And to tell you quickly what we do today,
which is that we now study the fiscal theory hypothesis
in a full fledged or more or less full fledged
new Keynesian model.
So we move to the framework,
which is standard in a macroeconomics
that we assume some degree of price stickiness
and that we equip the model
with a standard new Keynesian Phillips curve
and the standard new Keynesian IS curve.
And then the budget equation,
which we typically use to motivate fiscal theory,
thinking in place of a Taylor rule.
This is somewhat technical,
what I will present today,
but the technicalities are nothing new.
Actually we have covered the methodology in this lecture.
So I will not put too much emphasis
on how you deal with the equations
when I think that the principle, you know how it works.
And you just have to figure it out
by looking at the formula, which I provide on my slides,
then that should actually become clear to you
in detail at home and in principle already now in the lecture.
So how do we proceed?
Let us suppose that endowment is non-constant
as we had already had this when we covered
the non-constant real interest rate.
So the stochastic discount factors last time.
And let us suppose that instantaneous utility
is of a standard CRR type.
So we have a constant relative risk version utility here,
which is parameterized by parameter gamma.
So instantaneous utility is one over one minus gamma
times CT to the one minus gamma.
Therefore, marginal utility,
marginal instantaneous utility is just CT
to the minus gamma.
And when we use this type of utility function,
then obviously our necessary condition 109
will take the specific form that the marginal utilities
are now in here.
Usually we would have the marginal utility
of T plus one consumption here
and T consumption here.
This has negative exponent as you see.
So we get that the expectation of beta times the ratio
of the marginal utility,
therefore the marginal rate of substitution times,
well, basically the real interest rate
should be equal to one condition,
which is quite familiar to you.
I suppose we've had it many times in this lecture
and you will also have seen it in other lectures
on micro-based microeconomic models.
This equation 117 can be approximated in the normal way
by a linear approximation.
And that would then give that the log of beta
plus the expectation of gamma
times the log of the percentage deviations of consumption.
Here's CT minus C hat T plus one,
plus the nominal interest rate minus inflation
of the next year would be equal to zero.
And I use again the notation that a hat denotes
a percentage deviation from the steady state.
It is useful for later purposes
to look at this equation 117 here also in the steady state.
In the steady state,
we would suppose that consumption does not grow.
So we look at a no growth steady state
since we don't have any source of growth in this model.
Therefore, CT over CT plus one would just be one
and therefore this whole term parentheses here vanishes.
And we would also assume
that we have a zero inflation steady state.
So the price level in period T is the same
as the price level in periods T plus one.
So in the steady state, obviously everything boils down
to something very easy,
namely that beta times one plus I is one plus steady state
value of the nominal interest rate is just equal to one,
which is this relationship here.
And the Fisher equation of course tells us then
that R the real interest rate factor
is equal to the nominal interest rate factor
because inflation is zero in the steady state.
And therefore the agents don't expect any inflation
and real interest rates
and nominal interest rates are just the same.
So we know that R is equal to the inverse of beta
just as we had the very easy versions
of the fiscal theory model.
So that's the steady state relationship
here steady state value of the real interest rate factor
is equal to beta inversely.
And from this, it follows that the log
of the real interest rate factor
is equal to minus the log of beta.
And since the log of capital R
is approximately equal to R,
we know that the log of beta is approximately equal
to minus R.
So to minus the steady state real interest rate.
I use the symbol sigma here
to indicate the temporal elasticity of substitution.
And I use again that in equilibrium,
of course all output must be consumed
since we do not have any investment here.
So we will have X hat T is equal to C hat T
for all T's in equilibrium.
And therefore we can also write this condition,
the linearized condition of 117
as X T hat is equal to the expectation
of X hat T plus one minus sigma times.
And now we see here the nominal interest rate
minus the expectation of the future inflation rate.
So this is the real interest rate
minus its steady state value.
And therefore you quickly recognize
that this equation 118 is nothing else
but the new Keynesian IS curve, right?
It relates output X hat T
to the expectation of future output
and to the real interest rate
with some negative influence of course,
higher real interest rates typically leads to lower output.
The real interest rate is also written
as a deviation from its steady state level.
Okay, the standard name for this is New Keynesian IS curve.
And I like the idea which Cochrane had in his book
who says that well, IS stands for investment
is equal to savings
in sort of old fashioned Keynesian type models
but there is nothing about investment
being equal to savings in our equation 118 here
in particular since we do not have any investment.
So that's actually a very bizarre name IS curve
for this thing here in a model without investment.
He says, in fact, 118 describes in the temporal substitution.
And so he interprets IS,
IS as standing for inter temporal substitution
and that is indeed much better interpretation
of the meaning IS.
Okay, now you are probably aware
with the micro foundations
of the New Keynesian Phillips curve.
They are quite intricate and complicated.
We have typically this assumption of Calvo pricing in there
and then there is a model of monopolistic competition
with partially sticky prices where entrepreneurs
sometimes get a green light for adjusting prices
and sometimes they get a red light.
They are not allowed to adjust their prices.
Quite a funny story actually has very little to do
with reality but it has become extremely popular
in the New Keynesian modeling.
I will not spell all of that out
but rather move directly to the New Keynesian Phillips curve
assuming that you are familiar with micro foundations
of this New Keynesian Phillips curve
which after going through the micro foundations
is then derived as inflation rate of period P
is equal to beta times the expectation
of next period's inflation rate
plus some kappa times the output gap, right?
So essentially entrepreneurs are able to change
those entrepreneurs which are able to change
the current prices know that they may not be able
to raise prices in the next period.
So they try to anticipate future inflation
when they decide on today's prices.
This is the reasoning behind the New Keynesian Phillips curve.
And then of course if we have a positive output gap today
so rather high income today
then this also drives up current prices
so kappa we can safely assume to be a positive.
So we have now the two main ingredients
of New Keynesian models
or two of the three main ingredients of New Keynesian models
namely the New Keynesian IS curve
and the New Keynesian Phillips curve.
And now I do what we've done all the way through
this semester that I solve these equations
in the standard way.
In this case for the IS curve
I iterate the IS curve forward.
Have a look at the IS curve again
which is this one here.
You see this has a coefficient of one here.
So there's a unit root in equation 118.
Typically if the coefficient were greater than one
then it's clear we have to iterate forward
if it were smaller than one we would iterate.
No, sorry if it is greater than one
we would iterate backward
and if it were smaller than one we would iterate forward.
Here we have the borderline case
where the coefficient is equal to one.
And I just use the sort of degree of freedom I have
in that by iterating forward.
So I just replace X hat T plus one in the next line
by precisely the same expression again.
And then I get this line here
and I then substitute out one more time.
And I think actually I was just a little bit confused
because I think this is a mistake here.
This should be ET plus one of X hat T plus two
of course not X hat T plus one.
Otherwise obviously this thing would not have vanished
but we do this over time.
And then we arrive at this expression here
provided that this infinite sum converges.
What does this give us?
It tells us that the output gap or output itself
is below its steady state level.
So the output gap is negative
or output is below its steady state level
when current and future interest rates, real interest rates.
So this here is the real interest rate
and this is its steady state level.
When current and future real interest rates
are expected to exceed the steady state interest rate
on average, right?
So that's basically here an average
of all future real interest rates.
And when we expect them to be positive
or when we expect this average to be positive
then we would have that output is below
its steady state level at the converse is true
if this difference here is negative.
This means now that we have endogenized
output production if you like.
We do not have an exogenous fluctuations anymore
in endowment as we had it in the theory model
but rather we have output determined endogenously
and actually we have determined it endogenously
from the demand side.
So very much in the Keynesian tradition
because you can see in this dependency here
on the real interest rates
something like destined demand, right?
Which is bigger when the real interest rate is small
and which is smaller when the real interest rate is great.
So it's a demand determined way
in the Keynesian tradition to endogenize output.
As I have already said, R is not the real interest rate
because in this model the real interest rate
is not constant anymore
but R is just the steady state level
of the normal interest rate
provided that inflation is stationary
with a steady state value of zero.
The true real interest rate is the expectation
of IT plus tau minus inflation rate of T plus one plus tau
or that is the expectation in previous T
of a future real interest rate.
I could of course introduce for this year
a different symbol like writing RT
which would just be an invitational matter.
Actually it would mean that we add a Fisher equation
and define the real interest rate by the Fisher equation
but since this would just be the definition of a new symbol
it wouldn't add any economics to the model
and therefore we can just leave the model as a
well so far two equation model
and we will add one more equation
which is the budget equation of the government
but we do not need a Fisher equation actually
it would just serve us as defining a new variable RT
and as I say, we see this immediately
that the series is RT that we don't really need this.
Of course, we could also add a Taylor rule
but what we have learned in this lecture
is that this rule is not very informative
because we run into the problem of observational equivalence.
So the reasons why a central bank
conducts a certain monetary policies
or changes monetary policy parameters
like the nominal interest rate in a certain way
these reasons are probably just not identifiable.
There's observational equivalence
and therefore the Taylor rule also does not add
to our understanding of the model.
Rather what we will do in line
with what we have already done in previous lectures
is that we assume that the nominal interest rate
is an exogenous variable.
So it is a policy choice of the central bank
and we have no clue how the central bank sets policy rate
it is unexplainable from our point of view
what the central bank does there
and we just take it as given
and therefore as an exogenous variable.
So the nominal interest rate is not endogenous
but it is exogenous now since we have dispensed
with a Taylor rule.
We have iterated the new Keynesian IS curve forward.
We do the same thing now with the new Keynesian Phillips
curve which as you see here is beta T time is equal
to beta which is smaller than one times the expectation
of pi T plus one, sorry, not PT but pi T
is equal to beta times expectation of pi T plus one
plus then kappa times the output gap.
So we can again here substitute out the pi T plus one
by inserting this equation here
pushed forward in time by one period
and then we get this expression here
and if we do this infinitely often
then we arrive at the expression that IT today's inflation
rate is equal to kappa times the expectation of well
if you like discounted future output gaps.
So inflation is high when current
and future output gaps are expected to be high
and therefore we now also have endogenous inflation
from a goods market perspective,
namely from the perspective of what sign the output gap has.
This is now sort of a competitor to the endogenous
to the endogeneity of inflation in the fiscal theory model
because in the fiscal theory model
we had determined inflation from an expectation
of future fiscal surpluses essentially.
Here we don't do this, here we determine inflation
from an expectation of future production
of future output gaps, right?
So at this point the model is truly a new Keynesian model
and there is nothing about the fiscal theory in here.
We use the new Keynesian Phillips curve
to determine inflation.
But what is interesting of course is now of course
what happens when we merge the two explanations
when we also include in our model a third equation
which says that inflation may respond to current
and future expected fiscal surpluses.
And that is what we will do.
So we add now the government budget equation
as a third dynamic equation to a model
which otherwise so far is completely a new Keynesian model.
We assume that in the steady state inflation is zero
as I have already pointed out.
So the steady state relationship
of the government budget equation
is simply this relationship here, B over P,
so debt over price levels,
so real debt in the steady state is equal to surpluses
plus the price of debt times newly issued debt
that it was, right?
But in the steady state of course we assume
that old debt is the same thing as new debt
or the time indices here are missing.
And therefore we get this relationship in the steady state
which of course we can immediately solve
for the steady state value of surpluses.
The steady state value of surpluses is then R minus one
over R times the real value of steady state debt.
Now we are of course interested in the dynamics
of the model so we need to go to the quantities
which carry time index.
And these we would define here slightly differently
from what we have done previously
as BT being equal to the logarithm of real debt in period T.
So B time T is now the log of real debt in period T
and pi T is of course approximately equal
to the log of PT over PT minus one
and normal interest rate as we know is approximately equal
to negative the log of QT.
So when I use this notation here
then the budget equation of the government
which is equation 45 is approximately equal
to equation 121 here which I here still write in levels
so I take E to the power of the logarithms
and then we get the level variable of course here
which would be real debt in period T minus one
and here we would have the inflation rates
or the ratio of the two,
not the inflation rate levels
it would be the ratio of the price level in period T
over the price level in period T minus one
or actually the inverse of that would be equal
to ST plus E to the negative IT times E to the BT
so certain logarithm of real debt in period T.
Now we would also for the logarithms
have certain state levels in the zero inflation
as steady state which we would then denote as B for BT
state level of inflation is zero as I have already said
steady state level of the nominal interest rate
is equal to the steady state level
of the real interest rates since inflation is zero
and S is of course the steady state level of surpluses.
This equation here is highly nonlinear as you see
in the logs of real debt
so we take a linear approximation to 121 at this steady state
and that would then be of this form here
which you may verify at home.
From this representation here
which is just the first Oratina approximation
of equation 121 we may define again percentage deviations
from the steady state so B hat T would be BT minus B
for instance and as I had T I would define IT minus R
this is not a percentage deviation
but deviation in percentage points
but that is of secondary importance.
The important thing is that again we have a variable
which is measured in percentages.
So that's the deviation of the nominal interest rate
from its steady state level which happens to be the real rate
and S hat T I defined a little differently
or actually Cochrane came up with this idea
which is quite practical because S hat T is now defined
as the deviation of surplus from the steady state
so SD minus S but in relationship
to the steady state level of real debt
so divided by the steady state level of real debt
defining S hat T in this way
by dividing through the steady state level of real debt
is very practical because then we can approximate equation 45
so the government budget equation
by this expression here in equation 122
and you see that this equation here is completely free
of any steady state values.
This is why we have divided through by E to the B here
to get rid of all the steady state levels
we just have the deviations now of the steady state
in this equation here.
I write pi T here rather than pi T hat
because pi T and pi T hat is the same thing, right?
The steady state level of inflation was zero
so pi T is the same thing as pi T minus its steady state
and we don't really need a hat
but if you want to you can also think of pi T
as a deviation from the steady state.
So with this equation 122, the model is completed
so in its linearized form we have the new Keynesian IS curve
here in the way we have derived it
we have the new Keynesian Phillips curve here
and we spoke about that
and this here is now the government budget equation
in linearized form.
What we can do is that we solve equation 119
for the expectation of the future inflation rate
and then substitute in into equation 118
so we would get equation 123 here
and equation 119 stays virtually the same as before
as before just has been solved
for the expectation of pi T plus one.
The nice thing about this is now
that we have two equations here
which represent exactly the same two equations as here
but they have been written in such a way
that the period T plus one variable
is always on the left hand side
as an expectation being taken in period T
and then on the right hand side
we have the period T variables.
So that's sort of a standard form
to write this type of difference equations.
Of course 123 and 119 are not yet the full model
we also need to pay respect to equation 122
which we have used so far
but those two equations here I can write
in somewhat nicer form without changing anything
actually as equations 123 and 119.
Now you know that the expectations of a future variable
can always be written as the actual value
of the future variable plus some expectation error.
So the expectation error with respect to X
I denote as delta X T plus one
and that would be defined as X hat T plus one
minus the expectation of this variable.
And similarly for the inflation rate
we would have the expectation error here
delta T plus one pi is equal to pi T plus one
minus the expectation period T of pi T plus one.
We know that these errors here
these expectation and errors
are actually functions of the structural errors
in the model and one of the difficulties
in solving the expectation difference equations
is that we have to compute how the expectation errors
depend on the other type of truly structural errors.
So how the structural errors actually
give rise to expectation errors.
Okay, with this additional notation here
we write the system in matrix notation
as X hat T plus one.
So the expectation error
the expectation operator is gone here
B hat T plus one and B hat T
and then when you go through the matrices
which I have set up here
you see that this is a simple autoregression actually.
Here we have the same vector as we had it here
just lagged in time by one period
with this kind of matrix here
which describes the autocorrelation of the process.
And then we have the two shocks in our system.
The shocks are the interest rate and the surplus.
Both of them are exogenous.
We have surplus shocks and we have interest rate shocks
and they drive this system here.
So we have two shocks for three variables
and the way the system depends on the shocks
is described by this matrix here.
Plus we have the expectation of errors
which affect the first two equations
but not the third one.
And the third one there is no expectation error involved.
So this is a matrix valued representation
of the model we currently have.
And you know from the way we worked through the model
that here we have the Keynesian IS curve
and here we have the new Keynesian Phillips curve
and this is the fiscal theories budget equation.
Actually it is a usual budget equation
but the question is always
how do you interpret the equation?
As you know, we interpret it in the fiscal theory way
because we assume that the surpluses
are exogenously determined.
They do not adjust in response say to inflation
and output and the interest rate
but they are an exogenous process
in line with fiscal theory assumptions.
Now when you look at this matrix representation
of our model here, you may think that it is useful
to solve the model in two steps.
And the first step would be to first solve
just the first two equations
because it seems that the first two equations
are decoupled from the third equation.
Why is that so?
Why does it seem so?
Well, because here we have these two zeros.
So essentially in the first two equations
for x hat and pi hat, the coefficients
which are in this matrix here
only relate to the light values x hat t and pi hat t
but they do not relate to the third equation
to the variable of third equation
which is the b hat t or b hat t minus one.
This is because of those zeros back here.
And for the shocks, we also see that only
the interest rate shock plays a role
for the first two equations.
This is multiplied by sigma for the first equation.
Actually, it is not used at all
for the second equation here.
And the circular shock doesn't play a role
for the first two equations because again,
we have zeros here.
And then we have the two expectation errors.
So you might think now that the system is decoupled
and then we can simply determine x hat and pi hat
in this system in the way in which we do this
in new Keynesian models.
However, this is not the case.
And we will try to understand why this decoupling
which seems to be suggested by 124
does actually not take place.
So let us just try to solve the first two equations
without paying respect to equation three.
Perhaps one more verb on equation three.
Equation three is tied to equation one and two
by means of this negative R here.
We see this negative R is multiplied by pi hat t
which is a variable of the first two equations.
Therefore, we may think of b hat t
as some kind of appended equation
which we can of course easily solve
when we have a solution for pi hat t.
So we would use the first two equations here
to compute the solution for x hat t and for pi hat t.
And once we have the solution,
at least the solution for pi hat t,
we can use it in this process here
because we see that the negative R
would be multiplied with pi hat t.
And when we have the solution for pi hat t
then this makes our life here easier
and we can solve equation three.
Okay, so for this reason,
let us just first look at the first two equations
and they are just as written here in equation 125.
Now, if you recall how we solved
matrix difference equations.
We had a standard form which I write down here again
as z t plus one is equal to a times z t
plus b times epsilon t plus one
plus c times delta, so the expectation errors,
t plus one plus d times omega t.
The z t's are of course the endogenous variables
and they are being explained by their own past
and by the exogenous structural errors.
So in our case, it would be I t and S t
and by expectation errors.
And then this approach also allows
for exogenous forcing errors.
Omega t, which we just set to zero
if there are no such forcing variables.
I mentioned already that the expectation errors
delta are endogenous and typically depend
on the shocks epsilon
and if we have forcing variables omega
then they will usually also depend on the omega.
Now, the usual way to solve such a system
is that we compute the German decomposition
of this matrix A.
So what is a Jordan decomposition?
Suppose that A is some real quadratic matrix
of M by M, then we have M distinct,
oh no, and suppose the matrix has M distinct eigenvalues.
So the eigenvalues are all different,
which need not be the case for M by M real matrix.
But suppose there, then we can write A as T inverse C,
lambda T where lambda is just the matrix
which has on its main diagonal all those eigenvalues here
and the off diagonal elements, there's nothing.
And T is a non-singular matrix whose columns
are the corresponding eigenvectors of A.
So what we need to do is we need to compute
the eigenvectors of A and we need,
sorry, the eigenvalues of A
and we need to find the eigenvectors
because then we can put the eigenvectors
in this matrix T here and then we'll find
that A is equal to T inverse times lambda times T.
That's actually a standard linear algebra
but it is very useful as you know for solving the model.
Now we have here a case of just a two by two matrix.
So it's rather easy even to solve by hand
for the eigenvalue.
So look at our matrix A which has this particular form here
where we know that sigma and kappa parameters
greater than or equal to zero.
And now try to show that lambda two is smaller than one
and lambda one is greater than R
which in turn is greater than one.
So we have one eigenvalue which is greater than one
and we have one eigenvalue which is less than or equal one.
Now in principle, there's a borderline case
of lambda two being equal to one.
We can only show or you are supposed to show
this exercise that lambda two is less than or equal to one.
So it could in principle also be one
but we ignore this case and just study the main case
where lambda two is strictly smaller than one
and lambda one is of course strictly greater than one.
Under the Jordan decomposition which I just explained
we would have ZT plus one is equal to T inverse times lambda
times T times ZT plus C delta T plus one
plus D omega E if we have forcing variables.
Now what can we do?
We can pre-multiply this equation 126 here by a matrix T
and define that Z tilde T is equal to T times ZT.
Well, let us look what Z tilde T would look like in this case.
I do it here for the variable pushed forward
in time by one period.
Z tilde T plus one is then equal to lambda Z tilde T
because you see here is T times ZT
which gives me T tilde
and I have pre-multiply this whole equation by T,
by T inverse E, sorry, by T,
by T so that I have Z tilde T plus one here too
and this T inverse E of course vanishes.
So we get Z tilde T plus one is equal to lambda Z tilde T
plus T times well, C times the expectation errors.
So this has just been multiplied by T here
plus T times D times omega.
So omega is not really an important variable here
and just for the sake of completeness
so that you can see how these ones are solved.
I carry it with me in this equation here.
I now just define new quantities
and call these quantities here Z T plus one
tilde is equal to lambda Z T, Z T
and then delta T plus one tilde and omega tilde T
defined as these two terms here
after having been pre-multiplied by T.
Now, why is this helpful?
This is helpful because we know
that lambda is a diagonal matrix.
So now we have in fact decoupled equation one
from equation two or regardless of how many equations
were in the system which we have just analyzed,
lambda being a diagonal matrix would tell us
that each variable just depends on its own leg.
Each variable in Z to T plus one depends just
on its own leg and not on the legs of other variables.
This means that we can now solve each equation separately.
So take the i-th equation of 127
then we would have Z tilde i in period T plus one.
So just the i-th component of this vector here
is equal to the eigenvalue lambda i times Z
and there's an i missing here,
Z tilde i T plus delta tilde i T plus one
plus omega tilde i T.
So that's already the second typo I find in this lecture.
This must be delta i, sorry here,
this must be Z i tilde T and not Z tilde.
If the lambda i is greater than one,
we can write equation 128 in this form here
that Z tilde T is equal to one over lambda i
times the future value of Z i tilde T plus one.
So we can solve into the future
and now by iterating forward.
And that was the way we always used to nail down
the expectation errors and see how the expectation errors
depend on the structural errors of the system.
Well, here we have Z tilde T is obviously equal
to the expectation in period T of Z tilde T
because in period T we know already the true value
of Z tilde T and we also have that the expectation
of the expectation error is equal to zero
when the expectation error occurs in the future
because by definition we don't know
what the expectation error would be in the future.
So this allows us to iterate now in the following form
Z tilde T is one over lambda i times E t,
Z i tilde T plus one, sorry, minus omega tilde i t
and then using successive substitutions,
this leads to Z tilde T being equal to the negative
of the infinite sum from tau running from zero to infinity
over one divided by lambda i one plus tau
times the expectation period T of omega tilde i,
t plus tau.
This solution we can substitute into equation 129.
So perhaps I can see it again.
Here is equation 129.
We have now substituted this equation
or we have computed this here, Z tilde T
as this expression here.
So I can substitute it in here
and I can substitute it in here.
And my goodness, again, I see missing subscript i here.
This should of course also be i
and then I should also add it here and there.
So all the Z tilde T's apparently miss their subscript i
but that's just a typo
and does not affect the validity of the equations
I'm showing to you.
So I substitute in now this expression here
for the Z i tilde T and the Z i tilde T plus one.
And this looks a little messy.
What we have here, these are these kinds of things
which I don't wanna go through in all detail
because you can just verify at home that is correct
and probably it's not correct
with respect to some subscript which I have left out.
Well, perhaps I haven't.
In any case, if you do this,
you will arrive at that infinite sum over one
over lambda i tau plus one to the power of tau plus one
times the expectation of future omega's
is equal to the delta i tilde T plus one
divided by lambda i plus here discounted sum
of expected future terms omega i t plus tau.
So that's all just technique
and I encourage you to really follow this technique at home.
But what we see here is that the delta i T plus one
tilde is then obviously a linear or the linear combination
of the expectation of errors which is in here
is determined by this expression here in infinite sum
over one over lambda i to the power of tau
minus the expectation of errors
with respect to the omega here.
So we have one equation currently
which determines linear combination
of the expectation of errors.
That is not yet enough to determine both expectation of errors.
One equation we have, but we do not have a second one.
So the solution to the first two equations
in our three equation model
pins down the expectation of errors
or better their linear combination.
But since there are two expectation of errors
in this model which are related to the delta T tilde
by means of the matrices T, C
and then the delta tilde errors here,
we need now since there are just two equations
we need now another equation
to pin down both expectation of errors.
And this is why I say that the system is not decoupled
in the way in which one might think it is
because if we just look at the first two equations
in the system here and solve them
then we see that we cannot uniquely determine
the two expectation errors.
That doesn't work.
We can just determine a linear combination of the two
but not a single one.
Okay.
Yeah.
So we can pin down just one expectation error
it would need a second or in this case a third equation
to also lay down the other expectation of error
or we have two equations
for nailing down two expectation errors.
Well, you may say we still have a second equation
but the second equation has an eigenvalue smaller than one.
And therefore the corresponding equation
must be solved backwards.
See the problem was that for this matrix here
we had one eigenvalue smaller than one
and one eigenvalue greater than one.
So when we solve just this system here
we knew only one equation can be solved forward
because the eigenvalue has the right size
and the other must be solved backwards.
We have the property of saddle path stability here
but this does not help us nailing down
both expectation errors of this model.
Where am I going?
Okay.
And now the second system has an eigenvalue smaller than one.
So for this equation, we must solve backwards.
And here it is again, I forgot the index here.
This is now for some component J.
ZJ two in T plus one is equal to lambda J times Z.
Well, JT plus the delta terms with respect to J
and omega terms with respect to J.
And of course this equation we can again
solve by successive substitutions.
So until we reach this infinite sum here
which in this case would be lambda J
to the power of T times the sum of theta's and omega's.
So here the variable, this artificial
artificially combined variable of ZJ to T
depends on lagged forcing variables
and on current and lagged expectation errors.
Do we know these errors here, these expectation errors?
Well, most of them we do know
and then it is not a problem.
Then we would say the ZJ T tilde is well determined.
So here in the sum, we have the expectation errors
of the past because tau starts at one.
So the expectation errors in here start at T minus one
and then comes T minus two, T minus three.
So these are past expectation errors
which we know from the data,
which we know from our experience or wherever from.
These are things which are printed in principle
in the information set.
But as you see, the Z to JT
also depends on the current expectation error.
And this expectation error, we do not yet know.
So the equation 133 does not determine ZJ to the T
because it depends on some expectation error
which is unknown to us.
In fact, this could here be anything.
And if people believe that this is the expectation error
then actually we would have a rational expectation
which also justifies this expectation error here.
So this could be completely weird event
like say the expectation of sunspots occurring
which is why these equilibria
are typically called sunspot equilibria.
So even if people expected something completely irrelevant
like for instance, sunspots
which most people today would say are completely irrelevant
for what happens economically on earth,
this would still give us a rational expectations
equilibrium solution for the model.
But it would be a rational expectations equilibrium solution
which leaves ZJ to the T undetermined.
And since ZJ to the T is,
or perhaps I should first explain,
it is undetermined because here as the shock
we could have just anything.
So we can also have as a rational expectation solution
just anything as this Z to the JT here.
And since this is actually a combination,
a linear combination between two structural variables
in the model, we would come to the conclusion
that the whole model solution
for the two equation model we've been dealing with
is not determined.
To cut things short here,
what we see is that the fact that one eigenvalue
is smaller than one implies
that there are infinitely many solutions to the model.
So equation, the system of equations in 132
to which I will go back now is not a well-defined model
because it doesn't determine inflation and output.
Look at 132 again.
Oh, where was it?
Oh, there.
No, that's actually, yeah, that's also 132.
But I want to show you this actually in here.
No, in better even, well, okay.
In here, in equation 125 is probably better to use
for showing to you what happens.
125 is a system consisting of two equations
and two expectational errors, right, which are in here.
Or if you wish to put it differently,
125 is a system of two expectational difference equations
in the sense that the expectation of the future variable
comes up in one equation and here also actually
in the second equation,
let's say in a second variable,
which are output and inflation.
Now we can solve a system
with an expectational difference equation uniquely
only if we can solve forward
because only then we can nail down the expectation error.
You can determine uniquely what the expectation error is
and how it depends on the structural shocks of the system.
So for two expectational equations,
we would actually need two eigenvalues greater than one.
Then we would have a well-defined solution 225.
But as things play out in this model,
with this A matrix here,
we have one eigenvalue smaller than one
and another eigenvalue greater than one.
So we can nail down just one of the expectation errors
and not the others.
Right, we can just nail down one of the linear combinations
of expectation errors.
That doesn't matter what exactly we can or cannot nail down.
Important is that there is something left over
which is not nailed down at which we cannot nail down
by just looking at the solution of those two equations here
because as I said, there are two expectation equations
and just one eigenvalue greater than one.
So the model here suffers from indeterminacy
and does not uniquely determine us output and inflation
or the deviations from the steady state.
So that's the problem with this equation here.
It doesn't determine inflation and output.
So we do need to have a third equation
to determine inflation.
And which equation is that?
Well, in the New Keynesian model,
so the standard New Keynesian modeling approach,
this would be a Taylor rule.
The Taylor rule with coefficient for inflation
which corresponds to the Taylor principle
namely a coefficient greater than one,
which again corresponds in some sense
to eigenvalue greater than one.
In the standard New Keynesian approach,
this would be the Taylor rule
which determines inflation,
which completes the model in such a way
that we actually have equilibrium determinacy
that all the variables are determined.
So the Taylor rule is some hypothesis
which we make about the behavior of central banks.
And we have learned in this lecture
that this is not an innocuous assumption, this Taylor rule,
but rather it introduces a problem
of non-identifiability to our model.
And we don't actually know what the Taylor rule
helps us in determining when we cannot identify
what the Taylor rule does or does not do.
So what we do now is that rather than using a Taylor rule,
we use the budget equation of the government
as the third equation, which means that we move
from the standard New Keynesian approach
to the fiscal theory approach.
So in the FDPL approach, we would use equation 122
as the third equation,
which then helps us pin down the second expectation error.
And when you look at equation 122,
then you see there is b hat t here
and b hat t minus one here with coefficient of r
and r is greater than one
because it is the real interest rate factor.
So you may think that this equation,
you must actually be solved forward
because well, this here would suggest
a backward looking mechanism,
but with a coefficient which is greater than one,
we actually have to divide by r
and then we have to solve forward.
This judgment that this equation here
requires a forward looking solution
is however premature because we do not yet know
how the pi t here affects the dynamics of this equation 122
but t is also an endogenous variable
and also is affected by other influences,
which we have already dealt with
when we were looking at these systems here
or actually this system here,
this pi t plus one in here.
So pi t plus one is determined
by things which are beyond our reach
or perhaps beyond our reach,
but by other influences where we first,
we'd have to see how this affects the coefficient
which stands here in front of the b hat t minus one.
So what we do need to do is actually
that we really look at the full system.
We cannot just look at the third equation
and solve it on its own.
And this means that we have to compute the eigenvalues
of the full three-dimensional system 124.
That is somewhat hard to do just symbolically.
So I took specific parameter values here.
So for instance, I assumed a real interest rate of 1%.
So the real interest rate factor
is 1.01 and elasticity of substitution sigma equal to one
and kappa, the coefficient of the output gap
in the Phillips curve would be 0.25.
So I've basically taken these values out of thin air,
but I will in a minute make the argument
that for a very wide range of parameterizations,
we find qualitatively the same results.
So if I use these values here
and compute the eigenvalues of matrix A,
then I get eigenvalues lambda one is equal to 1.01
and lambda two is 0.61 and lambda three is 1.65.
So we have now two eigenvalues which are greater than one,
one here and one there.
And therefore we know we have enough eigenvalues
to pin down the two expectation errors
and explain them uniquely.
So the indeterminacy of the model is gone, right?
So we have a well-defined solution for our model now
which determines inflation and determines output
and determines the three, the two expectation errors.
The price level is of course determined
by the fiscal policy of the government
because we have now used a fiscal theory assumption here
where surpluses are exogenous
and the budget equation of the government
is the third equation of the model
replacing the Taylor rule.
Now suppose, or I don't go
and I hadn't even prepared it on these slides
as you see into the full solution of the model.
You can compute this along the lines of the theory
that I've just presented
by taking the Jordan decomposition and that.
So iterating and coming to your moving average
representations of the endogenous variables.
If we do all that and parameterize the model
with certain parameter values
like the ones I have just already discussed
then we can study the impulse responses again of the model.
So suppose we are in a situation
where the central bank intends to fight inflation
by raising the nominal interest rate
like we have seen or know that the central bank
thinks it is useful policy to do
when inflation seems to be out of control.
So for instance, the central bank would use
of market operations to buy government bonds.
No, in this case, they want to raise the interest rate
so they would sell government bonds.
Yeah, perhaps I just show you what happens.
So our shock is a shock in the nominal interest rate
because we have said the nominal interest rate
is the exogenous policy instrument.
That is sort of a shock like
the other settings surpluses are shock.
So IT is not an endogenous variable
but an exogenous variable
and the central bank raises IT
because they want to fight inflation.
So what happens with this?
Well, first thing you have to note is that
by raising the nominal interest rate,
output will go down
and it will stay lower than its state value
for a protected period of time, as you can see here.
And even at five periods time,
it is not quite back to normal.
Actually, it will never, yeah,
it will come back to normal
but it will never come, no, sorry, I should press this way.
Actually, it will come back to normal
just for one brief instant in time
and will end up here interestingly
with a very slight positive effect.
So this black curve here does not go to zero
as you may expect
but it goes to something positive.
And we'll explain later why this is the case.
It's actually not terribly important for two reasons.
One reason is that this effect is really small.
The other reason is it is not robust
to changes in the model.
So we should not care much about it.
Anyway, we have a negative reaction of output
which is exactly what you would expect
from the new Keynesian IS curve
because we raised the nominal interest rate
for a given expectations of inflation.
That would mean that the real interest rate also goes up
and actually we see this here.
Here's the real interest rate
with an increase in the real interest rate,
of course, output falls by means of the IS equation.
Let us look at this a little bit more precisely.
Why does the real interest rate go up
as much as we see it here?
Well, because there is the reaction of pi t
and pi t is this red line here.
Not yet the dotted line, this comes later.
This red line here gives us the reaction of pi t.
Or you see pi t adjusts in such a way that it, yeah,
doesn't adjust immediately, adjust in a staggered way
which is due to the fact that we have a new Keynesian model
here where we have some degree of price stickiness.
So not all entrepreneurs are allowed
to adjust their prices all the time.
Some have to stick with their old prices.
Therefore, there will be no inflation for them
or they won't be able to raise their prices.
But over time, everybody gets a chance to adjust prices
and then price just due to a new level.
What is interesting here is that again,
we have the Fisherian property.
The central bank raises the interest rates
in order to fight inflation.
But what happens is actually that inflation goes up
rather than going down.
So the effect here is exactly the opposite
of what the central bank had in mind.
So we see that when the central bank raises interest rates,
the model responses in this mixture of a new Keynesian
and the fiscal theory model are basically the opposite
of what the central bank would like to achieve.
Namely, they would achieve a temporary increase
of the real interest rate,
which is probably not the objective of the central bank.
They would achieve a temporary decrease in output,
something which very likely the central bank
would not like to have.
And they would achieve a permanent increase in inflation,
which also the central bank doesn't want to have.
So the responses are just the opposite
of what we would think a central bank tends to achieve.
Now, we can play with the parameter kappa here.
And you see that the speed by which inflation
adjusts depends on kappa.
That's not so very interesting actually.
Basically, it means that when inflation depends less
on output, so when kappa is rather small,
then the initial response of inflation is faster
because this negative output effect here
is placed less of a role for inflation.
And therefore we have a speed year adjustment of inflation,
but it's still the case that the adjustment is not complete
after the first period because of price stickiness.
And it is still the case that we have the Fisherine property
in here because we see that inflation rises
with the interest rate rather than who fall
with the interest rate.
In the long run, the increase in the inflation rate
matches exactly the increase in the normal interest rate
so that the long run effect of the real rate is zero, right?
So this is why the green line here goes back to zero
and that really goes back to exactly zero
since inflation and the nominal interest rate
move to the same increase.
I think the rest of this year,
well, perhaps I should still mention
that the sluggish response in inflation
is of course expected by the agents
because the agents are rational
and they know there is caliber pricing
and they know that a certain share of the entrepreneurs
is unable to change its price each semester.
So the rationality assumption guarantees
that agents actually expect the sluggish response
in inflation.
And this means that the real interest rate increases
and then subsequently decreases
during the adjustment process.
Now, when you read Cochrane,
then you read that he considers it a disappointment
that the model does not have this anti-Faschirian property
so that the model again yields the result
that an increase in the nominal interest rate
increases inflation.
I'm not sure it really is a disappointment.
First of all, it's just a result.
And we have to check or we have to cross check
with empirical data whether this result makes sense
or does not make sense.
As I had pointed out in the previous lecture,
the correlation of at least the ECB's main refinancing rate
and the inflation rate in the Eurozone is positive.
Clearly smaller than one,
the correlation is clearly smaller than one,
but it also was clearly positive.
This correlation.
And therefore the model produces exactly the result
or produces pretty much not exactly,
produces pretty much the result
which we see in the Eurozone.
Here is the graph again,
which you are already familiar with.
You see the ECB's main refinancing rate
and you see ECB's inflation in red here.
And basically there is a longer downward trend
and rather a positive correlation between interest rate
and between nominal interest rate and the inflation rate.
Also note that in this case here,
we really look at the policy instrument,
the ECB's main refinancing rate.
And this is very much aligned with the fact
that we had interpreted our IT now as the policy, right?
As the exogenous instrument the ECB is using.
Two more comments on the model responses.
The inflation response is now slower
than in previous models we have studied.
In previous models, there was just the lag of one period
and then inflation was where the interest rate was.
Therefore the inflation response is actually more realistic
because we know that inflation moves rather slowly.
And second, note that there is an instantaneous response
of inflation to arise in the nominal rate
as opposed to this one period lag, which we had had before.
So also in this respect,
the model is probably more realistic.
This one period lag was sort of not too convincing
that inflation could move only say after a year
or something like this if you choose periods to be a year.
Here we have the possibility
of an instantaneous response of inflation
to arise in the nominal interest rate.
But we know inflation is somewhat sluggish
and will adjust only over time to the higher value of inflation.
Now, why is there an instantaneous response of inflation?
In this model, that's actually a fairly complicated mechanism
which works through what is called the discount rate channel.
Namely, sticky prices or sticky price assumption
cause the real interest rate to rise temporarily.
Normal interest rate has a reason
but inflation can't move up as quickly
because we have sticky prices.
So this means that the real discount factor arises
because we have a higher real interest rate.
But a higher discount rate
decreases the present value of government surpluses.
And for some reason I have written here,
increase decreases, that's of course nonsense.
Please delete the increase here.
Oh no, sorry, it's correct.
The discount rate increase,
that's the way you have to read it.
The increase in the discount rate
decreases the present value of government surplus.
So that's fine, no type of thing.
Decrease the present value of government surpluses
and that we know from our thinking in the fiscal theory world,
if the present value of government surpluses goes down,
then we need to also have the present value of existing debt,
real debt go down,
which means that we need to have a rise in current inflation.
And that's how the instantaneous response
of inflation comes by.
It is essentially a fiscal theory effect
that inflation responds instantaneously
rather than only in the next period.
So intuitively, the higher debt service costs
from higher real interest rates
add here to the fiscal burden.
And this then of course has the same effect
on inflation as decline in surpluses.
Good, what we see in this model again
is that monetary policy can have indirect effects
on fiscal policy.
So there is again, monetary fiscal policy interaction.
The two spheres cannot be separated from each other
while the central bank is of course
not able to directly conduct fiscal policy.
It has indirect means of impacting
from the present value of expected surpluses.
The fiscal theory part of the model
have induced the surprising result
that decrease in output
does not induce a decrease in inflation,
which is the usual mechanism in a New Keynesian model
via the Phillips curve.
But here it is not the case, right?
That a decrease in output leads to a decrease in inflation
as you have seen from the impulse responses here.
Inflation is going down, sorry, output is going down
and inflation is going up.
So inflation and output move counter cyclically,
which actually I think is a weakness of this model here.
Why is that so?
Well, we still have the conventional reaction
which we know from the New Keynesian Phillips curve,
sorry, from the New Keynesian IS curve first
that an increase in the nominal rate
increases the real rate due to price tikiness
and therefore decreases output via the IS curve.
So this conventional reaction is still there.
But the decrease in output
does not decrease current inflation
via the New Keynesian Phillips curve
because inflationary expectations now rise by more
than the decrease of output.
So we have here the New Keynesian Phillips curve
and we have here decrease in output,
but it happens to be the case that in the model,
the expectation of future inflation rises
because the present value of the surpluses
in the budget equation of the government falls.
And that's the fiscal theory effect.
So this thing here goes up when this thing goes down
and the rise in expected inflation is greater
than the decrease in output
both multiplied by the coefficients kappa and beta, right?
So that the net effect on inflation is actually positive
rather than negative.
Now, last point here, as I have pointed out,
there is a tiny permanent effect on output,
which is well, unusual because typically
nominal changes, changes in monetary policy
do not cause permanent effects on real output.
So why is this so?
Well, actually this effect is due
to some long run output inflation effect
which we have embodied in the New Keynesian Phillips curve.
And you can see this most clearly by looking at its long run
that is to say at its steady state form.
The steady state form of the New Keynesian Phillips curve,
we have X hat is equal to one minus beta
over kappa times pi.
And since pi has risen permanently here,
there was a permanent increase in pi
due to a permanent increase in the nominal interest rate.
We have a permanently positive output effect,
which however is small
because it is multiplied by one minus beta
and beta as you know is quite close to one.
So much of the 1% increase in inflation
or much of a 1% increase in inflation
would carry over to an increase in output
to a permanent increase in output.
Still, you may say, well, why not?
There is something positive about it.
We have permanent effects of nominal policies here.
But there's a second drawback
because the effect is not robust to changes in the model.
For instance, if you suppose that current inflation
also depends on lagged inflation.
For instance, because there are laws in the country
which index wages or prices or rents
with respect to inflation.
So that current inflation would be dependent
on lagged inflation.
Then the UK and Philips curve would look like this.
Pi t is equal to theta times pi t minus one.
So this is the effect of lagged inflation
on current inflation, possibly due to indexation
plus one minus theta,
the expectation of future inflation plus kappa X hat t.
So this here is the part of the New Keynes and Philips curve
that we have dealt with so far in this model.
And if we have this version of the New Keynes
and Philips curves,
then the long run effect on output would be equal to zero again.
So there is no long run output inflation trade off anymore.
Good, this was the last slide of the lecture.
We are through with everything and time is actually
also used up, except that we still have time
for interaction.
May I ask you if you have things, questions, comments
you would like to discuss in interaction,
then please raise your hand.
In this case, I would suggest that we go to the break now
and meet after the break in the interaction
on the interaction link.
Yes, there is something, actually two comments.
Can you introduce, I don't understand this first comment by
do do and then I have a question.
Oh, schedule of the exam.
Yeah, I come to that later.
I have a question regarding the previous lecture.
Yes, John, please go ahead.
Or do you want to discuss this in the interactive meeting
after the break or do you want to pose the question now?
Let me stop the recording already.
Why do you type your answer?