Overview:
• AS-AD model as a system of stochastic difference equations: The “bread and butter” of modern macroeconomics
• The impulse response function as an inhomogenous difference equation – the propagation mechanism
• Solving a second-order deterministic difference equations after a permanent shock (a constant): Cookbook method
• Characteristic roots of deterministic difference equations / eigenvalues of the canonical (or companion) form as key to understanding dynamics
• The role of expectations and expectational dynamics in macro
by Prof. Burda
#macroeconomics #analysis #AS-AD #stochastic #difference #equations #eigenvalue
I hope you’ve all had a chance to visit the the Leopold’s review section for this for past week and for this week because it’s kind of important to to make the connection between the the long run and the short run and for the rest of this course we’re talking about the short run
We’re talking about uh um how impulses get transmitted into into effects into propagation into the business cycle and we started with the graphical analysis and today we’ll try to move into a mathematical version of the same idea and this this vanilla version is basically just to show you that you can generate
Using this idea of slutsky random shocks accumulating to effects that look like a cycle an irregular cycle is the way that macroeconomists think about the business cycle in general and therefore we have we can’t just talk about a shock there are different types of shocks there’s demand shocks there’s Supply shocks
Among the demand shocks so maybe a shock to monetary policy there could be a shock to fiscal policy it could be a shock to Consumer sentiment and the model that we’ve dealt with until now is very generic and and that’s we’ll we’ll keep it for the next uh for
The next uh hour and a half and then after that it gets more specific and more detailed and if you take more advanced courses in this area you’ll you’ll do more about this so this is I’m going to try to teach you a few Basics about what we’re
Called difference equations and how we think of this slutsky mechanism this impulse propagation mechanism that’s at the core of what we do in this course in in macro in general I call it the bread and butter of modern macroeconomics okay so it’s difference equations current values depend in some way on the past
And that’s because of some economic model that we think we understand and then the shocks are not inherently foreseeable for example we observed this year this year is a complete surprise this past year 2022 was a complete surprise to everyone um and the effect of that surprise is what
We’re sort of living through right now okay so I associated with this with the great Russian mathematician and Economist uh Eugene slutsky but of course it’s also been attributed to other people and it’s again what we do throughout the throughout the field I’m going to try to give you a cookbook
Approach to solving uh second order difference equations because that’s a fairly standard way of summarizing dynamic Behavior second order means the current value depends on its lagged value and another lagged value two lagged values and in quarterly data that seems to be a pretty good way
Of capturing what’s going on so if you actually like econometrics you could go out and try to estimate some lag dependence and you’d find generally the second order is a pretty good approximation it turns out to a lot of our models will point to a second order difference equation so it’s a
Starting point more complicated analyzes you need a computer you really need to throw the thing at a computer and let the computer Solve IT numerically usually and then produce some some summary statistics and thinking about the way the model behaves and if think about it we had these
Curves shifting that was the impact and then we thought well what happens after that well what happens after that is exactly what the the difference equation system would try to to predict okay so um at the end of this hour I’m going to try to show you that there are a couple
Of special summary statistics that summarize how these Dynamic systems behave and we call that the the characteristic root of the system so depending on the order of the the difference equation system there’ll be different number of characteristic routes and what those things look like will be kind of a diagnostic for how the
Model behaves especially if it’s explosive or not explosive so some models you shock them and the thing just just sort of goes out of control that’s not what we observe in reality so we’d like to have a discipline in our model that would prevent that from happening so that’s one kind of basic
Idea the other is how does the model converge does it converge in a loop or in a cycle or does it converge monotonically to its long run value and sometimes we’ve we’ve seen a little bit of that in the growth Theory we didn’t have Cycles yet we had pretty much monotonic convergence
To a to a long run so let’s review what we did last time last time was easy we were thinking about pictures right supply and demand thinking about the cycle as a deviation from some long run Trend and those deviations are caused by Supply shocks or demand shocks
Usually most of us think about demand shocks so think about the the recent uh Corona crisis mostly a demand shock people stayed home didn’t want to go shopping and um on the other hand there’s a feeling that may have been some Supply shock because workers didn’t want to work they were
Scared of going to work or they were staying at home weren’t being productive so it’s kind of a combination of those two but you know you can think about any sort of um business cycle think of the financial crisis 2008 2009 collapse of the banking system collapse of investment that’s a demand
Shock and we saw that in the data we saw the United States in Germany and France and in Italy we saw a decline in output and a decline relative to Trend and we saw a decrease in the inflation rate so it’s pretty easy to diagnose that kind
Of shock right and I tried to motivate that by just showing some some facts and it’s not necessarily the case that inflation and output always move together in the same direction so that would indicate that indeed we have to be a bit more sophisticated we have to think about Supply shocks and demand
Shocks and Supply shocks and this undergraduate model was kind of my basis for getting you interested in it and it’s a nice way of making some some clear distinctions between the way an economy reacts to different different events okay so the again the the recent year was really a challenge for
Economies and we think we think we have a very good idea of what’s going on but um it’s not uh it’s not enough to tell a story we want to have something that actually can can be used in new situations going forward So today we’re gonna we’re gonna spend a
Little bit more time talking about these stochastic difference equations and it’s always an implementation of slutsky’s idea the idea is as a random shock that we can’t necessarily predict maybe next year we’ll have this year we’ll have peace and the price of oil will fall and
We’ll be back to good old days uh 19 2019 or 2018. that’s a possibility but I can’t tell you what’s going to happen you can’t either but conditional on that assuming that that happens what happens that’s what economics is about trying to tell us that okay so again
Difference equations is the core of macroeconomic Dynamics there’s there’s dependence of current behavior on the past current mistakes we’ve made carry forward into the future so you signed a wage contract and you only thought the price level would rise by five percent and now we’ve got a ten
Percent change in the price level you made a mistake you work too too hard you know you probably would have if you could have Rewritten your contract you would have done something differently okay so those are mistakes that may also get propagated by the by the system
The same thing is true of investment investment is conditional on an expectation if I spend a half a billion dollars renovating these these buildings across the street that’s kind of conditional as an investor that there’s going to be some rate of return that will justify uh there’ll be some return
On that investment but if I’m wrong right if people just decide not to come to Berlin anymore and the students go somewhere else and study in another country or another city in Germany that would depress the demand for for the rent of those those buildings and that
Would be a kind of a mistake so a lot of what economics is about is kind of correcting mistakes and that’s the that’s where the Dynamics comes comes in so we’re going to try to think of that as a an Impulse response a response of a
Shock how does the shock get sort of work through the system and I’m going to try to motivate that now undergraduate model and we thought of that as kind of a demand in the supply system and the the outcome of this demand and Supply is output is deviation
From a trend and some inflation rate right that was kind of a there’s a lot of consensus on that model and you can take course you can take macroeconomics anywhere in the world and get that sort of basic insight and we kind of know where that long run inflation rate comes
From we know where the long run output comes from and we know kind of where the deviations of the trend come from but making more sense of it is is where the the um the course gets interesting in my view anyway okay so we you know we started talking about this uh
This picture so we start with the picture and we think about what that picture is supposed to embody the behavior of of Agents on the the demand side demand for goods and services final goods and services at at constant prices it’s kind of an easy definition of what real GDP
Is okay so that’s already incorporating the behavioral households the behavior of firms the behavior of government behavior of foreigners who are importing our Goods because that’s what matters and it’s also been behavior of us consumers and firms demanding output of other countries that doesn’t go into our GDP
So it’s a net right so for for a closed economy we can we can restrict our attention to C plus I plus G we wrote down this mathematical summary of those relationships and I tried to you know Leopold is going to go through this uh in the current in the coming
Week this is a really stripped down version but it’s kind of getting down to the to the basics that we’ll need to understand this Dynamics how the model propagates shocks so the first equation is the demand the demand curve there’s some sort of dependence of current demand for goods and services on the
Past and in two weeks we’ll explain a very very big component of that dependence there’s a dependence on the real interest rate right so the real interest rate is low people demand more goods and services companies invest more in equipment that gives rise to a to a potential negative relationship between inflation and
Um and demand because the interest rate in the second equation is determined by the central bank that’s a very common element of macroeconomics every country has a central bank unless you’re in Hong Kong Hong Kong is run by private Banks but in most countries it’s a it’s a central
Bank and the central bank has a rule that they react they want to fight inflation that’s the first term C1 and the second one is C2 they want to keep output not from collapsing so if output is low relative to Trend they stimulate they’ll cut interest rates if output is
Two is growing too fast they raise interest rates you combine those first two equations you have really what the the aggregate demand curve is in our in our course it’s a negatively related um it’s a negative relationship between output demanded and the inflation rate that we observe
Okay and that’s a pretty robust uh relationship it can be shocked by D so you can have a you know Schultz decides to give everybody a present 100 100 200 Euros we all have some extra income and we spend it that would be thought of as a demand shock holding inflation constant holding
Everything else constant more demand is um is forthcoming now that’s just one side of the scissors think of think of microeconomic supply and demand it’s the same idea the output that is demanded in an equilibrium has to be forthcoming so firms and workers have to combine their
Efforts to produce it and they have to be incentivized to do so and that’s the third equation and this is one of the most controversial relationships and macroeconomics is basically how economies respond to changes in inflation relative to the inflation rate they were expecting or had contractually embodied in their behavior
Okay so that that um three elements we talked about briefly the first element is this core inflation this inflationary expectations in the old days we used to call it inflationary expectations it’s more than that now it could also just be people writing contracts that bind a company to
Supply Goods at a certain price for two years workers have to work for a certain wage for two for three years and they get stuck in inflationary expectations that are no longer valid they can’t do anything about it okay so that’s kind of the that that sets the position of the supply curve
And then we’ve got this this notion that when output increases relative to its Trend firms raise prices workers can try to raise their wages that would give rise to a B1 that’s positive and then all that given there’s a shift that’s possible and that shift is what we’ve
Observed last year we observed these massive increases in gas prices for example 500 increases in gas prices had nothing to do with inflationary expectations or embodied inflation it had nothing to do with output relative to Trend so that’s the that’s the Joker if you like the supply
The supply curve and then just to wrap up the model we add this um this one version of thinking about this core inflation and I’ll talk about at the end of the era I’ll talk about what we what we also think about so if you take other
Courses in macro at this University or elsewhere um the idea is this is the anchor that kind of sets the position of the supply curve in this diagram that we looked at okay so Leopold will push we’ll solve this in the problem set I give you an
Example of a problem set the the problem says slightly different because it involves a Target inflation rate the Central Bank in this particular setup has a Target rate of inflation of zero just to make it easy for you to get your mind around it okay but in the problem
Set we think about what happens if a central bank decides to increase the target rate to from two percent that’s the alleged Target in the European monetary system or the United States two percent suppose they went for five five percent permanently what would happen what if they went to seventy percent
What would happen so that’s kind of what this model is going to tell us right the Central Bank permanently increases its Target um from here zero to some positive what we’d expect because of monetary neutrality in the long run the inflation rate will eventually also be that rate
Inflation rate will catch up eventually questions how does it catch up which curve shifts first which curve shifts second which which catches up so that’s really what the Dynamics is all about so I took this toy model from last week and I just did some simple substitutions
This is what you’ll do on the problems that I expect you to do this except your version will have a little Target rate of inflation it’ll be a little bit different a little bit a little bit you’ll have a little constant to carry around through the algebra okay but it’s really simple
Right you take the first off you have a core inflation rate it’s based on inflation and lagged inflation so you can substitute it out okay so that makes it easy for you to get your mind around and then you can combine it with the Taylor Rule and first difference the
Aggro demand curve takes current value minus lagged value and then you can eliminate inflation so you end up having the last equation output is dependent on past output depending on two periods past output and a shock okay we call that a stochastic difference equation inflation has been solved out okay
So that is that an accomplishment well it summarizes all the interesting action in the model we don’t we can look at inflation we can actually solve for inflation but we have a we have sort of a an equation an output alone and that can help us understand how this propagation mechanism works
On the other hand these these Alphas are functions of the deep parameters so no matter what we do in the next few weeks we’ll always be thinking about these deep parameters that’s the the foundation of the model and that’s what makes this Asad model an undergraduate model it doesn’t really explain very
Well what’s going on in the background behind the curtains behind the curtains are the micro foundations what justifies the upward sloping curve or the downward sloping curve and that’s basically what those those Alphas are about their function they’re complex functions of A1 A2 C1 C2 B1 and Theta
Okay and there you know if you know something about econometrics you know that there’s not a unique set of parameters that you can reverse engineer to get those Alphas there are many so it’s an identification problem theory is we have too many too many theories and maybe only a couple of
Values of Alpha One and Alpha two that the data tell us about so the challenge of our of our field is to try to find out what is the best model for explaining those values of alpha 1 and Alpha 2. and you can see that the shocks are actually complicated
As well look at the first this Epsilon is a mongrel it’s a it’s a it’s a it’s a mixture of many different things supply and demand so unless we know for a fact that Supply was not active during a period there’s no a prior reason to
Think that the the shock will have a positive or negative effect right it’s a it’s the same thing we talked about with inflation the inflation correlation with output can be positive or negative depending on whether the demand or the supply shocks were predominant so we really we’re starting it we have a
Bunch of data we have some interesting observations but the you know getting the right model is the challenge for for us for the next few weeks yeah why the supply shock is a level and the first difference yeah okay that’s that’s a great question the supply shock is is Shifting the
Supply of of goods and services at a given inflation rate and the demand shock does the same but the demand shock cannot raise the inflation rate permanently the only way you can raise the rate of rate of inflation permanently is what I kind of said it at the beginning
How does our model predict a higher inflation rate forever and where do they come from ultimately if you expect inflation we learned that earlier in the course it’s not enough what you need for inflation to be higher in the long run you need the money supply to be growing
At a higher rate forever or the central bank make sure that happens by having a higher Target rate of inflation so it’s the target it’s all about the target the target is the position of the demand curve in this in this world and it’s and if
You have a world of fixed exchange rates like in the Netherlands for example you’re fixed you know the or Germany relative to the Netherlands because we’re all using the same currency it’s about the other guy’s inflation so the Target that is there’s no target inflation if um
Croatia was fixing the exchange rate to the to the euro for a while right so when they did that basically they just took the Euro inflation rate as their inflation rate and that fixes their expectations but you know that’s for an international course we’re talking about a closed
Economy the central bank has the ultimate say on the inflation rate does everybody buy that I mean and that’s that’s why the only way you can get the inflation rate up permanently is to have a higher Target rate of inflation and we don’t have Target rate inflation changes here it’s it’s set to
Zero just to help you get your mind around this but later on we can do that we can add a Target rate of inflation which is not anchored and that’s why central banks are always being asked to commit to an inflation rate otherwise they have no credibility
Right so the whole discussion of leading to the European Central Bank was how can we how can we force the European Central Bank to have credibility it’s never existed before we’re inventing something from scratch people aren’t going to trust this money they’re going to expect Weimar Germany or something right
For the Ukraine in 1990 hyperinflation so you know so we start with we start with this um this setup the shift in a permanent shift in demand can only that leads to a higher rate of inflation permanently can only come from the rate of growth of
Money or from a Target rate of inflation that’s higher and we know from our first part of the course that’s the only way this happens inflation needs an expansion a permanent expansion of the monetary Aggregate and the only way that can happen is the central bank accedes to that and does it
Okay so that’s you know that’s the bridge all right so this looks awful right this is not something you’d want to do for an exam and I wouldn’t ask you to do that but you’d be expected to know that this reduced form the first equation is a reduced form
Expresses a very complex dependence that comes from economic theory and this is not great economic theory this is undergraduate Asad so we’re gonna I’m gonna try to burst out of the chains of of that in this in this course and I mentioned these this Supply this supply and demand shock come together in
A way that’s not it’s not it’s dependent on the model it’s going to depend on A1 A2 C1 C2 so depending on whether the central bank is really strict about inflation C1 is really high or whether it doesn’t care right C1 could be very low it’s going to affect the way this model
Behaves even this model this is like right driving a a two-cycle engine uh you know car like the trabant you know it’ll get you there but it’s not a great it’s not a great way to get there you want to have a nicer car if you if you want to have a car
Okay so the reason I do this is just to get you on the page so we know the next the next set of things we’re going to talk about is why that ad curve is negatively slow what is the what is the fundamental economic mechanism we
Already have a good idea of what is the motor mechanism for why prices don’t uh fully reflect um demand conditions why why do people have to make expectations and lock themselves into contracts why does that happen well there’s some good economic reasons to that to have for that to happen
People don’t want to pay attention to the price level every day they have to do their business right they want to have fun in life not paying attention I mean if you’re in a hyperinflation environment and if you’ve ever been in one but if if you the people spend a lot
Of time thinking about the exchange rate they think about inflation every morning they look at the exchange rate what’s the dollar exchange rate got to go under the run to the bank and trade my increasingly worthless money for for some dollars so I can secure my purchasing power but most people don’t
Have to deal with that most of us are lucky enough to live in a country where the inflation rate is low right so just to give you an idea of why why we do this so let’s let’s try to think about this stochastic difference equation more carefully so we have this
We have why thinking that as deviations from the trend YT is a function of YT minus 1 and YT minus 2. okay it’s a lag dependence so I’m going to show you a little trick to help us think about that it’s called the lag operator it’s an operator
If you hit it with this L it shifts it back by one period okay it’s just an it’s an arithmetic it’s an algebraic trick that will help us think about this past dependence so watch this I hit x t with l and it gives it a back shift it’s often
Called the back shift operator in Time series it’s a linear operator so if I take this L and I hit a linear combination of two variables with it it’s the sum of the two variables with weights given by the linear combination that’s what a linear operator means
And it also means if I hit if I hit a variable twice with the lag operator it’s like the square so the square of The Lager operator is a two lag okay Isn’t that cool so three L cubed L4 is the fourth lag it’s the thing about in the quarterly
Data they’ll be like the the same quarter a year back okay and why would I do that well because if you allow me to do that I can actually show you why this dependence is really deep this past dependence that comes through a lagged um difference equation system will give us
A very rich dependence on the path to some some extent shocks that happened many many years ago are still being felt today okay so this building was built in 1905. that was kind of a shock the Emperor of Germany the Kaiser came here and dedicated this building
Does anybody know who paid for this building not the Kaiser the Berlin Chamber of Commerce a bunch of businessmen paid for this building in 1905 and it was probably a surprise that they built it they built it we’re still feeling the effects today and this is because this is Capital stock
Capital stock doesn’t disappear after one period it hangs around for many many periods so the it’s one of the mechanisms of the propagation mechanism I’m talking about is the Persistence of the Capital stock like we learned in the first half of the course so the investment actions of 1905 are
Still around today of course it looks a lot better than it did when I came here because it was falling apart in 1992 this place was was a dump okay because the gdr didn’t take care of it they didn’t invest in it but more investment paint call some workers in
Put some put some stucco on the on the outside and it looks pretty good now you’re very lucky to be here other cohorts were not so lucky okay why am I telling this because you can express a lot of this backward dependence using the lag operator
I’m going to give you an example here’s an example of an ar1 auto aggressive first order difference equation remember I showed you a second order difference equation this is a first order difference equation right so this Alpha here less than one so thing has a has a certain stability to it
If you shock it it’s going to return to its initial condition eventually and I’m going to shock it with Epsilon Epsilon is a is a independently identically distributed random variable so it’s like slutsky okay and you start with Y zero so that was the the condition before the the building was built here
I can use the lag operator right this expression like that right so I’ve got YT on the left hand side I’ve got YT on the right hand side this implies that the past the infinite past has an effect on today’s value in addition to the current shock so the
Current shock gives us the the little the jump of today but all the past is just a accumulation of the lagged something and that lag something is exactly previous realizations of this shock that’s the slutsky idea we can show this formally one way to show it is just recursively
Substituting okay that’s trivial you should all be able to understand that without even be having to stand up here and talk to you about it right you just lag it by one period and insert that in the in the in the current period and then you do that
Again and again and again and there’s no end okay so this is what we call recursive substitution so what do we get as a result we get an initial condition going s periods into the past that could be 1904 and then we’ve got all these different shocks that have happened since 190 1904
1905 that may have been the building this thing and then we’ve got another shock then we had World War one and we had World War II and what we have today is basically accumulation of those shocks the stability of the structure is is the presupposition of course this is all to
Work but as long as the alpha is is the same as we go through time we have a moving average an infinite moving average if we push it back far enough but if we push it back um to T to S periods in the past we have a
Moving average of the intervening S Plus One shocks okay so that’s kind of what macro is going to be but you know that’s um if I push that back far enough the initial condition won’t matter right because if the initial condition is kind of is holding the thing down but
If we keep pushing it back eventually the weight applied to that initial condition will go to zero so literally slutsky was right in some deep sense what we’re observing today is accumulation of a lot of Randomness over time and that’s kind of one way of thinking about
About macro the other way to do this is to actually use long division okay so I mean if you you remember in high school or hopefully in grade school when you learn how to divide divide five by seven remember doing that okay well 5 divided by 7 is not a a
Round easy number to Express it’s a rational number but it’s got a it’s kind of weird right so you have to write it down before I did this when you were in third grade or something maybe maybe first grade probably seventh grade I don’t know
We can do the same thing with a lag operator and you can get the same outcome so this is kind of an aha moment for a lot of people we can use what’s called long division okay so take one minus Alpha L and divide that into one
Well it goes in one time leading term multiply one times one minus Alpha subtract that from one what do you get Alpha l how many times does 1 minus Alpha L go into Alpha l Alpha l subtract Alpha l minus Alpha squared L squared from that so you eliminate the leading term and
You get Alpha squared L squared repeat repeat repeat you get the same type of lag dependence on the as far back as you want to go back but infinite moving average it’s an infinite geometric moving average of these shocks this is a very simple example obviously okay but you see the logic
You can use long division to prove effectively or to demonstrate that this um this recursive relationship can be expressed in that way okay so that’s kind of an interesting fact kind of going to motivate our macro model again we can also push it a little bit we can push to a harder example
But let me just say one thing it does presuppose that Alpha is not too big Alpha has to be less than one otherwise this doesn’t work makes sense right if Alpha is greater than one the thing will explode the accumulation of random shocks that accumulate in a way that doesn’t
Deteriorate in any way that doesn’t get depreciated like the Capital stock right will eventually explode so kind of unpleasant result we want to have a stability condition imposed the obvious stability condition for this simple ar-1 is that Alpha is less than one in absolute value could be it could be greater than
Negative one it could be between 0 and negative one and it would still be stable it’d be kind of strange looking but it would certainly still work okay so these are it’s kind of an intuition for stability what if it gets more complicated what if
We have an ar2 system look at this this is exactly what we had before ar2 would have two lags L squared okay is this can you do this yes you can um it’s kind of ugly it’s ugly let me just show you how we get there we
Do the same the same trick we take that equation we write it using the lag operator now it’s got a polynomial in the lag operator it’s a second order polynomial what we had before was the first order polynomial now we have a second order polynomial in the lag operator because
It’s got L squared in it okay under certain conditions we can invert that too and the two ways to do it the dumb way the way I did it before recursively that’s a mess you don’t want to do that nobody wants to do that okay you could also use the long division method
But the logic is the same we’ve got a complicated moving average of the past values of Epsilon that’s what I want you to remember the inversion of this polynomial in the lag operator induces a very complex dependence it’s not trivial you can have all sorts of I’m going to
Show you in a second you can have waves you can have monotone convergence one shock that’s what we call an Impulse response and you watch how this one shock plays through the system and that’s all determined by the by this Alpha One and Alpha two the model’s economic structure
Still a moving average okay we can show that two different ways the first way is absolutely stupid you wouldn’t do that recursively substituting because you’d have to track all those complex products of Alpha One and Alpha two well you’re gonna have to do that anyway but I’m
Going to show you the easy way to do it okay so let’s just try this let’s try this hard way first let’s try the the long division it’s nasty it works you can do it but it requires a lot of pencil and a lot of patience you don’t want to do that
But I want you to remember that if you look carefully at this thing what does it look like it looks like a polynomial in the lag operator now acting backwards on this Epsilon so it’s a complicated lag structure of lagged shocks we can do better we can factorize it the first term
Is a polynomial in the lag operator we can write it as the product of one minus Lambda one times L and 1 minus Lambda 2 times L and Lambda 1 and Lambda 2 are defined defined as the sum makes Alpha One Lambda 1 plus Lambda 2 equals Alpha One
And minus Alpha Lambda 1 Lambda 2 is equal to Alpha 2. that’s in high school I learned that when I was in high school factorization that’s how you solve a quad complete the square solve a quadratic form quadratic equation okay but look at that why would
I do that well it turns out the 1 minus Lambda 1 and 1 minus Lambda 2 have the same logic of inversion that we had before we can do that much more easily you can invert them separately the only thing we have to worry about is whether these these Lambda 1 and Lambda
2s are big or small if they’re big that could be a problem if they’re big in a way that I’m going to explain to you that could cause instability remember I’m always worried about instability in my macro model I think blowing up right that’s the problem so we’re going to ask
We’re going to require that Lambda 1 and Lambda 2 are small in some in some sense before I required that Alpha was less than one in absolute value so it’s a similar logic a little bit more complicated it’s going to involve an absolute value of a number which we call the modulus
So that’s a word you have to remember okay so look if I can do that if I can write this inversion in a factorized way I’ve got two individual inversions because the inverse of a of the first term is equal to the product of the inverses of the second
And I can think of each one of those as hitting the shock so I get the geometric moving average but I get it twice I get a convolution so that looks really really complicated but it’s not it’s easy to just to formalize and it looks um
It looks so easy because if I understand the behavior of Lambda 1 and Lambda 2 I understand the behavior of this system so these this this is often called the characteristic root or the if you’ll see later it’s it’s in a similar way in a matrix form it’s
It’s the eigenvalue of the system it’s stability so we’re going to have to stare at Lambda 1 and Lambda 2 for a while first off I can solve for them if I know what Alpha One and Alpha 2 are I can solve for Lambda 1 and Lambda 2. you can do that
You should try to do that tonight you’ll have to use the quadratic formula to do that because we’ll involve solving a quadratic formula but that’s the cool part because it gives us a whole it opens new Vistas in terms of what the Dynamics can look like the Dynamics can have waves the dynamic
Dynamics can have monotone convergence and that’s why um slutsky was right even if I shock it once I can get wavy wavy reactions but if I shock it every period I’m really going to get some interesting action okay so I do the factorization we see that it’s a product of two inversions of
One minus the respective characteristic root and then if you multiply those out you can see what happens we’ve got it we get this very very elegant looking product of different powers of Lambda 1 and Lambda 2. is called a convolution of these infinite order polynomials in the lag operator
Okay so that’s kind of a that basically tells us that if I shock this this system in Period zero I’m going to get I’m going to get a reaction into the infinite future it’s going to be it’s going to be damped away or disappearing at some
Point but it will always be there we’re still hearing the noise of of the the hundred year war in Germany we are if you look at it look at The Villages in Germany that were Catholic Protestant fighting each other I mean when the when the when the Turks were
About to take Vienna in in the 17th century you know this is kind of we still hear the resonation of that today it’s one way of thinking about it economics is all about um things that happened a long time ago that are still bothering us in some sense or affecting our lives
It’s kind of the way it works okay so that’s sounds really philosophical but that’s the way it looks in from our perspective at macro the impulse response function so we can write we can write this as a way a model responds to a to a one period shock um
In some sense why dep is a sum it’s some infinite sum of of these shocks going off into the past you can think of this as a you should not think of this as a structural model this is a reduced form model but it’s still a way of expressing inter-temporal dependence of variables
In macro and if we do identify the shock and say this was a demand shock then we can look at this adjustment path through time and call that the effect of a demand shock so I I put a word here that sounds a bit terrifying but these lambdas I told you
These lambdas are not just um they’re not just normal numbers they can also have they can have complex conjugate form and that means that we have to again dust off the high school algebra and remind ourselves that there’s a the possibility of having a more complicated expression than just a real
Number that we encounter in everyday life which is a complex conjugate number so we have to remind ourselves what that is but let’s get there first let’s get there first again thinking about this univ variate macro case inverting the the formula we get this set of coefficients so that
Thing of that as the impulse response um the sequence of coefficients that summarize the reaction to a unit shock okay so to get there we have to we have to solve this difference equation we just solve it right now we just we just wrote it down we have a reduced form YT
Depending on YT minus 1 and YT minus 2 and we’ve got this um this potential shock that we’ll call it Epsilon zero think of this as a constant just think this is a constant to solve this we need a function of time that only depends on
The shock and Alpha 1 and Alpha 2 without any lags so we want a function of time and to solve this if we solve it we’ll have the the object we’re looking for we’ll be able to to describe um the reaction of the system would be able to describe its stability
Properties and how long it takes for a lag how many lags it takes for a for half of the shock to be dissipated Etc okay so we’re going to use we’re going to use a little bit of cookbook now so I’m going to show you just like we did
The hamiltonians it was kind of ugly but it’s really helpful extremely helpful there’s a cookbook way of solving difference equations so if you can remember these steps you’re fine and then you can see why this is a useful way of proceeding because you’ll understand why we we like to stare at
These lambdas These lambdas are incredibly important for describing the stability and the properties of the of the Dynamics okay so let’s take let’s take YT as a function of one slagged and twice lagged values and this constant so think of that as a if you like think
Of as a permanent shock permanent shock maybe an increase in the inflation rate the target inflation in the Central Bank whatever so step one what happens when all the Dust theoretically is settled what to what value does y converge and how do you figure that out
YT equals YT minus 1 equals YT minus 2. right that’s the the steady state okay so try that YT equals YT minus 1 YT minus 2 is equal to YP call it YP what’s the value of YP it’s that it’s Epsilon 0 minus a divided by 1 minus Alpha One minus Alpha two
It’s the steady state so all the dust settles what is y it’s YP that’s called the particular solution it could also be zero if Epsilon were zero then it would be it’d be zero but that’s not interesting okay step one step two involves invoking what I showed previously called the superposition
Principle if I know that one solution is a solution to a difference equation I know that the other is a solution then I can add the two together and that’s a solution to the difference equation okay so if I know that the particular solution is a solution then y minus y p
Must also be a solution so I need to find out what y minus y p is well doesn’t y minus y p sound like the deviation from the steady state isn’t that what we were looking for the business cycle we’re looking for the deviations around the long run so call that the
Homogeneous solution why is it called homogeneous well if you multiply that Solution by a constant it doesn’t change the solution it’s the same it’s scalable think of the YP as the location and the y-h the homogeneous is the is the is the additive that’s where all the action
Comes from the action comes from the homogeneous solution the particular solution is kind of boring it’s the steady state it’s like the growth stuff we were doing last semester or last year right this is this is where the action is the business cycle is going to come from the homogeneous solution
But to do that we need to know that it’s going to have that form it’s going to look like my famous characteristic roots raised to the power t this is a little bit technical but it’s actually kind of interesting Kappa One in capital look like K’s those are constants but this
This function of time means that it’s a special function of time it’s the root raised to the teeth power okay and there are two of them that’s because it’s a second order difference equation there are two components that you add those together and we just need to find out that that combination that
That combination of Kappa One and Kappa two that gives it but we also need to find out what Lambda 1 and Lambda 2 are but I told you what those were if I tell you what the alphas are you can tell me what the lambdas are okay man this must be a demonstration
Outside a lot of action yeah Yeah that’s one way of thinking about it exactly that’s a that’s a that’s that was kind of was hinting at I don’t want to push it too hard because it’s true for any system you know it’s true for any any model but in in this particular application yes the output Gap
The the relative to the trend that we’re not trying to explain in a business cycle we’re really trying to explain the cycle and not the trend we’ve already kind of explained the trend step three is the hard salute the hard part you have to find the the lambdas
Now I told you what Alpha was Alpha One and Alpha two you should be able to tell me what Lambda 1 and Lambda 2 are and to that you’re going to have to get your pencil and solve an equation okay and that’s what they look like that’s called the quadratic formula
Because once you take the definition of Lambda 1 and Lambda 2 in terms of alpha 1 and Alpha 2 and you try to solve for it using algebra you end up having a quadratic equation Lambda squared plus something times Lambda plus some constant equals zero and you got to solve for that
In high school depending on whether you went to high school in the United States or in Germany or some other country if you went to Turkey you probably learned the German Way there’s a different formula but it’s the same answer it’s just a different the Germans call it the PQ formula and
The Americans call it the ABC formula I would love to write this down because this is like the drama the drama is important so a times x squared plus b x plus C equals zero now when I was in high school I was in ninth grade I learned how to solve that problem
What is X a is a constant B is a constant C is a constant this is a quadratic equation this is like bread and butter in economics you know we’re we’re pretty close to that already but it’s going to be involving Alpha One and Alpha two
You can do it in this section but what’s the answer who wants to give it a shot you’re so close but you’re not quite there you started you start on the false on the wrong foot foreign no no he was right it’s got be at the
First at the beginning but it’s a minus B minus B plus or minus because B squared minus 4 a c divided by 2A very good okay there it is that’s the famous Infamous quadratic formula you have to learn this why because that’s the only way you’re going to figure out what lambdas are
That’s the only way you’re going to figure out what lambdas are I told you what Alpha One and Alpha 2 are you got to figure it out but this is this is already kind of interesting look at this remember I told you the the solution this homogeneous solution is powers of Lambda
So you already have if you have some intuition if Lambda is greater than one in some sense in some expanded sense we’re in trouble that thing is going to blow up so we we want the lambdas to be small in some well-defined sense because we’re taking powers to T and T
Goes as T get bigger and bigger the powers of t of something that’s greater than one are going to get really really big and that’s not what we want we want the model to to kind of damp down to something to converge so that and if in a sense the the size
Of Lambda 1 and Lambda 2 are essential and that’s why I keep harping on that and when I say the size I mean the modulus because sometimes if we’re unlucky and this thing is negative we’re going to have a complex conjugate number plus or minus something times I right and that’s
Don’t worry it just means the impulse responses have a cyclical cosine sine pattern okay all you have to remember is that when I saw for the Lambda I’m going to get something like this and it’s going to be potentially complex conjugate if B squared is greater than 4AC
Then it’s going to be real valued but if it’s not I can’t deny its existence it means that a shock is going to have a cyclical adjustment so that’s the cookbook and we will spend some more time thinking about this if you take any advanced courses in macroeconomics
You’re going to have to worry about the stability of a system and it’s going to involve these things which are in a matrix setup called the eigenvalues the eigenvalues of the dynamic system so you if you don’t like macro you can run away from macro but if you want to deal with
Macro and grab it you’re going to have to deal with eigenvalues so this is kind of an easy introduction to the stability of dynamical systems some of you may want to run away right now but you can’t run away from this course because it’s required but otherwise you know think about it macro
Is about Dynamics we need to understand that yeah yeah there’s a there’s a no no it’s it’s a good point I mean it’s a it’s a notational issue I mean it’s um think of YT now as describing output that we actually put our hands on okay so and then we
Um it’s got this YP which is a permanent part it could be a trend it could also be time dependent so in fact the modern the advanced approach would be to allow the YP to have a growth component but you have to strip that out first okay so this is yeah it’s um
That’s why I think I told you last time that I like to put the hats on and I didn’t I didn’t do that but if you like you can think of why H as being y hat okay okay so you know you can write the solution to this
Difference equation as literally as the sum of a particular solution and the homogeneous solution and that’s what I’ve got here first part Epsilon 0 divided by something that’s less than one we knew that and then plus this stuff that depends on time there’s no lag values in there anymore we’ve actually
Solved the difference equation that’s the solution so if you like that’s a that’s kind of an important an important achievement for the day it’s we’re able to write YT in this form now you asked you probably asking yourself where do these guys come from where do the kappas come from the kappas
Are just constants and they depend on the initial condition of the of the system so if there’s a period zero there must have been a period minus one one period back because the dynamic system always kind of marches forward but you can always go back and say what
Happened before that and there is no beginning of time here you just have to define the beginning and then ask okay well two periods before that was what was life like and then if I know those two I can actually solve for Kappa Kappa 1 and Kappa 2 are functions of
Those initial conditions that’s just to mention it right now you might be interested later okay let me just tell you what this means this this has incredible implications so we’ve written this like this we we know we have to look at the out the lambdas these are called characteristic roots of the difference
Equation and this is worth staring at for a second we see that you know we know we’re not really sure what Lambda looks like we know it could look like a normal number it could look like a complex conjugate number if we’re unlucky but actually if we’re unlucky we
May be lucky because we learned something and we also may have some unusual Dynamics we may have waves okay and the more advanced PhD level of course we’d actually talk about that in detail I’d ask you to talk about the trigonometric identities that give rise to that it’s not worth it the important
Thing is that it does imply cyclical Behavior and the reason I know that is because of its complex conjugate I can write that complex conjugate number as h plus v i I is the square root of negative one and that’s what makes it so crazy Loco right but Descartes and Euclid were clever
Enough to say well I can still think about these numbers even if I can’t imagine them I just put them in a two-dimensional plane and the horizontal plane I’ve got the real part the horizontal Dimension the horizontal axis and the vertical axis the is the imaginary part it’s that easy
So I’ve expanded my mind I can think in complex terms now because if I if I get off the x-axis I’m in the complex world I’ve got a complex Dimension to my number so if my if the solution to this problem involves complex conjugate numbers I’m
Going to have two of them and they’re going to be mirror images of each other around the x-axis those two dots it’s the complex conjugate if the number is real valued the com the imaginary part is zero then v v is zero and then we’re on the x-axis and it’s uninteresting okay
So just allow me to do that I take the Lambda I tell you they’re going to come in pairs because we live in the real world and the real world has only complex conjugate pairs as solutions to this equation so one is going to be the reflection of the other around the x-axis
Okay that’s why I say symmetric about the x-axis now why do I had to circle here what’s the point of the circle it’s a unit circle it’s a circle of radius equals one why would that be kind of interesting for thinking about Lambda exactly they should we want to have a
Convergence result suppose the imaginary parts were equal to zero then we would be on the x-axis we want to have those lambdas less than or equal to or actually just less than one in absolute value so that means on the left and on the right what happens if we have a complex
Conjugate value well we have to worry about that obviously it turns out if we’re inside the unit circle we’re stable we still have stability it’s that distance the distance to that point is what now see if you remember your high school your gr your grade school euclidean algebra or geometry
What is the distance from zero to HV thank you Plus a okay that’s the distance to the origin that’s going to be the stability condition so even if we have a complex conjugate root we can still have stability but the length of this modulus the modulus the distance to the
Origin has to be less than one I’m going to show that in a second it turns out this is because when you have a complex conjugate number and you write down this powers of powers of the complex conjugate number it’s like a cosine curve so the cosine curve has to be damped and
The distance has to be less than one if it’s not less than one it’s not going to be damped it’s going to be either cyclically cycling forever or it’s going to be cycling and exploding okay so that’s kind of a radical tour de force you know getting
Through this uh is is it’s just about comp Computing the distance to the origin and because it’s symmetric those distances are equal I don’t need to do it for one because of the same one is above one is below the distance to the origin is the same okay
That’s why and this is called the modulus the modulus is like an expansion of the idea of an absolute value but it’s an absolute value for a number that’s complex conjugate think of it as the distance okay so this is called the modulus the length of the euclidean distance or the arrow
Now if you really like this stuff now if you remember trigonometry you can express HV as a distance and an angle right if we’re going to blow our minds we might as well just really blow it up so we can use sine and cosine to think about the same numbers you tell me
HV I can tell you R Theta I can compute it I just go to my calculator and figure out what’s the cosine sine of theta um it’s defined as the run over the over the distance r or the rise over the distance r those are the definitions of sine and cosine
I’m going to give him my ticket to getting cyclical Behavior because if I take if I take um if I take the this way of writing the complex conjugate number I can also write it a third way in the Euler form and if I take powers of that
Powers of R times e to the i t Theta are just powers of that which are effectively powers of RT and powers are something that is cycling over time so it turns out that that’s the second line basically shows that that’s just a function of cosine and sine it’s a shifted cosine function
Stability will only depend on r R has to be less than one and R was what R was the distance to the origin so powers of that have to be shrinking as T gets bigger and bigger okay so I summarize this is what we got the lambdas can be expressed they may be
Complex conjugate numbers and if they are we can um still think about them as being powers of um of numbers that are real valued in the end and I can I can prove that if you want but it’s not really necessary you can you end up getting a shifted cosine
Wave if we have complex conjugate numbers otherwise we get something that’s damped over time and goes to goes to zero there were some questions somebody had a question no yeah the answer the answer to that question is the closer you are to one the more persistent the Dynamics are it’s a great question
So if in the limit if if R were very very small if we’re very close to the origin then the Dynamics would be very very uninteresting they wouldn’t be persistent but the solution if if R is is very very close to one that means that a shock will take a long time for
For it to die down for to to damp down and that’s that’s exactly why we stare at these lambdas because we care about the persistence we want a model that that look gives persistence that looks like the data and if you look like if you look at the the
Standard recession it takes it takes a while for the recession to go away right and like the one we’re about to have right now may take uh six to six to twenty quarters depending and that again these are all things that we can learn from looking at the at the
At these models okay so I’m trying to I Don’t Wanna I’m gonna spend too much time on this but this is extremely important um if you write if you if you take what I just derived and write it in terms of complex in terms of the the sign and the
Cosine functions you end up you end up getting something like this so so Kappa 1 and Kappa 2 could be complex conjugate but Kappa 1 and Kappa 2 with primes on them written as the sum and the difference of the of the original Capital One in Kappa
Two those are real numbers so you don’t have to worry we’re not getting a complex conjugate you know data we’re getting real value data but they come from this this solution which could potentially lead to complex conjugate numbers and I’ve already said we need we need our lesson one to have damped oscillation
So again if you if you this is not um exam material but if you really care about this a lot then you can actually use this this again this is reacting to a single shock a permanent shock you can see that we can derive an amplitude and the amplitude gets smaller over time
If R is less than one so it’s a damped oscillation in the case that we have complex conjugate roots you can think of a shift a phase shift and you can think of a frequency so if you let the shock play through how many times is across the origin in a in
A period of time in a year four quarters okay so this is just some extra information for you in case you’re interested so this kind of shows the the one to one correspondence between um between what we call the frequency the frequency domain and the time domain
And if you take a if you do a PhD in macro or in economics you would spend some time thinking about that more complicated because you can use we can use um many methods that are used in in Applied Physics for example to look at business Cycles to look at Cycles in in
Financial data or and the like okay so this is you know if you have complex conjugate Roots you’ll have something like this a one-time shock could lead to cycles and they would lead to Cycles forever if R is equal to one this is a particular case this is not going to happen
But it in principle it could happen so this would be the case of no no damping and you know if you if you’re if you’re interested you can use Matlab or Exile even to generate examples of a cosine sine function and change the frequencies Etc okay so let’s conclude this discussion
And I’ll give you some summary remarks it’s important to think about macro as a set of stochastic difference equations it’s kind of what we talked about last week with slutsky there’s an initial shock and there’s a there’s a culmination of of that shocks effects and if we have many shocks lucky slotsky
Said there wasn’t just one shock there was every period there was a shock and what we’re observing is an irregular set of movements of the economy but they have the cyclical component and that comes from the culmination of these shocks that’s exactly what we’d have here what
I discussed before was a single shock the reaction to a single shock okay and the question about persistence is very good because if the closer the underlying model tells us we are to the unit circle the more persistent the shocks are going to be the longer it’s
Going to take for the economy to digest the shock so even without the complex conjugate story that I had before you can still have Cycles and they’ll be irregular we saw that in the case of the ar-1 if you shock the ar-1 you’re going to get something like a cycle it doesn’t look
Exactly like what we would call our cycle everything I’m going to do in the next few weeks will be trying to figure out the most reasonable economic model that gives us this type of behavior again as simple as possible but not too simple and again this ultimate question will be
How persistent are the shocks how long lasting are the shocks it’s going to be asking about the size of the roots so again how big is this guy if it’s close to zero then shocks are not going to have any persistence at all please remember this cookbook
Okay you can do this in a lot of different contexts a lot of you may do end up doing microeconomics dynamics of of the industrial organization or you might do um you might do econometrics if you’re doing time series econometrics this is bread and butter do a lot of this okay
And remember this formula please okay so what does this mean for us the last 10 minutes I’d like to get ready for the next next week we’re going to talk about a real economic model that is not baby undergraduate Asad we’re going to actually do something serious
Okay but it’s always going to be recourse recurring thought in our heads we have to think about this implications for Dynamics if we write this down okay and we have the characterization with the roots Etc we study those characteristic routes we know where they are
This is I did an example for you so you can calculate yourself these are two routes that actually have no imaginary part they’re just real valued you can also have that okay um let’s go back to our let’s go back to our model again last time Asad after that ID okay
A d curve a S curve now I’ve done the substitution for you you can write this in a matrix form and this is what you would do if you took Professor ranka’s Advanced course for example or if you took there are many many courses on on monetary fiscal policy in Berlin that
You could take you can write down instead of writing everything in terms of Y and YT minus 1 and YT minus two you can write it in terms of just Y and pi and y minus 1 YT minus 1 and Pi T minus 1. so A first order Vector valued Auto regressive system
Contains the same information as before it’s actually a much more elegant way of writing it right you don’t have to have all those ugly quotients and but it’s the same information it’s expressing the same thing but now I’m putting the inflation equation directly into the object of of Interest which is this vector
Y t Pi t it does everybody understand that so I’ve I’ve captured the same information that I had before in this ugly ar2 now I’ve got it in an ar1 right this is kind of and again the same information should tip you off well maybe it’s better to do it like this
Maybe it’s better to do a matrix Matrix representation and you’re probably right why should I force myself to focus on a single variable write it in terms of this nasty look at look at the demand and Supply shock they don’t even look as ugly as before right they’re still there but they’re just
What’s how did I get from one to the other it’s easy I can just try to solve this I can invert that Matrix I could even use the lag operator if I wanted to right um in more advanced macroeconomics you do that you can write um
The vector YT and Pi t as a function of lagged values and the shocks so all I’ve done is just write it a different way it’s the same information as we had before okay and if I if I’m willing to invert that Matrix because that if if you look at if
You look carefully at this and those of you are a little bit mathematically inclined remember um I’ve got I’ve got it Android a matrix to make this work because when I started I started with this with this system I had a matrix on the left hand side hitting the
Contemporaneous values of YT and Pi T and the right hand side I have a matrix hitting the lagged values so to to write the reduced form I have to invert the left hand side Matrix and assuming it’s invertible I can pre-multiply both sides and write the whole thing as a as a
A form like this Okay so I could also use the lag operator I could use the lag operator on a vector that works too I’m not going to do it for you I did it last time I taught um last time I taught Yama and I almost had a revolution
I just wanted to demonstrate that it gives you the same result and you should just remember that it gives you the same result as we had before okay but the ultimate thing you should remember is we’ve got the impulses and we’ve got the propagation we’ve got exactly the same message except now I’m
Tracking inflation as well I’ve got inflation as well on the left hand side the eigenvalues of the Matrix um a0 are the same ones as the characteristic Roots as before so they’re they’re going to give me the same stability information even though it’s only regarding Y and not Pi now I’ve got the
System and it’s all about the stability properties of of that system so ultimately that’s a message to you if you go on to do more advanced stuff you’re going to think about the eigenvalues of a matrix but it’s the same logic as the characteristic roots of a of a
Univariate representation okay you have a question it’s a matter of taste to call them one or the other I mean I would prefer not to call them structural shocks but there are people who you know at some level they are we’re making a structural assumption because the supply
Shocks uh involve producers and and workers and the demand shocks involve households and and investment uh Decisions by firms so they they are almost they have a structural flavor but the purists might say not structured enough for my taste maybe I want to distinguish demand shocks coming from
Consumer uh financing access versus The Bank Central Bank policy interest rates so I mean yeah I mean for me there’s there’s structural for the Asad there are they’re also kind of structural right okay so we talked about the the characteristic Roots this is just a general set of remarks you need to pay
Attention to this in in a longer run sense if you if you care about macroeconomics there are lots of other uh details on these eigenvalues especially regarding their distance to the origin so going back to this the eigenvalues will have potential complex conjugate form and if they do you need
To take their distance to the origin as a measure of stability last few minutes talk about expectations we’ve talked about we talked about core inflation the anchor for the as curve right that’s the way I tell undergraduates the story is that inflationary expectations proxy for a
Lot of other things that are going on and um explain that the Phillips curve could have been stable for some period when people had stable expectations of inflation but as soon as those expectations become unhinged then the Phillips curve starts to move that’s the that’s the the standard macroeconomic narrative and
Everything is based on how those expectations are formed so I try to draw a picture for you this is this is the same picture I showed you yesterday last week okay positively sloped aggregate supply curve negatively slow sloped aggregate demand curve and now I’m putting Pi Bar there that’s the central bank’s Target
Inflation so I’m trying to make it more specific and this is a system in a state of resting a resting point in the system where people expect inflation and they expect correctly inflation so I know that because the as curve intersects the a Las curve and the ad curve at inflation
That they expect which is Pi tilde so that’s the resting point of the system this is this could describe the world in 2017 where inflation was low we had no pandemic demand was pretty stable people weren’t talking about Ukraine we weren’t talking about much of anything okay so
Now what happens if the Central Bank suddenly says to hell with that we’re going to shoot for five percent inflation instead of two percent just a thought experiment okay and suppose people didn’t get it at first so they wrote contracts expecting two percent inflation firms deliver expecting two two percent inflation they
Mark up on that and then all of a sudden we get this big surge in demand coming from a permanent increase in the central bank’s inflation Target this means that the Central Bank reduces nominal interest rates and stimulates the heck out of the economy okay we’ve seen this happen in the past
In various places in the world I’m not going to comment on it but this is what when you get there what do you see you see an increase in output so you see a boom and you see an increase in inflation depending on the slope of the aggregate supply curve
Now is that a resting point for this system who are The Losers after this happens anybody who are The Losers somebody must be I mean we have this big expansion of output inflation’s Rising I’ll give you a hint people were expecting inflation equals Pi tilde and they’re kind of stuck there they
Wrote contracts they agreed to wage increases based on an old incorrect inflation rate so that cannot be an equilibrium that cannot be a resting point for the system going forward if the Central Bank continues to Target inflation at a higher rate so what do you think is going to happen next
Again this is undergraduate pictures somebody made a mistake here right I mean I mean all the people who expected Pi Bar Pi tilde inflation conditioning the position of the as curve they worked Harder They produced more they they may have raised their prices a little bit but not by as much as we’re
Going to expect in the long run so what happens next people yeah very good okay I mean it’s it’s it’s a no-brainer that if people continue to observe this higher inflation they’re going to adjust their expectations so we had this we had this Mickey Mouse equation for
For core inflation in the model that’s a part of the people get it right and some people get it wrong but eventually everyone’s going to figure it out and those people who didn’t get figured out early are going to lose they’re going to lose a real purchasing power because
They can’t catch up what they had before so we’d expect the long run the long run if the Central Bank continues to Target a higher rate of inflation we’re going to actually have a higher rate of inflation Central Bank is going to do what it’s supposed to do
And people are going to figure that out and they may be mad they may actually try to get even more so the Dynamics could be a circle a loop around that long run it may not be a monotone convergence so the way inflationary expectations react is going to determine the
Characteristic roots of the system it’s going to be part of the story that’s why it’s relevant what I’m telling you right now so how we adjust to an inflationary surge a mistake because policy changed means that the Dynamics of how we get there could look like this could look like like this
It could if people are really really smart and figured out right away hey they’re just trying to they’re just trying to screw us I’m going to expect five percent inflation immediately then this curve shifts immediately and we get a much more direct movement to the new steady state so that’s why inflation expectations
Especially now right now what do workers think is going to happen do they think the European Central Bank is going to accept a higher rate of inflation for the next five or ten years they keep saying two percent had they proved it had they done what they would
Actually have to do to fight inflation if they really wanted two percent inflation we’d have much higher interest rates right there so now I’m getting real with you think about it so that pie tilde is the most important part of this model how do workers and firms adjust their expectations of
Inflation to changing events do they believe the Central Bank Central Bank promises two percent I don’t think so uh how much time do they think the central bank is going to need before they react that’s where macro gets really interesting we call this the question of whether expectations are rational or not
So I’ll stop in a second rational expectations is one of the driving still one of the driving Revolutions in this field do we assume people are really smart do we assume they’re stupid do we assume they can correct their mistakes because obviously it makes a difference makes a huge difference
So the rest of the course is about thinking about that and you’ll see that next time we meet okay I don’t like to assume people are ignorant I don’t like to assume that economic agents don’t get it I think they’re pretty smart the wisdom of the crowd so keep that in mind
One of the key shocks for us is going to be how an economy Works through a mistake how long it takes for it to happen keep that in mind okay sorry I tortured you with the characteristic Roots but it’s it’s important believe me and um Leopold will go through it in section
And you’ll have to solve you’ll have to work through this this macro model that this baby toy Asad model with that in mind okay have a nice week see you next time