Prof: All right,
well, today I'm going to talk about Social Security again.
There's going to be one more
discussion of Social Security later.
I'm going to defer most of my
plan until later, but I want to finish the
discussion so that we totally understand the subject.
It'll also allow me talk about
demography and introduce one of the most famous models in
economics called the Overlapping Generation Model.
So in the 1940s someone named
Maurice Allais, a French economist,
introduced the Overlapping Generations Model into
economics. He wrote it in French and it
was sort of rediscovered by Samuelson in the 1950s.
I'm not sure whether Samuelson
had read Allais. I think Samuelson may well have
read Allais, but anyway Samuelson--so these were 1947,
something like that, 1958 in which Samuelson
rediscovered it. And you'll see it's a very
basic thought and it seemed at first to challenge everything
that we've learned so far. So the idea of the Overlapping
Generations Model is that time doesn't have a beginning and an
end like we've assumed so far, time might go on forever.
Now, whether or not you believe
time-- whether there's scientific
proof that time goes on forever or scientific proof that the
universe has to come to an end, let's face it,
many of our institutions presume that time goes on
forever. The chief most important among
them is Social Security which we'll see is the easiest thing
to, the model's design to understand.
The idea of Social Security is
the pay as you go Social Security.
The idea of Frances Perkins was
that every young generation was going to give money to the old,
but they shouldn't worry so much about it because when they
got old the next young generation would give them
money. Obviously if you thought time
was going to come to an end the last young generation,
knowing that they were the last generation,
would refuse to give money to the old because they weren't
going to get anything back when they were old.
But then the second to last
generation knowing that when they got old that they'd get
nothing from the last generation's young they wouldn't
give anything either to the old, and working backwards like that
if everybody's rational and it's common knowledge that the world
is going to end nobody would ever participate in the Social
Security scheme. So it's clear that there's some
thought that the world might end, or at least there's a
thought that it's not worth bothering about the world
ending. So the Overlapping Generations
Model is meant to take that idea extremely seriously and imagine
life going on forever. So let's take the simplest
example where there's a generation that
begins--generations last when they're young and old.
So let's say we're at time 1
here, and there's a generation that's young and a generation
that's old. Maybe I'll write it a little
bit lower. Sorry about that.
So there's an old generation
and a young generation and the endowment of the old generation
is 1. The young generation has 3 when
it's young, 1 when it's old, 3 when it's young,
1 when it's old, 3 when it's young,
1 when it's old. So when you're young you have 3
apples. When you're old you have 1
apple. The next generation when it's
young has 3 apples. When it's old it has 1 apple
and so on forever. So this'll be T = 1 here,
2 here, 3 here, 4 here, etcetera.
So it goes on forever like that.
Now, what did Allais and
Samuelson both basically say? They both basically said,
look, everybody when they're young is incredibly well off.
They're working.
They're productive.
When they're old and retired
and feeble they don't have very much, but what can you do?
Where can you trade?
According to Samuelson what
does the old have to trade? It doesn't look like there's
any trade that can take place. So it seems like it would be
very helpful if the young would give the old something,
and when they got old the next generations young could give
them something. So the young could constantly
be making gifts to the old. That was Allais and Samuelson's
idea and it's very closely related to Social Security.
So Samuelson got a little
carried away with his idea and he said that Social Security was
the greatest and only true beneficial Ponzi scheme ever
invented because in a Ponzi scheme,
this has gotten a lot of attention lately thanks to our
famous Madoff. So a Ponzi scheme is basically
an investment strategy where you take the money that people give
you, and you buy yachts with it,
and then when they ask for their money back you tell them
that you've gotten great returns and you just hand them money
that new people have given you. So they think you've gotten
great returns. And when that second generation
of people wants their money back you tell them,
well, you've invested it brilliantly,
you've gotten great returns, but all you're doing is giving
them the money that the third generation is giving you,
and you keep going like that until finally you owe so much
money you can't find new people fast enough to pay the people
off and then everybody discovers the Ponzi scheme and it all
unravels. And the people at the beginning
have made out very well and the people at end who,
you know, the second to last generation they've lost all
their money and then lawsuits accumulate.
And it's quite interesting that
in the Ponzi scheme the very first generations that benefited
are not free because the whole thing was a scheme.
The guy Madoff running it knew
from the beginning what was going to happen.
And so those first generation
of people, there's no reason why they deserve to make their great
returns. They benefited from the Ponzi
scheme, so the last generations suing
Madoff are effectively suing the first people as well,
and so we'll see how the courts decide it.
But often the people who get
out of the Ponzi scheme early still are held liable and the
money is taken back, so we'll see what happens in
this case. But anyway, so obviously Ponzi
schemes are terrible ideas in general.
But Samuelson said well,
when life is going on forever there might not have to be an
end to the scheme. It just keeps going.
This guy gives to this guy.
This guy gives to this guy.
There's no reason for the Ponzi
scheme to end if you really think that the world's going to
go on forever, and therefore Social Security
is a great Ponzi scheme which is actually beneficial.
So Samuelson even wrote this as
a journalist for Newsweek and there's
many-- I happened to be a little boy
when he was writing this stuff and I remember some of the
articles and I've gone and found them.
And so he describes the
benefits of Social Security as a Ponzi scheme.
So this didn't turn out to be
quite right, but it certainly sounds plausible.
Another thing he said was he
said that in the Social Security--
money, you could think of as a Ponzi scheme,
money too, because you have worthless pieces of paper,
but you're willing to hold them when you're young,
you'll accept them for goods because when you get old you can
find the next guy who's willing to give up goods for your money,
and that guy is willing to take the money for it because when he
gets old he can find the next generation's young who's willing
to give up good for his money. So the money,
which is a worthless piece of paper,
never gets exposed as being worthless because there's always
another generation around to take it who thinks they're going
to be able to use it later. So these are fascinating topics
that Samuelson discussed, but I don't want to discuss
them in this class. I want to discuss Social
Security, and so I'm going to do a
variant of the model that Samuelson never thought about,
which I think is a much more realistic model.
So I'm going to do overlapping
generations with land, and you'll see that in this
model I'm going to add land to it.
It's, I think,
a more interesting model and I'm not going to discuss the
paradoxes of infinity because once you put land,
even though time goes on forever the paradoxes disappear.
So rather than spending all the
time on the paradoxes, and how they come about,
and do they make sense, I'm just going to add land from
the beginning and it will recapitulate in a nice way
almost everything we've done this semester and there won't be
any paradoxes. So what do I mean by land?
I mean suppose that land
produces 1, so this is land output, produces 1 every period.
So there's another output
called apples that just, you know, a tree that's going
to live forever, let's call it land,
produces 1 every single period. So here are the periods and
this is what's happening in every period.
Now, land seems to make the
situation much more complicated, but in fact it will turn out
that we can analyze this pretty easily.
So let me summarize,
again, the model. The model is that every
generation has 1 agent or a million identical agents,
let's say. Every generation has
endowments, so every generation t has endowment.
It's generation t so it's an
endowment at time t and at time t 1 equals (3,
1) and let's say they all have utility and the utility of every
generation t, which only depends on what they
consume when they're young and old let's say is log
x_t log x_t 1.
So everybody cares about
consumption when they're young, consumption when they're old,
they don't care about consumption any other time.
They begin with 3 apples when
they're young. They know they're going to have
1 apple when they're old. Everybody's like that for
generation t greater than or equal to 1, but generation 0
just has U^(0), so I can call that U^(t).
Generation 0 only cares about
consumption when old and that's that guy at the top.
But I also have to talk about
the land. Generation 0 owns the land.
So that's the economy,
very simple economy, but it looks much more
complicated than anything we've done before,
but it will turn out not to be, but it looks it at first
glance. So every generation has 3
apples when young, 1 apple when old except the
very first generation which has land in addition to his 1 apple
when old, and that land produces 1 apple
forever. So we have to figure out what
equilibrium is and then we have to look at Social Security.
And by doing this we're going
to understand Social Security much better than we did before.
So are there any questions
about this, what's going on? Then I'm going to try and write
down what equilibrium is and then solve it,
and then we're going to talk about Social Security.
Yep?
Student: Is U not
supposed to be >?
Prof: Oh,
well if you only can eat 1 good if I put log x_1 here
it would be the same thing. Remember if I take a monotonic
transformation, if I double the utility or take
the utility, e to the utility like that,
which is just equal to x_1 it describes the
same utility. The guy's not trading anything
off. There's only 1 good,
so more of the good is better for him.
It doesn't matter if we call it
log x or x_1 is the same thing, but that's a good
question. Let's call it log
x_1, make him symmetric.
All right, well,
how would we define equilibrium?
So there's financial
equilibrium, and then we're going to see if we can solve
this. So what's happening in
financial equilibrium? Well, it's kind of interesting
here. There's going to have to be a
price of goods every period q_t.
That's the contemporaneous
price of apples. There's going to be the price
of land every period, pi, let's call it
pi_t. So this is apple price.
So these are contemporaneous
prices. With contemporaneous apple
price, land price, what else do we need?
Well, we have to decide what
everybody's going to consume every period,
so generation t what they're going to consume when young and
when old. Then there's generation 0,
what they're going to consume when old.
What else do I need to describe
equilibrium? That's probably it.
And so what's the budget set
everybody's going to face? A budget set for t greater than
or equal to 1 is the set of all--let's call it young
consumption and old consumption. So generation t because it's
going to consume something when young and when old,
such that, what? What's the budget constraint?
Well, when they're young if
they want to consume goods they have to spend q_t to
consume goods. So this is young, call that Y.
And when they're old they're
going to have to consume, sorry, when they're young
they're consuming--what else would they want to do?
They might want to hold land so
we could call that pi_t and I better add
a theta here for their holding of land,
pi_t theta, that's how much land they hold.
Remember how we did this?
And then what have they got?
We'll they've got their
endowment, time t. This is time t, let's say.
This is generation t.
So their endowment is going to
be e^(t)_t, but that's 3,
so I might as well write e^(t)_t just as 3.
And what else have they got?
Nothing?
Everybody comes into the world
with just apples when young and apples when old,
so when they're young they've got 3 apples they can sell.
They can consume something,
so 3 - q. The stuff they don't consume
they sell and they can buy land with it.
Ah, god all this time this
never happened. Sorry about that.
And when they're old what can
they do? Well, when they're old at
q_t 1, times Z now,
are they going to bother to hold the land when they're old?
No, because they're going to be
dead and they don't care about their children.
All they care about is eating
as much as they can. So they're going to sell all
their land, so that's got to be less than or equal to,
what? And so here's the--what are
they going to get when they're old?
Well, they have q_t 1
when they're old, and their endowment,
of 1 what? They can sell their land
pi_t 1 times theta. Whatever they bought the first
time they can now sell when they're old.
Plus what else do they have?
So one last term,
what else is it? So see what they do.
Why would you buy land?
Well, because when you're young
you're so rich you don't want to consume everything when you're
young, so instead of consuming your
whole 3, you consume less.
You buy the land because the
land's going to be worth something next period.
So what is it going to give for
you next period? What do you get next period
from the land? Well, if you own the land you
get what? You get the dividend.
So the dividend we have to
multiply by the price, which on one of the early
classes I forgot. And the dividend is 1,
that's the dividend, but how many dividends do you
get? How many apples?
It depends on how much of the
land you had. So if you had theta units of
the land you get, you know, if you had 3 acres of
land you get 3 apples which you can sell for a price q_t
1. You also still have the 3 acres
of land which you can sell off when you're old.
So that's the revenue you get
by selling off the land and you also sell off some of your
endowment, maybe, and that's how you can
buy when you're old. So that's what the budget set
is of the young, of every generation.
And the budget set for
generation t=0 is simply x_1,
is simply Z, we'll call it Z such that what
does this guy do? Well, q_1Z has to be
less than or equal to, what does he have?
q_1 times 1,
plus he's got all the land. So he's got 1 acre of land.
That's all the land there was,
so he's got pi times--pi_1.
He's got all the land,
so I've normalized the land to be 1 acre.
So pi 1 times all land.
That's what that guy can do.
So equilibrium is,
x_t is best for generation t in budget set t and
x_1, so this is this and theta,
and x_1-- did I write theta?
Oh, I forgot to write theta in
the definition of equilibrium here,
so I have to write a theta t there also having to keep track
of how much land they're going to have.
This is t = 1 to infinity.
So the equilibrium is what are
the prices every period of apples and land?
What does every generation do
in terms of their consumption and how much land they hold and
how much does that very first generation consume.
It's obvious they're going to
sell all their land to consume as much as they can.
So the budget set,
so x_1 solves. So Z = x_1 and y Z
theta equals that, best for generation t in this
budget set, and Z = x_1 is best
for generation 0 in this budget set.
So to say it in words it's very
simple in words, and then we have this
mathematics and it looks complicated but we're just going
to say it's going to be very short to solve it even though it
looks very complicated. So the problem is this.
We've got generations who are
rich when young, poor when old,
there's land that lasts forever.
The only way people can save is
by holding the land. That's like holding stock.
That's holding something real.
So when they're young they're
going to take some of their extra endowment,
because they're so rich when young,
they've got 3 when young, and they're going to use it to
buy land. And when they get old they're
going to sell the land, and eat the endowment from the
land, the dividend from the land,
and sell the land and use that sale proceed to also increase
their consumption when old. So they'll be taken care of
when they're old because they're able to hold the land.
And the question is how do we
solve for this equilibrium? And so why is it--so just to go
back to Samuelson and all that, why is it so interesting?
Well, one of the reasons it's
interesting is that people when they're young have to think
about the price of land next period when they're old because
they know they're buying the land today.
They can see the price today,
but when they get old they have to,
you know, why are they buying the land today,
partly for the dividend next period when they're old,
but also for the resale value of the land.
So everybody is thinking to
himself, what's the value of land going to be next period?
And of course the value of land
next period depends on what the young in that period are willing
to pay, but they're thinking about what
they're willing to pay on the basis of what they expect to
happen the period after. So everybody has to think about
the guy after him, and what the guy after him is
thinking about what the guy after him is thinking,
and it looks very complicated. And we want to solve for an
equilibrium which everybody can rationally anticipate what the
guy in front of him is going to do which means rationally
anticipate what that guy is rationally anticipating the guy
in front of him is going to do, and you have to solve for the
whole equilibrium and see how it turns out.
Any questions about this?
It looks very hard,
but it's going to turn out to be very, very simple.
Yes?
Student: For generation
0 why haven't we added the dividend that he would get from
holding the land in the second period?
Prof: By the way,
I haven't said what happens to the dividend in period 0.
So actually I think that was a
good point. So he gets 1.
He also gets the dividend in
period 0, so I'm glad you asked that question.
So to answer your question,
remember the convention that we've made which holds in the
market and it's one of the reasons for the breakdown of the
market. One of the reasons why the
market seized up in the last year or two,
or the last year, is because when you buy a stock
like the land somebody has to give the money and the other guy
has to give back the stock, and the people buying and
selling are not actually meeting each other and simultaneously
transferring money for the ownership of the land.
One guy's in San Francisco and
the other guy's in New York and they're doing it through some
screen or something. So the physical asset isn't
quite changing hands. So you have to make a
convention about when do you say the deal has actually concluded.
So the convention is,
that we always use, is that if you buy the land at
time 0 you don't get the dividends--
at time t, if you buy the land at time t you don't start
getting the dividends until time t 1.
So if the young generation buys
the land at time 1 they don't get the dividend until starting
at time 2. So the very first dividend at
time 1 is going to go to the old guy at time 1,
which I had left out here. So the old guy does get a
dividend, it's the dividend at time 1 because he began owning
the land. So he had it before,
so he gets that piece of land. Even though he's selling the
land at time 1 he still gets the dividend at time 1,
and the generation that bought it at time 1 doesn't start
getting dividends until time 2. So that was an excellent
question, and it was an oversight of mine,
so exactly right. And that's how it happens in
real life. Of course, the length of time
might be 3 days or it might be 1 month.
It depends on the security,
what the settlement rules are, but there's always got to be a
break between when you buy the stuff and when you start getting
the dividends because it just takes time for the whole
physical process to happen. So they say t 3 is a very
common kind of settlement, or t 1.
That means that in 3 days or in
1 day, and so if you're desperate for
cash and you have to give-- so anyway, I won't get
into--we'll come back to this when we talk about the crisis
and what happened. So people who are desperate to
get stuff, it doesn't start coming for a little while,
so. Any other questions?
Yes?
Student: Should we
multiply the price of apples by q_1?
Prof: Should I what?
Student: Should we
multiply what >?
Prof: Absolutely.
Very good.
Any other comments?
All right, now how do you solve
this? Well, let's figure out how to
solve this. So the first thing you could
notice is that--so Fisher never thought of having infinite time
and never thought about Social Security.
Maybe he thought about Social
Security. I'm not aware that he had any
thoughts about Social Security. We didn't have Social Security
so it's unlikely he had been thinking about it.
So I don't think he thought
about time going on forever, but that doesn't mean his
methods aren't-- he died, by the way,
in 1947, I think, something like that,
so right around the time Allais wrote his paper.
So what is it that he said?
All his lessons are going to
hold true. The first thing he said is that
look, in every period there are qs on every side of things.
Here are q_t,
q_t, q_t 1,
q_t 1, q_t 1.
So if you just double--there's
no loss in generality by taking all the q_ts to be 1.
We have no theory of inflation
yet, because there's no money or anything.
So you might as well assume
we're measuring everything in terms of apples and take
q_t to be 1. So without loss of generality,
as they say, q_t = 1,
and of course that means we can divide every equation by
q_t. So pi_t divided by
q_t would go here and we'd have a 1 and a 1.
We divide this one by q_t
1, still it's an equation. We have Z less than or equal to
1 pi_t 1 over q_t 1 1 times 1 times
theta, so we just re-normalize all the prices.
So, re-normalize all nominal
prices in terms of apples. So one simplifying thing is we
can get rid of--so think of pi_t as the price of
land in terms of apples. So we just get rid of all the
qs here. Just assume that they're 1,
and so we have this equation. And now that means that
pi_t = price of land at time t in terms of apples at
time t. So that was a normalization we
did many times before. We said that we might as well
assume the contemporaneous price.
We can't figure out what
inflation is, so let's figure there's no
inflation. The price of apples is always 1
and we're going to measure the price of land in terms of
apples. So we get the same equation
that now looks a lot simpler. So the qs, we don't have to
really worry about. We're just worrying about the
pis every period. So what did Fisher say to do?
So we've got to worry
about--what did Fisher say to do?
What was his--whenever you have
an economy, a stock market economy like this,
what did he say to do? What was his advice?
He said turn it into general
equilibrium. How?
By doing what?
Yeah?
Student: Adjust the
endowments. Prof: So Fisher said,
Fisher's lesson, forget about assets by putting
their dividends into the endowments.
That's his first lesson.
So you see this is a good
summary of what we've learned so far, and a second lesson was
look at present value prices. So he says forget all these pis
and things like that. Just look at the present value.
So we're looking at
p_1, p_2,
p_3 ... where p_t,
price at time 0, let's say or time 1,
it doesn't matter, price at time 1 of an apple at
time t. So once you knew the presence
of the endowment then, I'm not going to write all
this, the endowment of all generations 1 and above it stays
the same, (3,1), (3,1),
(3,1), but the endowment of generation 0 is now 1,
1,1, 1,1, 1,1 forever. And the budget set,
everybody's going to have a budget set determined by the ps.
So now we're just one step from
solving this. So there's one more thing to
notice before we can solve it, so by symmetry,
so observe that by symmetry we can hope to find an equilibrium
with p_t 1 over p_t = p,
a constant. So every generation is the same.
The only relevant price for a
generation is the tradeoff between the price of goods when
they're young and the price of goods when they're old.
That's what Fisher said.
The veil of the stock market is
just a means of transferring wealth between when you're young
and when you're old, and really you should
calculate--all these guys have to calculate.
If I buy the stock now when I'm
young it's going to cost me a certain amount of money and I'm
going to be able to get a return.
So the real rate of return,
right, that they're all going to calculate is going to equal--
we'll, by putting in pi_t dollars today
they get out pi_t 1 the dividend,
1, tomorrow, right? So tomorrow they're going to
get pi_t 1 1. So if they buy 1 share of stock
it costs them pi_t and tomorrow they get pi_t
apples, we're measuring in terms of
apples, and in the future they get
pi_t 1 1 apple in the future.
So that's the real rate of
return. That's like 1 r,
1 r_t, but we're going to assume that
that's a constant. 1 r we're going to guess it's a
constant and that's just 1 over this p that I told you about
before, the ratio p_t 1, the present value.
So this ratio is just the
interest rate between time t and time t 1.
So Fisher says you don't have
to think about all the stock market and what the return on
the stock market and all that's going to be.
Really you're just trading off.
By doing all that calculation
you're figuring out what's the tradeoff between time t goods
and time t 1 goods. In the Fisher economy you don't
look at the stock market. You assume that everybody knows
the present value prices and therefore the tradeoff between
time t and time t 1 goods, and we're going to assume
that's a constant because the thing's so symmetric how else
could it turn out except a constant.
So now we're ready to solve it.
We've done all the tricks,
almost, to solve it. So what is equilibrium going to
be? So here's equilibrium.
It's going to be a very simple
equation. At every generation you're
going to have an old and a young.
So what is the total supply of
goods in every generation? How much goods are there?
There's 1 for the old apple.
All right, if you look at any
generation like this one at time 2 there's that one,
no not that one, a little less.
The one in the middle that's
the 1 for the old guy, then there's the young guy who
has 3, right, and then there's the apple that
the land produces. So this is young apples and
this is the dividend of apples. So that's how many apples there
are in the economy. So who's going to be eating
them? Well, there's going to be an
old guy eating apples and there's going to be a young guy
eating apples. So how much is the old guy
going to eat? Well, the old guy's going to
spend half his money, and how much money does he
have? Well, from his point of view
he's just trading off when he's young against when he's old.
So from his point of view he's
got 3 apples when he's young plus 1 apple when he's old,
but that's worth, to him, the tradeoff between
apples when he's young and when he's old is just given by p,
divided by p. All right, so this is the whole
trick. So we have to spend a minute
until this dawns on you why this is true.
And now what's the young going
to do? The young generation,
what's their income? Well, they only care about the
tradeoff between prices when they're young and when they're
old. So when they're young and when
they're old they don't care about land according to Fisher.
They don't have to think about
that. They just have to know the
price, which we're assuming is p for everybody,
the tradeoff between young apples which are worth more than
old apples. So to them they have the same
income 3 1 p. Now, these ps refer to
different time periods, but we're assuming the same.
Here's the young.
This is the young,
so this is divided by 1. So this is the young and this
is the old. And once I solve this equation
we'll have the whole equilibrium,
but we need to understand this equation [note:
the equation is: (1 half times (3 1 p) over p 1
half times (3 1 p) over 1) = 1 3 1 = 5].
So where did I get this
equation? So let's take any time period 2
like T = 2 for example. We know the total apples around
that can be eaten are 5. That's on the right hand side.
The old guy's apple,
the young guy's 3 apples and the land producing 1 apple,
5 apples in all. Now who's going to be eating
the apples? There are going to be old guys
and there are going to be young guys, eating apples.
So the old will be the
generation 1 guys. They're going to be eating
apples, and then the generation 2 guys are going to be eating
apples. Now, the generation 2 people
they've got 3 apples at time 2, so they're looking at 3,
but they've got expectations in their head.
They're going to think ahead.
What's the price of stock
market today versus tomorrow? Fisher says they do all that
thinking, they realize the tradeoff
between consumption at time t when they're young and
consumption at time t 1 when they're old.
That's given by the price p.
We've assumed there's a price p.
So they're going to say to
themselves, "Okay, I've got 3 apples
when I'm young, 1 apple when I'm old is not
worth the same in present value terms--
it's just 3 1 P because the old apple's not worth as much at
time t as it as time t 1, it's 3 1p.
I'm going to spend half of my
income and the price when I'm young I've assumed that's
one." So that's what the young are
doing. The old guys,
now the old guys what they're doing at time t depended on what
they did when they were young. But when they were young at
time 1, they knew there was a tradeoff
between time 1 and time 2 apples,
but we've assumed it's the same price tradeoff,
so we've assumed it's the same p, and so they did the same
calculation. When they were young they were
going to spend half their money, this time 1 generation is going
to spend half its money when it's young,
half its present value when young, divided by--
so when they were young they were going to do that,
but when they're old now they're looking forward to
spending half their present value when they're old,
and the price of apples when they're old is given by p
relative to when they're young. So they're going to spend 1
half times the present value of their income,
divided by the present value of the price of the old apple which
is the apple we're talking about because they're consuming when
they're old now. So their consumption plus this
consumption equals 5. Now, at least half of you must
be baffled, so ask me a question to see if we can get to the
bottom of this. That's it.
That's the whole equation.
As soon as we solve this we'll
figure out all the prices and everything in the whole economy.
And so we're at the end,
but this requires a little bit of thought, so go ahead.
Student: Shouldn't it be
4 1 p because there's the dividend, or maybe like 3 2 p?
Prof: No.
Student: I mean,
where does the dividend fit in? Prof: That's a good
question. The dividend fit in here
because, yes, there's a dividend.
Here's the dividend that got
produced, so there's an apple to be eaten from the dividend,
right? Now, you're saying--so it's a
very important question you're asking.
So it's correct what I wrote,
but it doesn't sound correct. So what he's saying,
the question is, what happened to the dividend.
This generation,
say, this young generation, either one, say the old
generation. The old generation,
when they were young at time 1 they bought land looking forward
to their old age. They sold the land,
and they got the dividend. How come that's not factoring
in to their demand? That's his question, right?
The answer is because that's
precisely the point of what Fisher did.
Fisher said,
yes, in the real world everybody is thinking to
themselves, like generation 1, "I'm young now.
I'm buying the land because
when I get old I'll be able to resell it and I'll also get the
dividend." But Fisher has thought ahead.
Fisher's saying if the guy's
going to think about pi_t 1,
that's what he'll be able to sell the land for,
and he's going to think about the dividend he's going to get,
and his rate of return is therefore divided by how much he
had to pay for the land today. That's his rate of return.
But see, the bottom thing is
how many apples he had to give up, pi_t there,
is how many apples he had to give up to get the land.
The numerator is how many
apples he gets next period after selling the land and taking the
dividend. So that ratio is the tradeoff
between apples today and apples in the future for him.
So if he says to himself,
what Fisher says, the only thing the guy cares
about is that ratio, that tradeoff,
which we're calling p, 1 over p.
That's the tradeoff he's going
to have in his mind. And so what is his income?
This guy began with no land.
Only 0 began with the land.
Everybody else buys the land
only as a means of getting to more consumption when they're
old. So the whole point of Fisher's
insight is you don't have to keep track of how the guy's
managing to get the payoff when he's old.
All he's doing is he's
recognizing a tradeoff of 1 r or 1 over p,
that tradeoff between consumption when young and
consumption when old and all he can do is turn his endowment
when he is young into more endowment when he's old at that
ratio, and it doesn't matter how he
does it as long as we have the right ratio.
So Fisher says forget about the
assets. Just keep track of the present
value prices and these will tell you the ratio of transformation
of goods when young to when old and that's all you need to know
to make a decision. That's the whole point of
Fisher. You don't need to think about
the assets. Now, after we get the ps we'll
go back and figure out what the price of land is.
Any other questions?
Yes?
Student: Can you explain
again why the price of the second part is just 1?
Why can you use 1
>? Prof: Yes,
because I'm taking p as the ratio.
So p, this p is here,
so it says if I had 1 apple at time t how many apples could I
get at time t 1, [correction:
1 over] p of them, right,
because this ratio--so for every apple I have down here I
can get [correction: 1 over]
p apples at time t 1. If the ratio of two prices is p
it doesn't matter what their levels are--so I might as well
as think of one of them as 1 and the other one as p.
I could think of the first one
as 2 and the second one as 2 p. That would be the same thing,
right? So if I thought of the first
price as 2 and this price as 2 p I'd put a 2 here,
a 2 here, and a 2 here and it wouldn't change anything.
Remember, we learned this the
very-- this is why the lessons of
general equilibrium they seem so obvious and then you put them in
a slightly different context and you realize how clever Fisher
was. If you double all the prices
you're not going to change anything.
It's the price ratios that
matter. So it's the tradeoff between
apples when young and apples when old that matter.
If you assume that tradeoff is
given by p you might as well assume that the first guy,
for all we care, he might as well assume that
he's measuring the prices when he's young in terms of 1,
and the prices when he's old in terms of p.
Those were very good questions.
This is a little confusing but
when we go back to the original equilibrium it'll be clearer,
I think. But so that's the only equation
that we have to satisfy. So we can solve that equation
now. Any other questions?
Let's solve that equation.
That equation is I'm going to
multiply by 2 p, so I'm going to get 10 p on the
right-hand side, right?
Because if I multiply by 2 p I
get 5 times 2 p is 10 p. On the left-hand side I'm just
going to get 3 1 p 3 p p squared.
So if I rearrange I just get p
squared--uh-oh, my usual problems here.
p squared - 6 p 3 = 0.
p squared - 6 p 3 = 0,
so p = 6 (and I'm using the quadratic formula which I assume
you know) - b squared - 4--. So 6 squared - 4 times 1 times
3, minus 12, over 2 = 6 - the square root of 24 over 2.
So the square root of 24 is a
little bit less than 5, so this is going to be a little
bit more than 1, sorry a little bit less than 5,
so this will be a little bit more than 1,
so the whole thing will be a little bit more than a half.
So let's say it's .55.
So here's a crucial step.
How did I know when I said 6
plus or minus, why did I take the minus?
Because if I had taken 6 plus
this I would have gotten a gigantic--well,
let's come back to that. So how did I know to take 6
minus that instead of 6 plus this?
So if this is the p then what
are the prices? What are Fisher's prices?
We've just solved for
equilibrium and these prices are going to be--we call this 1,1,
maybe .55, .55 squared, .55 cubed etcetera.
That's what Fisher says the
prices are. And we can now figure out
everybody's consumption. Y and Z, what's Y going to
equal? Y is this thing on the right,
1 half 3 1 p, so it's .355 divided by 2.
So that's 1.775.
And what's Z?
You should check.
I'm going a little fast here
for myself. 3.55 divided by 2 is 1.775,
so what is Z? It's going to be 1 and a half
plus--no, it's 3.5 divided by 1.1.
Student:
> are those numbers
> Prof: p is .5--I'm going
too fast for you, ah ha!
p is .55, right?
I just got p.
So if I want a C now,
go back to consumption, what's the consumption going to
be when you're young? It's going to be,
Y is going to be 3 .55 divided by 2 which equals that,
right, 3 .55 that's the guy's income and he consumes a half
when he's young, so that's 1.775.
And when he's old he's going to
have the same income, so 1.775 divided by p,
so it's going to be 1.775 divided by .55,
and that you can see--no, that doesn't look right.
Yeah, that divided by p and
that's going to be a little bit more than 3 and so,
in fact, if you solve it out it turns out to be 3.225.
Did I go too fast there?
I'm plugging in .55 there.
So it's 3.55 divided by a half.
That was that number,
1.775 divided by p which was .55,
so you can see it's a little bit more than 3,
because .55 into 1.7 is a little bit more than 3.
In fact, if you solve it out to
some decimal places it's that. So we've solved what everybody
does. Now we know what everyone's
going to do. The young are going to spend
1.775, are going to consume 1.775, they're going to consume
1.775 here. They have an endowment of 3
apples. They're not going to eat them
all. They're going to consume 1.775
of them. And then the old,
at the same time, what are they doing?
They're consuming 3.225,
but you notice that those things add up to 5,
so the consumption when they're young plus the consumption when
they're old-- so the consumption of this old
guy there is 3.225 and you add the consumption of that,
yeah, I went too high there, the consumption of the young
guy just under him is 1.775, the two of them add up to 5.
That exactly clears the market.
So at time 2 you repeat the
same thing, at time 3, etcetera.
So you see we've already
cleared all the markets except the one at time 1 which looks
more complicated, but we've cleared all the
markets. Now, what's the price of land
going to be? What's the price of land?
Nowhere to write that.
Let's write it here.
What's the price of land which
is going to be a constant? How do we figure that out?
How would Fisher say you figure
out the price of land? So the price of land at time 1,
say, what would Fisher say? Yep?
Student: Whatever the
young guy would pay for it? Prof: That's one way of
getting it. So what's he paying for it?
Student: Whatever he
doesn't spend on his consumption.
Prof: Right,
so from this equation he spent 1.775.
His income was 3,
so what's left over? What did he spend on land
therefore? Student: 1.225.
Prof: So the price of
land has to be 1.225. Now, that's not the way Fisher
suggested finding out the price of land.
What did he say you should do?
What's the fundamental theorem?
Student: Present value
of all the payments. Prof: And what is that?
So the land pays 1 apple every
period. Student:
> Student: So it's paying
1. Prof: So it's paying 1 p
1 times p squared 1 times p cubed 1 times p to the fourth,
right, because looked at from the point of view of time 1 you
get an apple next period relative to the apples today.
That's worth p.
An apple in 2 periods is worth
p squared at time 1 because an apple at time 2 is worth p
apples at time 1 and worth p squared apples at time 0.
So you just keep doing this,
but this is a perpetuity and so therefore it's equal to?
Student: 1 over r.
Student: We're going to
>. Prof: Yeah, 1 over r.
So it's equal to 1 over r,
and so what's r? So what's r?
How do we figure out what r is?
Student: If we know p
then we can find r. Prof: 1 over 1 r = p,
right, = .55, so therefore 1 r = 1 over .55
and r = (1 over .55) - 1. And so this is a little less
than 2 - 1 is going to be like .81 or something,
and you take 1 over .81 and you get the same number.
So Fisher solved everything.
I mean, the Fisher method
solves it all. Let's worry about time 1.
We haven't done that.
So what happens every period?
Every period like from 2
onwards, the young guys says, "Ah ha!
I've got 3 apples.
What am I going to do with
them?" He says to himself,
"Well, the price of the land is 1.225 so I could eat
some of the apples or I could buy some land."
And what does he decide to do?
He says, "Let me eat 1.775
apples and spend the rest of my money buying 1 acre of land.
Now, why am I doing that?
Because next period I know the
price of land's going to be the same 1.225 and I'm going to get
a dividend of 1, so I'm going to be getting a
rate of return (we just calculated) of 81 percent."
So this ratio up here,
this is another thing, this also this number is equal
to 1.225 1 divided by 1.225. That's also equal to 1.81.
Remember r we just calculated
over here. Where did I do r?
r was 81 percent.
That's the same number here,
81 percent. So everybody says to himself,
"Given that there's an 81 percent rate of interest I'm
happy to hold the whole unit of land because at that rate of
interest I'm just trading off consumption today for
consumption when I'm old at the rate that I want to."
And at time 0 the market clears
also. So we cleared the market for
every time 2 through infinity, and that was by this equation.
That was up here.
By picking the right p we know
every market from T = 2 onwards was clearing.
And Fisher would say by Walras'
Law we don't have to worry about time 1, that's going to clear as
well and sure enough it does. It's a little bit different now.
It's just the old guy,
but the old guy with his land which is 1.225 plus his dividend
of 1-- his land which is worth 1.225
and his dividend of-- hope I wrote down the right
price of land all this time. Oh, that would be bad.
So the old guy, what does he do?
He has his 1 apple plus he has
the land so he's going to consume--the old guy has
his--oh, I see. So what happens at time 1?
The old guy has his dividend
that he had before, so he's got the dividend of the
land because he's owned the land forever.
So he gets the dividend of 1,
so that's 1, plus he has an endowment of
one, so he's consuming 2 now. Plus he sells the land for
1.225. So that all adds up to 3.225,
and then if you add the young generation's 1.775 that indeed
clears the market at time 1. So the market's going to clear
in every single period, but we only had to solve it for
periods 2 and onwards which were all symmetric because by Walras
Law, according to Fisher,
once it clears from time 2 onwards without even bothering
to check we know it would have had to work at time 1 and sure
enough it did. So as I said,
let's summarize now what we've done and then we can start
drawing the lesson. So what we did is we started
with a complicated model with land and people having to look
forward and expect what the price of land was going to be
depending on what the next generation wanted to hold,
which depended on what they were going to think the
generation of that were going to hold etcetera.
Very complicated stuff.
And we saw that to solve it was
very simple. You just do the Fisher thing.
If everybody's rational you can
forget about the assets and the land and turn everything into
present value prices and put the endowments--
so you forget about all the assets and just put the
dividends into people's endowments and look at all the
present value prices, and the present values prices
by symmetry, we're assuming,
just grow exponentially, decline exponentially.
And then we can solve the one
equation and figure out what that price was,
the exponential number p that's to the nth power gives the nth
price, the present value price,
so solving for that p we then cleared the markets.
We found out what everyone's
going to do when young and when old,
and by plugging in now Fisher's formula,
the price of every asset is the present value of its dividends,
we figured out what the price of land was every period,
so we've solved for the whole equilibrium and sure enough it
clears. And in equilibrium everybody's
doing this calculation. If I buy land today I'm going
to get that rate of return on the land which corresponds to
the p. It's going to be 81 percent,
and so everything works out. So what's this got to do with
Social Security? Are there any questions about
this? I sense a little bit of
puzzlement still. You shouldn't be that far away
from understanding it, so let's hear a question.
What don't you don't understand?
Just point to an equation you
don't understand. Yes.
Good, brave of you.
Student: How do we show
that theta should equal 1? Prof: So Fisher says
that--the way I solved it is I ignored the assets.
So I didn't pay attention to
what the assets were. I just put the dividends in the
endowment. I didn't pay any attention to
what people were holding of the assets because Fisher says
forget the assets all together. Just do the present value
prices and augment the endowments.
And I found the present value
prices by getting this factor p, and then it was just p to the n
and I found what everyone was going to consume.
That was it as far as Fisher's
concerned, but then Fisher says once we've
found general equilibrium we can go back to financial equilibrium
and figure out what the price of land is,
which is the present value of the dividend,
so it's price is 1.225, and the step I left out,
which you're asking about, you can also figure out what
assets everybody's holding. So Fisher's saying--what assets
are they holding? Well, the guy,
he's consuming 1.775 here. We figured out the price there
of 1.225 so it must be that he's holding exactly 1 unit of the
asset, and so the asset market is clearing too.
But that's no accident.
Fisher's saying if you clear
all the markets doing the present value general
equilibrium stuff and you go back to the financial
equilibrium you're automatically going to be clearing all those
markets too. So I left out the step because
somebody anticipated all that and got me to calculate the
price of 1 in a cheating way, he said assume theta's 1 then
figure out how much money you're spending on the asset,
so we did that. What I should have done is done
Fisher's trick of figuring out the price of the asset,
which is 1.225 and then, of course,
we know the guy must have bought 1 asset in order to use
up his budget set, but that clears the market for
assets. But Fisher knew that was going
to happen. It always has to happen.
That's the beauty of what he
did. Yes?
Student: So in that
equation over there we have to assume p is less than 1?
Prof: Yes,
so the point is, thank you, I'm coming back to
exactly that point. So the point is when you look
at the present value of land it's going to be p p squared p
cubed p to the fourth .... The present value of land had
better be finite, so in other words p has to be
less than 1 otherwise the value of land would be infinite and it
wouldn't make any sense because in the very first old guy with
an infinite value of land would buy more apples than there
possibly were in the world. So you know that the real
interest rate has to be positive.
If the real interest rate were
less than 0, and you had some asset that paid a constant
dividend forever, that asset would have an
infinite value. So the presence of land,
which pays a constant asset forever, forces the real rate of
interest to be positive. So Samuelson and all his talk
of negative real interest rates, it can't really happen.
Land's going to pay some
dividend probably forever, so there's going to be a
positive real rate of interest. And so it means in Social
Security, if the young give up 1 to get 1
back when they're old they're always going to be losing
because there's a positive real rate of interest,
and so every generation has to lose.
So it's precisely the point.
Because of the presence of land
you know that the real rate of interest is going to have to be
positive. Fisher never made this argument.
He said it's impatience and
maybe if people are incredibly patient it could even turn out
to be negative, or if output was bigger
[correction: smaller] next period than it is this
period you could even have a negative rate of interest,
but not so when you have land with a constant dividend
forever. Then if the interest rate is
constant it'd better be positive, otherwise the land
would have infinite value. So it's a new argument for a
positive rate of interest. The land has to have a finite
value. Any other questions?
So when you solve for a couple
of these you're going to do it very easily,
I mean, after you've done it a couple times this will seem very
easy to you. I know it seems a little
confusing now, but let's just do a couple more
thought experiments. Suppose we do Social Security?
What will happen with Social
Security? How does that work?
What does Social Security mean?
Well, Social Security means the
young give the old something and until now we talked about it as
if the young could give the old part of their endowment.
The young pay taxes and the
taxes get handed over to the old guy.
That's pay as you go and we
talked as if it wouldn't change the equilibrium,
but we said at the same time that Social Security was the
most gigantic program any government anywhere in the world
has ever adopted and the giveaway was bigger than GNP for
a year, 17 trillion compared to 12 or
14 trillion. So clearly it's going to have
an effect on the interest rate. So we ought to take that into
account if we're doing a more careful analysis of Social
Security. So what would Social Security
do? How would I take into account
Social Security? Suppose every young person gave
1 apple to the old guy at the same time?
How would I figure out what
happened in the new equilibrium? How would the economy change?
What would I change and solve
differently?
Well, all I would do is I would
change this to a 2. Every young guy now only has 2
apples when he's young because he's given 1 of them to the old
and the old would have 2. So all the way through here I
would just change all this to 2,2, 2,2, 2,2,
2. That's what I would do.
That's the change.
Then I have to re-solve the
equilibrium. So how would that change?
What would I change in my one
equation? Well, the apples in the
economy, the young apples, the guy's only got 2,
but the old's got 2 and there's still 1 apple coming from the
land, so that's still 5.
But now every generation's
going to be behaving a little bit differently.
They're going to have 2 when
they're young and 2 when they're old.
This guy will have 2 when he's
young and 2 when he's old and otherwise it's the same thing.
So I just re-solve for the
equilibrium. So if I re-solve for the
equilibrium multiplying by 2 p I'll have 2 2 p 4 p multiplying
by 2 p. Sorry, I just confused myself.
I multiplied by 2 p.
I've got 2 2 p multiplying by 2
p. I have 2 p here,
plus multiplying by 2 p I have 2 p squared here.
Hope I'm doing this right.
And I've got 10 p on the right
like I had before. So now I've got 2 p squared -
10 p and that's 4 p, so it's still 6 p (it looks
like) 2 = 0. And so p = 6 or - 36 - 8 over 4.
And so that,
I hope I did that right, I hope you're checking it.
So that turns out to be p is
2.8, no. Social Security, p is .38 now.
So now it equals .38.
So 6 - square root of 28 so
it's working out pretty much, so .38 is the new price,
so therefore the interest rate 1 over 1 r = .38.
So what do you think happens to
the interest rate? So r now equals--before I write
it you can figure out what it is yourself in a second.
Do you think the interest rate
went up or down in the new economy?
What would Fisher have said?
Student: Up.
Prof: Up,
it went way up, and so it's actually 347
percent. So the interest rate went from
81 percent to--no that can't be right.
It went to 161 percent.
So the interest rate went up.
So the loss is even worse than
it seemed before. Remember, in present value
terms, when you're young you give up 1 and when you're old
you get 1 and so you lose the present value.
So the present value of that
trade is this plus that over 1 r.
Well, now that r has gone up
you're losing even more. So Social Security at the
current interest rates looks bad for every generation.
After you do the Social
Security and everybody understands it's happening it's
going to be even worse in terms of present value.
So Social Security,
again, everybody is giving something when they're young to
the old. So the guy at the very
beginning, at the very top, gains a lot.
Everybody else,
every other generation loses and you can compute the utility
after Social Security compared to before Social Security and it
goes from something like 2.7 to 2.3.
So there's a substantial loss
for everyone's utility except for the first generation.
On the other hand we rescued
the first generation. Now, there are two more
experiments. I'm not going to be able to
finish today, but I'm going to mention them.
Experiment one is suppose we
had more and more children every generation?
How would we take that into
account? Well, it's very simple to take
into account. The same thing with the trivial
change you can figure out what happens with more and more
children. So I'm going to go back to
(3,1). This will only take me one
minute. Sorry about this.
I know time's running out,
but let me just finish this story.
So if we had more and more
children in every generation, so every 30 years let's say the
population doubled, that's not such a high growth
rate per year, you'd have 6 and 2 here and
then it'd go up to 12 and 4. But, again, it's all
exponentially growing and the dividends would also be growing.
So this would be 2 and 4 and 8
and 16 etcetera. But as you will see in a
second--I won't do it this time. Next time you'll see that it's
very easy to solve for the new equilibrium.
You put a double here thing
because the young, there are twice as many young.
You just solve it and you do
the whole thing. And what happens is Social
Security isn't solved. Samuelson was in a way wrong
again. Even though there are two young
people for every old person--so every young person only has to
give up half an apple. You only have to give up half
an apple when you're young and when you're old you still get a
whole apple back. It sounds like now surely you
should gain, but the point is you don't because the rate of
interest gets higher. And then we're going to have
generations that alternate in size, but all these are very
easy to solve once you figure this out.
So on Thursday you have to
solve a problem just like that so you get the hang of it,
just one problem to do for Thursday.
