Hello
So we're just going to wait until 605 and Then we'll we'll get going
Okay, maybe let's get started.
If any, can anybody let me know whether or not you're hearing me. Okay, so I
think the best way to do this as everybody keep their microphones muted
and use the chat function, so if anybody is having trouble hearing me,
everybody sound okay. So,
okay, looks like looks like mostly everybody is good to go. Um, so let's
uh, let's go Do some announcements. So first off, everybody should have
received a score for assignment one and perhaps some comments from me. So I
commented on probably 90% of people's assignments. If you didn't get any
comments, it means that the assignment look good. You know, I didn't see
anything wrong with it. You gave a good faith effort, everything. There wasn't
anything obviously weird about it. And your your essay was was fine. Most of my
comments were sort of directing people when I saw things that were just a
little bit weird. I didn't know what you were doing. Or if you weren't giving
full faith answers to things I made comments. The answer to key to that, of
course is up. It's I think the assignments are another good resource to
study for the exam. So that's probably the next thing to speak about is the
midterm exam. So the midterm exam is next week
and the plan It will be a take home exam.
So I'm going to post the exam at six
o'clock.
You have 24 hours to complete it. And then you submit it online, there is
already a box on canvas, you can submit it through through the Assignments tab
with the dates and everything up and up and ready to go. As far as study
resources, so here let me share my screen.
So if I come into
modules
down here at the
bottom, I put up a sample mid semester exam. This was the actual exam that I
gave in this course in 2018. So second semester 2018. The problems here are
indicative of the type of problems that you will have.
So this is you know, basically How I like to write questions, essentially,
you know, you write, I write sort of cute little word problems like this. And
then there's multiple parts to each. So I basically finished the exam, there'll
be three questions, and I have written the exam so that essentially, if you
were to sit it, it should take you around two hours and 10 minutes, if you
were just going to sit down and do it straight through. Now, I realize, when
you have an open book, open note exam, most people don't do that. So it's going
to take you longer. My suggestion, though, is to sit down and just work
through the problems as if it's a regular exam, you know, take the three
hours to six to nine that you were going to be here, work through the problems
and see how well you go and then what you don't know the next day you can you
can fill in. As far as resources, there is sample solutions. So you want to
practice you can take the exam and then come back and look and see how you did
There is also
a video. So tristin has put up a video of the solutions. So he's solving the
problems for you. So you can go ahead and watch that if you're having issues.
In addition to that material tristin is going to hold a online office hours sort
of thing on Friday. So if you went to this is the announcement that's up here
on the top of class. So he's going to have a q&a host a live q&a on the EDD
forums which the live q&a on the forum so far, hasn't been going very well. Not
very many people have been using it. But he's going to do that and he is going to
hold a zoom session that you can come in live to ask questions. I'm also going to
hold a zoom session from three to four on the day of the exam, so right before
the exam so you can feel free to think By there to ask any,
any last minute questions.
But that's how it's going to go.
So essentially, we're basically not going to have any lectures for the next
two weeks. So next week will be the exam. And then the week after that is
the mid semester break. So we'll basically have two weeks with no
lecture. But of course, next week, you'll be busily working on the midterm
exam, I'm going to try to turn around the marking as fast as possible. Marking
online is a little bit more slow than marking with a piece of paper in front
of me, but I'll, I'll do my best. Ah, are there any questions about format of
mid semester exam? What's going to happen? Anything like that? If you have
a question, please? You know, type me a message right now and I'll answer it for
everybody.
I'll give
I'm just gonna sit here for a couple of minutes and let you guys have a chance
to think and call it your questions. But if you have a question, please type it
down. If you don't want to ask the question to everybody wanted
anonymously, you can just for you can, you know, you can select to just have
the message go to me. And then I can answer the question. Yeah, so that's
right. So you'll need to upload the solutions just like the assignment. So
if I let me share my screen again. So under the assignment tab, there's a
midterm tab. Now,
when you go into the midterm tab,
you'll see
well, yours is going to look different because here I can actually make it look
like yours by coming in here and selecting student view.
So let's see.
So it's not available yet, but when it is available, it'll be just like the
assignment. There'll be a box that you can upload a file
My, my
hope is that the file
will
at some.
Yeah, please mute your mics when you come in. Yeah,
so just like the first assignment, that's how you upload it. And again, the
marking will happen the same way. So I'm going to use the speed grader thing
that's in there. So you'll get comments and stuff on the exam from me and a mark
all through through Canvas. Um, what's the date to post the MIT exam paper, so
I'm going to post it at six o'clock on Wednesday, the eighth. So next week at
six o'clock right on time on canvas, I'm going to post it. I'm going to post it
under module. So
if you
if you go to the modules, there will be A module up here at the top that says
mid semester exam, the exam paper will be posted there. You can download it and
complete it. When you're done. You go to assignments and you upload it.
So that's the plan there.
Yep. And you get, you get 24 hours. So like I said, I'm writing the exam. So it
can be done in about two hours in 10 minutes. So, I mean, essentially, if I
were you, I would know you were supposed to be in this class at six o'clock
anyway. So I would get the exam paper at six o'clock and spend six to nine
working on it. If there's anything you can't figure out in that three hours of
working on it as an exam, then you know, you have the whole next day to look
through your notes and things to try to figure it out.
But that's that.
So if your feedback on the assignment was that you needed Longer responses. I
mean, the point is, is that I give partial credit. So if you if you just
give me an answer, and the answer happens to be wrong, you're going to get
zero. If you give me an answer, and it happens to be right, but there is no
discernible work or I can't, there's no way I can tell how you actually got that
answer. Um, then you're not going to get full credit because I don't know where
you got that answer. I don't know, if you just you know, threw darts at the
wall, or if you knew what was going on. So, you know, give me an explanation of
of what's going to happen.
Can we do handwritten
or scanning so handwritten is fine. typing it is fine. Whichever way you
want to get it to me is fine. The only thing I'm asking is try to get it all
into one one document. If you absolutely have no way to get it into one document
and it has to be you know, four pictures or something like that. That's okay. But
just, you know, look it over first and make sure it's legible. So I, you know,
I have, there's something like 97 people in this class, I have a lot of these to
look over. And so I can't, you know, spend extra time just trying to parse
through what you're writing. So please just look at the file before your
Sundays and again, think about it a little bit and just try to put it in a
nice form, you know, the easier it is for me to market the better mood I'm
going to be in, you know, the more extra points I'm going to give. So you want me
in a good mood. To go back to the previous question about length, you
know, if you're looking for length, the the sample exam is a is a good guide for
what I'm looking for. So let's see.
I go to
sample mid semester solutions,
you know,
a couple of sentences for each of these things plus a diagram is all you need?
You don't need to do a lot. But there has to be, you know, there has to be
some work, you know, I have to see how the diagrams working there has to be in
an explanation there. It doesn't have to be a lot, but it can't just be a diagram
and then no commentary.
And I realized that
can't see what I'm doing. I was looking at the
actual
thing right here. But if you can see what I do, right, I do,
you know. So for the first half, it says George claims that have fallen G is
clearly permanent, because consumption has been increasing. You see, George
claims This proves his point using your diagram is storage. Correct? Explain. So
my answer is this does not prove Georgia's claim, you know, there's my
hypothesis, and then I explain what's going on and give you a figure. So if
that's your answer, that's fine. If all it was is George's right, here's a
figure. That's, you know, that might be a little too much. I want a couple of
sentences for everything. Okay, everybody feeling good
about the midterm?
Hopefully.
Alright, uh, you know, though, they'll be more chances to ask questions. Like I
said, I'm doing my best to try to patrol the AED form and ask questions. We're
getting some repeat questions. You know, if you come into the AED forum and you
haven't been there in a while, it's sometimes it's good to scroll through
some of the questions that you haven't seen before. And just make sure that I
haven't just answered the question that you're asking. Because I've had a couple
of people where literally, I have a question. Somebody yesterday asked that
question. And then they ask the question again the next day and I just have to
refer them back. So you know, take some time look through it. I've tried to make
some pretty detailed answers on there. Yeah, but I'm always always happy to
answer those and I'm trying to do my best to keep up with them as they come
in.
So the style of the question, so the question is came in so what will the
exam questions be in the same style and difficulty? So the exam questions will
be in the same style as far as difficulty. Um, let me put it this way,
the the
questions won't be
as won't be more difficult per se.
But for example,
if we go back to let me share the
screen again,
if I go back to question two, for example, on question two on this exam,
I mean, what's easier just to look at
this part? Question two on this exam is basically Having you solve something
which is directly out of the book. So because this is an open note, an open
book exam, I can't give you a question for which you could just copy right out
of the book. So you shouldn't expect questions like question two, the
questions will more be akin to sort of like what you see with question one,
right where I'm going to ask you to draw diagrams. Now the diagram itself you
might be able to find in the book, but the diagram you know, that you're going
to find in the book is not going to be one for one related to what I'm asking
you about in the in the in on the exam. So the exam questions won't be harder
than these questions, but a question like question two, you're not going to
be given a question that is just you could find the answer straight out of
the book. I mean, that that would just be pointless. So think more question one
style,
then then question two, but you know, the math and the I'm going to ask you to
do question two is about the hardest math I would ask you to do on this exam.
So by far the hardest math I'd ask you to do then would be a question like
question two, I'm not going to do anything crazy. For the most part. It's
going to be very reflective of what we've seen thus far.
Yeah, so our other way to sum up is the book will be helpful, but you shouldn't
expect to be able to find the full answers in the book. Like that would it
would be pointless if that was the case? The exam questions will be slightly
different from anything you're gonna find in the book, but the book will be
very useful
to do most of it.
Okay, excellent questions.
Anything else before we move on?
Now all right, looks like we're good.
Okay, so, um, let me share my screen and we'll open up
the slides.
So
let me open up the chat, box two. It is
anyway. So I want to go through, we have two more things to go through. So
everything that is going to be accessible on the exam is just what's in
the first lecture or the first five lecture weeks slide slide decks. For
lecture six, if you look on on canvas, what I have up here is an exam guide. So
the plan for tonight is I'm going to finish off lecture six slides, which
will probably take us until about eight o'clock, and then the last hour, we'll
go through the exam guide and I can answer questions that you that you might
have still related to the exam, but not Go through the exam guide to also give
you sort of like emphasis about, you know, what are the key things I'm going
to be going to be testing on.
I guess
a main point here, like the stuff that I'm going to teach you tonight, and to a
lesser extent, some of the stuff we saw last week, because we haven't had
homework questions or tutorials over this material yet, you shouldn't expect
it to be a large part of the exam. So I may assess parts of it. But if I was to
assess it, if you look at, for example, a this the old exam, the one I had
before, the problems where I asked you to solve models, those were based on the
stuff that basically were in the first four weeks of lecture slides and that we
had tutorial questions and homework assignment questions over and then where
the RBC stuff shows up, is actually down here in this true false uncertain thing.
So I may ask you questions that relate to this new material, but I wouldn't ask
you to solve, say the RBC model. Because you haven't had a homework question
about it yet, you haven't had a tutorial question you haven't had a lot of time
to, you know, to learn how to solve the model. But I could ask you just sort of
a general question about, you know, what are these models? What are the
assumptions under them? So questions like that, but I wouldn't ask you to,
you know, do full on out solve these models. Okay.
So
without further ado, let's pick up for where we left off last time. And last
time, we were basically playing around with our first real business cycle
model, and
basically the real business cycle model.
The idea behind it right is
let's take a growth model. And let's see how much of business cycles we can
explain with a growth model. And essentially, what we did is we took the
Ramsey model, and we put it in discrete time. Now we did discrete time mainly
because we actually want to, you know, explain economic data economic data
comes in mostly quarterly frequencies. So you can think about this as a
quarterly model. So we wrote down the Ramsey problem as described in a
discrete time framework. But then we added one extra choice in the Ramsey
model, there was no labor supply choice. Basically, we assume that everybody in
the household basically just worked one unit, and they couldn't vary that. So we
added an extra dimension on to this where we said, well, what if people can
vary their labor supply? Because when we, you know, look at recessions, right,
we think about people varying their labor supply, you see employment go up
and down. And so that that's a feature that we want to capture. So we were
laying this model out. And basically,
last time we discussed about
this model actually turns out to be a non trivial thing to solve.
Mainly because
we had an extra wrinkle to where we have this expectations variables, and here we
have expectations of nonlinear things. And so that just made it much more
difficult to solve. So we saw one example where you could solve this
thing, but we didn't really learn how to solve it generically. So basically, what
we're going to do tonight is we're going to learn how to solve rational
expectations models. I'm going to give you basically, a very generic
instructions on on how this works. And the methods we're going to learn tonight
are basically going to be applicable, applicable to all models that are, for
the majority of models, we're going to learn for the rest of this class and are
going to be applicable to basically any business cycle model that you want to
study going forward. It's sort of the building blocks of how you build a
dynamic macro model. So the recipe for doing this so the recipe for solving
sort of any dynamic stochastic general equilibrium model basically is going to
have three steps. Now, the first step is something that we've already done with
basically well all the models that had rich decision making problems in them,
which is we solve first order conditions for household and firm decision
problems. So we already seen How to solve first order conditions. We sort of
glossed over in the Ramsey model, but we spent a bit of time thinking about it in
the diamond model and saw you know how to construct an Euler equation in these
things. So, for all DSG models, that's basically going to be the first step,
right? You write down a household problem, you write down firm problems,
you can write down government problems, too, and more sophisticated models, and
you solve for the first order conditions. And of course, these are the
conditions right, that have to be true on an optimal path, something that
maximizes utility or maximizes profits for the firm. Now, the reason we do
that, right is because we want these models to predict sort of people being
smart and making economic decisions. So that's what the first order editions
tell you, right? They tell you, you know, what are the optimal decisions,
you know, how do optimal decisions look? Now, the second thing we do is we log
linearize. These first order conditions around a balanced growth path steady
state. Now the idea behind this is pretty straightforward. So
let's um,
let me stop sharing and go to a second and look at the whiteboard behind me and
let's recall
the
the Ramsey model. So in the Ramsey model
and in the Ramsey model, what we were primarily interested in right is we
wanted to know where the steady state was. And then we wanted to solve for
that stable saddle path
that tells you how to get to steady state, right and the The reason we like
this stable saddle path is because it said, All right, you give me an initial
level of
capital, I can tell you
exactly what consumption is going to be, that's going to put me on a path that
goes back to here. Alright. And then we talked about all the dynamics and
everything in this model. Well, in Ramsey, this is really, really
straightforward. Because there's only two variables, there's just consumption
and capital. What we want to get to larger models that have more variables,
and more sort of state variables and things that can move over time, such as
we want to actually model government spending that is not just a static
variable that goes up and down, but something that has dynamics to it
dynamics, like K, or we want to model technology that also can have short run
dynamics. Once we do that, we're now no longer in a two dimensional world like
this. We're in a multi dimensional world. And so we still want to do those
sort of the the intuition of what we do In this graph, so what we're going to do
is we're going to say, well, we want to find the saddle path sort of things. So
if we have a big multi dimensional problem that's hard to solve, what we
can do is we can simplify that into a problem which we know how to solve,
which is a linear system of equations. And we do
that by basically
narrowing our focus down to a small area around steady state. So we say, all
right, if I forget about the dynamics, and I just think about what has to be
true on the balance growth path, if even if the model is really large, if a
balanced growth path exists, I can probably solve
for that steady state.
Now, I can't draw this picture because it's in multiple dimensions, but what I
can do is I can approximate sort of what this thing would look like. And the
easiest way to approximate it is by doing linear
approximations,
so notice that you know, we get really close right here. There would be a
linear line
like this.
would be rk dot equals to zero, log linear. And, of course, the seed outline
already is linear, you know, we can just approximate those things and get linear
lines that if you're close enough to steady state, they work pretty well. And
so that's basically going to be our, our, our trick is we're going to take
these models, we're going to log linearize them. And once we log
linearize them, we're going to teach you ways to basically find the saddle path.
Well, in most cases, it will be numerically, but you're going to be able
to come up with functions that basically do exactly what we did in these
diagrams, but do it in a higher dimension.
So that's going
to be that's going to be our goal. Now, why do we log linearize things? Well, we
log linearize just for the simple fact that, you know, for small deviations for
small shocks, right, logs and percentages are basically
interchangeable, right? So I can I can interpret You know, a one a one point
log deviation and say like a 1% deviation. So basically just allows us
to think about our models and percent deviation from steady state, which is
kind of a nice way to think about economic growth, right? I mean, you can
think about business cycles as being you know, there is a trend growth rates a
2%. And then the business cycle bounces you around
that.
And the log linearization is just going to say, let's just treat all these
things as just percentage change from that trend growth. So that's the basic
idea of what we're going to do and why we log linearize.
Alright, so coming back to
sharing my screen.
So we're going to log linearize around that balance growth path. And then we
have to figure out okay, how are we going to basically solve for the saddle
path thing, how are we going to make sure that we stay on the saddle path,
and how we're going to do that is we're going to invoke the rational
expectations hypothesis.
So the rational expectations
hypothesis is basically going to give us a condition about what has to be true
about our forecast errors. And given this, given what has to be true about
the forecast errors, we're going to be able to figure out a way in order to
write the model so that agents correct their errors, once new information comes
into set comes into play, that will basically be equivalent to always
putting them back on the saddle path. That's going to be the intuition that I
want you to keep in your head. And finally, once you've invoked this
rationalization, I bought this you're going to be able to write the model
endogenous choice variables as a function of the model state variables,
which are essentially going to be say, like, basically finding a function for
How do you stay on that path. So So that's our goal. That might sound super,
super complicated, but I promise you, it's less complicated than you would
think. And if we will look less complicated, you know, once we do it in
sort of a simple framework, and then the idea is just thinking about how it
scales up. You know, the underlying mathematics can get complicated if you
want to go down that route, but it doesn't have to be. And so I think as
long as you keep the Ramsey diagram in your head, you can you can kind of keep
a handle about what we're attempting to do. Alright.
So,
to do this, I picked a model for us.
That is a simple RBC model. And I've used the notation and everything that
goes with this model from a paper. So if you go back to
modules,
you're gonna end up with this paper right here. And so this is kind of a
very famous paper. It's a paper that you know,
basically, you know,
textbook reading for how to solve these type of models.
You can
read up on And follow along, basically all the steps I'm going to give you are
written in this paper. And to keep things sort of nice, I've basically
written our model in the same notation that that he uses here. And in addition
to this, this is going to be really nice because the way that this paper is
written, or the way that this model is written is also going to be the way that
the model should be written to work with the computer program that we're going to
learn after the midterm to solve these models. So as I said, most of the time,
the actual solution that you're going to come up with you can't do by hand,
you're going to have to use a computer. And the way you go about doing it, the
way you have to write the model in order to make the computer
solvent is basically going to look
just like this. So so that's going to be helpful. So that's why I'm using this
example explicitly, and kind of deviating a bit from the book because
the book is using sort of its own notation. I want to give you something
that's that's going to be easier for you to translate into other things. All
right. So first thing to note about the way this problem looks and what we're
going to be doing here. So first of all,
on the maximization problem, right, the goal, the
paper, or the goal of the of the problem is still the same. If you can think
about what we did in the Ramsey problem, right, we want to think about maximizing
utility over consumption into an infinite horizon. We add something extra
here where there's a labor choice. l comes in with a negative, the reason it
comes in with a negative is because of the fact that you know, we don't like to
provide labor. So there's disutility
of labor.
consumption is the normal CRA consumption. And beta here beta is just
this e to the negative rho, our rate of time preference. In most macro papers,
we don't use this either the negative row that's normally reserved for
continuous time models when we go to discrete time. models, we tend to use
beta beta is the one you'll, you'll always,
you'll always see.
And that's what we'll use this. So I'm going to pop beta in here as well, I
think actually on one of my tutorial questions there, we pop beta in as well
when it went to discrete time and the diamond model, but beta is just another
way of writing the rate of time preference. So all that matters, right,
of course, is this, this beta is less than one, so that this thing converges.
Now, the problem we're going to solve here is called the planners problem. Now
what makes it the planners problem is we're going to assume that the budget
constraint for the agent are the economy wide variables, right? Its consumption
and investment has to equal income, the capital accumulation equation and the
production function. Now, this is an RBC model with perfect competition. So the
model the whether or not you solve the planners problem, or whether or not we
wrote out a budget constraint that had wages and interest rates, rather than
You know, the actual production function and capital accumulation, regardless of
how we wrote it, we're still going to get to a Pareto optimal equilibrium. So
there's nothing for a planner to improve upon that the agents wouldn't come to
there that wouldn't argue to on their own. So you can switch the planners
problem, or do the, the individual agents problem, you get to the same
answer. Now, that's not always going to be true in these models, when we get to
models that are going to have market frictions in them, what the planner
would do, what a government would do, if they were all powerful turns out to be
different than what an individual would do. And the reason is, is because
individual has to take wages and interest rates as a given they don't get
to affect those where and a planners problem you think about them being able
to affect these things simultaneously. The only reason for me to go into this
is just to say that
you
can get different answers, whether it's Depending on how you set up the problem,
I believe we set up the problem different in the beginning. So if we go
back up here to the original model we looked at.
So this model
Oh, actually, oh, yeah, no, we did we set it. Well, we we set it up with wages
and routers, we actually didn't write out the, the, or the budget constraint.
But the idea is you could write out a budget constraint that had wages and
interest rates the same way we did with the original Ramsey model, or you can
write it like this, Either is fine in this way, because it's a perfect
competition model. When we get away from over competition, you can't always do
that. But we're going to do it right this way, because it's just easier the
first time around.
Okay, um,
this is going to be a slightly simpler problem than the one we solve last time.
The only stochastics goes stochastic part here is going to be this log at so
productivity now is gonna follow this AR process. where epsilon t is this white
noise shock. So that's what's also going to be different about this. And just the
Ramsey model is now we have this uncertainty, right. So TFP can move
around, and the agent has to take that into account when solving it. Alright,
so how do we go about solving this problem? Well, we go about solving this
problem, the same way that we've solved any of our problems is we look at a
Lagrangian and I think maybe it's best if we work through this together and
then we'll have the slides to look at so I'm going to stop sharing my screen and
we'll set up the problem and and go through together I should say before I
do this, is there are there any any questions
have I lost anybody?
yet?
Let me know if you don't like the color if the color I'm using doesn't show up
well on the video, please, please let me know and I can switch to a different
color, but I'm going to go with blue for right now. So hopefully you can see
blue.
Okay. So
the grungy and trick that you can do is that we briefly discussed with the
Ramsey model in continuous time, and we saw in full force in the diamond model
is even if you have an infinite horizon problem, or if you had a model that had
more than one period in it that are more than multiple periods and in overlapping
generations model, you can always simplify these problems down into two
period problems.
So, when you write a Lagrangian,
and I should say that this is, this is not the most mathematically sound way to
do this. There. More powerful techniques you can use. But as I said, I want to
give you one sort of toolkit to solve all the models. So we're always going to
take Lagrangian technique. But the idea is that even if you have
this complicated, you know, Infinite Horizon
problem, you can always break it down into just a two period problem. So what
do I mean by breaking it down into a two period problem? Well, first of all,
let's just think about
what the problem is
or what
let's let's just write the problem now.
So
the problem is, is we want to
maximize utility,
subject to
a budget constraint,
which is Essentially,
well here, let's, let's just substitute some
things in.
is
capital today
is equal to new capital or essentially
income
minus consumption, that's investment
plus
how much capital we have leftover from the period prior. Now the thing that's
different about the way we look sets up the problem even from way sometimes it's
presented in our book
and from what we saw actually in
the diamond model is that we're going to use this this notation where kt instead
of it being Katie plus one and Katie, we're going to do Katie and Katie minus
one
For our production function,
we're going to assume that
production
depends on
last periods capital stock. So the capital that you invest today, the new
capital you invest today, you don't get to produce output with it until
tomorrow. So it's only last periods capital matters for production today.
And the other thing we're going to do is rather than looking at a labor
augmenting TFP, we're just going to move the TFP parameter out front. It doesn't
really change much if you do it, but like I say, this is the way who sets it
up. And so I want to stick one for one with with his assumptions, so you can
have his paper to follow along as well. All right. So what do we have here? We
have an objective function, and we have a budget constraint. This says we We
want to choose consumption, we want to choose labor every period. And that is
subject to the fact that the economy has a constraint, mainly how much output we
can produce in any given period is governed by this, mainly how much
capital we have, and how much labor we choose to supply. And then it's governed
by how much capital we're gonna have next period is how much we have to
invest. You know, plus whatever hasn't depreciated. So when thinking about this
in a two period problem and Lagrangian, basically what you can do is for your
objective function is you can just break this into two periods mainly think about
in period one, you want to maximize this in period one subject to this budget
constraint. And the next period you want to maximize the period in the future
subject to this budget constraint, advanced one period of the future. If
you put that all in a big Lagrangian, and solve for all and differentiate with
all your choice variables, you're going to get the right answer. So what do I
mean by that?
Well,
Let's set up our Lagrangian.
Our Lagrangian is going to have
utility in period one.
So C T one minus theta, one minus theta minus one
minus omega lt
Plus,
it's going to have utility utility in period two.
But
utility in period two needs to be discounted,
and it's going to be discounted by beta.
So this is period two, so it's c t plus one
and
lt plus one.
Okay, so those are our two periods up here. Now what we need to do is we need
to think about what are the conditions
Well,
we have a Lagrangian multiplier. And the constraint for the first period is going
to be kt minus y t, which is 80 k t minus one. Or sorry.
Sorry, I could mess up mine.
We got messed up my signs here. I'll keep going. I
think I may have messed up my son.
Yeah. This should be minus kt plus a t
minus c t
plus
one minus delta
k t
Let me just make sure I
know my signs Right. Yeah.
So that's the budget constraint for period two.
For the next period, I have beta, I need to discount it. I have a Lagrangian
multiplier t plus one. And I can just advance the budget constraint forward
one period
and I forgot one more thing.
These are of course in the future. So these are expectations
of these things.
Okay, so This is a two period Lagrangian. So anytime you have one of
these problems, anytime you, I give you or you know you you pull up a paper
that, let me
share my screen for a second,
you pull up a paper that looks like this, right, you now know what to do.
Essentially, this trick will always work, you can always break the problem
into two periods. And take first order conditions of this thing, eliminate your
Lagrangian multipliers and you're going to end up with the first order
conditions for the whole problem. And this is pretty powerful. So for example,
something that I put up if you guys are interested in I put up some papers that
look at macroeconomics and pandemics and we can look at this paper scroll down
Look at what their model is. And look what their model looks like, right? It's
a utility function maximize two things over an infinite horizon. There's a
budget constraint. If you want to solve the model that they're looking at, in
this paper, you can literally use what we just did there, you can break it into
a two period problem,
write out the budget constraints, and follow the recipe we're going to follow.
So this this can, this can always be applied. And you know, this is pretty
much how you pull out any macro paper, regardless of what the model is, like I
say, I mean, this model is a simple macro model that has epidemiological
model plugged in tight inside of it. They're all going to have this basic
structure.
All right, is anybody
is Everybody with me about how to set up the Lagrangian, and what the intuition
is?
Okay, I don't see I don't see any complaints. Okay.
All right. So now what do you do?
Once you have this Lagrangian? Well, basically you just need to figure out
what your choice variables are. And once you've figured out your choice
variables, you find first order conditions with respect to those things.
Um, now what are the things you can choose in here? Well, you can choose
consumption. You can choose consumption next period. You can choose labor, this
period, you can choose labor next period. Um, you also can choose capital,
how much capital you're going to have next period. You can take first order
conditions with respect to all of these things. And the great thing is is the
Lagrangian multipliers will
keep tracking have everything for you and make
sure that you get to the right answer. So in some sense, if you don't know what
your choice variables are, you can just say, Well, what are all the things I
possibly could choose? Take first order conditions with respect to those things.
And the Lagrangian multipliers will simplify everything down. Now, it turns
out in this model that you actually don't have to do quite that many. All
you need to do, I believe, are actually choices with respect to four of the
variables mainly, you just need to think about what happens with consumption
today. Your choice with consumption today, consumption tomorrow, capital and
labor supply today. Those will be all that are necessary in order to solve the
problem. But like I say, if you're confused about that, and you take extra,
it won't matter as long as it actually is a choice variable. The Lagrangian
multipliers will will keep track of you will keep track of these things for you.
So what does this look like? So let me
share my screen again.
And let's look back at the slides.
So, as we saw, here's your Lagrangian. Hopefully it's exactly what I wrote on
the board. It looks to be the case that it is.
Oh, actually, it's Oh,
was almost the case.
I had one error on there.
All right. So that's that. Now, as I said, step two, you have your first
order conditions, you take partial derivatives with respect to the choice
variables and set them equal to zero and two is eliminate the Lagrangian
multipliers. So as I said, there are really only four for certain conditions
you need to take, which are electric is electric side correct. Now let's see if
the electric side is correct.
I think the lecture slide is correct. I think I was wrong. We look up here. I
should have had Katie and this should have been Katie minus one.
I think everything else
in the lecture slide is correct. Yeah.
This k t minus one lt k t minus one k t Yep. Lecture slide is right, I was
wrong.
Okay.
So, first order conditions, what do we do?
Let me
go back to the board and let's rewrite this a bit smaller. So we can take some
derivatives together.
So our Lagrangian
Lagrangian is
let's go. Yeah plus at a T minus one alpha lt one minus alpha minus ct plus
one minus delta k t minus one and plus beta at lambda t plus one minus eight t
plus one. plus a t plus one, k t l, lt plus one one minus alpha minus c t plus
one plus one minus delta k t.
All right. Um, if you want to know a trick
to know, how did you get your signs right and your Lagrangian multipliers?
Well, if you remember what your Euler equation should look like, it's going to
be obvious to you. And I'll show you why. So let's look at the first order
condition.
dl DCT.
What does this first order edition look like? Well, where are all my CTS now
there's a CT right there. So that's going to be c t to the minus sigma. And
then coming along here, coming along here, coming along here. Oh, there's my
next CT.
It's gonna be minus lambda t
equal to zero.
Now,
the reason you can tell whether or not you got your signs right is you always
want your lambda to have a negative sign mainly because it should always be the
case
that the marginal utility of consumption is basically equal to your Lagrangian
multiplier. This is a phenomena This is something that will come up over and
over and over again in these problems. So, as long as you got the negative sign
right there, then you set your problem up correctly,
it comes out the other way, then then you did something wrong.
Now,
do it with C t plus one, what do we get? Well, I get beta e t
c t plus one to the minus sigma
coming along through here I am With minus beta e t
lambda t plus one
equal to zero
or
e t, c t plus one to the minus theta is equal to
e to the lambda t plus one.
Notice I had beta and beta and a zero over there so I can divide those out.
Moving that over the other side.
So we're left with let's see.
All right. And,
finally,
take the derivative
with respect to
Katie
Nolan He's up there, my first Katie ends up right there, I get
negative lambda t.
coming across over here, no Katie's, I come over to here, I get
plus beta
et lambda t plus one alpha a t plus one, k t alpha minus one, lt plus one
and
plus one minus delta is all equal to zero. All right, from these three first
order conditions. If I get rid of my Lagrangian multipliers here, I can
basically recover an Euler equation. To see that, notice I have C T equals the
lambda here I have C t plus one equals to expectation lambda t plus one,
I have lambda t, e t of lambda t
plus one in there. So I can just plug those things in. I rearrange that thing.
And I'm basically an Euler equation is going to pop out.
Let me just go back to the slides for a second.
So I ran out of room here. But if we do the one for labor,
you're going to get
a fourth first order condition that's going to look like that.
Now,
if you start collapsing these things down, so like I say, eliminate
Lagrangian multipliers. So as I just said, right, you have lambda t,
lambda t plus one,
I know what lambda t plus one and lambda t equal. put those things together and
rearrange, and then notice that you can plug in a definition for the real
interest rate.
You're exactly going to recover an Euler equation.
Eliminate the Lagrangian multipliers
from 15 and 18.
So we have a lambda t here, so we can substitute that in. And we're going to
get a condition that basically tells you how to equate the marginal utility of
consumption with the marginal disutility of labor. So this is going to govern our
labor leisure choice. For the most part, though, what's really going to matter
for us is this equation up here, because this is the only dynamic equation this
equation is static. Meaning, you know, once you once you have the current
period variables, you you know how much labor you're going to choose. Alright.
If I rearrange those variables, I can write them
this way.
an Euler equation,
an equation for the real interest rate and an equation That basically relates
labor supply and the marginal utility of consumption. So basically tells me, you
know, how much do I hate, hate working? If I if I were to graph these things on
essentially what I've done
is I've created for myself
a labor supply labor demand model,
just like the one
that we discussed last time. The upward sloping labor supply curve basically
comes from that last condition. The downward sloping labor demand curve
comes from the diminishing returns
assumption on
the top left was production. But that's basically going to be the labor market
that's going to underline this model. All right.
We're at seven o'clock.
Let's, uh, let's take our first break. I'll give you a chance to digest that.
If you have any questions, please type them into the chat bar and we can stop
and go through this again because I
realized that
this step to this step to this step,
these are big steps. So, gather your thoughts, give me some questions and we
can walk through these things. So I'll start back up at 7707
Okay cool.
Okay
in the slide sending up the model mu What does k minus one and A zero mean.
So k minus one and a zero is to make the problem well defined. So think about
those growth rates, you have to have an initial level of capital, and you have
to have an initial level of technology. So think about what happens if you
started with zero capital. If you have zero capital or zero technology at the
beginning, then you have zero income. If you have zero income, you can't do
anything in the model, you can't consume because you have zero income, you can't
invest because you have zero income, you have nothing to invest. So in order to
make the problem well defined, there has to be some initial level of capital and
some initial level of technology. So that's all those are there for there
just to make the problem well defined. So I mean, in some ways, you know, the
way I kind of structure this class, is, you know, most of you, I mean, some of
you have come right from undergraduate economics degrees. And so you're very
familiar with a lot of these concepts, but many of you it may have been, you
know, seven or eight years since your last economics class. Some of you may
have Little economics training at all. So my my point to this is I'm going to
show you sort of what a well specified problems look like. But then I'm going
to try to simplify them down just enough so that you can work around with them
and play around with the intuition of it. And then the idea being that, you
know, if you're really, really interested in this stuff, you then have
a base to start building off of and go more into, you know, making the problems
wholly concrete and, you know, tying off all the loose ends. So,
I, they're there, but
for the most part, you can assume that I'm giving you problems where there is
an initial amount there is an initial amount of that. So that's what I have to
say about that.
Ah, okay.
So let's let's just go ahead and let's do all of these
just one more time. I think it's useful to see these a couple of times. Where
was that the only question there no other no other questions? All this is
crystal clear.
All right, well, we'll pretend as if it is even if it isn't. Alright. So again,
this is our Lagrangian, we've broken it up into a two period problem.
There are lots
of choice variables up here, the four that you need to do to solve this
problem, the minimum four are these four. As I said, if you did lt plus one,
or you did kt plus one, you tried to do those, the Lagrangian multipliers would
make sure that the problem still worked out. So you don't have to worry about
that. But these are the minimum ones you would need. Alright, so differentiating
with respect to CT, there are only two CTS up here, there's that CT, and
there's that CT right there. The derivative of that thing is CT
minus theta.
And this is minus c t times
lambda t. So that's c t minus theta minus lambda. marginal utility of
consumption minus lambda t equals zero c t plus one, we get beta e t c t plus one
minus theta.
And then down over here we get minus
beta at lambda t plus one equals zero for k t, we end up with
minus lambda t
plus beta e t lambda t plus one alpha a t plus one, k t alpha minus one, l t
plus one one minus alpha plus one minus delta
equals zero.
And for lt we have
minus omega up there.
And then we have plus lambda t at kt minus one alpha B. One minus alpha at k
t minus one alpha, lt minus alpha.
and that's it equal to zero.
So there are your four first order conditions. Now let's go about
eliminating Lagrangian multipliers. So
why don't we do that in a different color?
the fun of it.
This one is C T minus theta equal to lambda t.
This one implies
e t c t minus theta is equal to E t, lambda t plus one.
This one is going to be
oh geez.
Well, why don't we
must easiest way to do this, I guess we'll write down what I
kind of liked writing it the other way though. So,
switching around beta t t, lambda t plus one, alpha, a t plus one at alpha minus
one equals one, one minus alpha plus one minus delta and This one would be omega
is equal to lambda t, one minus alpha at kt minus one alpha, lt minus alpha. All
right, this equation, we can get rid of the lambda T's by substituting these in.
So maybe we can do that
over here and red looks like you guys can still see over there.
So what are we going to have here? We're going to have
see t minus theta is equal to beta times expectation of c t plus one minus theta.
And then, on top of that,
we have all this other
Oh, by the way, this is
there should be a bracket around that. My apologies, right? lambda t plus one
multiplies all. Now, all of the stuff in here, right? What is this? This is the
marginal productivity of capital minus depreciation. What is the marginal
productivity minus depreciation? Well, that's the real interest rate. And in
discrete time, that's actually the gross real interest rate. Meaning you can
think about this as big R t plus one is equal to one plus little r t plus one
where that's the interest rate you normally think about this is one plus
the interest rate. So if you think about, if you, you know, invested $100
at a 1% interest rate for one year, you would get back $1 plus your principal,
the 100 plus
one Your gross interest rate, so we just substitute that out to be a gross
interest rate.
If I
divide both sides by c t minus one, I end up with this equation, which is
clearly an Euler equation, right? That's clearly an Euler equation. It's got a
growth rate of consumption. It's got a rate of time preference. It's got the
interest rate. It's got intertemporal elasticity of substitution or relative
risk aversion, right, all the pieces that we need for an Euler equation, here
they all are. So that's, that's all that is. Now, we have to write it like this
because we have this expectation thing where we have this multiplication
problem with these expectations. So we can't just pull these things out. The
way we're going to get around this is we're going to get around this by by log
linearizing. But this is the point fitness is this is how you solve it, you
go like that eliminate your multipliers. This one is sort of trivial, right? It's
just a lambda t, you just plugged, plug that in there.
And you're able to write the economy like this.
So is that is that clear what the strategy is? Right, right, the
Lagrangian multiplier, take derivatives with respect to your choice variable,
set them equal to zero. Use those equations
to eliminate Oregon's multipliers
to basically find the equations of interest, which in this case, is always
going to be an Euler equation
and a labor supply decision.
Once we get those equations,
right, we can just write them down and we
have the startings of a fully fledged dynamic system where notice two things
here so we have YT over k t minus one and YT over Lt. These are just using the
sort of Standard properties of the Cobb Douglas production function right?
So if we think about
Cobb Douglas production
if you take the, you know, well
clearly right if you right you can do well here. Let's let's just do it.
Let's just do this one
back again.
So, notice right that
with
right, so that's the
Cobb Douglas production function
if I divide both sides by k T minus one
What am I going to get?
Well,
I'm going to get at k t minus one alpha minus one
lt one minus alpha,
which is
partway there to being the marginal productivity of capital, right? What
are we missing? All we're missing
is multiplying by alpha. So
it's just using the fact right that
when you have Cobb Douglas production, you can actually
substitute output back into it. And we saw the same thing for wages right? for
wages.
In the just like the simple one, right, we learned that
wages with Cobb Douglas production, oh, that
was equal to something like that.
Well, the exact same thing can be shown in this model, with wages being one
minus alpha over YT lt So it's basically just exploiting these properties to get
the model in terms of yt. Um, the reason we want to do that is, if you think
about sort of the endogenous variables you want to explain YT can be a much
more handy function to work with than a TLT. You know, it collapses a whole
bunch of other variables into one makes things a little bit cleaner. So it's a
it's a common trick that you'll you'll see us do throughout. Okay, so these are
the first order conditions from the household side of the economy. This is a
real business cycle model. So we still have perfect competition on the firm
side. So it's still sort of uninteresting, right? You just have real
wages and rental rates being renewed rated at their marginal products. So
that part we don't really have to
have to worry about.
So this is basically it. We've completed step one.
So if I go back
up to my recipe, what do we have to do? We had to solve the first order
conditions of the household and firm decision problems. Well, the only really
interesting problem was the household problem. We've solved that we now have
first order conditions.
Now, what do we do with this problem?
Well, now we want to log linearize, the first order conditions around the
balanced growth path steady state. And maybe here, it's a good time to remind
you of sort of the goal, right?
So if you look at this,
remember, right that we're attempting to get
something like our Ramsey diagram.
Right? This is this is our goal, which you can't see that but you will be able
to now
our goal, right is to get
to something like our Ramsey diagram,
right here. And what do
we have in that Ramsey diagram? Well, we had a C dot equals to zero and a k dot
equals zero. So we had basically an Euler equation that told us how
everything moved in z space. And then we had a capital accumulation equation that
told us how everything moves in k space. Now the extra difficulty that we're
going to have in this model
is there's actually going to be extra dimensions, right?
Because now it's not just capital that depends on output, and hence
consumption. But labor matters, too. So there's a third dimension. And it turns
out, we're going to have a fourth dimension because now there's going to
actually be dynamics in productivity as well. So we're gonna have a fourth
dimension that we have to wrangle with. But you can see we already have the
building blocks. So basically, we have an Euler equation, which would tell us
the dynamics here. And in order to simplify the problems, we can actually
solve it we're going to
have to linearize
okay. So log linearization essentially all logs authorizations are going to go
the same way. And they're, the concept is going to be straightforward enough.
Essentially, we're going to do a substitution for all of our variables.
And then we're going to do a first order Taylor approximation.
Now, the substitution is just going to be
a clever one, so that we can interpret our models as sort of percent changes
from steady state. So how are we going to do that? Well, first, I just want you
to notice sort of the progression of this rules of logarithms, right. So if I
had a variable x t, I could write that variable as the log of the steady state
plus
some value, which this value is going to end up being the deviation from steady
state. So if I have some variable x t, I can approximate that as the log of its
steady state plus some percent deviation change around it.
If I exponentiate, both sides of that,
I'm going to get the An equation that looks like this, where I have x t would
be equal to x times the steady state times e to the x t, where t is the
deviation from steady state. Now why this is a clever choice A clever
substitution you can make a clever way to write your variables is because the E
function
is zero. If you raise if x t is zero, meaning you're at steady state, then e
is equal to one. So e raised to the zero is equal to one. So therefore, x t would
be equal to steady state. So essentially, by writing my variable like
this, I've centered that variable around its own steady state. And now I can
think about deviations around it, I can then study this x t variable. And that's
essentially what we're going to do. We're going to say, for every variable
in your economy, let's just replace replace it with a variable that looks
like this. And then let's take a first order approximation of the economy
around This variable right here,
so it's just going to be a clever trick.
So what happens if we do that? Let's take that first order approximation now.
So let's define our variable to look like 22 here. And now let's take a
Taylor approximation just to first order one around x t equals zero.
Well, if we do that,
it turns out that either the zero is one, either zero is one, and then what
do we do for a Taylor approximation? Well, you need to take that variable and
minus it from the point that you're trying to expand around, which is zero
in this case. And so you're left with one plus that deviation from steady
state.
So we took something that was
nonlinear
first order differentiation, made it linear, and it made it linear in a very
straightforward way where now we can basically just read off that this is
deviation from steady state.
It turns out that it's
actually very Easy to start then applying these rules. So I'm actually
going to skip this and go to this thing. So basically the rules of logarithms,
the fact that we're using the E function to basically do all this means that the
rules of natural logarithms are actually going to be a nice guide, or simple
rules for how to do this. Now, of course, you could always just use the
Taylor approximation theorem that we learned earlier to think about
convergence. But you don't really have to you can learn just simple rules for
doing this. So essentially, if we wanted to think about log linearizing, our
production function, how would you go about doing this? Well, you make the
substitution that we said so we have K to the alpha, you replace eight kt, with
its steady state value times e to the k Tilda t,
where our capability here
basically stands for deviation from steady state.
If you do that,
you can then start simplifying mainly you can carry that Alpha through. So
you're going to have KSs raised to the alpha and e to the alpha k t. And then
you can basically apply the basic rules that we've learned, which is if you have
E times something times that x t variable, the first order approximation
is that is just one plus whatever was multiplied by it by that variable
itself. And that's basically it. So we've now taken K to the alpha which is
nonlinear. And we've logged linearized it in a straightforward way. And we can
use these rules over and over again. So let's think about how this works now for
our Euler equation. So let's let's do this together. I'm gonna go and do this
on the board.
Alright.
Okay, so we have to walk linearize these things.
If you think about deviation from steady state, on that side, fairly
uninteresting. It's.
So let's focus on on this side of the equation.
What we will do
is you are going to replace all of your variables
with
those expressions,
a little RT,
just like that.
And then you can basically just apply the rules that we saw on on that last
slide. Mainly, if we have something like this on, it's basically going to be as
if we as if we took logs of it, but multiplied by what its steady state
value was. So in steady state, when c t is equal to zero, c t plus one is equal
to zero RT plus one is equal to zero. We're going to have beta CSS, over CSS,
the minus sigma or minus theta minus theta.
RSS
and that's going to be multiple
by
one
minus theta
e t c t plus one.
Now this is in the new denominator so the rule of law is going to be a minus a
minus plus
c to the T
and then
plus
our Tilda plus one. Okay
on the other side of it
You're just gonna have one.
Now
how did our become our
so we're just making are saying substitution, right RSS e r tilde. I
mean, it just, we could call it big R, it doesn't matter. But the point is, is
that this is going to be deviation from studies day. So that's that's the key
takeaway from this is we're now taking all of our variables and turning them
into deviation from steady state.
Okay.
So if we simplify this all down, we're going to have one is equal to
beta r s s,
one minus theta
et CIE t plus one hat.
Hold on loss
t plus one
gotta be expectation T.
Okay.
Now, what you need to do to further simplify this is you can then think
about what the steady state relationships are going to be and see if
there is any steady state relationship that can help further simplify this
down. And but what I mean by that is, is you can think about, well what happens
if all of these things are zero. So right, so C t plus one, this is
deviation from steady state if your app steady state is equal to zero. So, let's
come over here.
And let's just note
at steady state, that thing implies one
is equal to Beta RSS.
So, if this is equal, if one is equal to beta RSS,
then
notice that beta RSS here is actually one.
And hence, it's going to drop out.
So at steady state, right one is equal to beta RSS, we already have a one here.
So we have one beta RSS one, times one.
And we're going to be left with
so you guys can see that
zero is equal to
E t, one minus theta, c t plus one
plus theta CT
minus
r jilla t plus one,
which if we
go back to our slides
is what we're left with right here.
So, noting that in steady state,
we get one is equal to beta RSS
and or one over beta is equal to RSS, then log linearizing, we're going to get
something that looks like this. Plugging in the fact that the steady state this
thing is equal to one that's equal to one
gets us
to the final equation. So that's, that's basically the strategy for log
linearizing. So figure out what the steady state relationships are so that
you can then reduce things. Take your log linearization apply all the
reductions you can and simplify down the equation as much as possible.
A couple things to notice
here the interest. So we also noticed that in steady state, one over beta is
equal to RSS. Um, this is normally a way to think about what the interest rate is
so in the interest rate in most of
these economies
ends up being pegged
by the rate of time preference in the economy. So before, right, we had this
notion that interest rates were always tied down by the marginal productivity
of capital. Well, now it's going to be the case that people's actual time
preferences are going to start interacting with that in steady state,
it's going to be the case right that how much capital you have and how much
therefore the marginal productivity of capital is actually going to be
determined by how much people like to save. And how much people like to save
is going to be determined at somewhat by their rate of time preference. And in
this model, and steady state, it's basically going to uniquely pin it down.
So it turns out that one over that that preference that one of our beta is
basically actually going to then pin down your interest rate. So notice it's,
you know, in the solo model
and all these other models where we were thinking about these things were pinned
down by themselves. That's because savings rate was exogenously given in a
model now where the savings is endogenous where the agents get to make
the choices now people savings decisions determine how much capital there is how
much capital there is then determines the interest rate. So these things get
jointly determined.
So we have that going for us, which is nice. Alright,
I'm in
your next homework assignment, which,
luckily will not be until after the midterm, you're actually going to get
have a chance to log linearize and go through a model from scratch. So that's
we're going to have have you do like I say, it's not as hard as it seems. It's
confusing when you first do it, but you know, once you start playing going
around and get a couple of equations and have a look at it, it comes out very
naturally. Alright, if we were going to log linearize every equation in this
economy in order to describe its dynamics, basically,
this is what you're going to end up with. So
we just saw how to do the Euler equation. On the first three equations
up here, though, are just these three equations here, log linearize, taking
the exact same steps. So log linearize, those three equations. In addition to
those three equations, we need the production function, we need the capital
accumulation equation, and we need this 80 variable which where does this at
variable Well, this 80 variable is going back all the way to here. This is the
the, the stochastic productivity process. So you have to log linearize
that as well log linearize on Of these things. And you're left with a system of
equations that look like this. Now, this is the dynamic system that basically
gives you what we saw in the Ramsey model. So this is your C dot. Well,
there's your C dot equation, there's your k dot equation. Now, of course, the
problem is right is we don't just have a C dot k dot this time, we now have an 80
equation that's dynamic. We have a production function that has dynamics in
it, we have an interest rate that now has dynamics in it. So we have lots more
variables that we have to keep track of.
So what do we do? Well,
now this slide is maybe summarize exactly
what I said. Right? We have a system just like in the Ramsey model to
characterize the dynamics of the economy locally around a steady state balanced
growth path. Um, the last thing we have to do though is is how do we solve this?
How do we actually figure out what that that saddle path is going to look like?
And, you know, what do we do with this expectation variable. And it turns out
that the expectation variable is going to be the way to sort of close the
model. This is how we're going to determine how to find that saddle path.
How to how to think about the system is how do we do this expectation variable.
So step three,
invoking the rational expectations, hypothesis. And step four, what does
that mean for signing a solution? That's where we're headed to next. So
conceptually, the idea behind rational expectations is that we want agents to
form model consistent beliefs. And what that means is that that expectations
operator that
we're putting in this economy is taken against the
assumption of the models, meaning that we're going to assume that the agent
knows the model and therefore can make a perfect expectation given all the known
stuff. Now remember, because the TFP process
is stochastic,
it's actually has an unknown part right? That innovation variable, the white
noise shock those shocks are unknown to the agents. So the agents can make a
mistake, meaning they can't perfectly forecast those shocks. But once a shock
is known, they can perfectly forecast the effects
of that shock.
Now, this is a
strong assumption, right? It's going to imply that the agent knows the structure
of the economy. And also is going to imply that they know the distribution of
the shocks that affect the economy. So they they don't just they don't know the
individual realizations, but they know the distribution. And more than that,
it's going to have this idea of the common knowledge of rationality right,
that everybody else knows this, so I don't have to worry about people doing
dumb things. I think everybody's just as smart as me. Um, so there's a lot of
assumptions here. So why do we study rational expectations? And why
is that the benchmark? Well,
I think naturally, the
natural benchmark you want to think about in any economy, right is what
happens is everything everyone gets it right. It is true that people are very
smart. And certainly if you study financial markets, you will see that
that people basically figure things out very, very quickly. People are very good
at figuring things out, especially when there's a lot of money on the line. And
so a lot of times rationality is a good way to describe people's behavior. Now,
that is not always the case. And we have a large literature of economics that
studies cases where there are repeated failures of rationality. So it's not
always that case. But the issue is, is that if you want to make an alternative
assumption, to solve your model, you need to then
justify why people would make systematic
mistakes. And if you're going
to do that, you need a lot of evidence to support the fact that your agents are
going to make these type of mistakes. Otherwise, you know, you're just saying,
Oh, these agents are behaving dumb, and I don't know why they're behaving dumb,
but here you should listen to you should look at my model and believe what it
says. That turns out to not be very convincing.
So if you think about the way these models are normally
solved, it's either you do rational expectations or you do expectations and
you compare it to some behavioral assumption about how these expectations
might be formed. But that behavioral assumption is backed up by a lot of data
such as you know, laboratory experiments or data on people's, you know, you've
actually surveyed people on the expectations, they hold about these
variables, and you have a model for how they form those expectations.
So that's basically the way we go
on another justification is you can think about rational expectations as an
as if justification, right? So you know, most golfers don't understand the
physics of what they're doing. But you know, physics will still predict what
they're doing very well.
So that's,
that's sort of the defenses of rational expectations.
Now, to show you
exactly what rational expectations is doing, it's actually going to be simple.
If we actually are simpler if we move away from the RBC model, because the RBC
model that we've just log linearized you can solve by hand
and I substitute it with a model that
a simplified version of the RBC model, it's going to have the same features as
it as far as the dynamic properties of the model. But it's going to just be a
two equations. simple case. So that's what I want to do. Next, we're going to
go to a two equations, simple case. And I'm going to precisely show you what
rational expectations and the solution means in that method, build intuition.
And then we'll jump back into the RBC model, which again, I said, you have to
solve with a computer. All right, so what's our simple model that's going to
have the same types of properties? Well, we're going to look at a simple asset
pricing model. So we're going to suppose that investors in the economy having a
choice between investing in two assets, y and b, y is going to be a risky asset
and B is going to be a safe one. The Safe asset is going to return R, which
we'll assume is known and fixed. So think about this as our RBC model except
the interest rate doesn't move. It's known and fixed. The risky asset pays it
down. dividend D and that dividend process is stochastic. So think about
this as income in this model income in the model is driven by TFP, and nothing
else. So that TFP follows an AR process. So this dt is just like that at process
we saw.
But
that's the only sort of uncertainty it's in here. So all you have is an agent who
they know what the interest rate is. They just want to decide whether or not
they want to buy an asset that pays this dividend stream, or they just want to
accept the the risk free rate. Now, if you think about what's going to happen
in this market for risk neutral investor, you can say any risk neutral
investor is going to want to equate the expected returns of investing in the
risk free asset versus investing in the risky asset.
In the risk free asset.
You're going to earn one plus the interest rate times PT. So by invest PT
dollars into the risk free asset I get my dollars back plus the interest rate.
If I go for the risky asset, what do I get back? Well, if I sell it next
period, I get the price of what I sell it for
Plus,
I get the dividend that that came out of it, which of course is stochastic,
right? It could have been a high dividend, it could have been a low
dividend, it could have been zero, because those are shocks that I can't
predict. I know the mean, but I don't know anything else. I could know the
standard deviation, but I don't know the individual realizations.
If I move one
plus r over the other side, rename this one over one plus r beta, I'm going to
get an equation that looks like this. And notice this equation is basically
going to have the same form as the Euler equation that we studied up here, right?
If I subtract CT over to this side, what am I going to have I'm going to have CT
is equal to beta CT plus one plus RT plus one where RT over there so gastic
instead I have PT, beta times etc. plus one plus this other stochastic piece. So
the structure of this, the dynamics of this economy are going to be very
similar to the RBC model. But the nice thing is there's only going to be two
dimensions, we have a price process and we have a dividend process. That's all
we have. Okay, this thing up here, this dynamic system, similar to the dynamic
system, we saw in the RBC model, you can call that the law of motion for the
economy, or sometimes people call it a temporary equilibrium condition. It's a
temporary equilibrium condition, because if I stick in some value for
expectations, here, it's going to spit out a current set of prices. Likewise,
if I went back over to here, if I stick in a value for c t plus one, I can stick
out I can pick out all the current values for all the other variables. If I
know that and I know you know what the level of capital is at the turn at the
current time. I know those two pieces of information, I can back out everything
that's going to happen in this period. Now
All those things that I backed
out in this period, they're all going to satisfy the first order conditions. So
they're a potential equilibrium, but they may not be the full equilibrium.
Again, this is where you have to think back to
the Ramsey model.
Right in Ramsey.
When we drew this diagram, we were able to pick points anywhere on here. And as
we said, right, the diagram told us points that all satisfied the Euler
equation, the Euler equation, told us how all the points moved. Now, there was
only one point that actually solved the problem for us, because we could rule
out all the other points all the other points eventually lead us places that
would violate some other assumption of utility
maximization.
He was only this one line that always satisfied all the
conditions for the problem into the infinite future.
So we're basically in
the same situation here where once you write down that system of equations, you
have a temporary equilibrium in that just like here, I can pick a point and
ask what happens next period, but it doesn't give me the full
solution.
Okay, so temporary equilibrium.
Another thing to note here is that this variable in front of the expectation
beta is greater than zero. So we say that this law of motion has positive
expectation of feedback. Now, what do I mean by positive expectation or
feedback? Well, if you expect prices to rise they do and vice versa. Positive
expectation feedback is a property of most asset pricing models and in most
asset markets, right. If I if I think a stock is going to be more valuable
tomorrow, I might decide to buy that stock to try to take advantage of those
capital gains, the action of me buying that stock, actually pushes up the
value. So your expectation
is partly confirmed. And the
price today,
you can also have negative feedback, which is somewhat sometimes what you
think about when you think about commodity markets, especially with
farming, where you have to, you know, think about what the prices are going to
be for your goods at the end of harvest.
Um, when you're planting,
if you think the price is high, you may choose to plant a lot. If you plant a
lot, and that whole crop makes it to harvest though the supply you're going
to have is a lot and because there's a lot of supply, the price will fall. So
your expectations can have negative feedback, negative feedback models, I
guess there are sort of more rare and macro most of what we think we study are
these positive feedback models.
Anyway, that's just a technical term.
All right.
So let's let's play around with this idea. Let's think about what it would
mean to say that you have naive expectations or versus rational action
affectations. And basically the the thing that I'm going to set up for you
here is going to be exactly akin to what we did with the Ramsey model where I had
that s dot equation that I just made up relative to
the Euler equation that we got from optimal decision making.
Right? We had two different ways you could think about savings behavior. And
we studied the model and saw what the outcomes were. And like we said, we like
the Euler equation one better because it gave us a unique prediction. Something
similar is going to happen happen here. So let's think about the case where
people are naive. What do I mean by people being naive? Well, let's assume
that you can solve this equation by saying that ATP, t plus one is equal to
PT minus one.
So just last periods
price. So that's how people form their expectations. Sometimes this is called
adaptive expectations. I look at whatever the price was last period. I
think that's going to be the same price in the future period. If I take this
price, and I plug it in to the temporary equilibrium can difference, I get an
equation that looks like this. So this is a fully formed system of equations,
right? I mean, I could simulate on this and see what would happen.
Um, so
this is an illustration of what you would get well, if you think about a
shock to the dividend process, a change in TFP that would be unintimidated. So
in the beginning, there'd be this big gap error in your expectations. But if
you look at what would happen as expectations are revised over time, you
would have systematic errors,
you would undershoot for a long time, and then you would overshoot the agent
would never correct and never be able to figure out the process of the price
process. Even though the fact the shock has gone away. The uncertainty is gone.
The agent never figures it out. Now, if you were thinking about this in terms of
like a competitive market, essentially this person is leaving a lot of money on
the table.
All right.
So what do we think about then under rational expectations? How do make
rational expectations precise? Well, as I said earlier, under rational
expectations, really what we're going to say is that the agent if they make
errors, those errors can't be for castable. So, to get a model to think
about rational expectations, what we're going to do is we're just going to
define the expectations variable in terms of the forecast error. So what we
say is is p t plus one minus the expectation of P t plus one has to be
equal to some variable a to t plus one, where a to T is any Martindale different
sequence, which is just a fancy way of saying Ada T is any unfor castable
sequence. As long as it is the case that my expectation of time t of a to t plus
one conditional on all the information I have is equal to zero, it's fine. It
doesn't matter what it is, is just says if I make errors with my forecast, those
forecast errors have to be unfortunate for castable. Okay, if we make this
assumption, we can substitute this assumption into the model. So P t plus
one minus a t plus one, rearrange that to get ETP t plus one, substitute that
into our temporary equilibrium equation and rearrange. And now we have an
equation in terms of P t plus one equals one over beta p, T, and so on and so
forth that is now driven by this new shock, which we haven't defined what it
is other than it's just some unfor castable sequence.
If we lag everything back one period,
we now have a two dimensional system in prices and dividends that looks like
this. So think about this as your capital accumulation equation and your
Euler equation.
And this system has
rational expectations embedded in it, right? we substituted out the
expectations and we substituted in the rational expectation assumption, which
was that if there are errors that there are mistakes made in this model, they're
completely unfair castable and I want you to think about another thing that
happens here. In a model like this, it turns out that you can read off the
dynamics in the economy by basically just looking at the coefficients in
front of the T minus one variables. Now, I'll show you formally why this works.
And how this happens. I mean, formally, we're going to be doing, I guess,
looking at the eigenvalues of A system of equations. For this simple system,
it's really easy. You can just read them off mainly, the thing in front of PT
minus one is one over beta. The thing I'm going to find a DT minus one down
here is rho. These are basically going to tell you the dynamics you know
whether or not our phase diagram whether or not there's a push or a pole. And
that price equation has a very nice property that now it's one over beta.
And remember, we said that beta was less than one. So that means one over beta is
greater than one. So what does this mean under rational expectations? Well, it
means that if you just pick A random price, the price process will explode.
Now in practice, we don't see asset
prices exploding. We see, you know, well determine prices, they fluctuate around.
But we think those fluctuations happen over here, not necessarily in here.
So what does this given us? Well,
this has given us a saddle path, it's given us dynamics for which there is
going to be a saddle path. We have one over beta and explosive dynamics price
processes now explosive, and we have this row which is between zero between
negative one and one,
which means it's stationary.
And we're basically back in the world of have the Ramsey model, you have a
stationary capital accumulation equation, and you have that explosive
Euler equation dynamics. Okay, so there are many ways to find the saddle path
here, Cochran and Uhlig. So, two papers that I put up, give you various ways you
can do that. It's useful to know but the simplest way To do this and the way that
we're often going to refer to in this class is we're going to look at
something that's called the method of undetermined coefficients, or it's a
fancy way of saying guessing check. Okay? So what we're going to do is we're
going to guess a functional form of the solution. And we're going to use the law
of motion to solve for the coefficients of that function, and then check to see
if our answer is correct. All right. So what do I mean by that? Let's Let's go
to the board.
Okay, so This is this is our equation.
So if I wanted to guess a solution to this, what I mean by guessing a solution
is I want to guess, a price process that doesn't involve expectations. And the
price process, I'm going to guess, is a process. That just depends on the state
variables in the economy. What's a state variable? Well, a state variable is any
variable that you don't get to choose over time that you don't get to choose
over time in the sense of remember our
our Ramsey model. When we thought about
what happens in a certain period of time when
something changed, right, you could always choose consumption,
right? You could choose any consumption you want it,
but in order to move capital, right, you had to first choose a consumption and
then capital would evolve slowly. Capital over here was our state variable
consumption was our choice variable.
In this case, This ends up being our choice variable and the dividend process
ends up being our state variable. So the solution we guess we just say, I guess
that the solution takes a linear form, and only depends on state variables. So
what are the two possible state variables here? Well, there could be an
intercept, a, meaning something that doesn't change with time. And then it
could depend on the state variable itself, dt, the dividend process, but it
depends on that with some coefficient B that I don't know what it is.
Okay.
So that's my guess.
just guessing it. Now,
I picked this guest cleverly, I know this guest is going to work out but this
is the guest I've chosen. All right. Now, let's do the same thing we did
under rational expectations. Were I Under the naive
right where I want to use,
I want to get rid of this expectation variable that I don't know what's up
here. I basically want to use my guess, to plug it in here to see if my guess is
correct. And the nice thing about using this method where I'm going to use a
guest to see if it's correct is, if my guess is correct. It should solve this
equation. And at the same time, it's also going to satisfy the rational
expectations assumption because my expectation will turn out to be true.
All right, what do I mean by that? Well, let's calculate the expectation of P t
plus one, assuming our guess is true. Well, that would be E t plus one, a plus
b
dt plus one.
If I
move the expectations operator through there, the expectation of a constant
is just a constant.
The expectation of dt plus one, well dt plus one is a state variable. And
I know what how that state variable
evolves, meaning it's going to be rho dt plus e t expectation of epsilon t plus
one. Now my expectation of time t u epsilon t plus one, that thing is zero.
So my guess
implies that the expectation of time T of t plus one is a plus v rho dt.
Now that I have that guess
I can plug the guest back into there.
Sorry, I ran out of space. So let me let me write this over again.
So PT beta, et PT plus one plus dt, but substitute my guessing that beta A
plus B rho dt
plus dt.
Notice if my guess
is true,
then these equations have to equal each other. Right?
Because I've assumed that PT is equal to A plus B gt. And on this side of the
equation I plugged in my guests for if that's true, what it means for et PT
D plus one.
So essentially, right think about this in terms of rational expectation case. I
if I if this is my forecast, it could be wrong, but the Only thing that was wrong
by was that epsilon t plus one which my expectation of it was zero. Otherwise, I
know all this stuff. So that has to be what my expectation is. And if this
thing has to be true that it has to be equal to this again, if this is true, I
can then just look at combining like terms and solving mainly a must be equal
to beta A. So combining those pieces, and B must be equal to be rho plus one.
How do I get that? Well, B times dt over here, and I hope sorry, there's a beta
over here sorry. And then I have beta v rho plus one after you make out the dt.
So b is equal to beta B rho plus one. So what can solve this? Well, the only
thing that can solve that is a is equal to zero. This one, we're going to get b
is equal to one minus beta rho equal to one and b is equal to one over one minus
beta rho. So, a is equal to zero p is equal to one minus beta rho that means
my guess solution is dt is equal to one beta rho, that
right there
is my ra solution.
So, essentially what we have done here, if this is correct, I'm pretty sure it
is. We can check it in just a second is it tells me how prices should adjust
anytime DT moves And notice, remember I told you right the two things that
mattered those two eigenvalues that mattered, right? Were beta and rho, they
end up showing up in here. So basically, this is telling you basically, anytime
you see a dt, how do you adjust your dynamics to stay on that saddle back?
Alright, how can we check this?
What's the solution? Well,
plug back guests in
at PT plus one of that guess would be et
dt plus
one or one minus beta rho, which would be rho dt, one minus beta rho there
therefore p t with equal to beta rho dt one minus beta rho
plus dt.
Alright, so I'm just take my expectation, plug it into the law of
motion. That's gonna be beta rho dt one minus beta rho plus dt. factor out a dt,
and I'm going to get beta rho over one minus beta rho plus one.
All over dt.
Give me some more room over here.
And I'm going to have PT is equal to beta rho plus one minus beta rho over
one minus beta rho, all multiplied by dt.
beta rho minus beta rho gives me p t is equal to one over one minus beta rho dt.
So, sure enough, my guest turned out to actually solve the equation.
And this trick
will work almost always. And notice that the assumption that we plugged into it
was was basically using that same rational expectation notion, right as we
just we guess, a solution for bt plus one. We had an error term up there, when
we did this expectation, right for this dt plus one, there's an epsilon t plus
one, but my expectation of that is zero. So we say All right, here's the What our
forecast can be, it could be wrong by that epsilon t plus one term. But
otherwise, we're using the structure of the model. And then we just back out
with the correct structure should be. And that turns out to solve for us what
the rational expectation solution is. And this turns out to solve for us an
actual closed form solution in this case of
essentially what that saddle path is.
So, if I go back
to the slides, this just walks you through, Oh, I forgot the gamma, make
gamma be one but same same process applies. Now, when there's a shock, only
in the first period, do you experience a forecast err, then after that, your
expectation, one period ahead and the current price are spot on. So your
forecast error ends up being zero. The only time you make a forecast error is
in that initial innovation. happens. So when the initial shock happens, that's
on forecast. So I make a forecast error. But every period after that, the agent
knows how the economy is going to unfold. And they perfectly pick it up.
Now, there's no more systematic error.
way to think about that, in terms of our Ramsey diagram. So Ramsey of courses is
in discrete time, but if you think about
or in continuous time,
but think about what we did when we thought about changes, right? So let's
think about an increase in the rate of time preference, right? So if there's an
increase in the or one of the, I can't remember the increase or decrease the
rate of time preference. I think it's got to be an increase in the rate of
time preference. C
equals zero shifts over.
And what happens?
You jump down immediately.
So there's a big revision.
And then for that big revision,
there's a smooth path back to steady state.
That's exactly what we're attempting to do. But now Where are those revisions
occurring? The revisions are actually occurring in people's expectations. So
first period, boom, expectations move. After that, you're on the saddle path,
glides back in.
So that that's the intuition. So So keep that in your mind when we're We're doing
these things. Alright, so we're, we're past our breakpoint, where at 810, we
basically finished RBC, we're just going to look at, we're just going to look at
a couple of impulse responses. This is basically just, you know, tracking what
happens through shocks through the model. We're going to spend a lot of
times of solving this in the computer. So I don't think it's crucial that we
cover that now we've kind of covered the things. So let's take a five minute
break. You guys gather questions that you have from that, and then we'll go
over those questions and then we'll start with going through the exam guide.
So I'm good.
So I'll be back at 815. Again, type your questions and we'll be ready to go.
Okay,
all right. Looks like we got
a couple of good questions here. So I'm going to start with how do we get a
equals zero from from that.
So let's uh
so the relationship we had was
a is equal to beta times A.
Beta
is some number between zero and one.
Well,
yeah, should be some number between zero and one.
So, right if I divide a on both sides, I could get one is equal to beta. Well
That's not true.
So the only way this equation can hold the only number that makes a this
equation true is zero,
right? If A is equal to zero, this equation is true.
This equation could be true. If beta was equal to one,
then a would be undetermined. We couldn't figure out what a is.
But the way this
equations written, the only way this equation can be true is if a is equal to
zero.
Yeah, so let's,
let's look at the rational expectations, graphs again. So I realize I actually
change these up because I didn't like the way they were written. But they're,
I mean, so essentially, what I'm, what I'm showing you here is the data, the
data, you can't see that sorry.
Alright, so explain the rest of the dotted line. Here is the expectation. So
you can think about, you know what happens when the shock hits,
when the shock hits,
the price goes up. But agents don't update their beliefs, right beliefs stay
exactly the same because they have to wait for data. In the next period, they
revise their beliefs up, but now price has gone up even farther. And so that
generates another forecast error. Next period, they revise their beliefs up to
that point, but now price has gone up even farther, and so on and so forth.
And then over here, they're not revising their beliefs down fast enough, and so
they're getting negative forecast errors. In this graph, I've kind of
drawn them offset, because really, what should happen is these two lines could
be right on top of each other if I drew them the same way. But the idea is to
think about going this direction. So essentially, in the first period, your
expectations are that there's not a shock. So your expectations is that
prices at steady state, but there is a shock. So that opens up a big forecast
there but In the next period, you have perfect expectations about what's going
on here. So every period, say going this way is basically the way to think about
it. Agents are perfectly expecting this is the price. And this is the expected
price in that period. They're perfectly getting it right.
If I,
your right to be confused, and that the timing of these two graphs is different,
this graph is showing the real time so the expectation on the actual price in
each period, if I did that same timing in this graph, this dotted line would
lie right on top of here. So I just wanted to show the fact that it mirrors
it so I should have added lines to show that this is the case but if I wanted to
do them to say this would be like back one, and I had them going the other way
around the other time, and I don't know it was maybe these aren't the most
helpful grass, but that's what I was trying to get out of it.
And then,
I guess first question, can we explain the the Ramsey diagram just for the
break? So let's, let's do the Ramsey diagram one more time. This is very
important because it is definitely a Ramsey diagram will be on the midterm
100%. And like I say, I've been using it all night for intuition. So hopefully,
most people have a good handle on it. But if not, let's let's review. Right,
so what the Ramsey diagram do for us, is it.
It tells us
it's a phase diagram that tells us how capital and
consumption move along the optimal paths
from the Ramsey consumer problem, right. So what we were graphing
Is the
we want here we want
right?
So we had a an Euler equation, a first order condition from household
maximization. And we had our capital accumulation equation. And what the
diagram tells us right is it basically plots out for us all the dynamics that
are implied by this. Now, the Euler equation right is a first order
condition. So it is not the solution. It is just a condition that has to be true
in the solution. So, just like with the Euler or the Lagrangian method that I
showed you tonight Right, we've got first order conditions. And these first
order conditions tell you what has to be true
on a solution
that solves that Infinite Horizon problem. But it's just tells you what
has to be true between two periods. So it's a condition that must be satisfied
for the solution. But it isn't the solution, just like this, this tells you
a condition that must be satisfied on the solution, but it isn't the solution.
Same with this, this tells you what must be satisfied, but it isn't the solution.
Now, we can learn a lot from these equations, though, mainly, for C naught
equals zero and k naught equals here where these two things intersect. There
is a steady state in this model. There is consumption there is capital, which
means that there's no more movement in this model, you're at a steady state,
and that will solve the maximization problem why? Well, it solves both
of these equations.
Now, if we want to think about dynamics in here,
we found out that these two equations
imply what are called a saddle path dynamics
where
there are two arms,
a stable arm, and an unstable arm.
And these trace out all the dynamics everywhere else in the economy. So, for
example, if you were to
pick
some capital and consumption level here,
these arms would tell you exactly
the trajectories, it would take, mainly, you would move this one and then you
would move that way, and then you would go over here. Now, this is exactly how
this trajectory right here, this satisfies the equation that satisfies
this equation. But we can rule this out as a solution
to the Infinite Horizon
problem, because we can look at where this
trajectory ends up. This trajectory ends up basically at Capital equals to zero
and if you ever get capital equals to zero, your income goes to zero
consumption goes to 00 consumption can't maximize consumption.
So we can rule that out.
Likewise,
if you think about points down here, these take you to zero consumption, zero
consumption can't maximize utility. So that has to be ruled out. Turns out, the
only points on here that can't be ruled out are the points along the saddle
paths that take you to steady state. Right. So the nice thing about these
diagrams, it allows us to solve that Infinite Horizon problem. Remember, you
had to solve that maximization problem from time t to infinity. That's actually
solved. The saddle path is what solves it. Now, rather than using the math to
figure out what was going on, we could then just draw this picture and then
think about how do you move
on to the saddle path when things change.
So that's basically what now we want to do once we're in the process. rational
expectations models that were in these
RBC models or New Keynesian models that we're
building now, except the problem has just gotten harder because now we don't
just have first order conditions for consumption and a capital accumulation
equation. Now we have additional things. We have technology shocks that are
additional state variables, we have labor supply decisions, which are
another household choice variable that has an optimization condition that has
to be satisfied. When you add these things up. Right now we don't have a two
dimensional graph, we have a four or five dimensional graph.
And now we can't draw these pictures anymore.
Now we actually have to do the heavy lifting of actually solving a
mathematical problem that can actually describe what the saddle pads are doing
for us.
And basically, tonight,
you solve by hand or at least you watch me solve by hand,
the first one of those.
Okay, we'll come back and draw some Ramzi dice. grams again, when we go to
the review, but are there any other questions about sort of the material
we've seen
tonight?
Okay, well, maybe more questions will come up.
As we go through the exam review, let me share my screen with you.
And let's go to the exam guide.
Okay, so, exam, what do we got? It's going to be 24 hours, open book open
note on there's going to be a mix of short answer and multi part questions.
So well look, look to the practice, mentor, it's going to be your best
guide. It's not going to be exactly the same again because that was a exam that
you know, people had two hours they sat in the classroom and took it's not going
to be exactly the same, but it's going to look very, very similar on partial
credit will be awarded. So please show your work. You know, show me what you're
thinking, if you leave stuff blank, I can't give you any credit. So you know,
give me your best shot, especially when you have 24 hours to do this. Don't
leave anything blank, try and try on every problem. What is assessable?
Technically, every word that has come out of my mouth assigned in reading in
the textbook given in tutorial or in any assignment may be assessed. So
technically, everything's fair game, but I'm going to primarily focus on the
models that we have had tutorial assignment questions on. So solo model,
are you going to have to be able to solve it? Yes. Ramsey diagrams, are you
going to have to be able to draw these things? Yes. Are you going to have to be
able to think about how to model shifts of those curves in there? Yes, you are.
The endogenous growth
model are you going to have to know how to solve For the growth rate and the
endogenous growth model, Yes, you are. Basically any of those models that we've
had you play around with that you spent significant amount of time studying,
like those are going to be on the exam. So my advice to you is, if you don't
think you can replicate exam or assignment questions or tutorial
questions, you can't do those problems at all, then you're in trouble. You
should be able to be able to answer problems that are similar to that. Like
I say, some of the especially some of the tutorial questions especially are
some of them are are pretty hard. They they won't be as hard as that, like I
say, Look to the the midterm for an assessment of how hard it's going to be
but but that's mainly what I'm going to focus on. So when you're thinking about
studying for this, you know, what's the best place to start the tutorial in the
assignment problems. So go back, try to do the tutorial problems again. See how
you go watch the tutorial videos. Look at my answer key for the question. See
if you can figure out, you know what was going on what you didn't know. And of
course, you can always take the mid semester, the practice mid semester
exam, see how you do. If you do very well on the mid semester exam, you can
go through that exam without looking at any of your notes or anything that I'd
say you're going to have no problem on the exam that I'm going to set
topics
from chapter one solo model. thing is that, you know, I'm definitely going to
test you on, you know, key assumptions of the model, you got to know what the
model is doing, you know, what's exogenous what's endogenous. Know to
know how to solve for steady state values, know how to use the Solow Swan
diagram and how to do some analysis in it. I have transition dynamics and how
to graph time path endogenous variables. actually haven't used that very much in
lecture or tutorial. So I probably actually won't do much on on four. You
should know what this is, but you don't have to do much with it. More, I think
focus on number three here. And certainly number two, you're going to
have to know how to do a little bit of algebraic manipulations on this. And
like I said, you're going to have a lot of time to work on this. And you know,
you have all the resources of the internet at your disposal. So, you know,
it shouldn't be it shouldn't be a Herculean task to get these things done.
To reiterate, though, the one thing you do not have at your disposal, you're not
allowed to actually ask another person for help, right? Other than me, you can
ask me questions, but you can't ask anybody else. So you can use whatever
resource you want. But you can't actually ask another person, that would
be considered cheating. Do not do it.
For the Ramsey model, again,
know the key assumptions of the model. I wouldn't ask you to solve the household
problem and go and how do you derive the Euler equation, I just want you to know
how to work with basically, what you see on
the board.
You should know how to manipulate the Euler equation In how to manipulate this
thing, how to find steady state capital in this, you should know something, you
should definitely know how to draw these diagrams and how to analyze these
diagrams. Um, you should know, for example, that the C dot equals zero line
always has to be to this side of the top of the k dot equals zero line. Right, we
spent a lot of time talking about the fact that the golden rule level of
capital, the thing that's on the top of here is not reached in the Ramsey model
because of, you know, the rate of time preference that people are impatient.
And we talked about these things, these are facts of the model that you're going
to need to know and that might be assessed.
The diamond model, same thing, we spent a lot of time looking at this in class,
we spent time on tutorials working on this problem. As you can see in my
sample midterm there was a problem on this. So key assumptions of the model
you need Know how to solve the household decision problem, how to get from
household savings to the aggregate law, law of motion for capital. And you
probably want to know something about, you know, the arguments about dynamic
inefficiency in this whole idea that even though we have perfect competition,
right, it can be the case that the equilibrium is in credo optimal. That's
kind of a very important point, right? That it can be the case in this economy
that you can have over savings. And because there's over savings, and
because people don't internalize or don't live long enough to internalize
the fact that they're imposing a burden on somebody else on this problem never
gets solved, right? It's they're always looking at it from their own point of
view of, oh, there's a lot of capital around interest rates are low. That
means I have to save a lot to have the consumption. I wanted my retirement and
so that I save a lot, because I saved a lot of interest rates continue to be
low. You get stuck in this in this problem. That problem
doesn't arise
in the Ramsey model though because they're infinite In its infinite horizon
model, so the agent has multiple periods to make this adjustment they can
internalize those those adjustment costs were in that two period problem they
can't. So knowing something about that is good and will be likely assessed the
simple endogenous growth model so you know, know the key assumptions of the
model know how to solve for the steady state know something about what the
model predicts
growth empirics um,
you know, we went over a lot of different theories if you read the book,
there's a little bit more background information on this stuff, you know, the
main theories to explain the evolution of income from prehistoric times know
something about the Kraemer paper. I never asked you in any tutorials or
assignments to solve the Kraemer model so you don't need to know how to solve
it, but you should know something about it. You know, if we go and look at it,
the practice midterm, you can see sort of the type of stuff solutions that
might show are the type of answers I might ask. So like in this true false
and uncertain thing, you know, I asked about primer so I say true, false or
uncertain. crimina argues that technological growth rates differed in
Eurasia, the Americas and Australia because different availability of
domestic edible plants and animals effectively a doubt each continent with
different levels of initial capital stock. Now, all of you of course, it's
screaming in your head right now that is clearly false. That's not what Kramer
said. Kramer was about population density and land size explains different
rates of technological growth. The domesticated animals thing that was
Jared Diamond argument from the book guns, germs and steel. So that's
basically the type of question I'm asking like, no, something from those
slides, no, sort of something about the different theories. As long as you know
something about them, you should be able to answer at least true and false
questions about those parts.
Okay,
and then from the real business cycle models
again, since we haven't Had assignments over this yet or tutorials over it, all
I'm going to ask you to know is concepts. So, you know, what is real
business cycle theory? What is the whole goal behind it mainly trying to explain
a business cycle using only productivity shocks and government spending shocks.
So basically, you know, what can Ramsay model teach us if we just put
stochasticity in these two variables, but otherwise, it's a perfect
competition model where government intervention has no purpose. So you
know, know something about the underlying assumptions. And then how
these models are usually solved, right? I gave you the recipe right? We've gone
through the recipe. Where was it? up here, solve for first order conditions,
log linearize and vote graphs and expectations you should know something
about this recipe. You know, I should be able to ask you a question that I've
asked in the past is just you know, what is rational expectations mean you define
rational expectations. For me,
now, I'm probably not going
to ask a question that's specific, because all of you could go to the
Wikipedia article and copy that down. But I, I'm clever, I can come up with
ways of asking these questions to test whether or not you you've been paying
attention in that sort of way, so you don't have to, you don't have to be able
to solve these models, but you should know something about them. So that's,
that's basically it. That's, that's the, the outline and sort of the key topics.
Again, like I said, if if I was constrained on time for studying, I
would start at the tutorial questions, I then would do the assignment. I then
would go back and look over all the slides, you know, that that would be my,
my, my order of studying. Um, and again, you know, so as to not kill yourself by
you know, beating your head and doing this thing for forever. I suggest that
you, you know, sit down on Wednesday night. And you know, spend the two or
three hours just trying to work through the problems right there without looking
at any other resources and just see how far you get. And then you can
reevaluate, because if you go and you start checking your books, looking at
slides for every little thing, you could spend 10 hours on this thing, and not
really up your marks that much. So that's my, my best advice is study for
the exam as if you're going to be in a classroom sitting there with no notes at
all. And then try to set that exam conditions and that's the way to make
sure that this thing doesn't turn into the 24 hour hell for you. Anyway, so
that's, that's my suggestion. Are there any, any questions, any problems people
would like me to do any? You know, if there's one one thing you'd like me to
review, what would it be? Again, this is not your only chance to review right?
You can still ask questions on Ed tristin is going to hold a zoom session
on Friday and I will have my normal zoom office hours leading up before the exam
Yeah, you're you're completely allowed to write on paper and submit photos My
only, you know my preference is is that it's all in one document one document
will make my life much easier in order to keep track of things if you
absolutely do not have the technology to make one document out of this thing and
you have to submit multiple photos that's fine but think think before you
submit right the better mood I am in when I'm marking your assignment the
easier I can you know read your your paper the happier I'm going to be and
the more likely I'm going to be to be saying like oh, well, they don't really
know what they're talking about, but I'm going to give them points for that
anyway, if I can't read what you're saying, I'm just going to be pissed and
I'm going to mark everything on so you know make it nice for me and I'll be
nice to you. Yeah, I mercy points will be given for clear formatting. Like I
said, the easier I can read the better mood I'm going to be in. So and and I
think you'll find I'm, I can be a very lenient grater in the sense that, you
know, I want I want to see you guys try. I mean, I understand that this material
is hard. And it's even more hard that we're not in a classroom, we're having
to do these videos. There's technical glitches that happen all the time. Like,
this is not ideal circumstances. So I understand that and I want to work with
you guys. So that's why I think it's very important to, you know, give me
Show me the work show me Show me what your thinking is. Because if I can see
what you're thinking, then I can give you points because sometimes I can see
like, Oh, I can see why they got this wrong. This is actually my fault because
I didn't explain this very well or I could see why they got that I can give
points for these things and then also helps me you know, figure out how to
correct things as we go. So, you know, give me good information, and I will,
you know, try to be as fair as possible. Thank you. Um, any any other questions
for we call it a night
I'm reading the textbook.
I've tried to set the class up so that you can pass the class only looking at
my slides and the material I give you. So I, you know, like I said, my order of
operations for studying would be tutorials, first assignment, second
slides, third book would be fourth. So the book is helpful because, you know,
it gives more in depth detail to some of the stuff we've talked about. And you
know, it lays out some of the things but you know, if you've been watching all
the lecture videos if you came to lecture when we still, if you happen to
be in Australia at that time, and you've been following along on the slides, you
probably don't need to read the book. So I've tried to make it so that just the
slides and the stuff that I give you on canvas is enough. But some people find
the textbook helpful. I think it is a good textbook, it's written well, so if
it helps you do it, but you know, don't you don't, you don't have to, you know,
if you've never if you've never open the textbook in this course, once you can
still pass for sure.
Okay, well, I'm just going to leave zoom on and I'm going to just hang out here
for a bit. So if people want to hang around and ask questions you can
otherwise you're free to go
So Kraemer didn't argue that output per capita did not grow before the
Industrial Revolution. He argued that output per capita, and technological
growth was growing all the time. His argument was more about that, you know,
you sort of have to hit escape velocity, right. So there there are sort of
multiple things that have to go because you have those Mousavian dynamics. So is
sort of the argument that, you know, when you, when you've got more income,
you've got more people, more people allowed you to create, you know, be
luckier and get new inventions. And if you got new inventions that would give
you more income, and then you could get more people. And you sort of had to get
all those things, right, right in order to keep technology progressing, but he
very much thought that income for capitals were rising well before the
Industrial Revolution. I mean, that's Part of his whole argument, right is he
says that, you know, before the year 1500, based on population density and
land size, you can basically read off what income per capita was in different
parts of the world. And he says, Europe and your Eurasia, you know, Europe and
Asia, have the highest income per capita in the world. They were higher than what
was in North and South America. And those were higher still than what was in
Australia. And if you want to go into the islands of Australia, Tasmania and
Flinders Island, they were even higher, they were even lower still in those
places. So it was he was not arguing at all that there was no growth before the
Industrial Revolution. It's just more that, you know, actually, right, we're
all we're all living the curse of exponential growth right now. But just
like with epidemics, right? economic growth is a bit like an epidemic. It if
you have just a small difference in growth rates, right, you go from a very,
very, very Mild increase in growth, which is what you can describe from 1
million to 300 years ago, you up that growth rate only just one or 2%. And you
go from a mild flu season to a global pandemic. Right. That's, that's the
power of exponential growth. And so he absolutely did not argue that there was
no growth, it was just slower before. And once you sort of hit escape
velocity, everything goes.
So so that's the idea.
Um, yeah, I should say,
you know, definitely go have a look if you are interested in sort of what the
economics field is talking about. epidemiological and macro stuff. So this
paper is kind of cool. This paper is cool. I'll start putting these up. I'm
actually really surprised at the speed at which people have been able to put
these papers out. But a lot of these papers, I mean, they're more complicated
models than what we're learning about here. But I think you'll be pleasantly
surprised in how familiar a lot of the things are and you know how much more
you can get out of these things just with the little bit of background that
we've already built in this class. So hopefully you find that's the case.
All right, it doesn't seem like there are any more questions. So I think I'm
gonna stop the video here. So thank you guys all for participating and well
we'll be in touch up until the
midterm.
So happy studying