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