi t and the
00:46:56.000 --> 00:47:05.000
exogenous shock theta t and then i can linearize this whole expression and get the expectation in
00:47:05.000 --> 00:47:13.000
a linear approximation the expectation conditional on period t information of pi tilde t plus one
00:47:15.000 --> 00:47:22.000
where pi tilde t plus one is pi t plus one minus its state value is equal to alpha times beta times
00:47:22.000 --> 00:47:33.000
pi tilde t plus beta times theta tilde t and as i said as an exercise derive this equation
00:47:33.000 --> 00:47:43.000
12 the linear equation from this non-linear equation here i have highlighted this equation
00:47:43.000 --> 00:47:50.000
here in yellow because this will be one of two core equations which characterize our model so
00:47:50.000 --> 00:47:56.000
we very often analyze this equation 12 here which is a dynamic equation which is a difference
00:47:56.000 --> 00:48:02.000
equation actually it's an expectational difference equation because the expectation of pi tilde t
00:48:02.000 --> 00:48:09.000
plus one is related to the actual pi tilde t and period t and i'll show you in a minute how to
00:48:09.000 --> 00:48:16.000
solve these type of expectation difference equations if you don't know that anymore i haven't
00:48:16.000 --> 00:48:21.000
heard that before and but it's one of the two building blocks of this model so it's a very
00:48:21.000 --> 00:48:27.000
important equation and therefore it's important that you know how to derive it and do exercise
00:48:27.000 --> 00:48:37.000
this exercise on the slide here there's another exercise which you may do or may not do it is not
00:48:37.000 --> 00:48:43.000
as important as the previous exercise but it may be interesting for you because what we currently
00:48:43.000 --> 00:48:51.000
do in this model is that we linearize in terms of deviations from the steady state so this where
00:48:51.000 --> 00:48:59.000
was it x tilde variable here which is just the level of a variable minus the level at the steady
00:48:59.000 --> 00:49:08.000
state of this variable that's a perfectly valid way to do the linearization but today almost 30
00:49:08.000 --> 00:49:16.000
years after Lieber has published his paper in 1991 it is more common macroeconomic models to
00:49:16.000 --> 00:49:21.000
see linearization not in terms of absolute deviations from the steady state like we have it
00:49:21.000 --> 00:49:29.000
here x t minus x but in percentage deviations and to understand the relationship between
00:49:30.000 --> 00:49:35.000
linearizations in absolute deviations and linearizations in percentage deviations
00:49:35.000 --> 00:49:42.000
you may also check what happens in the model if we linearize in terms of percentage deviations
00:49:42.000 --> 00:49:49.000
from the steady state the percentage deviation of variable x from its steady state level i denote
00:49:49.000 --> 00:49:58.000
by x hat x hat would be x t minus x so this is x tilde here in the numerator of this ratio it's
00:49:58.000 --> 00:50:08.000
x tilde t divided once again by the steady state level which then gives this year the
00:50:08.000 --> 00:50:15.000
interpretation of which then makes it a percentage deviation from the steady state so x hat t would
00:50:15.000 --> 00:50:24.000
tell us by how many percentage points is x of its steady state level and we can also do the
00:50:24.000 --> 00:50:30.000
linearization in terms of these percentage deviations that's actually very similar to what
00:50:30.000 --> 00:50:36.000
we would do if we take absolute deviations because we would write f of x t is equal to the
00:50:36.000 --> 00:50:42.000
steady state value f of x and now we take sort of the same linear approximation that we had
00:50:43.000 --> 00:50:50.000
before just that we divide by x here and multiply by x here so that these two things would actually
00:50:50.000 --> 00:50:56.000
cancel right so if you cancel out these two x's here then you have exactly the same expression
00:50:56.000 --> 00:51:03.000
for the linear approximation of the non-linear equation that i had used previously with x tilde
00:51:03.000 --> 00:51:13.000
up here but now we leave the two x's in here so that this thing here is x hat and then we have
00:51:13.000 --> 00:51:20.000
to multiply by x times f prime of x rather than just by f prime of x which however is not really
00:51:20.000 --> 00:51:26.000
in any way a big difference or need not be a big difference because this is also steady state
00:51:26.000 --> 00:51:35.000
value so again this thing here is a constant so we would have f of x is or the the approximation
00:51:35.000 --> 00:51:41.000
is f of x at the steady state f at the steady state plus the percentage deviation from the
00:51:41.000 --> 00:51:49.000
steady state value times x and f prime of x and the same holds true for the second equation where
00:51:49.000 --> 00:51:58.000
we have two arguments in the function g g at x t y t would be equal to g at the steady state x y
00:51:58.000 --> 00:52:07.000
plus then x hat t times x g x so here again the partial taking at the steady state plus y hat t
00:52:07.000 --> 00:52:18.000
times y g y taking at the steady state now why doesn't Lieber do this here um well um perhaps
00:52:18.000 --> 00:52:24.000
it was easier to derive the results in a linear approximation with the variables being absolute
00:52:24.000 --> 00:52:31.000
deviations than percentage deviations um that would only have legitimate way of doing it
00:52:31.000 --> 00:52:36.000
perhaps he was shied away from using percentage deviations which were already in the market which
00:52:36.000 --> 00:52:41.000
were already using other models at the time of his writing this paper and by the fact that
00:52:41.000 --> 00:52:50.000
here we have theta is equal to zero in the steady state so we cannot express a percentage deviation
00:52:50.000 --> 00:52:56.000
for theta because we are not allowed to divide by zero right this expression here for theta for
00:52:56.000 --> 00:53:04.000
instance would give us theta t minus zero divided by zero and that's not defined so we cannot use
00:53:04.000 --> 00:53:09.000
the percentage deviations for shocks which have an expected value of zero because the expected
00:53:09.000 --> 00:53:16.000
value of the shock is the steady state value of this variable of course it would be possible to
00:53:16.000 --> 00:53:24.000
linearize the model say in terms of pi hat so using the percentage deviation here and theta tilde
00:53:24.000 --> 00:53:33.000
so using an absolute deviation here that's also a possibility um the exercise here actually
00:53:33.000 --> 00:53:38.000
want to draw your to your to your attention the fact that we can use different variables
00:53:39.000 --> 00:53:44.000
differently defined variables to do the linearization in and then after you have
00:53:44.000 --> 00:53:53.000
understood that there are two things which you might check so show that we get the equation
00:53:53.000 --> 00:54:05.000
et of pi hat t plus one is equal to alpha times beta times pi hat t plus beta divided by pi
00:54:05.000 --> 00:54:14.000
theta tilde if we use this type of linearization where we have pi hats and t tilde which means
00:54:14.000 --> 00:54:21.000
that the coefficients in the linear approximation equation differ from the coefficients we had in 12
00:54:21.000 --> 00:54:31.000
right in equation 12 we had here just a beta um we had alpha beta here like uh we have it in the
00:54:31.000 --> 00:54:40.000
other equation but here we had just a beta and here we had have beta divided by pi so the difference
00:54:40.000 --> 00:54:45.000
is not as one might expect with this coefficient even though here we have changed the variable
00:54:46.000 --> 00:54:50.000
but the difference is with the coefficient uh theta tilde which is actually unchanged because
00:54:50.000 --> 00:54:58.000
we had theta tilde before too uh since now the coefficient needs to be defined it needs to be
00:54:58.000 --> 00:55:05.000
divided by pi so needs to be divided by the steady state value of the inflation factor
00:55:07.000 --> 00:55:12.000
why is that so that the difference in the coefficients pops up here rather than here
00:55:12.000 --> 00:55:21.000
well actually that's just a little trick um the difference was indeed in these uh coefficients
00:55:21.000 --> 00:55:27.000
here so it was here there was a pi in this expression but there was also pi in this expression
00:55:27.000 --> 00:55:32.000
because this is the same variable linearized or transformed to a percentage deviation in the same
00:55:32.000 --> 00:55:38.000
way so basically we had the steady state pi here we had the steady state pi there where they belong
00:55:38.000 --> 00:55:42.000
because we have used the different variable to do the linearization in but then we could divide
00:55:42.000 --> 00:55:51.000
the whole equation by pi and then the pi ends up as changing the coefficient of the theta tilde here
00:55:53.000 --> 00:55:59.000
yeah so you may show that the linearized equations are actually slightly different
00:55:59.000 --> 00:56:05.000
if we use different variables and that may have um implications of the results that we derive
00:56:06.000 --> 00:56:15.000
um also please beware that the x hat variables x t minus x divided by x is approximately equal
00:56:15.000 --> 00:56:20.000
to the difference of the logs which gives it very simple uh interpretation and which actually makes
00:56:20.000 --> 00:56:28.000
it very relatively easy to evaluate these things here empirically and use them in the empirical
00:56:28.000 --> 00:56:35.000
work by just taking the log of a variable well minus something which is constant so that's
00:56:35.000 --> 00:56:46.000
uh really easy to use in say linear vector autoregressions uh i told you to get many exercises
00:56:46.000 --> 00:56:57.000
today um now take uh equation 11 and 4 um so previously we had used the policy rule for
00:56:57.000 --> 00:57:04.000
the central bank equation 11 along with 5 and i had shown you what the result is here i show
00:57:04.000 --> 00:57:10.000
you what the result if you substitute out the interest factor the nominal interest rate factor
00:57:10.000 --> 00:57:18.000
r t from equation 4 and then do the linearization you should get if you linearize in terms of
00:57:18.000 --> 00:57:27.000
deviations this expression here until the t is equal to minus c by divided by r minus 1 squared
00:57:27.000 --> 00:57:35.000
times alpha pi tilde plus theta tilde t that you may verify at home and
00:57:36.000 --> 00:57:40.000
check that the result yeah check that the result is true
00:57:43.000 --> 00:57:51.000
this equation here is not as important to our analysis um as is equation uh what was it 12 and
00:57:51.000 --> 00:57:56.000
will be equation 13 which comes here so the two equations i have highlighted and because this is
00:57:56.000 --> 00:58:05.000
what we call a static equation so if we have a solution for pi and a solution for theta tilde t
00:58:05.000 --> 00:58:10.000
then it is easy to calculate what m tilde t is but m tilde t does not feed back into
00:58:10.000 --> 00:58:15.000
our dynamic equations in the model and therefore it's not so important to us to know exactly what
00:58:15.000 --> 00:58:22.000
m tilde what is important to us is that we solve the highlighted equations and one of these equations
00:58:22.000 --> 00:58:29.000
was as i told you an expectation of difference equation namely this here just in pi t with some
00:58:29.000 --> 00:58:37.000
exogenous disturbance theta tilde and the other equation you get when we linearize equation 2
00:58:38.000 --> 00:58:44.000
and this is then a conventional difference equation in b tilde so here we have b tilde t
00:58:44.000 --> 00:58:53.000
uh which is related to b tilde t minus one so this makes it a difference equation and then there are
00:58:53.000 --> 00:58:59.000
also the pis in here which makes the equation kind of complicated and then there is the theta tilde
00:58:59.000 --> 00:59:07.000
so the shocks in there the theta tilde t the theta tilde t minus one and psi tilde t psi was the
00:59:07.000 --> 00:59:16.000
physical exogenous shock which hit the fiscal policy rule so this equation here is not easily
00:59:16.000 --> 00:59:21.000
solved and equation two is not easily solved so we'll do it together now in the lecture but when
00:59:21.000 --> 00:59:30.000
we have a solution for equation 12 and 13 then we have solutions for b and for pi and for theta
00:59:31.000 --> 00:59:36.000
either solution for theta and obviously with these solutions we can easily compute them until
00:59:37.000 --> 00:59:42.000
is of little interest so we won't speak much about this equation but make sure please that
00:59:42.000 --> 00:59:49.000
you do this type of linearization at home and verify that my equation 13 is correct you see
00:59:49.000 --> 00:59:57.000
that equation 13 involves certain coefficients which i haven't defined yet phi 1 phi 2 phi 3
00:59:57.000 --> 01:00:06.000
and here again a phi 2 these are defined in this form here so phi 1 is a
01:00:07.000 --> 01:00:16.000
fairly complicated expression already of steady state quantities s is phi 2 which is also some
01:00:16.000 --> 01:00:24.000
expression of steady state quantities s is phi 3 actually there would have been a phi 4 here we i
01:00:24.000 --> 01:00:32.000
could have written phi 4 here but phi 4 is very close to the definition of phi 2 so we can easily
01:00:32.000 --> 01:00:39.000
write the phi 4 as phi 2 divided by alpha and i have used this in order to limit the amount of
01:00:39.000 --> 01:00:47.000
new notation in this model here and i just have three coefficients to deal with phi 1 phi 2 and
01:00:47.000 --> 01:00:58.000
phi 3 okay so we started out with a model which had six equations and six endogenous variables at
01:00:58.000 --> 01:01:03.000
least after we introduced the two policy rules it was a model with six variables and six equations
01:01:03.000 --> 01:01:11.000
six endogenous variables six equations we have now reduced the number of variables in our model
01:01:11.000 --> 01:01:16.000
and the number of equations that's legitimate thing to do we can always use simple equations
01:01:16.000 --> 01:01:20.000
to substitute out certain variables from the more complicated equations and then we are left
01:01:20.000 --> 01:01:26.000
with the complicated equations and the reduced number of variables to do so here we are left
01:01:26.000 --> 01:01:32.000
basically with two equations namely two linear equations 12 and 13 the two highlighted equations
01:01:33.000 --> 01:01:41.000
and they have just two endogenous variables b and pi the thetas where are they thetas and
01:01:41.000 --> 01:01:46.000
size here are exogenous variables so we we don't need to bother about them but in terms
01:01:46.000 --> 01:01:52.000
of endogenous variables we only are left with b and pi and the other endogenous variables like
01:01:52.000 --> 01:02:02.000
ct and m tilde t and rt have been substituted out here's the system again both of these equations
01:02:02.000 --> 01:02:10.000
this is equation 12 and this is equation 13 now written just one on top of the other to show you
01:02:11.000 --> 01:02:17.000
if you haven't realized it so far that this system of equations is recursive recursive in
01:02:17.000 --> 01:02:23.000
the sense that we can actually solve or it seems so but it is the case we can actually solve the
01:02:23.000 --> 01:02:31.000
first equation without knowing what the solution for b tilde is so we can solve for pi tilde first
01:02:31.000 --> 01:02:36.000
right using this equation here that's difference equation i will show you in a minute how to solve
01:02:36.000 --> 01:02:41.000
this difference equation the solution of this difference equation will give us an expression
01:02:41.000 --> 01:02:48.000
for pi tilde and then we can in principle substitute in the solution for pi tilde for
01:02:48.000 --> 01:02:54.000
the pi tilde t here and pi tilde t minus one there and then we have a second difference
01:02:54.000 --> 01:03:02.000
equation in just one variable namely in that and we can principle solve this variable of that
01:03:02.000 --> 01:03:12.000
and i will not do this because the solutions for the pi tilde and the pi tilde t here are infinite
01:03:12.000 --> 01:03:19.000
sums of uh sharks and this gives a very nasty expression actually here if i substitute in the
01:03:19.000 --> 01:03:26.000
solution solutions from this equation in here and then solve rather i will teach you in the
01:03:26.000 --> 01:03:32.000
second step after having shown to you how this equation is being solved how we can solve the
01:03:32.000 --> 01:03:39.000
whole system of difference equation in just one step so that will be our approach to
01:03:39.000 --> 01:03:47.000
deriving the solution for the level of it and all of that is as i said rather technical but
01:03:47.000 --> 01:03:57.000
at the end of our technicalities you realize that that there is actually the fiscal theory
01:03:57.000 --> 01:04:04.000
of the price level as sort of the solution of this system of difference equation so we move
01:04:04.000 --> 01:04:10.000
back from the math then to the economics of the model and i will show you that this model can be
01:04:10.000 --> 01:04:20.000
partitioned in four different regimes and we will deal with two of these regimes a little bit more
01:04:21.000 --> 01:04:28.000
detailed and in greater detail and see that one regime is sort of the standard regime where
01:04:29.000 --> 01:04:36.000
you are used to do your economic thinking in and the other is the fiscal theory of the price level
01:04:36.000 --> 01:04:41.000
regime which is perhaps less familiar to you but which is as legitimate as an interpretation or
01:04:41.000 --> 01:04:49.000
as a possible solution of the model as the standard regime so we need all the technicalities in order
01:04:49.000 --> 01:04:56.000
to understand how those four regimes in this model unfold two of which have reasonable economic
01:04:56.000 --> 01:05:06.000
interpretation okay so this is why we now have to deal with the technique of how to solve a
01:05:06.000 --> 01:05:14.000
difference equation and i should perhaps first clarify what it means to solve a difference
01:05:14.000 --> 01:05:21.000
equation of this form here i suppose all of you have already solved difference equations
01:05:22.000 --> 01:05:30.000
previously in your studies this is actually not a particularly difficult difference equation but
01:05:30.000 --> 01:05:39.000
it's also not the most simple one because it is a difference equation which has some heterogeneous
01:05:39.000 --> 01:05:46.000
part here so would be distinguished in terms of difference equation between homogeneous
01:05:46.000 --> 01:05:53.000
equations and heterogeneous equations homogeneous equations are equations which just are equations
01:05:53.000 --> 01:05:59.000
in one variable which is which has a time difference in it like this pi tilde here
01:06:00.000 --> 01:06:07.000
but heterogeneous equations also have some kind of form external influence in this case
01:06:08.000 --> 01:06:18.000
case exogenous shock so this here is not a ordinary not a homogeneous difference equation
01:06:18.000 --> 01:06:28.000
anymore sorry i said heterogeneous is the expressions in homogeneous that's in homogeneous
01:06:28.000 --> 01:06:35.000
this equation here and the second thing which makes this equation a little unconventional is
01:06:35.000 --> 01:06:43.000
that we do not relate pi tilde t plus one to pi tilde t but we relate the expected value of pi
01:06:43.000 --> 01:06:50.000
tilde t plus one uh two pi tilde so these are technically not the same variables obviously
01:06:50.000 --> 01:06:56.000
the expectation of pi tilde t plus one is related somehow we would specify how pi tilde t but
01:06:56.000 --> 01:07:01.000
currently it's not the same variable here this is an expected variable and this is a variable which
01:07:02.000 --> 01:07:08.000
has realized so they're not the same variables i'll show you here with that if you don't
01:07:08.000 --> 01:07:16.000
know it yet but the solution of such a difference equation means that we find an expression for
01:07:16.000 --> 01:07:26.000
pi tilde t which consists only of fixed coefficients and exogenous terms like
01:07:26.000 --> 01:07:35.000
theta tilde and possibly the time as some additional variable so we will describe pi
01:07:35.000 --> 01:07:42.000
tilde t as a function of time and of exogenous shocks with the coefficients which come up here
01:07:42.000 --> 01:07:50.000
in this this difference equation and the solution has the property that if i substitute the solution
01:07:50.000 --> 01:07:56.000
into this equation for pi tilde t and for pi tilde t plus one then this equation here holds
01:07:56.000 --> 01:08:04.000
that it's true that we satisfy this equation a solution also has the property that it is a
01:08:04.000 --> 01:08:11.000
general expression which can be used for any time index down here so it's a solution for
01:08:11.000 --> 01:08:20.000
pi tilde t as it is a solution for pi tilde t plus some s for instance some some uh integer s
01:08:23.000 --> 01:08:30.000
where you then just have to exchange in this expression which is the solution any t for t
01:08:30.000 --> 01:08:36.000
plus s and that is the same then it's also correctly describing the pi tilde t plus s
01:08:36.000 --> 01:08:39.000
okay um
01:08:42.000 --> 01:08:50.000
review of system of linear of systems of linear difference equations we don't start with a system
01:08:50.000 --> 01:08:55.000
yet i mean we have a system here consisting of two linear different equations as i already pointed
01:08:55.000 --> 01:09:04.000
out but we'll start with just one equation which is the equation 12 which i just discussed
01:09:05.000 --> 01:09:13.000
um and i show you how to solve such a difference equation if it is an
01:09:13.000 --> 01:09:19.000
expectational difference equation and the way to do this is actually very easy because we say
01:09:19.000 --> 01:09:25.000
if it is a rational expectation as the mathematical expectation operator is always
01:09:25.000 --> 01:09:32.000
meant to indicate a rational expectation then we know that the difference between
01:09:32.000 --> 01:09:39.000
the true variable pi tilde t plus one and its conditional expectation in the previous period
01:09:39.000 --> 01:09:45.000
so e t of pi tilde t plus one the difference between this is something which cannot be
01:09:45.000 --> 01:09:52.000
forecasted at all uh so it is some error term which has an expectation of zero
01:09:52.000 --> 01:09:59.000
and nobody has any clue to how this eta tilde t plus one will develop in the future so we get
01:09:59.000 --> 01:10:06.000
a new equation here 14 which says that pi tilde t plus one is equal to the conditional expectation
01:10:06.000 --> 01:10:13.000
of pi tilde t plus one conditional on information plus something which cannot be forecasted
01:10:13.000 --> 01:10:21.000
something which has an expected value of zero so something which is an error term this error term
01:10:21.000 --> 01:10:28.000
as i say cannot be forecasted so it is serially uncorrelated often we just say this white noise
01:10:29.000 --> 01:10:34.000
here it need actually not be white noise because the variance of the eta t plus one need not be
01:10:34.000 --> 01:10:40.000
constant over time but it must be true that the eta tilde t plus one is serially uncorrelated
01:10:40.000 --> 01:10:44.000
because if it were not serially uncorrelated then we would be in a position to forecast
01:10:45.000 --> 01:10:54.000
the next value of eta of the the eta process so it is serially uncorrelated like white noise is
01:10:54.000 --> 01:11:04.000
serially uncorrelated and this then enables us to substitute out in our difference equation
01:11:05.000 --> 01:11:14.000
the expectation of the conditional expectation of pi tilde t plus one or taking it the different way
01:11:14.000 --> 01:11:22.000
and the way it is written here to substitute out the true pi tilde t and replace it by its
01:11:22.000 --> 01:11:30.000
expectation in period t minus one of pi tilde t plus some eta t and this is what we will now do
01:11:31.000 --> 01:11:41.000
so we used successive substitutions of equations 12 in order to derive an expression for
01:11:41.000 --> 01:11:49.000
the expectation of pi tilde t plus one conditional on information available in period t
01:11:49.000 --> 01:11:57.000
so our difference equation 12 had this from here et of pi tilde t plus one is equal to alpha beta
01:11:57.000 --> 01:12:06.000
times pi tilde t plus beta times theta tilde t so what can we do we can now say well this pi tilde
01:12:06.000 --> 01:12:15.000
t here is actually by equation 14 on the previous page equal to the rational expectation of pi tilde
01:12:15.000 --> 01:12:23.000
t in period t minus one so et minus one of pi tilde t plus some expectation error i will call
01:12:23.000 --> 01:12:32.000
these shocks here the etas the expectation errors of this particular equation so what you see in
01:12:32.000 --> 01:12:39.000
parentheses here is just equal to pi tilde t and then we have again plus beta times theta tilde t
01:12:39.000 --> 01:12:52.000
okay so what can we do now we have here et minus one of pi tilde t et minus one plus pi tilde t
01:12:52.000 --> 01:13:00.000
is by virtue of equation 12 so by virtue of this equation here just lagged one period
01:13:01.000 --> 01:13:09.000
it is equal to alpha beta p tilde t minus one plus beta theta tilde t minus one
01:13:10.000 --> 01:13:17.000
so in place of et minus one pi tilde t minus one pi pi tilde t i can introduce here alpha beta
01:13:17.000 --> 01:13:25.000
times pi tilde t minus one plus beta times theta tilde t minus one which is just the
01:13:25.000 --> 01:13:29.000
lagged form of equation 12 which i have now substituted in for the expectation here
01:13:30.000 --> 01:13:40.000
plus beta times theta tilde t and then i can go on like this i can again say well pi tilde t
01:13:40.000 --> 01:13:47.000
minus one can be replaced by the expectation in period t minus two of pi tilde t minus one
01:13:47.000 --> 01:13:54.000
so expectation of the same variable but taking in period t minus two one period previously
01:13:54.000 --> 01:14:03.000
plus some shock eta t minus one so this pi tilde t minus one here is again replaced by this term
01:14:03.000 --> 01:14:11.000
parentheses here and these terms here are left unchanged so from beta tilde t minus one to beta
01:14:11.000 --> 01:14:23.000
t we have the three terms here okay now again we can say et minus two of pi tilde t minus one
01:14:23.000 --> 01:14:30.000
we can replace using equation 12 just lagged two periods so this equation here which is equation
01:14:30.000 --> 01:14:41.000
12 lagged two periods gives us then instead of et minus one et minus two pi tilde t minus one
01:14:41.000 --> 01:14:49.000
we can also write alpha beta times pi tilde t minus two plus beta eta tilde t minus two
01:14:50.000 --> 01:14:58.000
which i have used here so this expression here is replaced by this expression here
01:15:00.000 --> 01:15:06.000
and the plus eta t minus one is left unchanged as are these three terms left unchanged here
01:15:06.000 --> 01:15:14.000
okay if we continue to do this many many times say for instance we do this s times
01:15:15.000 --> 01:15:22.000
then we derive an expression which is this is equal to alpha beta to the power of s plus one
01:15:23.000 --> 01:15:30.000
times pi tilde t minus s so you will see here we have the alpha beta alpha beta alpha beta
01:15:30.000 --> 01:15:35.000
actually raised here to the power of three alpha beta raised to the power of three times
01:15:36.000 --> 01:15:42.000
index t minus two so exponent three here index minus two here so this would
01:15:44.000 --> 01:15:50.000
lead to alpha beta raised to s plus one and then index t minus s here
01:15:50.000 --> 01:16:00.000
and then we have a sum which always has a coefficient beta beta beta beta is sum of the
01:16:01.000 --> 01:16:07.000
t tilde's of the theta tilde's right here's theta tilde t here's theta t t minus one here's theta
01:16:07.000 --> 01:16:14.000
t t minus two and we just have to be aware that there is no alpha beta in front of the sum here
01:16:15.000 --> 01:16:24.000
but here for theta tilde t minus one we have this alpha beta which has to be multiplied in here
01:16:24.000 --> 01:16:32.000
and for this beta times theta tilde t minus two we have an alpha beta here and an alpha beta here
01:16:32.000 --> 01:16:40.000
which is a coefficient to this thing so we have some sum which involves a alpha beta to the power
01:16:40.000 --> 01:16:49.000
of j when theta when it is multiplied by theta t minus j and the beta is some constant coefficient
01:16:49.000 --> 01:16:58.000
in front of the sum here and then similarly the etas also sum up in such a way we have eta t here
01:16:58.000 --> 01:17:07.000
and we have eta t minus one here but the eta t minus one is of course multiplied by alpha times
01:17:07.000 --> 01:17:14.000
beta sorry by alpha times beta squared I mean alpha beta times alpha beta and the eta t is
01:17:14.000 --> 01:17:24.000
multiplied by alpha beta so the sum will take the form sum for j going from one to s over alpha
01:17:24.000 --> 01:17:33.000
beta to the power of j times eta t plus one minus j okay so this here is our solution or almost our
01:17:33.000 --> 01:17:41.000
solution for the expectation of pi tilde t plus one conditional period t information
01:17:44.000 --> 01:17:49.000
we're not quite where we want to get to and the expectation is a little more complicated than it
01:17:49.000 --> 01:17:55.000
need be we will simplify it a little bit on the next slide but first thing to note is now the
01:17:55.000 --> 01:18:05.000
following the expectation of period t plus one inflation depends on previous inflation
01:18:05.000 --> 01:18:14.000
experiences this in general makes sense when people expect inflation in the coming period
01:18:14.000 --> 01:18:22.000
they may say well we have had certain experiences with inflation in the past and these experiences
01:18:22.000 --> 01:18:31.000
we use in order to build our expectation for the future in some sense you may think that people
01:18:31.000 --> 01:18:35.000
say we have had some experience with how our central bank deals with inflation
01:18:36.000 --> 01:18:43.000
or dealt with inflation in the past and we will make use of this information in order to form
01:18:43.000 --> 01:18:49.000
our expectation on inflation in the future because we think we have learned something about how our
01:18:49.000 --> 01:18:57.000
center bank handles inflation in Germany it is often said that the hyperinflation which
01:18:57.000 --> 01:19:07.000
Germans experienced in 1922-1953 led to a particularly hostile attitude towards inflation
01:19:07.000 --> 01:19:15.000
so that central banks in Germany after the hyperinflation have always tried to circum
01:19:16.000 --> 01:19:23.000
to prevent inflation by all means by almost all means i would say not by all means but
01:19:23.000 --> 01:19:29.000
usually we have had central banks which took a rather aggressive stance on inflation and tried
01:19:29.000 --> 01:19:37.000
to convince the public that something like 1922-1923 where many many many people lost a great part of
01:19:37.000 --> 01:19:44.000
their fortune due to inflation the reason at the time being was that many people had
01:19:44.000 --> 01:19:51.000
signed up to government debt during world war one they wanted to be patriotic they wanted
01:19:51.000 --> 01:19:57.000
to support the military expenditures of the government they wanted to contribute to a
01:19:57.000 --> 01:20:04.000
german victory in world war two so many actually gave all of their savings to the government and
01:20:04.000 --> 01:20:13.000
acquired government bonds which were denominated in some nominal units a mark was the currency at
01:20:13.000 --> 01:20:22.000
at the time being and then in 1922 and 1923 there was this hyperinflation and a government bond which
01:20:22.000 --> 01:20:28.000
had been worth a thousand marks or 10 000 marks which was huge some for people at the time being
01:20:28.000 --> 01:20:39.000
in 1914 15 16 17 18 found this money to be completely valueless completely devalued by
01:20:39.000 --> 01:20:45.000
inflation since the price level had risen so strongly that essentially they didn't get any
01:20:45.000 --> 01:20:51.000
money back from all the money they had given to the government some years earlier so inflation
01:20:51.000 --> 01:20:59.000
experience in the hyperinflation of 1922-1923 made a huge impression on germans at the time and this
01:20:59.000 --> 01:21:06.000
has perhaps carried over to later generations because even today germans are perceived the
01:21:06.000 --> 01:21:12.000
european union or worldwide as being particularly hostile towards inflation however some people
01:21:12.000 --> 01:21:18.000
today argue well forget about the bad experiences 100 years ago and and we know that central banks
01:21:18.000 --> 01:21:23.000
don't lose things like these anymore and why should the germans tolerate a little more inflation
01:21:24.000 --> 01:21:29.000
the european central bank has just a couple of weeks ago issued a statement that perhaps we
01:21:29.000 --> 01:21:35.000
should be more tolerant towards inflation and the european central bank would perhaps tolerate
01:21:35.000 --> 01:21:42.000
inflation beyond the 2 percent threshold that they have exercised in their monetary policy
01:21:43.000 --> 01:21:49.000
so far it would average us out over some long time horizon so there are signs that the european
01:21:49.000 --> 01:21:56.000
central bank actually tries to make the public get used to higher levels of inflation in
01:21:57.000 --> 01:22:04.000
the future and the question is whether this would be well received by the german public but whatever
01:22:04.000 --> 01:22:13.000
the reasoning and the concrete plans of the ecb may be their reasoning may perhaps be that
01:22:13.000 --> 01:22:24.000
previous um explanation experiences like a hyperinflation 100 years ago um have less value
01:22:24.000 --> 01:22:34.000
in forming today's inflationary expectations if it is already such a long time ago so perhaps
01:22:34.000 --> 01:22:40.000
people at the ecb think what germans by now are not as worried about hyperinflation anymore
01:22:40.000 --> 01:22:45.000
as they were in the 1970s when the experience was still just 50 years ago and many people still
01:22:45.000 --> 01:22:52.000
lived who had actually lost all their fortune due to the hyperinflation so it makes sense to suppose
01:22:52.000 --> 01:23:01.000
that this um reliance of current inflationary expectations on previous inflationary
01:23:01.000 --> 01:23:12.000
um experiences um has an alpha beta product term here which is smaller than one right so that would
01:23:12.000 --> 01:23:21.000
mean that very old inflationary expectations are not so important anymore after a long time period
01:23:21.000 --> 01:23:28.000
has passed because if alpha beta is smaller than one then when it is raised to the power of s plus
01:23:28.000 --> 01:23:33.000
one then this converges down to zero so it becomes smaller and smaller and the old inflation
01:23:33.000 --> 01:23:42.000
experience fades away in memory if alpha beta is smaller than one we may actually go a step
01:23:43.000 --> 01:23:51.000
farther and say um perhaps it is the only economically reasonable interpretation of
01:23:51.000 --> 01:23:58.000
this equation 15 here to say that alpha beta must be smaller than one because suppose that
01:23:58.000 --> 01:24:07.000
alpha beta is greater than one then this would mean the farther away in the distant past
01:24:07.000 --> 01:24:14.000
and inflationary expectation is the more influence it has on today's forming of inflationary
01:24:14.000 --> 01:24:21.000
expectations i don't think that any serious people would argue that this is the case that
01:24:21.000 --> 01:24:26.000
inflationary expectations we may have made during the german empire prior to world war one for
01:24:26.000 --> 01:24:32.000
instance where sometimes we had inflation sometimes we had deflation but not at the uh no inflation
01:24:32.000 --> 01:24:38.000
at the hyperinflation level then these things become ever more important the more time passes
01:24:38.000 --> 01:24:44.000
that would be completely unreasonable to say so the only economically reasonable interpretation
01:24:44.000 --> 01:24:52.000
of this equation here is that this equation holds if alpha beta is smaller than one an absolute
01:24:52.000 --> 01:25:05.000
value okay good um let me just uh finish um this here um or not quite and now i think i will finish
01:25:05.000 --> 01:25:12.000
and then i will leave you for a break um if it is true that alpha beta an absolute value is smaller
01:25:12.000 --> 01:25:19.000
than one and we let s go to infinity then obviously this term here vanishes right this
01:25:19.000 --> 01:25:27.000
converts to zero so we are left with these two sums here uh and s as the upper limit of the sum
01:25:27.000 --> 01:25:35.000
is then infinity so we get that the expectation today of inflation tomorrow is beta times this
01:25:35.000 --> 01:25:46.000
infinite sum here of weighted shocks beta some weighted uh interest rate uh shocks and this sum
01:25:46.000 --> 01:25:55.000
here of weighted inflation uh expectation errors right it does were expectation errors then we
01:25:55.000 --> 01:26:01.000
also have an weighted infinite sum of expectation errors both of these sums have the property since
01:26:01.000 --> 01:26:07.000
alpha beta is smaller than one that uh shocks which were experienced long ago are not so
01:26:07.000 --> 01:26:14.000
important anymore but uh shocks which are more in the recent past uh they have some influence
01:26:14.000 --> 01:26:21.000
and higher influence than the shocks which are some time ago so this is what we call a backward
01:26:21.000 --> 01:26:29.000
looking expression uh the the expectation which of course always is forward looking the expectation
01:26:29.000 --> 01:26:40.000
today of tomorrow's of period t plus one inflation um is a backward looking uh expectation in the
01:26:40.000 --> 01:26:46.000
sense that the expectation is formed by looking at the experience we have made with past shocks
01:26:48.000 --> 01:26:55.000
both with past fundamental shocks so shocks to the normal interest rate and with expectation
01:26:55.000 --> 01:27:00.000
errors we have made in the past so it's completely reasonable that people may form their expectations
01:27:00.000 --> 01:27:05.000
in that way that they take into account what kind of errors have we made in the past or perhaps we
01:27:05.000 --> 01:27:10.000
should learn from our errors and what times what kind of shocks have occurred in the past and
01:27:10.000 --> 01:27:15.000
we we know how these shocks have impacted on inflation so we take the shocks into account
01:27:15.000 --> 01:27:20.000
with some decreasing weights over time it's a perfectly reasonable way to form
01:27:21.000 --> 01:27:28.000
expectations if alpha beta is smaller than uh one um and this way of forming expectations is
01:27:28.000 --> 01:27:36.000
then called a backward looking expression yeah i have basically said uh this year now comes
01:27:36.000 --> 01:27:44.000
sort of the conclusion um moving to the solution for pi tilde rather than the conditional
01:27:44.000 --> 01:27:51.000
expectation of tomorrow's pi tilde that we have derived so far which was uh here um here we now
01:27:51.000 --> 01:27:59.000
derive what is a pi tilde t plus one and that is of course easy because we can just use equation
01:27:59.000 --> 01:28:10.000
for t we know pi tilde t plus one is the expectation in period t of pi tilde t plus one plus some eta
01:28:10.000 --> 01:28:18.000
tilde t plus one so essentially we take this expectation here and we add some one one eta t
01:28:18.000 --> 01:28:25.000
plus one then we have the solution for pi tilde t plus one not the conditional expectation anymore
01:28:25.000 --> 01:28:32.000
so we add this one plus eta tilde t plus one here otherwise this is the same expression uh here
01:28:32.000 --> 01:28:41.000
and then we see well this eta tilde t plus one seems to fit in very well into uh this um uh into
01:28:41.000 --> 01:28:50.000
this expression here by just adjusting the indexes a little bit and letting the index j start at zero
01:28:50.000 --> 01:28:58.000
rather than j start at one you see here we have um we lag the whole equation sorry i forgot to
01:28:58.000 --> 01:29:04.000
mention that if we lag this whole equation here then basically we have to uh by one period then
01:29:04.000 --> 01:29:14.000
we basically we have to replace the t plus one here by a t and the t here by a t minus one obviously
01:29:14.000 --> 01:29:20.000
so here the t plus one becomes a t here the t plus one becomes a t here the t becomes a t minus
01:29:20.000 --> 01:29:30.000
one and here the t plus one becomes a t so we have t t minus one t here and now watch out for
01:29:30.000 --> 01:29:37.000
j is equal to zero that's a new component of the sum we have j equal to one here before so one
01:29:37.000 --> 01:29:43.000
more components added to the sum with j equal to zero for j equal to zero we have alpha beta to
01:29:43.000 --> 01:29:51.000
the power of zero is one we had a one here no alpha beta here and then eta t minus j so eta t
01:29:51.000 --> 01:29:58.000
minus zero is eta t the eta t plus one here became an eta t because we shifted the uh solution by one
01:29:58.000 --> 01:30:05.000
time period so this fits well into the sum and therefore we have here the solution for the
01:30:05.000 --> 01:30:12.000
expectation of difference equation and for our uh inflation not expected inflation anymore for
01:30:12.000 --> 01:30:21.000
our inflation in terms of previous um interest rate uh shocks uh in an infinite sum with weights
01:30:21.000 --> 01:30:28.000
which decrease geometrically and conditional on expectation errors that we have made in the past
01:30:30.000 --> 01:30:39.000
so actually inflation in this particular solution of the model depends on past values on past
01:30:39.000 --> 01:30:45.000
interest rate shocks and on past measurement errors if the alpha beta is smaller than one in
01:30:45.000 --> 01:30:55.000
absolute value okay i stop here for 10 minutes break we meet again at uh two uh at two o'clock
01:30:56.000 --> 01:31:04.000
um if you have questions and uh would like to raise them um please do this via the chat function now
01:31:05.000 --> 01:31:13.000
if you want to interact in the meeting at the end of this lecture so when i finish the next part
01:31:13.000 --> 01:31:18.000
of the lecture you may also indicate there's not the break in the chat and i will read the chat
01:31:18.000 --> 01:31:26.000
when i come back from the break uh so that i can uh sort of um reserve some time uh for interaction
01:31:26.000 --> 01:31:32.000
if that is required by you okay i will pause the recording here and we meet again in 10 minutes time
01:31:35.000 --> 01:31:37.000
and then there is something in the chat
01:31:43.000 --> 01:31:48.000
oh yeah as somebody told me that i'm not recording thank you very much for ending that
01:31:48.000 --> 01:31:54.000
to me i'm correct that now um once again uh we consider the case alpha beta um we consider the
01:31:54.000 --> 01:32:03.000
case alpha beta is smaller uh than one now we will go to the case where alpha beta is greater
01:32:03.000 --> 01:32:08.000
than one and that may easily be the case that alpha beta is greater than one because beta is
01:32:08.000 --> 01:32:14.000
the preference parameter is typically very close to one even though smaller than one but easily 0.99
01:32:14.000 --> 01:32:21.000
or something like this and alpha is the response of the central bank um with respect to inflation
01:32:21.000 --> 01:32:29.000
so um may well be that the central bank decides to raise the nominal interest rate by more
01:32:29.000 --> 01:32:36.000
than the percentage point increase in inflation if it aggressively fights inflation or if it
01:32:36.000 --> 01:32:44.000
adheres to the Taylor principle in uh uh in new Keynesian models so if alpha is just a little bit
01:32:44.000 --> 01:32:50.000
bigger than than one and beta is close to one then we have already this case here this is what i
01:32:50.000 --> 01:32:57.000
already said we're not recording here it was once again now what would we do with our difference
01:32:57.000 --> 01:33:04.000
equation uh 12 if alpha and beta is greater than one then the standard way to solve the difference
01:33:04.000 --> 01:33:15.000
equation is to rearrange it um after dividing the difference equation by alpha beta so in this case
01:33:15.000 --> 01:33:23.000
we would get pi tilde t is equal to one over alpha beta times today's expectation of the future
01:33:23.000 --> 01:33:34.000
uh inflation rate pi tilde t plus one minus one over alpha teta tilde t so uh just divide the whole
01:33:34.000 --> 01:33:40.000
equation 12 by alpha beta and rearrange to have now i tilde t on the left hand side and the
01:33:40.000 --> 01:33:46.000
expectation or future expectation on the right hand side of the equation then we have an equation
01:33:46.000 --> 01:33:52.000
where the coefficient here in front of the expectation is smaller than one because one
01:33:52.000 --> 01:33:59.000
over alpha beta is smaller than one let's indicate by this little smaller one here is smaller than
01:33:59.000 --> 01:34:07.000
one if alpha beta is greater than one so this way to write the equation now reflects forward
01:34:07.000 --> 01:34:16.000
looking behavior today's inflation depends on what people expect inflation to be in the future
01:34:16.000 --> 01:34:23.000
right which is also possible if people expect that inflation will be higher in the future then
01:34:23.000 --> 01:34:30.000
prices will rise in the future and prices may already rise today anticipating what happens in
01:34:30.000 --> 01:34:39.000
the future okay so we can solve this a type of inflation again and again by successive
01:34:39.000 --> 01:34:46.000
substitutions but now the time direction of the successive substitutions is different
01:34:47.000 --> 01:34:54.000
because now we substitute into the future rather than into the past and in order to do this i
01:34:54.000 --> 01:35:00.000
have to remind you of the law of iterated expectations which you have hopefully covered
01:35:00.000 --> 01:35:05.000
in your econometrics and statistics lectures at least and not Thomas Ziedler is doing this in
01:35:05.000 --> 01:35:10.000
the advanced econometrics one lecture and estimation of the inference which you have
01:35:10.000 --> 01:35:18.000
probably taken last year the law of iterated expectation says that the expectation conditional
01:35:18.000 --> 01:35:28.000
on t of an expectation conditional on information in t plus tau of a variable is equal to the
01:35:28.000 --> 01:35:37.000
expectation conditional just of the information in period t of the same variable so basically the
01:35:37.000 --> 01:35:44.000
reading of that would be typically we have more information in period t plus tau than we have
01:35:44.000 --> 01:35:50.000
information in period t because we don't forget information but new information is coming up so
01:35:50.000 --> 01:35:55.000
typically the information set of this expectations operator here is greater than the information set
01:35:55.000 --> 01:36:03.000
of this expectations operator so if we take an expectation with respect to information set of
01:36:03.000 --> 01:36:10.000
period t of a greater information set then obviously we just don't know what new information
01:36:10.000 --> 01:36:17.000
may have come up for the new information set of period t plus tau so we can just use the information
01:36:17.000 --> 01:36:24.000
in period t for computing this conditional expectation and that is then the same thing as
01:36:24.000 --> 01:36:30.000
just expecting with the information set of period and e the variable x t plus tau plus j
01:36:30.000 --> 01:36:38.000
so we can just do away with this e t plus tau operator inside and information and expectation
01:36:38.000 --> 01:36:44.000
which has a smaller information set that's what the law of iterated expectation expectation says
01:36:45.000 --> 01:36:52.000
and this is what we will now use to solve the model forward for the case alpha beta greater than 1.
01:36:54.000 --> 01:37:01.000
So we start out with the equation which we have rewritten in this form today's inflation depends
01:37:01.000 --> 01:37:09.000
on today's expectation of tomorrow's inflation plus or minus the interest rate shock.
01:37:10.000 --> 01:37:21.000
So what can we do we can then say well pi tilde t plus 1 which is expected here is exactly what
01:37:21.000 --> 01:37:32.000
we have in the parentheses here so the et is unchanged and the et applies to pi tilde t plus 1
01:37:32.000 --> 01:37:39.000
but we know pi tilde t plus 1 is precisely given by this equation here shifted one period
01:37:39.000 --> 01:37:48.000
in the future so pi tilde t plus 1 is 1 over alpha beta times et plus 1 pi tilde t plus 2 minus 1
01:37:48.000 --> 01:37:55.000
over alpha tilde t plus 1 and this is what i have written in here so this replaces the pi tilde t plus
01:37:55.000 --> 01:38:06.000
1 here minus 1 over alpha tilde t is unchanged so we have this thing here and then we can move on
01:38:06.000 --> 01:38:12.000
like that of course here have we have the expectation conditional on t plus 1 information
01:38:12.000 --> 01:38:18.000
of pi tilde t plus 2 we leave the expectation conditional period t plus 1 information
01:38:20.000 --> 01:38:28.000
unchanged and replace the pi tilde t plus 2 by this expression here shifted two periods in the future
01:38:28.000 --> 01:38:36.000
so in this case we would get these parentheses here where an expectations operator conditional
01:38:36.000 --> 01:38:44.000
on t plus 2 information comes up and the pi tilde t plus 3 comes up and then you see what happens
01:38:44.000 --> 01:38:52.000
here we get a series of tetatilis tetatilde t plus 2 tetatilde t plus 1 tetatilde t with different
01:38:52.000 --> 01:38:59.000
coefficients there's 1 over alpha here there's 1 over alpha here but multiplied by
01:38:59.000 --> 01:39:10.000
1 over alpha beta by this thing here and then there is a 1 over alpha here but multiplied by
01:39:10.000 --> 01:39:20.000
1 alpha beta squared here and there so that when you just think about how these expressions here
01:39:20.000 --> 01:39:27.000
build up we arrive at 1 over alpha beta to the power of s of the expectation conditional
01:39:27.000 --> 01:39:37.000
t information of inflation pi tilde t plus s since the expectation in t of the expectation of
01:39:37.000 --> 01:39:43.000
t in t plus 1 of the expectation in t plus 2 and so forth is the same as just the expectation in
01:39:43.000 --> 01:39:52.000
periods t by the law of iterative expectations so you can forget about these and this and all the
01:39:52.000 --> 01:39:59.000
other future expectations operators and you just get today's expectation of pi tilde t plus s
01:40:00.000 --> 01:40:07.000
multiplied by a coefficient which is smaller than 1 in absolute value this 1 over alpha beta raised
01:40:07.000 --> 01:40:14.000
to the power of s so we can already see when s goes to infinity then this thing here goes to zero
01:40:14.000 --> 01:40:21.000
so essentially this thing here will vanish at least if inflation is not exploding faster than
01:40:21.000 --> 01:40:28.000
geometrically right so this thing will disappear when it disappear when we let s go to infinity
01:40:29.000 --> 01:40:37.000
and therefore we are left with just this infinite sum here and we let s go to infinity so we are
01:40:37.000 --> 01:40:44.000
left with minus 1 over alpha the infinite sum of 1 over alpha beta to the power of j expectation
01:40:44.000 --> 01:40:55.000
today of theta tilde t plus j just as we have written it here so in this case the inflation
01:40:55.000 --> 01:41:05.000
today depends on the expectation of future interest rate shocks only it does not depend
01:41:05.000 --> 01:41:13.000
anymore on past expectation and errors like we had in the other solution of the equation for alpha
01:41:13.000 --> 01:41:22.000
beta smaller than 1 and it does not at all depend on past inflation experience it just depends on
01:41:22.000 --> 01:41:27.000
what we expect for the future so it's a completely different solution which we have here
01:41:30.000 --> 01:41:38.000
this is also a perfectly rational expectation so the bottom line of these two solutions which we
01:41:38.000 --> 01:41:45.000
have now derived for two different calibrations of the model calibration with alpha beta smaller
01:41:45.000 --> 01:41:52.000
than one and calibration with alpha beta greater than one is that inflation is set in response to
01:41:52.000 --> 01:42:00.000
past data if alpha beta is smaller than one absolute value and we call this so-called passive
01:42:00.000 --> 01:42:10.000
policy whereas inflation is set with respect to future developments if alpha beta is an absolute
01:42:10.000 --> 01:42:19.000
value greater than one and we call this an active policy the terms passive and active may perhaps
01:42:19.000 --> 01:42:27.000
not be completely clear but the way you can explain them to you is that an active policy
01:42:27.000 --> 01:42:34.000
is not constrained by past and current data so in some sense it is a sovereign policy the central
01:42:34.000 --> 01:42:38.000
bank is just doing what it wants to do what it finds appropriate to do in view of what it expects
01:42:38.000 --> 01:42:45.000
in the future but it is not constrained by things which have happened in the past and in that sense
01:42:45.000 --> 01:42:55.000
the central bank can be active in setting its inflation so this may also be framed in saying
01:42:55.000 --> 01:43:02.000
that the center bank is not constrained by the decisions of other agents it is just doing what
01:43:03.000 --> 01:43:11.000
things is correct to do and sets the inflation rate actively in the case where alpha beta is greater
01:43:11.000 --> 01:43:18.000
than one an absolute value whereas this case here where alpha beta is smaller than one is passive
01:43:18.000 --> 01:43:23.000
since the expectations market results equilibria in the past periods play a role
01:43:24.000 --> 01:43:28.000
for central bank decisions and for the equilibrium inflation rate today
01:43:32.000 --> 01:43:38.000
formerly we always have this property that the difference equation can be solved either backward
01:43:38.000 --> 01:43:47.000
or formed forward but of course in an economic sense you always have to ask the question whether
01:43:47.000 --> 01:43:54.000
a certain representation is economically sensible a representation which would imply that
01:43:54.000 --> 01:44:01.000
the important of a very distant observation so very distant in the past would increase that
01:44:01.000 --> 01:44:07.000
importance would increase the greater the distance is is in most cases of course not
01:44:07.000 --> 01:44:12.000
sensible because we would think that the farther away people and if things have happened in the
01:44:12.000 --> 01:44:20.000
past the less important they are today there may be exceptions to this rule of thumb because there
01:44:20.000 --> 01:44:29.000
may be processes with unbounded and possibly accelerating growth so these are called explosive
01:44:29.000 --> 01:44:37.000
processes and they may occur in particular when we talk about inflation for instance but they are
01:44:37.000 --> 01:44:46.000
exceptions the usual non-exceptional case is that sensible to solve difference equations
01:44:46.000 --> 01:44:53.000
backward if they have stable eigenvalues so that today's value depends then on past
01:44:53.000 --> 01:44:58.000
values only like we have done it in the first version of solving equation 12
01:44:58.000 --> 01:45:08.000
and difference equations which have unstable eigenvalues so where the coefficient in front
01:45:08.000 --> 01:45:15.000
of the lecture is greater than one need to be solved forward which means that we rewrite the
01:45:15.000 --> 01:45:21.000
equation and what was an equation where a variable in period t was related to a variable in period t
01:45:21.000 --> 01:45:27.000
minus one with a coefficient greater than one becomes then an equation where variable in period
01:45:27.000 --> 01:45:36.000
t is related to a variable in period plus one so in the future with a coefficient smaller than one
01:45:36.000 --> 01:45:44.000
in absolute values these forward type equations are of course equations which apply that the
01:45:44.000 --> 01:45:50.000
solution depends then on future values or variables so the solution depends on expectations
01:45:50.000 --> 01:46:00.000
and you know that central banks regularly try to control expectations that they try to anchor
01:46:00.000 --> 01:46:06.000
expectations because they think that inflationary expectations have feedback mechanisms to the actual
01:46:06.000 --> 01:46:10.000
inflation if people expect much inflation then it is likely that the actual inflation will also
01:46:10.000 --> 01:46:18.000
be rather high are there any questions concerning the solution of
01:46:19.000 --> 01:46:20.000
expectation of difference equations
01:46:26.000 --> 01:46:34.000
if this is not the case and i don't see any here no if this is not the case then i will move on
01:46:34.000 --> 01:46:41.000
and teach you how to solve systems of linear difference equations because we here have a system
01:46:41.000 --> 01:46:47.000
of a linear difference equation and that will present the way these systems are solved in a
01:46:47.000 --> 01:46:54.000
more general way than just with two equations i will talk about in linear difference equations
01:46:55.000 --> 01:47:02.000
for the time being i will call these n linear difference equations or we just take linear
01:47:02.000 --> 01:47:08.000
difference equations which do not involve expectation terms to keep things simpler since
01:47:08.000 --> 01:47:12.000
we have just learned how to deal with expectation terms this can of course also be applied in the
01:47:12.000 --> 01:47:18.000
setting of a system of difference equations but we don't need to go to the expectation part again
01:47:18.000 --> 01:47:24.000
since we have just covered so suppose that we have n linear difference equations here
01:47:25.000 --> 01:47:33.000
so we have variables x1 x2 and xn which are different variables so they can be i don't know
01:47:33.000 --> 01:47:39.000
consumption and interest rate and inflation debt or something like that all observed at a point
01:47:39.000 --> 01:47:47.000
in time t right x1 t x2 t xn t are the period t observations of n different variables
01:47:48.000 --> 01:47:55.000
and in some kind of dynamic system these difference equations are related to past
01:47:55.000 --> 01:48:05.000
values of these variables i have written it here just as a system where the previous
01:48:05.000 --> 01:48:11.000
period variables play a role so just the t minus one variables of each variable but variable x1
01:48:11.000 --> 01:48:20.000
depends on its own past in t minus one x1 t minus one on variables two past in periods t minus one
01:48:20.000 --> 01:48:28.000
on variables three past in period t minus one and that all the way to variable n's past in period t
01:48:28.000 --> 01:48:38.000
minus one i also allow for some exogenous variable r1 t which is a variable which
01:48:39.000 --> 01:48:43.000
which comprises everything else which may have happened in period t but which is not explained
01:48:43.000 --> 01:48:49.000
by the model so it's exogenous to the model and the same structure applies to all the other variables
01:48:49.000 --> 01:48:57.000
so all of these variables depend on all the past variables in period t minus one so on all different
01:48:57.000 --> 01:49:03.000
variables in period t minus one with certain coefficients and therefore the coefficients which
01:49:03.000 --> 01:49:10.000
can be all different are indexed accordingly a11 here a12 for the first equations always the row
01:49:10.000 --> 01:49:16.000
index one and then the column index is indicating the variable to which it refers and then the second
01:49:16.000 --> 01:49:24.000
rank the row the row index is of course a2 is of course two so we have a21 a22 a2n and so all the
01:49:24.000 --> 01:49:35.000
way through to a n1 and ann here the reason why i have written this just as a first order linear
01:49:35.000 --> 01:49:43.000
system is that you can actually write any system which has more than just one period lag so which
01:49:43.000 --> 01:49:50.000
has a second block block of x variable say in period t minus two or t minus three or t minus k
01:49:50.000 --> 01:49:55.000
or something like this you can always rewrite this system as just a first order system so there's
01:49:55.000 --> 01:50:04.000
no loss of generality such a system is called homogenous if r i t if these residual terms here
01:50:04.000 --> 01:50:11.000
are equal to zero for all i's and for all t's so for all the variables one to n
01:50:12.000 --> 01:50:20.000
and for all period t's otherwise if just single r for one i or for one t is different from zero
01:50:20.000 --> 01:50:29.000
then the system is called inhomogenous now there is a general property in solving
01:50:29.000 --> 01:50:35.000
difference equations which says that the general solution of the inhomogenous
01:50:35.000 --> 01:50:41.000
equation so the solution where the r of some r is different from zero that the general solution
01:50:41.000 --> 01:50:46.000
to the inhomogenous equation is always equal to the general solution of the corresponding
01:50:46.000 --> 01:50:54.000
homogenous equation plus one particular solutions or any particular solution of the
01:50:54.000 --> 01:51:02.000
inhomogenous equation all i want to say here is currently that for the time being we can
01:51:02.000 --> 01:51:09.000
ignore the r's set them all to zero and first derive the general solution of the
01:51:09.000 --> 01:51:15.000
homogenous system of equations and then for the inhomogenous system you need to find a particular
01:51:15.000 --> 01:51:23.000
solution that's a specific special technique which we will not spend much time on and in order to
01:51:23.000 --> 01:51:30.000
find then the general solution of the inhomogenous equation as the sum of the solution the general
01:51:30.000 --> 01:51:34.000
solution of the homogenous equation plus a particular solution of the inhomogenous equation
01:51:34.000 --> 01:51:45.000
okay um the easiest thing to deal with these systems of equations is to write them in matrix
01:51:45.000 --> 01:51:53.000
notation so we would write xt which is a vector of all the x variables in periods t is equal to
01:51:53.000 --> 01:52:01.000
a matrix a which is quadratic and comprises all the small a coefficients here times the vector x
01:52:01.000 --> 01:52:08.000
in periods t minus one plus the vector of r's of the residuals in period t and the definitions
01:52:08.000 --> 01:52:17.000
of the vectors you get here x one two t two x nt gives us vector xt this is the matrix a which
01:52:17.000 --> 01:52:23.000
collects all the coefficients and this is the vector r which collects all the residual terms
01:52:23.000 --> 01:52:30.000
which we've seen on the previous slide as i said we now focus on the solution to the homogenous
01:52:30.000 --> 01:52:39.000
system so the system which has this the matrix representation xt is equal to matrix a times xt
01:52:39.000 --> 01:52:50.000
minus one it looks really simple to solve this system xt is a to xt minus one and in principle
01:52:50.000 --> 01:52:55.000
it is not but but you'll see that it's more complicated than you think when you first see
01:52:56.000 --> 01:53:04.000
this easy equation here so how can we solve such a system well we start with an informed guess
01:53:05.000 --> 01:53:15.000
so we guess that the solution to this system here takes the following form mainly there is
01:53:15.000 --> 01:53:25.000
a real number lambda some real number lambda which we have to find such that the vector xt
01:53:26.000 --> 01:53:37.000
can be written as a vector w of real values w1 w2 wn multiplied by lambda to the power of t
01:53:38.000 --> 01:53:44.000
so this product here is a scalar product essentially lambda t is a lambda to the power
01:53:44.000 --> 01:53:51.000
of t is a scalar and this here is a vector so each component in this vector is multiplied by
01:53:51.000 --> 01:53:58.000
lambda to the power of t and the guess that we make is that for some lambda and by the way it
01:53:58.000 --> 01:54:07.000
must be a non-zero lambda because it is clear that there is a solution for lambda equal to zero
01:54:07.000 --> 01:54:12.000
lambda equal to zero would give us a solution to the system here because then xt would be zero
01:54:13.000 --> 01:54:17.000
and xt minus one would be zero and then obviously that we have a trivial solution to this
01:54:18.000 --> 01:54:23.000
equation but the solution is trivial and of no interest we are looking for lambda which is
01:54:23.000 --> 01:54:32.000
different from from zero so we guess that the solution for xt is of this form vector w times
01:54:32.000 --> 01:54:38.000
scalar lambda raised to the power of t and we guess the solution because we know already that
01:54:38.000 --> 01:54:43.000
other people have solved this equation and we know that this is the correct solution as i will show
01:54:43.000 --> 01:54:54.000
you and that it is actually a general solution of this system or almost but this you would usually
01:54:54.000 --> 01:54:59.000
not know and if you now ask me why do i guess this then just because we already know it is the
01:54:59.000 --> 01:55:07.000
solution and i'll show you now what we do is that we insert the presumed solution w
01:55:08.000 --> 01:55:15.000
times lambda to the power of t in the equation for xt here and xt minus one here
01:55:16.000 --> 01:55:22.000
so we get w to the power times lambda to the power of t is equal to a times w to the power
01:55:22.000 --> 01:55:28.000
to two times lambda to the power of t minus one which is xt minus one of course this is why the
01:55:28.000 --> 01:55:34.000
lambda for xt minus one is raised only to the power of t minus one now since we know that the
01:55:34.000 --> 01:55:41.000
lambda is different from zero since the solution is not interesting we can divide this equation
01:55:41.000 --> 01:55:50.000
here by lambda to the power of t minus one then the lambda cancels here completely and then we
01:55:50.000 --> 01:56:00.000
just have one lambda so we get a matrix a times vector w minus lambda times vector w is equal to
01:56:00.000 --> 01:56:08.000
a zero vector the zero here is a zero column vector of yeah it's a column vector of zeros
01:56:08.000 --> 01:56:18.000
with n entries of course so this here we can also write as matrix i a minus lambda times the
01:56:18.000 --> 01:56:25.000
identity matrix i times w is equal to zero and you will now recognize that this is an eigenvalue
01:56:25.000 --> 01:56:33.000
problem problem so what we are looking for is all the eigenvalues of matrix a because these
01:56:33.000 --> 01:56:39.000
would be exactly those lambdas here and the w's which are the corresponding eigenvectors
01:56:40.000 --> 01:56:49.000
for a particular eigenvalue lambda here typically a n by n matrix a has
01:56:49.000 --> 01:56:59.000
n eigenvalues or let's say at most n distinct eigenvalues there may be less than n distinct
01:56:59.000 --> 01:57:06.000
eigenvalues some eigenvalues may have double or triple multiplicity so there may be that some
01:57:06.000 --> 01:57:13.000
of the eigenvalues occur more than once and then the number of distinct eigenvalues is of course
01:57:13.000 --> 01:57:21.000
lower than that but this is the principle for finding a solution find eigenvalues
01:57:21.000 --> 01:57:28.000
for this equation here and find the eigenvectors for this equation which already shows you
01:57:29.000 --> 01:57:38.000
that the solution is not a unique solution since there are typically many different eigenvalues at
01:57:38.000 --> 01:57:44.000
most n different eigenvalues for the lambdas here and even in the case of
01:57:46.000 --> 01:57:50.000
multiplicity of certain eigenvalues we would not have a unique solution
01:57:53.000 --> 01:57:54.000
okay
01:57:59.000 --> 01:58:05.000
for non-singular matrix aim there are n distinct eigenvalues lambda one lambda n
01:58:06.000 --> 01:58:14.000
and there are n distinct eigenvectors which we then call w superscript one w superscript n
01:58:15.000 --> 01:58:20.000
which then correspond w superscript one corresponds of course to eigenvalue lambda one
01:58:20.000 --> 01:58:25.000
and w superscript n corresponds to eigenvalue lambda n
01:58:25.000 --> 01:58:36.000
now these would constitute solutions of our system of our difference equation whenever
01:58:36.000 --> 01:58:42.000
i take an eigenvalue lambda one and the corresponding eigenvector then we solve
01:58:42.000 --> 01:58:48.000
this equation here and thereby it's clear we solve this equation and therefore we have solved
01:58:49.000 --> 01:58:56.000
this equation we found a solution of this equation in this form here where x t would be
01:58:56.000 --> 01:59:01.000
eigenvalue raised to the power of t times corresponding eigenvector that would be a
01:59:01.000 --> 01:59:08.000
solution of the system but it is not yet the general solution the general solution would
01:59:08.000 --> 01:59:17.000
be an expression which describes all the solutions at once so for a non-singular matrix A the general
01:59:17.000 --> 01:59:23.000
solution of the homogenous difference equation is then any linear combination of the basic
01:59:23.000 --> 01:59:29.000
solutions which i have just discussed i haven't used the term yet here but what i mean by basic
01:59:29.000 --> 01:59:35.000
solution is take one particular eigenvalue and the corresponding eigenvector and you would say
01:59:35.000 --> 01:59:42.000
this is one basic solution take a different eigenvalue say lambda n and multiply by the
01:59:42.000 --> 01:59:49.000
corresponding eigenvector that's another basic solution but all of these are just particular
01:59:49.000 --> 01:59:56.000
solutions to the to the difference equations the general solution would be a linear combination of
01:59:56.000 --> 02:00:10.000
all the basic solutions so if we set x t is equal to some c1 just any arbitrary real number c1
02:00:10.000 --> 02:00:20.000
times w1 lambda 1 to the power of t plus then any real number c2 times w2 lambda 2 to the power of t
02:00:20.000 --> 02:00:29.000
plus plus plus some arbitrary number cn times wn lambda n to the power of t
02:00:30.000 --> 02:00:38.000
then we have the general solution so this expression here for x t would also solve
02:00:38.000 --> 02:00:46.000
the difference equation the homogenous difference equation for if you just lag this expression here
02:00:46.000 --> 02:00:54.000
by one period and try then t minus one rather than t's right introduced into insert in the
02:00:54.000 --> 02:00:58.000
difference equation then you'll see that there's still a solution of the difference equation since
02:00:58.000 --> 02:01:05.000
all the basic solutions for solutions of the difference equations so we can have arbitrary
02:01:05.000 --> 02:01:12.000
real numbers which define the linear combination of the basic solutions here and these arbitrary
02:01:12.000 --> 02:01:19.000
linear numbers are c1 up to cn so we collect them in one vector c which i define here as column
02:01:19.000 --> 02:01:24.000
vector so this is why i write it as a row vector and the transpose to save base on the slide
02:01:26.000 --> 02:01:35.000
what does this mean if we have some starting value x not well x not where t is equal to zero
02:01:35.000 --> 02:01:45.000
right this would be c1 times w1 so first eigenvector plus c2 times w2 eigenvalue
02:01:45.000 --> 02:01:56.000
n vector plus plus plus cn w n the n eigenvector all the lambdas have disappeared for t equal to
02:01:56.000 --> 02:02:02.000
zero because lambda to the power of zero is one so we can't forget about the lambda when we compute
02:02:02.000 --> 02:02:11.000
the expression for x not in time period zero we can write this also in more compact notation so
02:02:11.000 --> 02:02:20.000
this equation here we can write in more compact notation as x not is equal to w times c where
02:02:20.000 --> 02:02:28.000
w is just the matrix which collects all the eigenvectors and c as i said is the vector of coefficients
02:02:28.000 --> 02:02:40.000
here so therefore we know that the coefficients c which we have here are just w inverse c times
02:02:40.000 --> 02:02:51.000
x not so just solve this equation here for c and then you get precise concrete numbers
02:02:51.000 --> 02:03:03.000
for the admissible c's namely c is the invert matrix of eigenvectors times x not that's the
02:03:03.000 --> 02:03:11.000
starting value now is this in conflict with the effect that here i said that c1 up to cn
02:03:11.000 --> 02:03:18.000
are arbitrary real numbers because here it seems there are currently specific real numbers no it's
02:03:18.000 --> 02:03:24.000
of course not in conflict because here we have added one further restriction we have added that
02:03:25.000 --> 02:03:30.000
the solution to the difference equation so the general solution we have here in equation 18
02:03:31.000 --> 02:03:39.000
that this solution shall meet some observed value x not some observed starting value
02:03:39.000 --> 02:03:50.000
if it is the case that the solution shall match the observed starting value x not in period zero
02:03:50.000 --> 02:03:57.000
then obviously we cannot take any arbitrary real numbers for c anymore but we have to take
02:03:57.000 --> 02:04:06.000
those real numbers which make sure that in period zero indeed the x not is
02:04:07.000 --> 02:04:14.000
represented by the solution that we have here and this puts particular restrictions on the value
02:04:14.000 --> 02:04:23.000
of c namely c must be equal to w inverse times x not so suddenly we have determined c values
02:04:23.000 --> 02:04:30.000
when we provide the system with a vector of starting values then the c's are not arbitrary
02:04:30.000 --> 02:04:36.000
anymore but then the c's are uniquely determined it's just one combination of the c's which then
02:04:36.000 --> 02:04:45.000
yields this x not value here you may ask do we know that matrix w is indeed invertible yes we
02:04:45.000 --> 02:04:54.000
know that the eigenvectors are always linearly independent of each other so the matrix which
02:04:54.000 --> 02:05:01.000
consists of all the eigenvectors as columns this matrix is always invertible
02:05:05.000 --> 02:05:12.000
now a solution is stable we call a solution of such a difference equation a homogenous
02:05:12.000 --> 02:05:20.000
difference equation stable if it converges to zero as t goes to infinity so if the limit for t
02:05:20.000 --> 02:05:27.000
going to infinity x t is equal to zero then we call the solution stable
02:05:30.000 --> 02:05:42.000
and this means for x t to be stable we need to either have that the value lambda i is smaller
02:05:42.000 --> 02:05:53.000
than one an absolute value or that the c j's are zero for all lambda j's which are greater or equal
02:05:53.000 --> 02:06:02.000
than one so if you look at this expression here we always have lambdas raised to the power of t
02:06:03.000 --> 02:06:08.000
and we know these lambdas explode when a lambda is greater than one absolute value
02:06:09.000 --> 02:06:12.000
so in this case the solution would go to
02:06:14.000 --> 02:06:22.000
in this case the solution would go to plus or minus infinity therefore a stable solution
02:06:22.000 --> 02:06:31.000
for x e where x t converges to zero we get when two or one of two conditions is satisfied
02:06:32.000 --> 02:06:38.000
either all the lambdas are smaller than one absolute value then lambda to the power of t
02:06:38.000 --> 02:06:46.000
for all the lambdas goes to zero and obviously the solution will be zero or we have some lambdas
02:06:46.000 --> 02:06:54.000
which are greater than one absolute value but the corresponding c is equal to zero so that the whole
02:06:54.000 --> 02:07:01.000
term here cancels from the beginning onwards and we have a lambda eigenvalue of a which is greater
02:07:01.000 --> 02:07:08.000
than one but we choose a c which is equal to zero and then obviously the eigenvalue doesn't
02:07:08.000 --> 02:07:16.000
play any role anymore so in this case we also would have stability two conditions either the
02:07:16.000 --> 02:07:21.000
eigenvalue is lambda the eigenvalue lambda is smaller than one absolute value all the
02:07:22.000 --> 02:07:27.000
c j is zero for those eigenvalues which are greater or equal to one
02:07:27.000 --> 02:07:38.000
note that some of the eigenvalues may be complexed valued so we can compute their
02:07:39.000 --> 02:07:48.000
absolute value of course now if an eigenvalue is smaller than one absolute value then the
02:07:48.000 --> 02:07:55.000
influence of the starting values which is embodied in c i decreases with increasing t obviously
02:07:55.000 --> 02:08:01.000
because the lambda to the power of t becomes smaller and smaller so that would be economically
02:08:01.000 --> 02:08:06.000
sensible to say that the influence of starting values so values long ago very distant in the past
02:08:07.000 --> 02:08:15.000
would decrease over time however if the lambdas the eigenvalues are greater or equal to one
02:08:15.000 --> 02:08:22.000
for some j then there may still be a sensible backward looking solution if the starting values
02:08:22.000 --> 02:08:30.000
ensure that the corresponding c j is equal to zero and this is a property which you may have
02:08:30.000 --> 02:08:36.000
run into or should have run into when you were introduced to settled path stability in say the
02:08:36.000 --> 02:08:42.000
analysis of the ramsey model in your dynamic macro models then you know that a way to
02:08:44.000 --> 02:08:50.000
determine initial consumption for a given capital stock the way to determine initial consumption
02:08:50.000 --> 02:08:55.000
so the starting value for consumption was the requirement that the it's stable eigenvalue of
02:08:55.000 --> 02:09:02.000
the ramsey system of difference equations that this had to cancel so that this had to have a
02:09:03.000 --> 02:09:05.000
coefficient c j equal to zero
02:09:09.000 --> 02:09:14.000
okay i won't read the exercise to you because i think you have had enough exercises into this
02:09:14.000 --> 02:09:21.000
lecture let me ask you since time is actually already over are there any questions which you
02:09:21.000 --> 02:09:27.000
would like to discuss now in interactive mode so that we leave this meeting here
02:09:27.000 --> 02:09:39.000
and go to the interactive meeting or so to the zoom meeting