## 11th lecture Introduction to Advanced Macroeconomic Analysis

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the holy grail of macro is trying to

find the model that delivers a nice

internal propagation of shocks not

something we just made up okay welcome

back happy new year and I hope you

weren't too terrified from the by the

last lecture it was kind of technical

but that's kind of the tone of the rest

of the course

we need to think really hard about

dynamics and the rest of the course and

to do that we need to have some basic

command of these instruments these these

tools we call them difference equations

I'll repeat some of that today and

please look at the slides it's it's

important and please visit andreas in

the section ok so today we're gonna

we're gonna basically review what we did

last time and I'm gonna introduce you to

the first serious model of business

cycles what we did before this AAS ad

the model was kind of a toy model just

to get you thinking about the the

mechanism by which shocks are propagated

into into cycles shocks that are

completely random in nature and possibly

unpredictable even from period to period

can still be transferred or translated

into waves irregular wave so this

Slutsky idea is very important in

macroeconomics today to do that we even

gave you a problem set that has a

different set different model so the

model and the problem set which you

really should do is the so called

multiplier accelerator model of Paul

Samuelson that was introduced in the

1930s so Paul Samuelson one of the

greatest economists of the 20th century

the guy who introduced mathematics to

the way we think about macro and micro

he basically tried to show that a very

simple model that Keynes had put on the

table in 1936 could be translated into

these cycles he was thinking about

deterministic cycles but he still had

some interesting insights so the idea

what the problem set which andreas will

discuss in detail is basically taking

these ideas which applied to the ASAT

model which is a mod

framework again we can put it on paper

and we can also put it on the computer

which I will do in a second and you can

do it in a very fancy way that involves

a lot more mathematical muscle these

ideas are all making recourse to the

same basic notion of a stochastic

difference equation a shock to

difference equation okay so that's a

extremely important thing so important

that I will review it briefly again okay

so I warn you this is not to be ignored

I'm glad a lot of people here on this

awful 8th of January this weather is not

great it's not it's really perfect for

studying ok it's perfect to make you

think hard about macroeconomics so we're

gonna do that quickly and then I'm going

to talk about expectations formation so

in the AS ad model the toy model has a

big problem which is it has a very

mechanical way of thinking about the way

people expect the future and one of the

biggest revolutions in macroeconomics

was in the 1970s brought about by the

work of Milton Friedman and Edmund

Phelps and their revolution was thinking

about people's ability to actually think

as well as or better than economists

themselves the idea would be is simply

how arrogant of economists to think they

can cook a model and exploit it for

the individual agents ability to adapt

their behavior to that policy intention

ok so this is kind of a mixture of what

Milton Friedman and Edmund Phelps talked

about which is the stability of the

Phillips curve with some ideas that

Robert Lucas and Tom Sargent in the

1970s picked up on and they continue to

accompany everything we do in

macroeconomics today even though we have

a what's called a modified or new

Keynesian approach to macroeconomics and

thinking in all schools today even the

most the most neoclassical of approaches

admit some role for rigidity in price

setting

to everyone we need to think about that

because any model that we write down

should be able to withstand the

criticism that agents should not make

persistent mistakes in that model okay

so that's why expectations and taking it

one step further rational expectations

it's an important way of thinking

disciplining the modeling strategy

remember modeling is like devising a

lens to look at the data lens to look at

the world and if you have a lens that

has got a couple of scratches on it it's

not going to work very well you need to

always think about lenses are imperfect

because they'll filter out certain

things that we don't want to see certain

types of radiation we don't want to have

in our eyes that's kind of the way

lenses work okay so the rational

expectations criticism is simply a model

should be robust to people figuring out

how the economy works as we think the

economy works and I'll give you some

mint some information on how we do that

operationally there are three basic

forms of rational expectations then I'll

take you into the world of

macroeconomics some data okay because

what drives us in this in this world is

data and we want to be able to replicate

with our models cycles that look like

cycles and the data okay so we want to

see output that moves you know up and

down but we want to see unemployment

that moves in the opposite direction

over the cycle we want to see inflation

that maybe it's mildly but not

completely procyclical maybe even

dependent on the period you look at I

say that with caution because in the

1970s inflation output moved in opposite

directions but over the course of the

last hundred years output inflation tend

to move procyclical II in the same

direction and a lot of other things for

example right now there's a lot of

discussion in the world about what's

happening with Trump okay so I mean I'd

love to talk about Trump Trump was a

shock to be elected was a shock it was

also kind of a shock that he came down

so hard on China with these with these

threats of import duties he's actually

followed through with it this was a

surprise this was a shock the impact on

World Trade is a shock and it hasn't a

shock on demand not just in the United

States but also

here in Germany okay so we need to

understand that also and these shocks

give give rise to cyclical fluctuations

and again one way of understanding that

is to is to reduce the sources of shocks

so for the first class of models we'll

look at we'll only think about models

that have one type of shock and then

we'll expand the notion to different

types of shocks we already have in the

supply shock so if you want to make a

quick judgment of what's going on in the

world today you need to understand the

facts you need to understand a good way

of looking at the facts and the the

quick-and-dirty model i gave you the toy

model gives us an interpretation of the

Trump thing it's a shock on demand it's

a shock to demand for US goods because

China is now putting retaliatory tariffs

on the United States and it's cutting

down the demand for us goods and

services at the same time the stock

market is crashing okay we know from the

data from the stylized facts that the

stock market is procyclical it's a

leading indicator so we need to have a

model that can interpret the stock

market as a pro cyclical leading

indicator so these are all really

interesting robust things that a model

has to pick up if you want to take a

model seriously it's really easy to

write down a model I mean anyone can do

that now anyone who has any sort of

mathematic background the question is is

it okay is it decent can it withstand

the the strong winds of falsification as

many scientists throw facts haking

explain that can you see this do you

this model as a training ground for the

New Keynesian model and it's called the

stochastic growth model so this is the

this is the link the bridge to the

previous part of the course the second

half of the course about macro macro

fluctuations the first part of the

course was about macro growth and the

bridge would be some meant like a model

in which instead of continuous-time we

have discrete time and we have shocks to

that model in every period so that's

making this bridge you can think of this

model as having all the interesting

aspects of the first part of the course

optimizing agents okay

and adding some other new bells and

whistles that make it even better and in

trying to account for stylized facts

okay so that'll be the introduction and

next week we'll do the stochastic

Grossman the so called RBC or real

business model in detail okay we'll only

spend a couple of hours couple sessions

on that because in my view the real

business cycle model is augmenta Bowl

okay it's it's basically a basis for

understanding the New Keynesian models

of the latest generation and if you go

on to take courses with Professor vine

cut you'll see that there's a lot more

you can put into this model modules you

could Park in the model that make the

model even more interesting and speak to

regularities in the data we have ok so

last time we introduced stochastic

difference equations I call this the

bread and butter of macroeconomics okay

you really need to understand this if

you want to take any further courses

have any other further understanding ok

stochastic difference equations is the

way we work it's like having a hammer

and you need a hammer to do a lot of

things perhaps not everything but a lot

of things when you're a carpenter you

need a lot of tools in addition but you

and we talked about the impulse response

function as a modern way of describing

the dynamic response of a system to a

one-time shock full not in full

knowledge of the fact that have you

shocked the equation or the system of

equations every period you get even more

interesting dynamics ok and I gave you a

cookbook approach now andreas will

repeat this cookbook approach in a

slightly different way to make you aware

that in the literature there are two

different ways of discussing this

problem some of you may be coming from

time series analysis you know is anyone

out there who's taking time series ok

just so many people ok sometimes there

are lots of people sometimes there's no

one there are different ways that they

have the same idea it's just a slightly

different tack ok I want you to

understand that both tacks lead to the

same outcome ok and we're going to talk

about the difference equation in the lag

operator

as a way of getting to this this

solution of the difference equation

solution of a difference equation is the

expression of the current variable YT as

a function of time only and some

constants okay that's an important way

of thinking about it because the the the

difference equation itself expresses YT

as a function of YT minus 1 and YT minus

2 and maybe further lags if you had a

higher order difference equation here we

want to be able to solve the equation as

a function of time only so it's a it'll

be basically you'll express this YT as a

constant plus powers of the

characteristic roots of the difference

equation the polynomial of the

difference equation okay so that's the

that's the understanding we're gonna

have for this thing and that if you

think of powers of time means that if as

time goes to infinity those

contributions to the to YT must get

smaller and smaller if they don't the YT

is exploding so the the fundamental

primitive requirement for an

unconditionally stable difference

equation is that those characteristic

roots all have to lie inside the unit

circle gave you solve them and as roots

to a polynomial in a characteristic

equation as andreas will show you those

roots have to be outside the unit circle

but the basically the lambda t I have

are the powers or the are the same

solutions that andreas will derive in

verse one over okay so that's a very

easy way of understanding what we're

about to do and then last time we also

went back to the AS ad model and we sort

of looked at this shocking the shocking

of the AS ad model and what happens how

you can get cycles i'll give you an

example of that in a second okay and

finally a very important this is a

generalizable approach so even though i

just discussed a very simple toy model

that had everything just perfect it

reduced to an AR an auto regressive

second order difference equation second

order difference equation that was easy

to sort of play with at least from my

perspectives relatively easy you know

you can actually get even more fancy and

in real life you might have several

equations okay it turns out that you can

put this in a so called companion form

or canonical form and that allows us to

to evaluate the eigenvalues of a matrix

and get the same results okay let me

return quickly to that model this is

again this is stuff that you may have

with macroeconomics very interesting a

way of understand it's a very general

sort of you know for people don't go

further than a bachelor's degree this is

the way to go but you know our objective

in this part of the course and this

course is to put life into a1 a2 c1 c2

try to understand what these constants

are okay so we want to go beyond that in

this course but I just need that to fix

ideas to get you on the same wavelength

we have a demand side the demand side

consists of a an aggregate demand

equation which is a positive function of

the previous values as an automatic sort

of persistence hardwired into the model

and then you've got a 2 which is

negative which expresses the negative

dependence of aggregate demand on the

real interest rate okay so the real test

rate is the nominal interest rate minus

the inflation rate and then you've got

this shock this demand shock which can

be anything it can be a shock to

government spending it can be a shock to

tax policy it could be foreign demand

for your goods it could be a tariff

shock or whatever you like now the

second equation is the Taylor rule the

Taylor rule expresses how the Federal

Reserve System or the European Central

Bank the Bank of China reacts to the

policy initiatives that give rise to its

interest rate policy to policy drivers

so central banks don't like inflation

okay

and they don't like it so when interest

rates when inflation is high central

bank's react to inflation by raising the

interest rate by more than one to one

it's kind of called the Taylor principle

C 2 is positive because central bank's

also don't like output to get out of

control okay so they want to lean

against

this is bait these are basically

assumptions about the policy rule that

the Federal Reserve with a Bank of

England or the debunked if thought well

sorry the the European Central Bank

follow the BOC de France is part of the

European system of central bank's I'm

still getting over the euro it's 20

years later I still can't remember that

the Bundesbank it exists on paper but

it's really just a part of this European

system of central banks that's the

demand side so that's the ad curve the

AAS curve is the next curve which

basically says that and we'll go into

this in detail in the rest of this

course that the inflation the behavior

of inflation is a function of three

things

it's basically inflation of what people

have expected it to be in the past and

we call that core inflation some people

call it inflationary expectations a very

particular view the second part is the

sensitivity of price setting to the

output gap so in the economy remember YT

is the output gap now I told you that

already this is the percentage deviation

from out from from the solo trend and

when YT is positive that means the

economy is is operating above capacity

or above average output and that means

the firm's tend to raise prices and that

tends to translate into higher prices at

the stores and also means that workers

have negotiating power asks for higher

wages they gets these get passed on in

higher prices these are all the things

the third part of the aggregate supply

story is the supply shock these are the

things that happen given the first two

sand economics we condition everything

all the time conditional on people's

expectations inflation conditional on

the output gap what happens if OPEC gets

its act together and raises the oil

price by a hundred percent okay what

would happen if Trump in in in March

raises the tariff another 25% on Chinese

goods what's going to happen to

inflation in America it's probably going

to go up given the output gap and given

the inflation that people expect okay

that's just a simple

this is a toy model believe me it gets

really interesting when we start

foundations of these behavioral

equations again this is under graduate

undergraduate stuff the last part is the

closure of the model so how do we get

inflation and this is a simple way for

you to get this quickly I'm doing this

for people who maybe are coming from

maybe even a hard science background or

people are coming from literature or

something I one of my one of my best PhD

students and best master student was

actually coming from you know Gama

mystic so there's hope for everybody out

there right now you like this stuff you

have to like it otherwise it's just

crazy to even be sitting here right now

you have to like it and you can see that

I like it cuz I've been here for I've

been doing this for 35 years I'm really

into this stuff right okay so the sec

the last equation is a simple just to

put it on a plate for you it's not the

whole story but one nice way of thinking

about corn core inflation is that it's

kind of a mixture of people getting it

just right and people just completely

morons they're looking backward and they

just take the last observation of

inflation as their expectations of

inflation today okay we call that auto

regressive so the extreme case would be

and you want to think about it if you

think about we try to I mean this

particular set up I restrict to an open

interval so so theta can't take the

values 0 or 1 but if theta did take the

value 1 it would be like a completely

backward looking

agent thinking about inflation okay on

the other hand if the agents are super

smart and they they have them they know

exactly what the price of everything is

in every second they have all the

information and it's also about not

having the information then basically

you would get the inflation rate right

every period so you can imagine in that

version of the model as theta goes to 1

I think I said miss book before theta

going to 0 was the case of the Deaf

experienced theta goes to 1 you have

almost perfect foresight people get it

exactly right and in those cases the the

aggregate supply curve is is the entire

story there is no other story the model

is determined by shocks the supply side

you can see that if you want by going to

the model and playing with yourself ok

so these are the definitions again I put

this in in so you could remind yourself

what these things are again this is a

toy model just to get you going I urge

you now on this point of the course

where you've got some you know some sort

of interest in getting through the

course and also getting getting

mastering these techniques play with

this as just after you solve the problem

set with ondrea's ok so the nice thing

about the toy model this toy model is

that it's easy to solve ok you can do it

in a couple of substitutions you don't

have to use the matrix form that I did

before ok ok so you can do the

substitution and then you get end up

getting this last equation which is

simply visible second order difference

equation and it's in homogeneous because

you have this constant which is not even

constant every period it's a different

value because it's shocked ok so in the

rest of the this the next 5 minutes

we'll talk about how this difference

equation second order depends when when

epsilon is constant in fact we'll make

it constant and then look at the

behavior and then we'll ask the question

ok suppose we take the shock and let it

be one in one period and then take it

away and you'll that's the most

interesting part how does the system

damp to some to its steady state ok we

call that a reduced form or univariate

reduced form and you should be convinced

yourself that epsilon is not white noise

remember white noise is very important

concept for us to macro it means

independent and identically distributed

random influences now if you don't know

more about the economy it's probably a

good idea to assume that the behavior of

Trump is kind of like white noise

okay but you know we can actually

predict some of what he's doing

you can predict somewhat of Mario

Draghi's doing and the policy makers in

general so you know in general we want

to these are things that we haven't

anticipated okay but in this particular

model the reduced form of that set up in

this form epsilon T is not white noise

it's actually a complicated it's a

mixture of two random variables one is

moving average of the demand shock and

the other is the supply shocks you

should look at that but it doesn't

really matter because we're just looking

at epsilon T as a constant and looking

the behavior of the system in response

to an isolated shock and that's what we

get when we we're looking at we look at

impulse response functions okay and this

extends to everything we do in macro so

invader and Macri can you can actually

ask a computer program to do this for

you it's not like you have to do it

yourself but you can actually ask a

computer program to plot the impulse

response function for the next 200

periods to a single shock to your model

depending on how you fix you put it

together it may look quite different

okay so here's the equation we want to

solve and now I'm setting epsilon t to a

constant epsilon zero it's the shock in

period zero okay so that shock were to

persist forever how would the how would

the equation respond to that okay so you

already see well there's lots of

possibilities it may just explode it may

explode in the positive direction it may

explode in the negative direction it may

cycle it may go up and then down and

then converge monotonically okay that's

the interesting thing about second order

difference equations you have a plethora

of possibilities okay I gave you a

cookbook to talk about that and the

cookbook is you find the particular

solution we call that the steady-state

in English okay and in theory we call it

the steady-state so let's just say okay

nothing else happens no further shocks

whereas Y gonna end up it's gonna be

epsilon 0 divided by 1 minus alpha 1

minus alpha 2

that's easy that's the particular

solution and the general solution to

this difference equation is equal to the

sum of solutions to this difference

equation okay and one of them is good

call the particular solution the other

is the solution around the difference of

the actual variable YT from its steady

state so we're decomposing the behavior

of our dynamical system into

fluctuations around the steady state and

the steady state itself so once you've

identified this is a generalized

principle so once you've identified the

particular solution you can look at the

deviation as also being a solution to

the equation that it is a homogeneous

solution because if you scale up YT in

that solution it doesn't matter okay

it's independent sorry if you scale up

the variable not YT yeah

YT right you scale it up doesn't matter

so it's independent of the constant has

no constant term that's why it's called

a homogeneous solution okay

so you need to solve for those lambdas

that's what I did last time and

basically the the equation is is simply

a sum of weighted averages and the

weights are constant Kappa 1 and Kappa 2

they have to be solved for and then you

need to know these lambdas and then you

raise those Lambs to the teeth power

okay and you can see basically how it

works if lambda 1 and lambda 2 are

greater than 1 in some sense then you

can have a problem if lambda 1 and

lambda 2 are real valued and if one of

them is real value the other will be

real valued then you have no problem

understanding so lambda 1 equals 0.9 is

great because lambda equals point 9

raised to the 200th power is going to be

a small number it's going to come close

to zero so the thing will not explode

and the same thing is true for the other

one but if they're complex conjugate

numbers you may need to refresh your

knowledge of complex conjugate numbers

they come in pairs and they can be

expressed in the polar form which means

a function of the cosine and sine of a

variable times T ok and if you do that

you

to think a little bit harder but it's

it's not a hard problem it's just a it's

just a a translated or displaced cosine

function and the most important thing is

the distance of that root to the unit

circle and that thing is greater that

distance is greater than one then you

still have the explosive problem you

have a way that's exploding basically

and I'll show you an example of how that

could work in a second okay so to do

that you easily get the lambdas and we

did that last time so I'm not going to

review all the slides I'm going kind of

slow 2x I really want you to get this

stuff getting these roots okay

these these these lambdas that you're

going to be raising to the teeth power

as you move through time will be a

function of your original alpha 1 and

alpha 2 so it is possible that you get

complex conjugate numbers if the the

object under the radical under the

square root sign is less than 0 you got

a you it's not a problem

means a mathematical challenge okay

you're gonna have a complex conjugate

set of roots that you have to deal with

but then that's not a that's not an

issue you can rewrite the thing and you

get a kook basically a displaced cosine

function okay so again this is just

repeating this slide is a repeat from

last time if you want to find Capital

One and Kappa 2 it's just a function of

the initial conditions then it makes

sense because where you where you wind

up depends on where you started if you

start from zero in period 0 and period

minus 1 or in period minus 1 and period

minus 2 then basically there's no

baggage that you're carrying forward but

if you haven't then basically where you

wind up as a result of this epsilon

shock will be different okay so that's

that's an important way of remembering

what's going on ok and it resolves as a

the slides if you haven't done that

already very important or do the

dynamics are very interesting ok so we

get this equation and we have the

particular solution which is the steady

state and then you've got the deviation

the actual variable what the actual

current realization of YT the value of

YT the the deviation of YT from its

steady state is equal to the second part

which is the homogeneous solution and

those roots can be outside the unit

circle they can be inside the unit

circle if they're outside the unit

circle that means that the and yeah if

their complex conjugate that means that

the waves are getting bigger and bigger

it's like being a tsunami that's getting

worse and worse and worse and it's it's

not a good model we don't observe that

in the reality so you shouldn't be

playing with that type of model we'd

like to have models that converge to the

steady-state because that's what we do

okay so let me give you an example of

and I'll put this if you'd like I can

post this on the on the on the website

so it's simply I'd this isn't this is it

this is a no-brainer this is like taking

a spreadsheet and doing this you know

when you get when you when you get to

like this you use better software you

can use MATLAB and stuff this is like

taking this with Excel okay so basically

I've plotted for some reason it's not

let's try it again

yeah okay so this is what happens if you

plug in a displaced cosine function and

there are interesting things that you

can derive you've we've already shown

last week that you can get this angular

frequency by looking at the roots given

by the roots of the equation the damping

is the most important thing that's the

that's the the modulus of the root if

that thing is less than one then you're

convergence so in this particular

example the modulus of the roots I

looked at is point nine five so it's

going to be slow damping it's going to

take a long time for the for the the

oscillations to go away not not visible

to you my they're still there but

they're really small okay but the first

twenty to forty periods you still have

significant wave-like motion and the

displacement is a function of the

initial condition okay so that's it's

really an interesting and here there's

no phase shift is it this is a response

to a negative shock okay now I'll give

you another example of how the ASAE

translates into that same form okay so

this is it this is a model the ASAE

model I've just put in values for a1 a2

b1 theta okay and basically you see you

get something like a and this is the

response not only of output but of

inflation okay so that actually I've

taken the out solution for output and

I've used that to solve for the

inflation so you can have this you can

also derive it directly using the the

the companion form of the reduced

reduced form okay this is just to show

you an example so if I were to change

and you can change it at your pleasure

any of those parameters and you can get

different amplitudes for the waves you

can get more dissonant more damping or

less damping and alike okay so this is

just a way of showing you how important

it is that the underlying parameters of

the model affect the dynamics that come

out

towards okay

so it's important that you look at that

stuff and get get a handle on what the

roots mean what it means to have complex

conjugate roots this is just the

beginning so when I start talking about

a serious model that we use in macro it

turns out that the unconditional

convergence is not necessary okay so the

we'll be we'll be dealing with models

that have some roots that are outside

the the unit circle okay so we need to

deal with that problem but you need to

understand this to get to that point one

discussion is that agents expectations

can also be wrong and if they're wrong

that can also be thought of as a shock

agents may also just have exogenously

RIA Dov frenzy okay so this is something

economists are more and more interested

in the possibility of irrational sort of

surges of irrationality how that would

affect the dynamics of the economy this

is related to what we think of as a

shock okay but but to understand that

last statement you need to go back and

understand why the the fact that agents

get it wrong is kind of like a

disturbance it's like it like making a

mistake okay and this has to do with the

with the Phillips curve which we

discussed last year in the last in the

first part of the last end of the of

last year okay and we said that

basically the reason why the Phillips

curve is downward-sloping

and the reason why the supply curve is

upward sloping is because agents have

expectations of prices and it and wages

that that may not be right so when wages

and prices deviate positively from their

expectations agents may be tempted to

produce more to work harder and when

they make these mistakes you know this

is an issue and maybe we need to think

hard about those those expectations and

those expectation errors so agents were

and the old fashioned models that

started with Phillips and moved into the

to the 1960s I gave that example

the United States exploiting the

Phillips curve and then getting burned

it's a great example of what Milton

Friedman said is that if you if you kind

of fool agents then you can you can

extract more output or you can push the

economy beyond its productive limits

okay but the problem with that is it's

based on the notion that people make

mistakes and you can continuously

persistently fool those agents okay and

this is not a great basis for policy

that was the criticism of Milton

Friedman and later Robert lute Robert

Lucas and Thomas Sargent so it's still a

part of this this notion of having

shocks okay but the shocks can also be

these expectation ulm estates okay and

again think about the Phillips curve

again doing this to remind you of how

important this is and how in the in the

grand scheme of economic history history

of economic thought actually how agents

help how economists have thought about

people making decisions and like this is

Phillips curve that he wrote about in

1958 okay you can see there's pretty

pretty convincing negative semi

logarithmic or logarithmic relationship

between inflation wage inflation and

unemployment during the gold standard

this is the time when inflationary

expectations were probably constant

because everyone in the world was

thinking about gold and the development

of the money supply and the if you no

matter what version of neutrality you

believe in was kind of governed by this

gold standard so it was it was pretty

reasonable to think that people believed

that the inflation rate at the time

which was fairly low possibly negative

because the gold supply wasn't keeping

up with without put was not changing

very much so Phillips discovered this

you know he discovered this in 1958

looking at old fashioned data didn't

even look at the Great Depression okay

so you know he put it on the table got

it published in economica and everyone

said wow this is great this is the key

between old-fashioned Keynes 1936 and

the neoclassical view that the supply

side wins in the end and we need to look

at the production function okay so this

is what got it all going

is what this is the story behind the

debate we called the neoclassical

synthesis and you know Milton Friedman

and Phelps criticism of that in the

nineteen late 1960s and early 1970s okay

now I'd like to do this I'd like to give

you the history lesson a lot of my

colleagues don't believe in this but I

think this is important you need to know

that you know people like these guys are

really top of the game and they thought

and a lot of them are kind of considered

old hat right think about that you spend

your whole life I mean there are some

physicists who did that there are more

economists in the graveyard of bad ideas

okay so you know take a look at these

guys these guys were out to show that

you know a simple-minded application of

and I'd even put you know even put I

didn't even put Phillips up there even

though it's his curve you didn't get a

Nobel Prize he actually seen his article

he said it's just a regularity it's an

empirical regularity it's not the basis

of policy but in America and in UK and

in Germany people started thinking this

is a basis for policy and the greatest

example is the United States Kennedy

gets elected president cuts taxes

Johnson gets into the Vietnam War they

start printing money in America to pay

for it they borrow from the rest of the

world okay putting a lot of stress on

the international financial system

inflation starts to rise and then Nixon

gets elected in 1968 and says okay time

to raise taxes and slide right back down

the Phillips curve okay but it's obvious

that that's that was just not gonna work

and Friedman Phelps picked this up right

away this is you're relating a rate of

change in a nominal variable to the real

stuff you're violating all the ideas of

neutrality of money that Hume talked

about back in lecture seven or eight

okay and they were right it took a while

but they got it right because the the

relationship did deteriorate and this

was a time when countries around the

world were printing lots of money

creating lots of money and the economy

was moving fast and people's

expectations were being ratcheted up

because they knew the governments were

going to for the most part try to avoid

unemployment by creating more demand and

financing financing that with money

creation so the 1970s was a big

challenge for economists and Friedman

and Phelps actually got it right okay

so Milton Friedman is famous address the

American Economic Association in 1967

published in 1968 basically said we need

to think about micro foundations the

rest of this course is about micro

foundations so I'm going to spend a lot

of time talking about people's decisions

to consume and save and I'm going to

talk about people's firm's decision to

set prices and that's going to drive

what we think about inflation and the

supply side that's about as far as I can

take you in this course but later on you

can do more and more you can think think

different types of households inequality

and the rest therefore expectations are

key so how do we model expectations in

of perfect perfect information agents

knew all the prices so you can't you

know but that's not the way it works

people signed contracts they take a year

they take two years to play out people

can make mistakes okay so this is

because it's costly to do that it's

costly to renegotiate every day your

wage it's costly to renegotiate prices

every day with your suppliers okay with

your customers it's also aggravating you

know one of the most famous of use of

this is Julio Rotenberg who said that

firms can aggravate their customers if

they raise their prices and it's true

when you go to your favorite bar and you

go in there you see the price of beer is

5 euros a glass used to be 350 maybe it

goes down to 2 you say well gee I did it

it's kind of it raises the tension level

raises the blood pressure of some of the

clients customers ok so this is an idea

of why we have in at least in in reality

we have something like stickiness of

a in the toy a si D model we had this

disc or inflation a trend you know

inflationary expectations we called it

this this we we've made it you can also

make you could do it

currently you could make it a wait this

is a different formulation so I'm gonna

call this you notice I use a different

notation just to get you to think about

it I've got a weighted average of last

periods value of the same core inflation

plus the new information we learned by

looking at the actual value this is

different from what I had before okay

this is cut this could be this could be

like a type of Bayesian learning you

know I I have this Bayesian prior I get

a draw of inflation and then adjust my

expectation given the given the prior

using Bayes rule it's a different and a

lot of people have tried to model

inflation and expectations using that

okay so this is a different so you could

put that in your model that gives you a

different type of dynamic it's also kind

of interesting okay again but it kind of

assumes that agents the theta of the

agents is fixed it's written in stone

which is crazy right if you go to

Venezuela if you have that type of rule

you're you're toast

okay Venezuela's 50 percent inflation

per year per month

this is ridiculous you'll never make it

you know so so you have to have a

reasonable view of how that can change

over time so here's what economists you

criticism because the original I told

you in my core inflation equation before

theta equals zero just assumes a

backward-looking completely you know an

inflation in the future the criticism

was that this type of exportation

expectation formation was assuming that

people didn't understand the way the

world works and that just seems wrong

okay because unions I don't know if you

know this but trade unions go to

economists and get forecasts of

inflation they have economists some of

you may go work for EJ Mehta be a great

job they have lots of money and they

they pay their economists like they pay

them the banking sector may be a little

bit less and they used they used their

knowledge to give them hey this is what

we think inflation is gonna be you

better at least get that for your

members otherwise you can look like

you look stupid okay so this is why

Lukas and Sergeant and others insisted

on rational expectations so we should at

least impose the discipline of the model

on the the discipline that we assume

agents to to be using on the model

itself okay it's not the same things

perfect foresight because we do have

shocks and shocks mean they're

unanticipated events for whatever reason

okay so here's the great hero of this

way of thought so Robert Lucas thought

Tom Sargent who also got a Nobel Prize

later on and with in his case he's also

affiliated with the so called Lucas

critique which you may have learned

maybe with me or Lutz Blanca in your

macro at the Humboldt University Lucas

said that macroeconomic policy has to

keep in mind that people's expectations

also are conditioned on a regime of what

they think the government is doing so if

the government is fighting inflation

hard with a steep Taylor rule it's going

to lead to people having different

anticipations

than if they thought this is a pushover

type of central bank that are

accommodating everything okay so the you

know really deep a deep point and the

Nobel Prize is completely deserved so

rational expectations something you

should probably write down your three

and they they're kind of different

levels of stringency in our thought and

the most the most frequently used one in

model solving is the strong form so we

take the strongest possible version we

just assume that agents are as smart as

we are so the model has to at least

replicate in their expectation what the

model on average generates okay so

that's called the strong form rational

expectations hypothesis so many of you

took my class in in and The Bachelor of

course this is kind of I mean I'm now

giving you the different the different

possibilities of rational expectations

it's much more complicated than then I

led you on to believe the intermediate

form would mean that agents have a

subset of information that they can

observe which may not be the same

as you have as an economist and they use

a conditional they form a conditional

expectations based on that information

so it might be better than your

information because maybe they have

better information than you have or

maybe they don't maybe you actually do

believe you know everything but the

agents don't know everything and they

still do as best they can basically

means agents are doing the best they can

give an IT the information set ok the

weaker form and a lot of people use this

in finance just means that people don't

make systematic mistakes so if you do

finance you do finance one of the most

important things in finance is the

efficient market hypothesis which means

that on average the rate of return on

stocks okay they you know the the the

persistent or predictable part on stocks

is is close to zero people who trade in

the markets you always hear about these

guys have done very well and the market

and you know trading some strategy but

in general you're only getting one side

of the story you're not getting all the

failures the people have lost tons of

on average don't yield better than the

market rate of return ok so unless the

only reason they would yield more than

the market rate of turn is if you're

bearing more risk ok so it's kind of the

the cap M version of this would say that

if you do have positive excess returns

in the market that's because you're

accepting to bear risk yourself and that

means you're it's a trade ok so the weak

form would say basically that for

example the first difference is in the

log of stock prices are unpredictable ok

it's like the random walk hypothesis for

asset prices right this is a very good

example of the weak form of rational

expectation it just means that agents

are processing information in a way that

in fact ways we may be they're using you

know fancy machine learning techniques

ok things that we don't do in this class

ok and they're just basically not making

any systematic mistakes they may make

mistakes but on average they don't ok so

let's review a few facts about

expectations before we get going because

we have to start thinking about a model

that has micro foundations

and rational expectations and some

clearer view of what's driving it okay

so I'm really pushing you hard now okay

we're gonna we're gonna think about a

random variable so agents are gonna have

to deal with some randomness in the

economy and they're gonna be taking

decisions based on a future that is

unknown and they gonna use information

that's available to them in period t so

generally we think about a probability

law little F defined on the set of

states of the world big ass okay so you

can think of the probability of

observing little s and then you can take

expectations of that and get an

unconditional expectation or you can get

a canoe can say I know I know something

about the past which helps me predict s

not exactly but some loose sense and for

that we can make a following general

statement okay suppose you know I know

some of the past and let's say let's say

that's let's say that's J and then I

know I I is basically J plus some new

information okay and then I'm going to

ask you what is the expectation of a

random variable X conditioned on the

bigger information set okay and what is

the expectation of that expectation

conditioned on the smaller information

set well the answer is it's going to be

the expectation conditioned on the

smaller information set that's called

the law of iterated expectations that's

going to help us think about the way

agents form expectations moving to time

away but we're going to come back to

this idea later on in the course when we

implication of the decisions for future

actions

because you take a decision tomorrow

the day after tomorrow okay okay that

and Reyes will talk more about

expectations in the next in the session

after this one okay so let me just

summarize and then we'll talk about our

first real serious macro model

so why is the ASID

model so lousy okay I have lots of

colleagues who really hate it some

people don't even teach it okay I have

this feeling a lot of you may not go

into economics a lot of you may not even

go into macroeconomics so I think you

should take something with you and this

is pretty good you can do a lot with

this okay shifting the curves and at

thoughts okay but the problems you want

to go further in macro this is not

enough demand is not micro founded in

that model we just assert there's some

relationship with the interest rate I

don't we don't know why okay

supply is not based on clear principles

on how firms set prices or how workers

negotiate wages expectations formation

is is basically ad-hoc the for the

perfect four side case the limit is kind

of implausible even if you do something

like that if you if you push people to

be very very rational if you let the the

theta parameter go to one you end up

getting output being it's going to be

some sort of autoregressive process but

only based on supply shocks okay so this

is not a really good theory of

propagation what we call internal

propagation so the the the holy grail of

macro is trying to find the model that

deliver us a nice internal propagation

of shocks not something we just made up

up and a lot of people will criticize

what I'm about to show you it's just

made up okay so the idea behind macro is

to keep pushing the envelope how far can

we drive the model with things that look

like things that are driving agents

decisions so a good example is the

financial crisis we know the banks had

something to do with that

so before nineteen before 2008 models

didn't have banks now models have banks

okay so if you do more advanced work in

this will have a banking sector the same

thing is true in the 1970s models didn't

have a seer

supply-side okay they didn't have a rule

for for capital stock it was kind of a

Mickey Mouse setup so you know the

responsive of macro to the to the real

world is basically a response to these

types of challenges that come up okay so

we're going to look at more micro

foundations we're going to deal with

rational expectations we're going to put

in a plausible internal propagation one

internal propagation mechanism is simply

the fact that capital that is installed

today affects the production

possibilities tomorrow right we learned

that in solo and Ramzi if I give you

more capital today you have more capital

tomorrow okay but that's not enough we

probably need something like people's

price setting behavior how many how many

agents are in the position to set prices

tomorrow

what are the costs of setting prices

changing prices tomorrow okay so the

next the next half hour I'm gonna stop

some data I want you to take some other

stuff with you if you leave macro

forever I want you to remember these

things okay macro is about growth Lucas

said growth is the most exciting thing

in macro because you just you have to

realize why you know if we can't explain

why Honduras and South Korea started at

the same place in 1960 and Korea is so

far ahead of Honduras today if we can't

explain that we're in trouble okay

that's why I spent the first half of the

course talking about growth and then

then there's fluctuations okay so even

though we think growth is most important

we know that politicians and you care

ignore the business cycle if you have a

if you have a if you can if you have a

lot of patience you can ignore it but

most people don't have patience and we

you know we're still kind of worrying

people care so much about the cycle is

the risk of unemployment is that the

jobs

because they get voted out of office in

a democracy that's a possibility maybe

there might be a revolution in a in a

dictatorship it's not really clear to me

why these fluctuations are

important if you think about the

variance over the over time or this is

the picture I showed you a long time ago

this is the evolution of GDP normalized

force I you know history etc it's a

historical series I've got some big

fluctuations they're not associated with

the business cycle associated with wars

ok and maybe a big you know the big a

few financial crises along the way but

in general the last 4050 years of of

economic development has been pretty

pretty stable but still we care about it

so I guess the idea and this is log

scale so these these little blips you

know have the same implication as usual

blips here it's easy it's easy to

conclude that actually output was more

volatile in the 19th century when we had

the gold standard than it is today still

people care about the cycle so I put a

microscope on that date and I look at

Germany between 1991 and ITA I see a lot

of action see a lot of fluctuations and

we see a one big shock which was the

financial crisis okay and you say well

the financial crisis wasn't Germany's

problem is the United States but it

larger output loss than America did and

we have to understand that because

exports the United States declined

dramatically also the credit markets

freezed up banks didn't want to lend

internationally anymore

a lot of American banks in Germany

didn't want to learn anymore and this

propagation of the cycle across national

boundaries was was quite extensive so

this is we're gonna learn about now so

again the ways to do this you should

know already this is like you just take

a take an HP filter and apply it to the

data and then take the difference of

that this is what you get okay so you

can see that these these these are an

absolute constant price euros so the

cycle is getting bigger because the

economy is bigger right so it's it's

it's really a it's disingenuous to say

that Germany's at recession was so bad

because 200 billion euros

decline in in in in value in GDP and

value-added a constant prices that's not

really fair you have to think about in

terms of percent because macro

economists know that what matters is the

percentage change big economies have big

fluctuation small economies of small

cons fluctuation okay so we need to we

need to get that out of her mind so you

might want to normalize by something my

normal you know one way to normalize

would be just to take the take the

deviations of the logarithmic trend

that's easy you could also do what burns

and Mitchell did so you remember what

Burns and Mitchell did if you didn't

you're gonna learn burns and Mitchell

economists from the 1920s and 30s in the

United States that thought there's a

science behind business cycles I'm going

to take the cycle that I showed you

before and I'm gonna chop it into pieces

boom and bust and I'm gonna take each

one of those pieces and I'm gonna treat

it as an observation of a cycle and then

I'm gonna take averages of those cycles

okay so the interesting idea in fact

they even they were even willing to

compress time in the cycle so they're

willing to look at you know cycles of

sixteen quarter duration and cycles of

40 quarter durations being equivalent

and just normalizing on the length of

the cycle to one okay most people don't

do that nowadays but even if you don't

you get you get an interesting set of

pictures I'm going to show these to you

and this is taking eight oacd countries

over 50 years quarterly data okay so

this is not sigh this is not this is

model independent this is stuff you can

take with you no matter what you do in

life okay you don't have to be a macro

economist you just have to be someone

who cares about the world to be

interested in this what do you see here

you see the private consumption

expenditure okay this is over the

average cycle this is ten quarters

before the peak of the cycle and this is

ten quarters after the peak of cycle

okay and I haven't I have indeed trended

these data so you see the trend after

every cycle things get better okay into

the next downturn

and there's a sharp decrease at the

point when the recession starts and you

have a slow recovery this is the average

behavior over eight countries some

countries have a few deviations from

that but the important thing is this is

an average behavior and that's kind of

what we're interested in these models

because like Luca said business cycles

are kind of all alike they have a lot of

similar dominant characteristics this is

consumption remember y equals C plus I

plus G Plus and X ok this is I okay this

is C and this is I lots of volatility

investment this is investment spending

only 20 to 30 percent of GDP huge

volatility so write that down investment

expenditures highly volatile if anything

brings down Germany in the next few

quarters it's going to be investment

spending okay it might be investment

spending in the value added chain for

the UK because we're getting brexit it

might be the chain with the United

States because we're investing to build

cars for Americans they don't want to

it could be uncertainty about been

building vinda beta or power lines okay

so it's a lot a lot of it this is really

a different type of behavior it's also

procyclical this is government spending

this is government purchases goods and

services on average no correlation with

the cycle surprising we ought we often

think government's actually try to act

against the cycle well maybe they did

this is an average maybe sometimes they

did and sometimes they didn't this is an

average so there's no systematic

tendency of government spending to get

ahead of the cycle or fall behind it

this surprised me this is probably a

publishable result this is real money so

if anybody wanted to say something about

that what would you say if you were a

banker

loans this is I mean again six quarters

before the peak of the cycle what do you

see happening it's not just slowing down

it's flattening out and remember the

balance sheet of the consolidated bank

banking sector has liabilities money

balances on the liability side on the SS

side it has loans and investments so

what's happening here is that banks stop

lending they're slowing down their

credit giving credit extending credit up

to six this is this is an average so you

can imagine how how different many of

these countries are this is you know

again the idea behind burns as

Mitchell's you're taking this average

tendency so real money balances real

money which is m1 or m2 / u we used m1

divided by P the price level okay is a

leading indicator so this is something

you might want to be able to capture in

one of your models okay here's some

other interesting indicators look at

look at nominal interest rates

procyclical with slightly lagging stock

indicator is basically the isn't is the

the nominal stock price index divided by

the CPI

okay that's why it doesn't have a trend

okay some interesting interest rates

long-term look like the short-term rates

and the differential the spread and in

interest rates tend to invert when the

recession is on the way okay in fact the

zero point is when the the long-run

interest rate is below the short run

interest rate so these are important

robust stylized facts that we want to

care about here's some other ones that

might interest you

productivity is procyclical on average

but it drops sharply in the unset of a

recession the first two quarters after a

recession it drops okay

so that's gonna be a challenge for us to

explain employment we hope that

employment drops employment is obviously

procyclical but it's also a lagging

indicator unemployment is also a lagging

indicator real wages don't do much real

wages kind of just keep on going but

they're they're kind of smooth and the

wage share wage share is the you can

think of this is a very important

variable this is kind of a proxy for the

the ratio of wages to nominal

productivity of workers just take W

times L divided by P times y and write

it as W divided by Y P or P y divided by

L this is kind of a this is a way of

thinking about wages in relation to

productivity we see that actually in a

recession the wage share rises it goes

up okay

the profit share is that is the converse

or the the complement of the wage share

be one minus this so profits tend to

rise in in the boom time but then they

fall when the recession starts to loose

so capital takes a hit and recessions

so we could summarize this by saying and

again this is summarizing other things

that we didn't discuss directly

recurring cycles are irregular so the

Slutsky approach is a good idea you know

magnitude is small relative to average

GDP as the lucas idea that most of the

action is the growth trend but we still

get this these these cycles they seem to

look similar but if you think about it

if we could just organize our society we

could probably avoid the welfare costs

of these fluctuations but we don't

people get mad they vote people out of

office etc consumption investment of

procyclical the current account is there

for counter cyclical and government

spending doesn't look like much of

anything okay

investments more volatile private

consumption and government consumption

are less volatile

okay so this key investment is a driver

of the cycle we think an investment is

driven by you know animal spirits

to the future it's a very important part

of the of the theory and a lot of other

variables move with the cycle as well

some move procyclical some counter

like financial variables and some are

lagging like the labor market variables

okay so these are the things that are

gonna guide the next two blocks of the

course the two models that we're going

to look at okay so now I have a nice

picture for you to break up the monotony

a little bit see some of you are falling

asleep hmm so who are these dudes if you

don't remember the guy in the right eye

I feel sorry for you okay

he's not very important anymore but in

fact I kind of miss I kind of miss him

who the other two guys anybody so I only

show a Nobel Prize winners in this

course right this was a reception for

two Nobel Prize winners huh no that's

not come on I think Friedman was on it

he was still alive but he was pretty old

okay these two are the fathers of the of

the model I'm gonna present you now this

is fin Kidd 'land and Edward Prescott

okay fin kid Linda's Norwegian and

spends a lot of his time at the at the

business school what's it called Bergen

and Prescott is it Arizona okay they

used to both be Minnesota guys I think

it was also at Carnegie Mellon okay so

they got the Nobel Prize in 2004 for

their work on the real business cycle

model the RBC stochastic growth model so

again this is the bridge to the first

part of the course take the first part

the Ramsey cores the part and think of

taking the Ramsey model chopping it up

into discrete periods and shocking every

period with a shock and only one shock

shock to the production function now why

did they do that

so they basically in their famous paper

in econometrics to they said is it

possible to generate the cycles that I

showed you before without any recourse

to money without any reference to a

financial sector and they were probably

fascinated by the fact that in the gold

standard you know the financial sector

was kind of he had this gold standard so

that the Fed didn't have there was no

Fed there was no central bank in the

United States until 1913 okay so the

gold standard

you still had cycles right so this is

kind of interesting fact some of the

most impressive development of the world

economy in the Western economy was in

the years in 1852 eighteen to 1910 okay

so these guys were kind of driven by

this fascination with the possibility

not necessarily the assertion that it

was so but is it possible to generate

cycles in a model without any banks or

money at all so imagine a world where

money's neutral people get it right like

we have the theta going to two-to-one in

our baby toy model how would that look

so the source the shock that will be

propagating this model is technology

it's a lot of techno people like this

you know technology we we tend to think

there are other things that drive models

you know maybe consumer optimism maybe

shocks maybe but also maybe banking and

financial shocks so we can expand the

model but right now we're going to take

a model with just one shock and see how

far we can push it that was the idea

behind kid land and Prescott and can we

generate cycles that look like the ones

I just showed you because that was the

genius of the that's why I think this is

a great and it also the fact that people

still use the basic set up even in the

New Keynesian paradigm is tribute to

these two guys okay so that's why I

showed you the picture before

so you have some respect for these guys

okay so let me just give you some some

words then I'll write down some

equations I already gave you the

historical stuff so the real business

cycle is kind of like the Ramsey caste

Koopmans model because it has agents

with it with an infinite horizon so we

have really smart agents in this model

they have rational expectations they

don't know the future perfectly but they

have a pretty good understanding of how

it works the model is discrete time

every you know T equals one two three

there's no two point five or seven point

nine or something it's it's basically T

is from this set of integers we have

supply shocks where the supply shocks

yeah and these supply shocks are shifts

to the production function okay now one

of the weaknesses of this set up is that

these shocks are persistent and they're

set up so they had to cheat a little bit

to make this work I'm saying this

because it's being filmed and maybe

someday Prescott will click on my

website and take a look at this because

he knows this but it's a fact that to

get this model to work really well you

have to assume that the technology shock

has a bit of persistence in it and that

is what we call a deus ex machina you're

assuming kind of what you want to prove

but the idea was still so good that we

let this would go okay so keep that in

mind there's a little reference now this

is where gets a little bit complicated

the model has once you have it give that

one shock it has a trajectory back to

the steady state but it's a unique it's

a unique trajectory there's only one so

in the sense of Ramsey it's a little bit

like jumping on that saddle path and

coming back to the steady state so if

you remember we had these complicated

paths with the arrows and stuff and

andreas told me I shouldn't show that to

you because it's it's boring it's

exciting because that's work that's what

this is coming from okay you have a

unique trajectory given one shock the

trajectory to the steady-state is unique

so this is what we call saddle stability

and this will have an implication for

the roots of the system that we will

write down so keep that in mind that's

why it's important to understand

characteristic the eigenvalues of the

matrix and the roots etc okay

and I already gave you historical so

when so repeat this it's really so it's

a lovely story okay so I should tell you

one thing Ramsey Ramsey had the central

planner and she was in charge of

everything

remember the central planner she was

really smart she did it she solved this

using the pantry Aegon's maximum

principle well in this setup we're gonna

have decentralized markets so we're

going to decentralize the system and it

turns out that because of the way we've

cooked the model it works and it turns

out also that the the social planners

optimum if we wanted to solve it would

give us the same answer as the

decentralized outcome okay again that's

because the the first welfare theorem

applies in this model I've the model we

cooked in such a way that preferences

are normal preferences with geometric

discounting non time consists in

consistent preferences technology gives

rise to constant returns to scale

markets are competitive firms are price

takers workers or wage takers okay there

are no externalities and everyone's got

the same information set so there's no

possibility that we would deviate from

the first will for a theorem of

to get going and again I promise you if

you go on to do this for a living

you will deal only with economies where

the first welfare theorem fails and the

government has a role to play

almost by construction as soon as you

have monopolistic competition firms are

setting prices you don't have a

competitive market anymore

okay so just keep that in mind

now a good model you need to write down

what you have do you have you have

households households have budget

constraints firms employ the labor of

the households and the firms have

technology and the firms use labor and

they use capital where do they get the

labor they hired from the households

where did they get the capital they also

hire it from the households so this is a

representative agent model okay now one

of the biggest criticisms of

macroeconomics in the past 20 years is

that we spend too much time staring at

these kind of models because we think

that you can summarize all the

interesting incentives by looking at a

single agent and if you think about it

what what kind of interesting stuff

could we have if we had a rich agent and

a poor agent and the rich agent had all

the capital and the poor agent had to

work ok that's the next step those are

called heterogeneous agent models ok but

right now you have to understand this

and this is already involving a fair

amount of elaborate mental gymnastics ok

so I'm going to write down those

incentives the preferences I'm going to

write down market structure I'm going to

derive optimal behavior and then I'm

going to derive an equilibrium concept

that's how we're going to proceed so for

the rest of this presentation and also

next week we're going to think about

additive periodic utility so in each

period the agent has consumption has

utility over consumption and leisure and

leisure is L sorry leisure is 1 minus L

so think of L is the fraction of the day

that think of L as fraction of day that

you work so 1 minus L is the amount of

leisure you can enjoy so the second

argument is bounded from above by 1 by

construction ok

so we're gonna have labor supply in this

model we're gonna try to explain

employment ok and we're going to use the

following function it's a logarithmic

function of consumption in real terms

and it's going to be a power function

with ADA as a an interesting parameter

which is going to summarize the

elasticity of labour supply then we're

going to have cobb-douglas production so

cobb-douglas production has lovely

properties it may not be the perfect

production function it's a good

starting place for us it was a starting

place for Kidman and Prescott okay so

we'll start there it has it has a

beautiful form look at it and we're

gonna have this thing called Z which is

the state of Technology so if you shock

Z holding K and L constant you get more

output and that that sets the model in

motion because in that in that moment

agents have incentives to invest more

because product productivity is higher

workers may have incentives to work more

because maybe the wages are higher so

that's what kind of what's gonna give

rise to the cycling's well similarly if

Z goes down people kind of interested in

saving investing anymore they're not

interested working they take a break

okay so that's kind of the that's the a

very very loose summary so here are the

ingredients okay we're going to start

with it this is what expected utility

looks like once you've accepted what I

said before it's gonna since we're

dealing with discrete time we don't use

an integral anymore we use the summation

of utility in each period beta is the

familiar with from the OLG model beta is

less than one so that thing is defined

okay so you get positive utility in

every period from consumption and you

get you get negative utility from

working hard more you work the less

leisure you have so this is this already

imposed some interesting structure on

the model I didn't put in imposes intra

temporal separably not just

intertemporal separable ax t it imposes

in truck temporal seven separate bility

the margin utility of consumption is

independent of how much you're working a

lot of people think that's kind of a

kind of a lousy assumption but we don't

have very good information or ideas on

how there are newer ways of thinking

second equation is the capital

accumulation equation the so-called

Goldsmith equation and note that

depreciation is partial if Delta were

equal to one then basically I'd have to

start from scratch like Robinson Crusoe

every year you'd have to need to decide

how much of my wheat do I plant it how

much do I eat okay well this

you can store it okay so if the capital

stock is moving through time and that's

Katie all right and then we have the

cobb-douglas production function and now

look what I've done I've given ZT a life

of its own ZT is now what we call a

stochastic process okay means that ZT is

random so I know it ZT is today but I

don't know what its gonna be tomorrow I

can I can possibly use the past to

predict CT and the way Kaitlyn Prescott

started this is they set it up like this

okay so basically ZT is equal to the

power of ZT minus 1 and if you take logs

that's going to be 1 minus Rho times

that power it's going to be Rho times

that log of that power sorry the log of

that value of z2 minus 1 plus 1 minus

Rho times the steady state of Z call it

Z bar plus stuff that may depend on time

so if we take the time trend out it

disappears and you've got this shock and

epsilon is white noise so that's the

stuff that really is purely

unpredictable as we go through time

nobody knows that whether that's going

to be next period okay that's again

that's a fundamental implication of

these models that again regardless of

what process you're talking about a

shocked difference equation is important

so this is our this is a great example

of the shock difference equation in the

log of technology this type of

technology has a special name we already

discussed it in the first part of the

course we called it do you remember type

of technical progress or three types

Harrod neutral technical progress okay

it's important write it down okay it's

labor augmenting it it makes people look

like they're getting you know bigger

more powerful to it and by the virtue of

that even though they capture all the

effects of their increased size

that's why it's called labour augmenting

technical progress as opposed to capital

augmenting so called solo technical

progress or Hicks neutral technical

progress in the cobb-douglas function

case they're all isomorphic okay

remember that I mean make some of you

nod and say I get that many of you don't

know what I'm talking about just

remember that it's important in the

complexes it doesn't matter but if you

try any other production function it's

gonna matter it's gonna matter so we

need to keep we need to keep pushing

this I've only got two minutes I'm just

gonna set set it up next week I'll

actually write down the optimization

problem we're thinking about a again

and because we don't have any money in

this economy we only have one good the

prices are easy the only price is the

intertemporal price so I'm producing

schmooze today the wage is in terms of

schmooze so W is a real wage R is the a

number of schmooze I get in period t

plus one if I put a unit of shmoo to

work in the capital stock tomorrow just

like Ramsey okay you notice mu is it's

the the single good in this economy in

every period there's just one good and

we eat it we smoke it we drink it we put

it in our cars to make it run it's just

the aggregate Hicks bundle of the of the

economy that's produced by the

production function okay it's a single

good economy so this is a really simple

this is but I believe it's it's simple

but it's complicated that the amazing

thing is out of this simplicity you can

get complexity so if you go further then

once you've mastered this you're in

great shape okay so if you go into a PhD

in some foreign country this will be

great starting you will already have a

leg up on many of your competitors but

believe me this is just the start the

last part is very important firms are

owned by the households okay so this is

like Bill Gates is my brother so on

average the US economy benefits from all

the wealthy guys and believe me if you

do the math even if Bill Gates were my

brother

if you were all my brothers too it

wouldn't make much of a difference okay

I mean make a little difference but I'm

I'd like to have a piece of the u.s.

profit share but it's just minutes if

we're reasonable we're talking about an

average family so you've got people

families that are better off families

and worse off we're describing the

average family in this economy the

assumption is that the firm's own the

capital so katie is the vehicle for

savings just like in the OLG model

except you're saving forever now you're

saving for your children it's a single

it's a representative family kay is the

way you save l is the way you work so we

have labor in this economy unlike ramsey

this is a an accretion if you like it's

an expansion of the model and we don't

have money okay so the to summarize

preferences technology market structure

optimal behavior and then finally what

is the equilibrium what is going to

describe the situation where workers

supply the labor they want to at the

wage they face firms demand the capital

to use in production at the expected

rate of interest or rate rate of

interest they face okay so with this

i'll let you go so it's really important

that you spend a little time on this you

know and spend some time with andreas

this week okay thank you for your

attention