WEBVTT

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Prof: I think I'm going
to start.

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So this is really the beginning
of the finance part of the

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course.

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So far we've reviewed general
equilibrium, which I said Fisher

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invented or reinvented in order
to do finance.

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And as you remember the main
conclusions from general

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equilibrium are first that the
market functioning by itself

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without interference from the
outside,

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in other words a situation of
laissez-faire,

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leads to allocations that are
Pareto efficient.

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So they're in some sense good
for the economy and good for the

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society.

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They don't maximize total
welfare.

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That's not even a well-defined
thing as we saw last time

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because how can you measure,
how can you add one person's

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utility to another.

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It doesn't even make sense.

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So economists at first were
wrong to think of that as the

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criterion for good allocations,
but there's another better

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definition of efficiency that
Pareto invented,

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called Pareto Efficiency,
and the free market achieves

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Pareto Efficiency at least if
there are no externalities and

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there's no monopoly.

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So that, lesson number one,
was taken to mean that the

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government shouldn't interfere
in the free market,

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especially shouldn't interfere
in financial markets,

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and that's something we're
going to come to examine.

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The second lesson we found was
that the price is determined by

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marginal utility.

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It's not determined by total
utility.

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So it may be that water is much
more valuable than diamonds

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because it does a lot more good
for everybody and for the world

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as a whole than diamonds do,
but the last drop of water,

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really most people have as much
water as they need,

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the last drop of water is not
doing that much whereas the last

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diamond is a rare thing and not
many people have them.

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So the last drop of water is
worth less than the last equal

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weight of diamonds and therefore
water is much cheaper than

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diamonds even though water's
much more valuable as a whole

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than diamonds.

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The price of things depends on
their marginal utility.

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A third implication of what we
did is that there's no such

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thing as a just price.

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It depends on what peoples'
utilities are and how much they

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like it.

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It depends on how much of the
good there is.

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That's why diamonds are priced
less than water [correction:

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that's why water is priced less
than diamonds].

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And it depends on how wealthy
people are.

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If you transfer money from
people who don't like apples

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compared to tomatoes,
to people who like apples a lot

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compared to tomatoes,
the price in the free market is

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going to reflect more the latter
class of people than the former

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because they've got the money to
spend,

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and so the price of apples is
going to go up relative to the

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price of tomatoes.

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So those are the three basic
lessons of general equilibrium.

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The first one about
laissez-faire has a huge

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implication for whether there
should be regulation,

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but the second pair of
implications,

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what determines the price and
how price changes as you

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redistribute wealth and so on,
and no just price,

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that set of ideas,
you'll see, is also going to be

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very important for finance.

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So those lessons seem clear.

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Some of those lessons were
understood already by Aristotle

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as we said.

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So the ancients understood
supply and demand,

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at least a little bit of supply
and demand,

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and yet as soon as they moved
from apples and oranges to

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finance,
they all got hopelessly

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confused.

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So Aristotle said,
"Interest is

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unnatural."

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I could go through a lot of
people and what they said,

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but I'm going to just leave it
at a few quotations.

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The Bible says interest is
terrible.

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Judaism frowns on interest.

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Christianity frowns on interest.

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Islam frowns on interest.

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All the great religions of the
world crystallizing,

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obviously, some of the most
important thinking of the time,

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frowns on interest,
so just to remind you of a few.

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So why do they frown on
interest?

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Well, the idea is that you do
nothing, the lender does nothing

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and he gets back more than he
lent to begin with.

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He's making a profit without
having exerted any effort

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whatsoever.

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So, Sulinay Middleton said,
"In trade both parties are

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expected to gain,
whereas in lending at usury

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only the usurer could
profit."

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So in Deuteronomy in the Bible,
so this is the Jewish Bible,

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is says, "Thou shalt not
lend on usury,"

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(that just means interest).

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"Thou shalt not lend on
usury to thy brother.

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Usury of money,
usury of victuals,

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usury of anything that is lent
upon usury, "--that's all

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terrible.

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Of course, "unto a
foreigner thou mayest lend upon

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usury, but unto a brother thou
shalt not lend upon usury."

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So the Jews could lend to
Christians but not to each

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other.

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So the Christian Church
outlawed usury,

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called it a mortal sin.

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Luther, for example,
says, "For who so lends

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that he wants it back better or
more, that is open and damnable

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ocker.

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Those who do that are all
daylight robbers,

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thieves and ockerers.

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Those are little Jewish arts
and tricks."

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So there was this antipathy
towards usury,

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and because Jewish moneylenders
were able to lend to Christians

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there was an antipathy to Jewish
moneylenders which we're going

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to come to when we talk about
Shakespeare.

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So Muslims also forbid lending.

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In fact, even today it's
illegal to charge interest in

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Islamic law.

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So in my hedge fund we tried to
raise money,

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and there's lots of money in
the Middle East,

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and most of it,
by the way, ten years ago,

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almost all Middle Eastern money
was invested in U.S.

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Government bonds and U.S.

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stocks, nothing else like in
mortgages, for instance.

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So I went to Saudi Arabia and I
met a bunch of brothers of the

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King, the eldest brothers of the
King, and I suggested they

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invest in our hedge fund.

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And they actually became sort
of interested,

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and so we had to write up a
complicated contract.

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Now, you know a mortgage pays
interest,

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so if you invest in the hedge
fund and the mortgage is paying

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interest it looks like they're
getting interest,

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and so that wasn't going to do.

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So we had to write a very
elaborate contract which

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disguised the fact that interest
was being paid,

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and it had to be overseen and
blessed according to Sharia Law

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by a holy person who was going
to verify that there was no

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interest.

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Now he charged a fee which was
a percent a year which looked an

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awful lot like interest,
but anyway so.

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So the point is all these
religions have banished interest

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despite the fact that they
themselves were involved in

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interest,
and lending, and borrowing.

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A world can't function really
without lending and borrowing

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and the charging of interest.

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So these religions that forbade
it at the same time knew that it

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was going on and allowed it to
go on and sometimes participated

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in it.

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But the point I'm trying to
make is that there was vast

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confusion,
and even today there's

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confusion because still today
the Jewish law doesn't allow for

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interest between Jews,
and--there's a charade that

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goes on there just like there is
in Islamic law,

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and just like there is--still
frowned upon by the Christian

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Church.

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So it's a hard subject to
understand, and why is that?

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Why is it that it's so
confusing, and how should we

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understand it?

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Well, Fisher cut through all
this extremely simply,

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and the way he did it was he
said, "Let's just think

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mathematically then we won't get
so tied up in all these

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religious complexities.

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Just let's do something
mathematical and concrete."

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So suppose that we consider a
problem, which is the one I'm

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going to work with the rest of
the class.

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Maybe I better do it over here.

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So let's say that there are two
agents and two goods.

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So the two goods now are
X_1 and X_2.

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So Fisher's first insight is
that let's think of

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X_1 and X_2
as apples,

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but apples today and apples
next year,

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Fisher said--although they're
both apples,

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exactly alike,
there's no difference between

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these apples,
the apples this year are

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different goods from the apples
next year.

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So let's move away from apples
and tomatoes to apples this year

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and apples next year.

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So I'm not going to call them
goods X and Y anymore.

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I'm going to subscript them by
time.

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These are both X because it's
the same good,

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but they're different goods
because they occur at different

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time periods.

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So Fisher said we can
incorporate time simply by

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having different goods.

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So of course people,
he said, are going to have some

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utility of consuming today
versus consuming tomorrow,

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and let's say this utility is
(log X_1) (1 half log

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X_2) for Mr. A.

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So I'm going to come back to
this in half an hour and explain

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why Fisher thought that this
half made sense.

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You see, this Agent A likes
good 1 a lot more than good 2.

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So Fisher would say that's
because Agent A is inpatient.

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An apple is an apple,
but if you get it now it's

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worth more to you,
it gives you higher utility

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than getting an apple next year.

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"This is a law of human
nature,"

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he claimed,
which I'm going to come back to

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later,
and that's why when you write

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down the utility function
there's a discount factor,

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which we're going to add--a
discount factor--

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which discounts,
reduces, the utility you get

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from future consumption.

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So let's say (U^(B) of
X_1 and X_2)

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= (log X_1 log
X_2).

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So B is more patient than A is.

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B actually doesn't discount the
future.

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A does discount the future.

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So A is impatient,
relatively impatient,

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and B is patient.

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They have endowments,
so E^(A) let's say the

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endowment is (1,0).

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Say it is (1,1) and E^(B) let's
make that (1,0).

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But now Fisher wants to talk
about finance and he wants to

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talk about stocks,
and bonds, and interest and all

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kinds of things.

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So he says, "We've talked
about good with no problem.

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We can talk about goods today
and next year with no problem,

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let's talk about stocks."

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What is a stock?

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Let's say there are two stocks,
stock alpha and stock beta.

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What are stocks?

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I mean, they're pieces of paper
that you're trading,

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but they give you ownership of
something like a factory or a

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company or something,
and what good is the company?

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Well the good of the company is
that it's going to produce

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something.

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So let's say that the stock is
going to produce something in

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the future.

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So we'll call the production of
the stock--so what is the

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future?

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There are only apples in the
future.

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So let's say
D^(alpha)_2 is 1 and

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D^(beta)_2 = 2.

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Fisher says,
you can tell a lot of stories

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about what this stock does,
and what its method of

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production is,
and what kind of managers it

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has and a lot of stuff like
that,

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but in the end people care
about the stock because the

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stock is going to produce
something,

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and the value of the stock is
going to come from what it

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produces.

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So D^(alpha)_2 is
what people expect the output of

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the stock to be next year which
is the last year we're worrying

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about,
and D^(beta)_2 which

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is 2 is what people expect the
stock to produce next year.

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And we're going to assume that
perfect foresight here.

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So Fisher says, "Well,
in general people's

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expectations might be
wrong," but let's start off

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with the case--
people anticipate something,

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surely they're looking ahead to
the future when deciding whether

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to buy the stock.

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We've got to assume something
about what they think.

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Let's suppose they actually get
it right and they know what the

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price of the stock is next
period.

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So what's going to happen?

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Well, we can define an economy
and presumably the interest rate

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and the stock prices and all
that are going to come out.

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Now I should mention,
by the way, I forgot to say

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this, but as I write this down I
suddenly realize I forgot to

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mention it.

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There are other theories of
interest too.

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Another famous one was Marx's
Theory of Interest.

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So this is to be contrasted
with Fisher.

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What did Marx say?

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So in my youth when I was your
age it was very fashionable to

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be a Marxist.

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You had to study Marxism
basically.

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If you wanted to talk to women
you had to know about Marx.

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So anyhow, I dutifully went off
and read Marx.

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And so what's the idea of Marx?

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The idea was that he imagined
an agricultural economy where

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you plant stuff today and then
the output comes out tomorrow.

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So you put corn in today,
corn comes out tomorrow.

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So it doesn't require much
effort to plant the corn.

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You have to buy the corn.

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So the capitalists would buy
the corn, but planting it didn't

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require much effort.

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However, harvesting it,
picking the cotton,

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picking the chocolate,
picking all that stuff takes a

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lot of effort.

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So in the end you'd get a lot
of output.

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Now when you pick the output
you'd have to pay workers in

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order to pick the output.

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So Marx imagined that there was
a wage that was arrived at by

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the struggle,
class struggle,

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between the capitalists and the
workers over the subsistence

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wage.

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So the subsistence wage was
something that resulted from

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this huge class struggle,
and over time maybe it would

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rise as workers got stronger,
but it was still always quite

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low and the subsistence wage was
what the workers would get.

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And what's left over,
which was the surplus--so the

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output being more than what was
put in, was the surplus.

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Part of the surplus would go to
the subsistence wage.

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The rest would go to profit.

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And so if you look at how much
was put in to begin with,

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you get all the output back
out, the same amount of corn you

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put in plus some extra you have
to give to the workers,

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and extra that the capitalist
gets back as his profit.

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The fact that the capitalist
has done no work at all,

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he's just bought the corn,
let someone else plant it,

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let someone else harvest,
paid all those guys virtually

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nothing at the beginning and a
lot at the end,

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he's gotten profit for doing
nothing just like when you lend

15:07.811 --> 15:12.811
money,
so the profit divided by the

15:12.812 --> 15:19.872
initial outlay,
that was the rate of interest.

15:19.870 --> 15:22.800
And so Marx said that a
capitalist, he could put his

15:22.799 --> 15:26.129
money in his bank or he could
run this farm and make profit

15:26.133 --> 15:26.883
this way.

15:26.879 --> 15:28.679
So the money,
interest in the bank would have

15:28.677 --> 15:31.047
to turn out to be the same as
this rate of profit otherwise

15:31.046 --> 15:32.636
he'd put all the money in the
bank.

15:32.639 --> 15:35.629
And if it was smaller the banks
would have to give higher

15:35.634 --> 15:37.884
interest in order to attract
depositors.

15:37.879 --> 15:40.379
So the capitalist's
profit--rate of interest was

15:40.376 --> 15:43.456
determined by the rate of profit
and the rate of profit was

15:43.460 --> 15:46.490
determined by the struggle
between capital and labor.

15:46.490 --> 15:49.420
So we've got these religious
figures and great philosophers

15:49.422 --> 15:50.942
saying interest is terrible.

15:50.940 --> 15:53.610
We've got this great
philosopher-economist saying

15:53.610 --> 15:55.780
it's the result of a class
struggle,

15:55.779 --> 15:58.399
and now we've got Fisher,
actually Marx was pretty

15:58.403 --> 16:00.473
mathematical,
but now we've got Fisher

16:00.471 --> 16:03.051
turning it into a simple math
problem and saying,

16:03.048 --> 16:05.468
"Let's reason out the math
problem and we'll have the

16:05.469 --> 16:07.719
answer to these questions,
and it'll turn out to be quite

16:07.719 --> 16:09.519
different from what all these
guys are saying."

16:09.519 --> 16:14.809
So here's his economy that I
just described.

16:14.808 --> 16:17.878
The Fisher example,
not literally an example he

16:17.881 --> 16:20.221
gave, but similar to one he
gave.

16:20.220 --> 16:22.910
So he said, "All right,
what happens in this economy?

16:22.908 --> 16:24.808
Let's just be very
commonsensical.

16:24.808 --> 16:31.718
What we need to find out now is
financial equilibrium,"

16:31.720 --> 16:38.280
so financial equilibrium is
much more complicated seeming

16:38.279 --> 16:42.869
than we had before,
because we care about the

16:42.870 --> 16:43.440
prices.

16:43.440 --> 16:48.070
So now I'm going to use (q) for
prices, a q for contemporary

16:48.073 --> 16:50.593
prices,
q_contemporary,

16:50.587 --> 16:54.827
so the price you pay today to
get the apple today.

16:54.830 --> 16:57.860
q_2 is the price you
pay next year to get the apple

16:57.863 --> 16:58.473
next year.

16:58.470 --> 17:03.190
They're contemporary prices.

17:03.190 --> 17:05.740
And of course people are going
to decide what they want to do,

17:05.740 --> 17:08.340
X^(A)_1,
what they're going to end up

17:08.344 --> 17:11.164
consuming X^(A)_2,
X^(B)_1,

17:11.164 --> 17:14.434
X^(B)_2,
but now we've got a more

17:14.431 --> 17:16.081
complicated world.

17:16.078 --> 17:21.438
There's stocks to be traded and
there's the price of stocks.

17:21.440 --> 17:25.500
So P--pi_alpha,
I'm running out of letters.

17:25.500 --> 17:26.830
I'm going to switch to a Greek
one.

17:26.828 --> 17:33.808
This is the price of stock A,
stock alpha.

17:33.808 --> 17:39.408
And pi_beta is the
price of stock beta.

17:39.410 --> 17:43.640
And then we have to know how
many shares are they going to

17:43.640 --> 17:44.160
hold?

17:44.160 --> 17:47.720
Well, it's going be theta^(A),
A's going to hold a certain

17:47.724 --> 17:50.444
number of shares of alpha,
and theta^(A)_beta,

17:50.440 --> 17:54.450
A's going to hold a certain
number of shares of stock beta,

17:54.450 --> 17:57.850
and B's going to hold a certain
number of shares of alpha and a

17:57.849 --> 17:59.769
certain number of shares of
beta.

17:59.769 --> 18:02.399
So we want to solve for all of
that.

18:02.400 --> 18:06.650
Now, I should have said at the
beginning if these are trees

18:06.648 --> 18:10.018
producing apples there was an
initial stock.

18:10.019 --> 18:11.859
People owned a certain number
of trees.

18:11.858 --> 18:17.858
So let's say
theta-bar_alpha^(A),

18:17.859 --> 18:24.289
so this is the original
ownership of alpha.

18:24.288 --> 18:26.508
Let's just say that's,
I'll make up some number.

18:26.509 --> 18:28.919
I might as well use the same
number I thought of before.

18:28.920 --> 18:32.320
Let's say that's 1,
and let's say

18:32.320 --> 18:36.360
theta-bar^(B)_alpha is
0,

18:36.358 --> 18:41.848
and let's say theta-bar^(A) of
beta is a half,

18:41.848 --> 18:47.258
and theta-bar^(B) of beta is a
half.

18:47.259 --> 18:52.569
So the original economy is more
complicated than before because

18:52.570 --> 18:54.370
we've added stocks.

18:54.369 --> 18:55.839
We characterized the stock.

18:55.838 --> 18:58.058
We said a stock is a very
complicated thing.

18:58.058 --> 19:00.998
A company is very complicated,
depends on managers and

19:01.000 --> 19:04.330
processes and there's all kinds
of stuff you think about when

19:04.329 --> 19:07.739
you think about a stock,
but really at heart all people

19:07.742 --> 19:11.412
are trying to do is forecast
what are they going to produce.

19:11.410 --> 19:14.440
And so we're going to make it
simple mathematically and say

19:14.440 --> 19:17.680
let's say we know what they're
going to produce next period.

19:17.680 --> 19:18.570
So let's say it's a tree.

19:18.568 --> 19:20.768
Everybody knows the alpha
tree's producing 1 apple.

19:20.769 --> 19:23.589
The beta tree's producing two
apples.

19:23.588 --> 19:30.608
Alpha happens to own the only
alpha tree.

19:30.608 --> 19:34.788
A owns the alpha tree,
excuse me, and A and B own half

19:34.794 --> 19:36.694
each of the beta tree.

19:36.690 --> 19:40.260
So that's the original economy
and the equilibrium is going to

19:40.258 --> 19:42.038
be,
what are the prices going to

19:42.040 --> 19:44.700
turn out to be of X_1
and X_2,

19:44.700 --> 19:47.740
what are the prices of the
trees going to turn out to be,

19:47.740 --> 19:51.350
how much will people consume
and how many shares?

19:51.348 --> 19:55.058
Because alpha A began with all
of tree maybe he'll sell his

19:55.057 --> 19:59.017
shares of tree and end up with
not having a tree in the end.

19:59.019 --> 20:01.739
So we have to see where they
began, the stock ownership to

20:01.739 --> 20:03.409
begin with and where they end
up.

20:03.410 --> 20:05.780
So we have to solve all of that
and it looks way more

20:05.778 --> 20:08.858
complicated than before,
and so complicated that you can

20:08.861 --> 20:11.431
see why people might have gotten
confused.

20:11.430 --> 20:14.310
But according to Fisher it's
going to turn out to be a very

20:14.309 --> 20:17.339
simple problem in the end once
we look at it the right way.

20:17.338 --> 20:21.698
So are there any questions
about what the economy is and

20:21.703 --> 20:25.993
what are the variables that
we're trying to explain?

20:25.990 --> 20:26.710
Yeah?

20:26.710 --> 20:29.890
Student: Sorry,
I can't read what that says

20:29.894 --> 20:31.264
over theta alpha A.

20:31.259 --> 20:32.539
That's original ownership?

20:32.538 --> 20:35.908
Prof: Original ownership
of stock alpha.

20:35.910 --> 20:38.820
And this is original ownership
by B of stock alpha.

20:38.818 --> 20:44.228
This original ownership by A of
stock beta, and this is the

20:44.230 --> 20:49.740
original ownership by B of stock
beta, original ownership of

20:49.736 --> 20:50.666
alpha.

20:50.670 --> 20:51.170
Thanks.

20:51.170 --> 20:52.380
Yes?

20:52.380 --> 20:54.680
Student: Having defined
all of these could you redefine

20:54.675 --> 20:55.425
what D stands for?

20:55.430 --> 20:56.680
Prof: D is the dividend.

20:56.680 --> 20:59.440
That's the output that we can
all--thank you.

20:59.440 --> 21:00.920
I should just write this down.

21:00.920 --> 21:10.460
This is the anticipated
dividend, which is the output

21:10.460 --> 21:20.920
since that's the end of the
world of stock alpha in period

21:20.919 --> 21:22.019
2.

21:22.019 --> 21:26.499
And D^(beta) of 2 is the
anticipated dividend of stock

21:26.501 --> 21:28.111
beta in period 2.

21:28.108 --> 21:31.018
So it's 1 apple we're getting
out of the alpha tree,

21:31.017 --> 21:33.237
2 apples out of the beta tree,
right?

21:33.240 --> 21:34.850
That's what the tree's good for.

21:34.848 --> 21:36.418
We can look at how beautiful it
is.

21:36.420 --> 21:40.000
We could talk about how much
the owner's actually watering

21:39.999 --> 21:40.689
the tree.

21:40.690 --> 21:42.530
We can talk about a lot of
complicated stuff,

21:42.534 --> 21:44.764
but in the end all we care
about is how many apples we

21:44.756 --> 21:45.926
expect to get out of it.

21:45.930 --> 21:48.860
All the other stuff goes into
helping us think about how many

21:48.863 --> 21:51.263
apples we're going to get out of
it in the end.

21:51.259 --> 21:52.989
So we cut to the bottom line.

21:52.990 --> 21:56.360
What are the apples we expect
to get out of tree,

21:56.359 --> 21:59.589
1 from the alpha tree,
2 from the beta tree.

21:59.589 --> 22:01.629
Someone else had their hand up.

22:01.630 --> 22:03.710
Student: I had it,
but you answered it.

22:03.710 --> 22:06.990
Prof: Any other
questions about this set up?

22:06.990 --> 22:10.670
So we're returning to first
principles here,

22:10.667 --> 22:12.547
very simple example.

22:12.548 --> 22:14.978
When there's ever a big
confusion about something

22:14.977 --> 22:17.757
important it's always good to go
to first principles.

22:17.759 --> 22:21.229
There was a chess player when I
was young named Mikhail Tal who

22:21.230 --> 22:23.640
was a world champion for a
little while,

22:23.640 --> 22:27.560
and he said that every two or
three years he'd go back and

22:27.561 --> 22:31.141
read his original introductory
textbooks on chess.

22:31.140 --> 22:33.720
So we're going back to the
first principles.

22:33.720 --> 22:36.700
How would you define
equilibrium here for a financial

22:36.698 --> 22:37.498
equilibrium?

22:37.500 --> 22:41.950
Well, the first thing is just
common sense.

22:41.950 --> 22:43.260
What are people doing?

22:43.259 --> 22:46.769
At time 1 what can they do?

22:46.769 --> 22:52.389
They can spend money,
so I'm going to look at the

22:52.386 --> 22:58.586
budget set for Agent i,
and i can be A or B so I don't

22:58.589 --> 23:01.749
have to write it twice.

23:01.750 --> 23:07.430
So he's going to say to
himself, let's say A is he and B

23:07.430 --> 23:09.290
is she, i is he.

23:09.288 --> 23:13.208
He's going to say to
himself--let's say i will say,

23:13.211 --> 23:17.291
"How much does it cost me
to buy goods?"

23:17.288 --> 23:20.268
Well, the cost of apples is
q_1 times

23:20.271 --> 23:21.311
X_1.

23:21.308 --> 23:23.658
That's how many apples I might
end up with.

23:23.660 --> 23:26.060
Now, how much does it cost me
to buy shares?

23:26.058 --> 23:29.648
It's going to be
pi_alpha times how

23:29.653 --> 23:33.583
many shares I end up with,
theta_alpha,

23:33.575 --> 23:38.145
plus (pi_beta times
theta_beta).

23:38.150 --> 23:41.280
So I'm buying goods,
I'm buying alpha shares and I'm

23:41.284 --> 23:45.034
buying beta shares and this is
how much I have to spend to get

23:45.032 --> 23:47.002
the holdings I want of each.

23:47.000 --> 23:49.610
Now where did I get the money
to do that?

23:49.608 --> 23:54.278
I got the money to do that
because I started with my

23:54.279 --> 23:58.859
endowment of goods which was
E^(i)_1,

23:58.858 --> 24:01.798
which in this case for A was 1
unit,

24:01.798 --> 24:05.068
for B was also 1 unit,
and then I also had shares to

24:05.069 --> 24:06.929
begin with of these stocks.

24:06.930 --> 24:09.180
So I had ([pi_alpha
times]

24:09.180 --> 24:12.010
theta-bar_alpha)
(pi_beta

24:12.009 --> 24:13.939
theta-bar_beta).

24:13.940 --> 24:16.570
In period 1 that's what I had
to do.

24:16.568 --> 24:20.858
I wanted to buy apples,
shares and I had shares to sell

24:20.861 --> 24:22.531
and apples to sell.

24:22.529 --> 24:25.389
So that's what i did.

24:25.390 --> 24:28.860
So of course if X_1
is bigger than the number of

24:28.857 --> 24:32.627
apples i started with that means
i has bought apples because he

24:32.632 --> 24:35.312
ends up with more than he
started with,

24:35.308 --> 24:37.498
so that meant he must have been
buying apples.

24:37.500 --> 24:39.230
If theta_alpha is
more than

24:39.234 --> 24:41.984
theta-bar_alpha it
means that alpha [correction:

24:41.983 --> 24:43.723
i]
bought shares of stock alpha.

24:43.720 --> 24:47.960
Theta_alpha is less
than theta-bar_alpha

24:47.963 --> 24:50.053
it means alpha [correction:
i]

24:50.049 --> 24:52.279
sold shares of stock alpha.

24:52.279 --> 24:55.749
All right, now in the second
period what happens?

24:55.750 --> 25:01.220
Well, in the second period we
have q_2 times

25:01.218 --> 25:02.868
X_2.

25:02.868 --> 25:05.688
The shares are going to be
worthless in period 2.

25:05.690 --> 25:06.720
So no one's going to buy them.

25:06.720 --> 25:07.840
Why are the shares worthless?

25:07.838 --> 25:11.578
Remember that when you buy a
share of stock the dividend

25:11.577 --> 25:12.527
comes later.

25:12.528 --> 25:14.398
You don't get the dividend
immediately.

25:14.400 --> 25:17.220
So someone buying stock in
period 2, it's too late to get

25:17.222 --> 25:17.982
the dividend.

25:17.980 --> 25:22.310
It's already gone to the owner
who bought the shares in period

25:22.306 --> 25:22.586
1.

25:22.588 --> 25:25.218
So the buyer of a stock
[doesn't]

25:25.223 --> 25:28.933
gets the dividend for a month
or something.

25:28.930 --> 25:32.530
So next period's dividend is
still going to go to the buyer

25:32.528 --> 25:34.868
in period 1,
that's why it's valuable to buy

25:34.866 --> 25:37.316
shares in period 1 because you
get next year's apple.

25:37.318 --> 25:41.478
So by next period you can buy
the tree, but the world's coming

25:41.476 --> 25:42.086
to end.

25:42.088 --> 25:43.578
That tree's not going to do you
any good.

25:43.578 --> 25:44.978
It doesn't produce any more
apples.

25:44.980 --> 25:46.800
So nobody's going to bother
buying shares.

25:46.798 --> 25:48.098
I don't have to bother with
them.

25:48.099 --> 25:49.849
The prices are zero.

25:49.848 --> 25:54.748
And so what income do people
have in period 2?

25:54.750 --> 25:58.890
Well, they've got the
contemporaneous price times the

25:58.893 --> 26:03.993
apples that somehow they find on
the ground or that their parents

26:03.993 --> 26:07.663
are going to leave them when
they get old.

26:07.660 --> 26:10.900
So that's their endowment of
apples, but what else do they

26:10.901 --> 26:11.301
have?

26:11.298 --> 26:14.018
They've got more apples than
that.

26:14.019 --> 26:16.269
What else do they have?

26:16.269 --> 26:17.109
Student: The dividends.

26:17.109 --> 26:17.719
Prof: The dividends.

26:17.720 --> 26:19.050
So what are the dividends?

26:19.048 --> 26:22.768
Well, you bought
theta_alpha to begin

26:22.772 --> 26:25.932
with so that's
D^(alpha)_2.

26:25.930 --> 26:29.660
So if you bought the whole tree
then you've got all the

26:29.655 --> 26:32.205
dividends, and similarly with
beta.

26:32.210 --> 26:33.520
Theta_beta times
D^(beta)_2

26:33.520 --> 26:34.610
[correction:
theta_beta times

26:34.605 --> 26:35.735
D^(beta)_2 times
q_2.

26:35.744 --> 26:37.264
And, theta_alpha
times D^(alpha)_2

26:37.257 --> 26:38.737
should also be multiplied by
q_2].

26:38.740 --> 26:39.170
So that's it.

26:39.170 --> 26:42.600
So the budget set is a little
more complicated.

26:42.599 --> 26:44.779
So that's the budget set.

26:44.779 --> 26:47.579
So it's got 2 equalities
instead of 1 equality,

26:47.582 --> 26:50.692
so already things look a little
more complicated.

26:50.690 --> 26:57.350
Now, so an equilibrium is going
to have to be that i chooses

26:57.354 --> 27:02.444
(X^(i)_1,
X^(i)_2),

27:02.438 --> 27:06.198
theta_alpha--
I can write,

27:06.204 --> 27:12.454
theta^(i)_alpha,
theta^(i)_beta,

27:12.448 --> 27:24.098
that's all the choices he has
to maximize U^(i) subject to

27:24.096 --> 27:27.976
this budget set.

27:27.980 --> 27:31.260
So A's going to pick what
shares to hold,

27:31.259 --> 27:33.639
how much to consume today,
then, of course,

27:33.640 --> 27:36.610
looking forward A's going to be
able to figure out what he's

27:36.614 --> 27:38.484
going to end up consuming
tomorrow.

27:38.480 --> 27:43.240
All right so,
and now in equilibrium we have

27:43.237 --> 27:49.647
to have that (X^(A)_1
X^(B)_1) has to =

27:49.653 --> 27:54.303
(E^(A)_1
E^(B)_1).

27:54.298 --> 27:58.388
And then we do the shares,
theta^(A)_alpha

27:58.393 --> 28:03.453
theta^(B)_alpha has to
= theta-bar^(A)_alpha

28:03.450 --> 28:06.830
theta-bar^(B)_alpha,
right?

28:06.828 --> 28:12.188
The stock market has to clear
and theta^(A)_beta

28:12.185 --> 28:17.915
theta^(B)_beta has to
= theta-bar^(A)_beta

28:17.917 --> 28:21.017
theta-bar^(B)_beta.

28:21.019 --> 28:25.279
So in period 1 the demand for
apples has to equal the supply

28:25.275 --> 28:28.225
of all the agents,
but now what's the last

28:28.231 --> 28:29.171
equation?

28:29.170 --> 28:30.220
This is a little trickier.

28:30.220 --> 28:31.750
What's the last equation?

28:31.750 --> 28:36.880
X^(A)_2
X^(B)_2 =

28:36.882 --> 28:42.872
E^(A)_2
E^(B)_2 ....

28:42.869 --> 28:44.849
Is that it?

28:44.849 --> 28:45.309
No.

28:45.309 --> 28:46.639
There's something else.

28:46.640 --> 28:50.050
Student: Plus dividends.

28:50.048 --> 28:53.318
Prof: The total
consumption of apples is going

28:53.324 --> 28:56.414
to be the apples that they have
on the ground,

28:56.410 --> 28:58.600
but also the ones that were
picked off the trees,

28:58.599 --> 29:00.199
so these dividends.

29:00.200 --> 29:04.350
So it's going to be the total
dividends which are

29:04.353 --> 29:09.633
[(theta-bar^(A)_alpha
theta-bar^(B)_alpha)

29:09.632 --> 29:14.332
times that tree--
D^(alpha)_2]

29:14.328 --> 29:22.048
[(theta-bar^(A)_beta
theta-bar^(B)_beta)

29:22.051 --> 29:26.111
times D^(beta)_2].

29:26.108 --> 29:29.538
So just to say it in words,
it's exactly what we had before

29:29.535 --> 29:33.015
except we have to take into
account in addition to the goods

29:33.019 --> 29:35.679
market clearing,
we have to take into account

29:35.681 --> 29:37.511
that the stock market has to
clear.

29:37.509 --> 29:41.779
And in the end demand for goods
has to equal the supply that

29:41.775 --> 29:46.255
people had in their endowments,
but also what the companies are

29:46.257 --> 29:47.267
producing.

29:47.269 --> 29:49.679
These companies are producing
output, apples.

29:49.680 --> 29:54.710
And so that's part of what the
consumption is going to be in

29:54.714 --> 29:55.914
the economy.

29:55.910 --> 29:57.010
Are you with me here?

29:57.009 --> 30:00.569
It's a good time for questions,
maybe.

30:00.569 --> 30:01.479
Yes?

30:01.480 --> 30:04.700
Student: Could you just
explain again why we don't take

30:04.699 --> 30:06.809
stocks into consideration in
period 2?

30:06.808 --> 30:09.838
Prof: So in period 2 you
might wonder,

30:09.838 --> 30:14.358
pi_alpha is the price
of the stock at period 1,

30:14.359 --> 30:15.909
stock alpha in period 1.

30:15.910 --> 30:20.220
Pi_beta's the price
of the stock beta in period 1.

30:20.220 --> 30:23.140
How come I didn't write down
the price of the stocks in

30:23.143 --> 30:26.503
period 2 and keep track of what
they're holding in period 2?

30:26.500 --> 30:30.240
And the reason is that when you
buy a stock you're buying it not

30:30.244 --> 30:33.104
for the dividends at this same
moment in time.

30:33.099 --> 30:34.229
You don't get those dividends.

30:34.230 --> 30:36.340
The guy who already had it gets
those dividends.

30:36.338 --> 30:39.178
When you buy the stock you're
buying it for the future

30:39.180 --> 30:42.500
dividends you can get,
and I've assumed the world's

30:42.503 --> 30:46.613
going to end after 2 periods
because nobody's utility cares

30:46.614 --> 30:47.894
about period 3.

30:47.890 --> 30:52.280
So if you buy the stock in
period 2 it's too late for you

30:52.279 --> 30:53.689
to get anything.

30:53.690 --> 30:56.720
There are no dividends because
you can only get them in period

30:56.721 --> 30:57.671
3,
and there won't be any

30:57.667 --> 30:59.437
dividends in period 3,
and if there were you wouldn't

30:59.441 --> 31:00.251
care about them anyway.

31:00.250 --> 31:01.820
So the stock's worthless to you.

31:01.818 --> 31:04.448
So the price of the stock in
period 2, of both stocks,

31:04.446 --> 31:05.386
is going to be 0.

31:05.390 --> 31:08.040
So there's no point in putting
down what people are buying of

31:08.038 --> 31:09.628
the stocks or selling or
anything.

31:09.630 --> 31:11.150
It's just not worth anything.

31:11.150 --> 31:14.410
But in general you're right,
and we're going to be more

31:14.405 --> 31:17.835
complicated later when you look
at your income from having

31:17.843 --> 31:19.173
bought the stocks.

31:19.170 --> 31:22.010
You'll have as your income the
dividend from the stocks plus

31:22.011 --> 31:24.711
the resale value of the stock,
because you could sell the

31:24.710 --> 31:25.770
stock next period.

31:25.769 --> 31:28.629
But I just know the resale
value's going to be zero because

31:28.634 --> 31:30.764
you're in the last period of the
economy.

31:30.759 --> 31:37.399
And I just want to keep it very
simple this first time.

31:37.400 --> 31:40.390
Step by step you'll be able to
keep very complicated things in

31:40.394 --> 31:43.294
your head but not right at
first, so any other questions?

31:43.289 --> 31:44.369
Yes?

31:44.368 --> 31:48.428
Student: Why are there
endowments in period 1 and

31:48.432 --> 31:50.502
>?

31:50.500 --> 31:54.440
Prof: You mean why do
people have endowments today and

31:54.441 --> 31:55.231
next year?

31:55.230 --> 31:56.920
Because you could think of the
endowment,

31:56.920 --> 31:59.190
for example,
as--here we've got apples,

31:59.190 --> 32:01.340
but usually the endowment is
your labor,

32:01.338 --> 32:05.618
so you can work this year--your
most important endowment is your

32:05.618 --> 32:07.248
energy and your labor.

32:07.250 --> 32:08.440
So you've got it this year.

32:08.440 --> 32:12.440
Next year if you're still alive
that's new labor that you have.

32:12.440 --> 32:15.370
It's a different good so it's a
second endowment that you have.

32:15.368 --> 32:18.498
So I don't want to get caught
up in labor and all that and get

32:18.496 --> 32:21.366
involved with Marx again,
so I'm just going to talk about

32:21.365 --> 32:21.925
apples.

32:21.930 --> 32:24.970
You have an endowment of apples
when you're young and next year

32:24.970 --> 32:27.080
somehow you're going to have
more apples.

32:27.078 --> 32:30.508
So you might have thought that
the only apples next year come

32:30.511 --> 32:32.631
from what the firms are
producing,

32:32.630 --> 32:35.970
but I allowed for the
possibility that people have

32:35.971 --> 32:39.041
apples too just like their labor
next year.

32:39.039 --> 32:42.669
Other questions?

32:42.670 --> 32:43.380
Yeah?

32:43.380 --> 32:45.830
Student: We then have to
define E_2^(A) and

32:45.826 --> 32:48.186
E_2^(B) in terms of
first period endowments,

32:48.190 --> 32:52.900
or is that something implied in
the equation?

32:52.900 --> 32:56.440
Prof: I should have
written this more carefully

32:56.435 --> 32:56.965
maybe.

32:56.970 --> 33:04.770
E^(A)_1 and
E^(A)_2 is that,

33:04.773 --> 33:15.073
and this is E^(B)_1
and E^(B)_2 is that.

33:15.068 --> 33:19.628
So as he was suggesting back
there I've assumed for person B

33:19.628 --> 33:21.868
that he's got an apple now.

33:21.868 --> 33:23.978
We aren't modeling what
happened to get us here.

33:23.980 --> 33:25.370
The guys got an apple today.

33:25.369 --> 33:26.739
They both have an apple today.

33:26.740 --> 33:30.630
Somehow A's also going to have
another apple tomorrow that he's

33:30.628 --> 33:34.138
going to find under his doorstep
somehow that isn't being

33:34.141 --> 33:35.711
produced by the tree.

33:35.710 --> 33:40.480
And maybe you can think of it
as labor that he's going to have

33:40.476 --> 33:41.566
next period.

33:41.569 --> 33:44.399
All right, so that's it.

33:44.400 --> 33:46.940
Fisher says as soon as you
write down the economy

33:46.939 --> 33:50.059
mathematically all sorts of
things are going to occur to you

33:50.060 --> 33:53.180
which if you're talking in words
about justice and injustice

33:53.182 --> 33:54.772
you're going to be lost.

33:54.769 --> 33:59.239
So what can we get right away
out of this?

33:59.240 --> 34:00.550
What can we get right away out
of this?

34:00.548 --> 34:05.048
Well, the first thing is,
how would we define inflation?

34:05.049 --> 34:12.279
What is inflation?

34:12.280 --> 34:15.230
What's inflation in this
economy?

34:15.230 --> 34:16.660
Assuming we've got the
equilibrium,

34:16.659 --> 34:19.129
which we're going to get soon,
we're going to calculate it,

34:19.130 --> 34:20.820
but right now we don't what the
numbers--

34:20.820 --> 34:23.420
you know we've got a bunch of
equations and stuff.

34:23.420 --> 34:24.950
We don't know what
X^(A)_1 and

34:24.949 --> 34:27.259
q_1 and q_2
are going to turn out to be,

34:27.262 --> 34:28.912
but we're going to find out
very soon.

34:28.909 --> 34:32.279
But before we find out,
assuming we've gotten those,

34:32.284 --> 34:33.944
what will inflation be?

34:33.940 --> 34:40.990
What is inflation?

34:40.989 --> 34:41.719
Yeah?

34:41.719 --> 34:45.039
Student: Is it how much
the ratio of the price of the

34:45.043 --> 34:46.343
dividend has changed?

34:46.340 --> 34:47.480
Prof: Well,
we're talking about inflation.

34:47.480 --> 34:50.370
When you talk about the
Consumer Price Index,

34:50.367 --> 34:53.057
inflation, what are they
talking about?

34:53.059 --> 34:53.609
Yeah?

34:53.610 --> 34:54.670
Student: It's the rise
of q_1 and

34:54.672 --> 34:55.022
q_2.

34:55.018 --> 34:57.938
Prof: So inflation is
just q_2 over

34:57.942 --> 34:59.262
q_1,
right?

34:59.260 --> 35:01.080
So that's the price of apples
today.

35:01.079 --> 35:02.389
That's the price of apples next
year.

35:02.389 --> 35:04.509
If the price of apples next
year is bigger than this year

35:04.514 --> 35:05.354
we've got inflation.

35:05.349 --> 35:06.839
If it's lower we've got
deflation.

35:06.840 --> 35:10.950
So already the model,
you're talking about inflation.

35:10.949 --> 35:12.409
What else?

35:12.409 --> 35:16.529
What's the next most obvious?

35:16.530 --> 35:20.400
Well, I think I'm going to skip
a bunch of stuff and get now to

35:20.402 --> 35:21.342
the key idea.

35:21.340 --> 35:27.930
The key idea is arbitrage.

35:27.929 --> 35:30.949
So Fisher says,
"People have

35:30.949 --> 35:32.649
foresight."

35:32.650 --> 35:34.750
They're anticipating what the
dividends are going to be.

35:34.750 --> 35:39.700
They understand that you can
talk about how beautiful the

35:39.697 --> 35:41.747
tree is,
and how much you like the

35:41.746 --> 35:43.596
owners, and how much they're
watering it,

35:43.599 --> 35:47.429
and whether they have a good
plan for irrigation,

35:47.429 --> 35:50.979
and whether they did well in
college and stuff like that,

35:50.980 --> 35:53.650
but in the end all you care
about the trees is how many

35:53.648 --> 35:55.328
apples they're going to produce.

35:55.329 --> 35:58.189
So knowing that,
can we say something about

35:58.186 --> 36:01.926
pi_alpha pi alpha
versus pi_beta?

36:01.929 --> 36:04.079
In equilibrium what's going to
have to happen?

36:04.079 --> 36:07.299
There is going to be some
connection, and what's the

36:07.300 --> 36:08.880
connection going to be?

36:08.880 --> 36:12.260
You've got two trees.

36:12.260 --> 36:12.990
Yeah?

36:12.989 --> 36:14.719
Student: The ratio
between the prices would be the

36:14.719 --> 36:16.139
ratio between the
>?

36:16.139 --> 36:19.229
Prof: Right,
so pi_alpha is going

36:19.231 --> 36:23.751
to be pi_beta times
D[^(alpha)_2 over

36:23.751 --> 36:32.801
D^(beta)_2]--
alpha will be better as long as

36:32.804 --> 36:36.224
the--
hopefully I've got that in the

36:36.215 --> 36:37.105
right order.

36:37.110 --> 36:40.590
And so in this case
pi_alpha is going to

36:40.588 --> 36:44.918
equal a half pi_beta,
because alpha's producing half

36:44.920 --> 36:47.690
the dividend that beta's
producing.

36:47.690 --> 36:50.360
So obviously it's going to turn
out to have half the price.

36:50.360 --> 36:53.110
That's the fundamental
principle.

36:53.110 --> 36:55.540
We're doing it in the most
trivial case,

36:55.539 --> 36:59.129
but it's the most fundamental
principle of finance that if

36:59.132 --> 37:02.732
you've got two assets and
they're basically the same up to

37:02.726 --> 37:06.506
scale then their prices have to
be the same up to scale.

37:06.510 --> 37:11.110
Who's going to bother to buy
alpha if it costs the same

37:11.114 --> 37:15.554
amount as beta when it only
produces half as much?

37:15.550 --> 37:16.600
Yes?

37:16.599 --> 37:18.779
Student: Is that the
same thing as saying that their

37:18.780 --> 37:20.210
yields will converge,
that equation?

37:20.210 --> 37:22.150
Prof: Well,
it is something like that,

37:22.153 --> 37:24.323
but that's a word that we
haven't defined yet,

37:24.320 --> 37:26.450
so we're going to define it in
the next class.

37:26.449 --> 37:29.739
So it's something like that,
yep.

37:29.739 --> 37:31.699
Any other?

37:31.699 --> 37:35.589
So that is a very simple thing.

37:35.590 --> 37:56.640
Suppose after finding the
equilibrium I added a third

37:56.637 --> 38:16.467
asset that paid 1 dollar in
period 2 next year.

38:16.469 --> 38:22.459
Now, it would have a price
of--added a third asset gamma,

38:22.460 --> 38:24.440
so pi_gamma--we'd
have to solve for the

38:24.443 --> 38:28.633
equilibrium pi_gamma,
and is there some word that I

38:28.626 --> 38:29.676
could use?

38:29.679 --> 38:35.029
So gamma is an asset that pays
a dollar in period 2.

38:35.030 --> 38:37.990
It's like a bond promising a
dollar in period two.

38:37.989 --> 38:42.639
The price of the bond would
then have to be what?

38:42.639 --> 38:51.069
1 over (1 i) where i is called
the nominal interest rate,

38:51.070 --> 38:58.750
so we've got inflation is
occurring in the model.

38:58.750 --> 39:00.580
If I added a bond,
which I didn't bother to do

39:00.583 --> 39:03.113
because it's just yet another
thing I'd have to write down,

39:03.110 --> 39:06.450
I could have had a third asset
which pays a dollar.

39:06.449 --> 39:09.239
The others are paying off in
apples.

39:09.239 --> 39:12.469
This one's paying a dollar and
its price today,

39:12.469 --> 39:16.579
if you pay 80 cents today it's
like saying I'm paying 80 cents

39:16.583 --> 39:18.443
today,
I'm getting a whole dollar

39:18.438 --> 39:18.918
tomorrow.

39:18.920 --> 39:24.790
So it's like a 25 percent rate
of interest, because another way

39:24.793 --> 39:30.103
of saying it is that 1 over
pi_gamma is 1 i.

39:30.099 --> 39:31.609
I put only pi_gamma
in today,

39:31.610 --> 39:34.190
I get 1 out tomorrow so I've
gotten back not only the

39:34.193 --> 39:36.733
pi_gamma I put in but
something extra,

39:36.730 --> 39:39.860
that's 1 the interest rate.

39:39.860 --> 39:43.070
So this world is going to have
an interest rate in it.

39:43.070 --> 39:46.200
It's going to have inflation in
it.

39:46.199 --> 39:51.409
With me so far?

39:51.409 --> 39:53.189
Let me add one more thing.

39:53.190 --> 39:54.490
I could come back to this.

39:54.489 --> 40:00.409
So I said that if you take
theta_alpha less than

40:00.405 --> 40:06.835
theta-bar_alpha it
means you're selling the stock.

40:06.840 --> 40:11.000
So I'm going to allow people to
go even further,

40:11.001 --> 40:14.191
theta_alpha less than
0.

40:14.190 --> 40:19.970
So let me just write that
again, theta_alpha

40:19.967 --> 40:24.327
less than
theta-bar_alpha means

40:24.327 --> 40:26.287
selling alpha.

40:26.289 --> 40:29.279
Theta_alpha less than
0 is doing a lot more than

40:29.275 --> 40:29.795
selling.

40:29.800 --> 40:33.360
You don't have it to begin
with, so what are you selling?

40:33.360 --> 40:36.320
Well, the mathematics is
telling you that over there

40:36.315 --> 40:39.035
theta_alpha's going to
be negative.

40:39.039 --> 40:42.249
Instead of getting extra
dividends you're going to be

40:42.248 --> 40:46.068
giving up dividends because it's
going to reduce your supply of

40:46.072 --> 40:46.692
money.

40:46.690 --> 41:01.160
So theta_alpha less
than 0 is called selling short--

41:01.159 --> 41:04.979
I don't know which one I've
lost, but it's got to be bad--

41:04.980 --> 41:18.670


41:18.670 --> 41:21.120
if you can still hear
me--you're selling short,

41:21.119 --> 41:24.949
this means.

41:24.949 --> 41:27.429
So you're selling something you
don't even have.

41:27.429 --> 41:32.139
It's also called naked selling.

41:32.139 --> 41:47.759
It's also called making a
promise without collateral.

41:47.760 --> 41:51.660
So I'm going to,
for now, allow for that.

41:51.659 --> 41:54.129
So we're not taking into
account that anybody's

41:54.130 --> 41:54.830
defaulting.

41:54.829 --> 41:56.609
If you take
theta_alpha negative

41:56.614 --> 41:59.054
it means your income in the
future is going to be reduced

41:59.054 --> 42:01.624
because you're going to have to
deliver the dividend because

42:01.623 --> 42:03.673
you're going to have negative
dividends,

42:03.670 --> 42:06.630
which means effectively you
take out of your endowment those

42:06.632 --> 42:08.192
dividends and hand them over.

42:08.190 --> 42:11.990
So it's as if you always keep
your promises.

42:11.989 --> 42:15.059
So this model so far,
the Fisher model,

42:15.059 --> 42:17.969
assumes no default,
no collateral.

42:17.969 --> 42:21.649
We're not worrying about any of
that stuff, and of course that's

42:21.648 --> 42:23.458
going to be a critical thing.

42:23.460 --> 42:28.760
So you see something's happened
that we never had happen before.

42:28.760 --> 42:31.980
In the past you traded money
for a football ticket.

42:31.980 --> 42:35.130
You gave up something you
wanted you got something that

42:35.130 --> 42:36.180
you also wanted.

42:36.179 --> 42:37.959
It was a trade of value for
value.

42:37.960 --> 42:40.050
Everybody agreed the two things
you traded were equally

42:40.052 --> 42:40.482
valuable.

42:40.480 --> 42:43.160
If you take
theta_alpha negative,

42:43.159 --> 42:46.529
by taking theta_alpha
negative that becomes a negative

42:46.530 --> 42:48.940
number here so it allows you to
spend more.

42:48.940 --> 42:51.900
You can buy more goods by
taking theta_alpha

42:51.904 --> 42:52.524
negative.

42:52.519 --> 42:53.339
That's negative.

42:53.340 --> 42:56.750
That means this can be more
positive and still satisfy this

42:56.753 --> 42:57.523
constraint.

42:57.518 --> 43:01.078
So by selling a stock short
you're promising to do something

43:01.083 --> 43:02.053
in the future.

43:02.050 --> 43:04.000
You get more money now you can
eat more now,

43:04.003 --> 43:06.553
and then of course you have to
consume less in the future

43:06.548 --> 43:08.638
because you have to pay back
your promise.

43:08.639 --> 43:11.309
So you're exchanging something
valuable,

43:11.309 --> 43:13.889
you're getting money,
something valuable in exchange

43:13.889 --> 43:16.769
for a promise which is worth
nothing until the future when

43:16.771 --> 43:18.391
you deliver on your promise.

43:18.389 --> 43:20.949
When you buy the stock you're
buying part of the tree,

43:20.951 --> 43:23.031
but the tree's doing nothing
for you now.

43:23.030 --> 43:25.290
You're doing it because it's
going to be valuable next

43:25.286 --> 43:25.666
period.

43:25.670 --> 43:27.740
You're actually not physically
owning the tree,

43:27.742 --> 43:30.452
you're owning a piece of paper
that gives you a right to half

43:30.447 --> 43:31.797
the dividends of the tree.

43:31.800 --> 43:34.160
So you're getting something
that's only good because it's a

43:34.164 --> 43:35.554
promise you think is being kept.

43:35.550 --> 43:39.010
So, so far for the next few
lectures we're going to ignore

43:39.012 --> 43:42.722
the fact that people get very
nervous when they give something

43:42.717 --> 43:46.177
up that's valuable in exchange
for something that's just a

43:46.179 --> 43:47.029
promise.

43:47.030 --> 43:49.390
So a critical thing has
happened here.

43:49.389 --> 43:51.819
So we've kept the same
mathematics except we've

43:51.815 --> 43:54.185
surreptitiously added this huge
assumption.

43:54.190 --> 43:56.870
Now Fisher said,
"Having done that,

43:56.873 --> 43:58.873
what can you realize?"

43:58.869 --> 44:00.889
This is the most important
insight.

44:00.889 --> 44:03.319
He said this model it looks so
complicated.

44:03.320 --> 44:05.640
It looks like now we have
vastly more equations.

44:05.639 --> 44:10.619
No wonder Marx and all those
religious zealots were getting

44:10.615 --> 44:11.555
confused.

44:11.559 --> 44:15.799
We can simplify it all and be
back to where we were before and

44:15.800 --> 44:17.540
yet talk about finance.

44:17.539 --> 44:32.389
So Fisher introduced the idea
of present value prices.

44:32.389 --> 44:36.399
So he said look,
when you buy a stock what are

44:36.404 --> 44:38.104
you really doing?

44:38.099 --> 44:39.529
This is the principle of
arbitrage.

44:39.530 --> 44:42.220
He says when you buy a stock
you're saying to yourself,

44:42.222 --> 44:43.622
I'm giving up money today.

44:43.619 --> 44:47.059
Now, money today is consumption
because I would have used that

44:47.057 --> 44:47.507
money.

44:47.510 --> 44:50.090
If I didn't buy stocks I would
have bought apples today and

44:50.090 --> 44:50.670
eaten them.

44:50.670 --> 44:53.800
So when I buy a stock I'm
giving up apples today.

44:53.800 --> 44:56.890
I'm getting the stock which is
then paying me dividends

44:56.891 --> 44:59.481
tomorrow,
which whatever they are I'm

44:59.476 --> 45:02.026
selling off,
I'm getting a profit out of the

45:02.030 --> 45:04.820
stock tomorrow and I'm ending up
with apples tomorrow.

45:04.820 --> 45:07.100
Maybe I'm just eating the
dividends straight off the tree.

45:07.099 --> 45:11.449
So when I buy a stock I'm
really giving up apples today

45:11.452 --> 45:13.872
and getting apples tomorrow.

45:13.869 --> 45:16.729
And no matter how I do it,
whether it's through stock

45:16.733 --> 45:19.103
alpha,
or through stock beta,

45:19.099 --> 45:23.949
or through a nominal bond it's
got to be the case that all

45:23.945 --> 45:27.455
three ways,
or all 50 other ways you could

45:27.458 --> 45:31.338
imagine doing it have to give me
the same tradeoff.

45:31.340 --> 45:32.970
This is your yield you were
talking about,

45:32.972 --> 45:33.772
the same tradeoff.

45:33.768 --> 45:36.578
The amount of apples I
effectively give up today in

45:36.577 --> 45:39.997
order to get apples tomorrow is
going to be the same no matter

45:40.003 --> 45:41.243
which way I do it.

45:41.239 --> 45:44.899
If it weren't the same,
if alpha's price was more than

45:44.898 --> 45:47.798
a half of beta's nobody would
buy alpha.

45:47.800 --> 45:50.950
In fact they would start
selling alpha.

45:50.949 --> 45:52.969
So that's why this assumption
is so important.

45:52.969 --> 45:53.449
What would they do?

45:53.449 --> 45:56.139
Not only would nobody would buy
alpha, but they start selling

45:56.135 --> 45:56.355
it.

45:56.360 --> 45:59.350
They'd say well,
alpha is so expensive,

45:59.351 --> 46:03.921
let's say it's the same price
as beta, I can sell alpha.

46:03.920 --> 46:07.230
With every alpha I sell I can
buy stock beta,

46:07.226 --> 46:11.656
and so I haven't done anything
today, but in the future I've

46:11.663 --> 46:14.673
got stock beta which is paying
me 2.

46:14.670 --> 46:17.130
I owe, because I sold stock
alpha short I owe 1,

46:17.134 --> 46:20.284
so I'll pay off the 1 I owe and
I'll still be left with 1.

46:20.280 --> 46:23.450
I'm making an arbitrage profit,
and so I'm not going to stop at

46:23.449 --> 46:24.829
selling 1 share of alpha.

46:24.829 --> 46:27.099
I'll sell 2 shares of alpha,
then 3 shares of alpha,

46:27.099 --> 46:30.959
then a million shares of alpha,
and everybody'll be selling

46:30.956 --> 46:34.806
alpha short to buy beta and the
market for alpha will never

46:34.813 --> 46:35.483
clear.

46:35.480 --> 46:37.510
So that's why the prices will
have to adjust.

46:37.510 --> 46:40.080
And so it has to be in
equilibrium,

46:40.079 --> 46:42.969
the price of alpha's exactly
half the price of beta,

46:42.969 --> 46:48.409
which is to say,
in short, that if you solve for

46:48.407 --> 46:54.657
this equilibrium you can solve
for an equilibrium where

46:54.657 --> 47:01.487
P_1 = q_1 is
the price today of an apple

47:01.485 --> 47:07.385
today,
and P_2 is the price

47:07.389 --> 47:16.329
today (that's why it's called
present value price) of an apple

47:16.333 --> 47:18.243
next year.

47:18.239 --> 47:22.419
So if you've got this
equilibrium by working your way

47:22.416 --> 47:26.296
through,
by figuring out what the price

47:26.300 --> 47:28.450
of alpha is--
so the stock,

47:28.447 --> 47:30.917
for example,
you want to figure out what the

47:30.920 --> 47:31.610
stock is.

47:31.610 --> 47:36.300
Suppose the stock of alpha,
suppose the price turns out to

47:36.302 --> 47:37.292
be a half.

47:37.289 --> 47:41.799
Then by paying a half today you
can buy stock alpha,

47:41.800 --> 47:45.780
which is going to pay you a
whole dividend.

47:45.780 --> 47:48.970
So the price,
therefore, of an entire--oh,

47:48.972 --> 47:50.222
let's do beta.

47:50.219 --> 47:54.169
Suppose the price of beta is a
quarter.

47:54.170 --> 47:57.120
Suppose we happen to find out
that the price of beta is a

47:57.121 --> 47:57.651
quarter.

47:57.650 --> 47:59.410
Then what's P_2?

47:59.409 --> 48:04.729
How much do you have to give up
today in order to get an apple?

48:04.730 --> 48:07.410
Well, by paying a quarter
today, that's the price,

48:07.405 --> 48:10.405
by paying a quarter today
you're getting 2 dividends.

48:10.409 --> 48:13.259
So by paying a quarter today
you're getting 2 dividends.

48:13.260 --> 48:17.330
If you paid to get 1 dividend
you'd have to pay an eighth

48:17.327 --> 48:17.907
today.

48:17.909 --> 48:22.999
So the price P_2
would be an eighth in that case.

48:23.000 --> 48:26.590
So by piercing through the veil
of the stock market you can

48:26.585 --> 48:30.045
always figure out what you're
effectively paying today in

48:30.050 --> 48:32.400
order to get an apple next
period.

48:32.400 --> 48:34.740
And that price which was just
computed would be the same

48:34.744 --> 48:37.394
whether we looked at it from the
point of view of going through

48:37.387 --> 48:38.937
stock beta,
or through stock alpha,

48:38.936 --> 48:40.026
or through the nominal bond.

48:40.030 --> 48:41.600
It would always have to give us
the same answer.

48:41.599 --> 48:45.889
So we know, from the financial
equilibrium, we can deduce what

48:45.887 --> 48:50.107
P_1 and P_2
have to be, the present value

48:50.106 --> 48:50.876
prices.

48:50.880 --> 48:58.660
And so effectively,
furthermore,

48:58.661 --> 49:12.971
stocks effectively just add to
the endowments of goods.

49:12.969 --> 49:17.119
So we can now consider another
economy.

49:17.119 --> 49:22.339
So let's consider the economy
E-hat, so the hat economy.

49:22.340 --> 49:27.170
So U-hat-^(A) of X_1
and X_2 is the same as

49:27.166 --> 49:30.346
it was before,
U^(A) of X_1,

49:30.347 --> 49:33.837
X_2,
U-hat^(B )of X_1 and

49:33.840 --> 49:37.140
X_2 is the same as it
was before,

49:37.139 --> 49:41.749
but endowments now
E-hat^(A)_1

49:41.748 --> 49:46.948
E-hat^(A)_2 is going
to be what?

49:46.949 --> 49:51.449
Well, A over here began with 1
unit of each good,

49:51.449 --> 49:57.169
but A also owned all of stock
alpha and half of stock beta.

49:57.170 --> 50:00.720
So all of stock alpha pays 1
dividend in the future,

50:00.719 --> 50:04.139
so really A effectively has
claim on two apples in the

50:04.141 --> 50:08.011
future and another half of beta
which is another apple in the

50:08.014 --> 50:13.174
future,
so really A's initial endowment

50:13.172 --> 50:15.862
of goods is (1,3).

50:15.860 --> 50:17.270
How did I get that again?

50:17.268 --> 50:21.168
I said it was 1 apple to begin
with he could anticipate having.

50:21.170 --> 50:26.350
He knew he owned all of stock
alpha which pays 1 apple,

50:26.349 --> 50:28.059
so that's another one that's
really his,

50:28.059 --> 50:30.849
and then in the future he's
going to get half of the

50:30.851 --> 50:33.891
dividends of stock beta,
and half of 2 is also 1.

50:33.889 --> 50:40.389
So he's got 3 apples in the
future.

50:40.389 --> 50:43.709
And E-hat^(B)2,
well his 1 doesn't change

50:43.711 --> 50:47.031
today, but what's his claim,
effectively,

50:47.034 --> 50:49.614
on dividends in the future?

50:49.610 --> 50:55.250
Student: 1.

50:55.250 --> 50:57.430
Prof: 1, thank you.

50:57.429 --> 50:58.769
Somebody answered that.

50:58.768 --> 51:03.678
So we've now reduced the
financial equilibrium to a

51:03.684 --> 51:08.704
general equilibrium,
the same kind of economy we had

51:08.697 --> 51:09.777
before.

51:09.780 --> 51:12.750
It's just that we had to
augment the endowments to take

51:12.753 --> 51:15.843
into account that people own
stuff through the stocks.

51:15.840 --> 51:19.370
So what's the equilibrium of
this economy?

51:19.369 --> 51:22.439
This has a simple general
equilibrium.

51:22.440 --> 51:24.570
So what is it?

51:24.570 --> 51:26.430
How do we solve for equilibrium?

51:26.429 --> 51:31.289
Well, take P_1 = 1
and we'll solve for

51:31.293 --> 51:32.953
P_2.

51:32.949 --> 51:35.489
So let's just clear the first
market.

51:35.489 --> 51:39.349
How do you clear it?

51:39.349 --> 51:40.009
You're with me here?

51:40.010 --> 51:42.970
It's a standard general
equilibrium, the same kind we've

51:42.972 --> 51:44.322
done many times before.

51:44.320 --> 51:47.120
So see if I can do it.

51:47.119 --> 51:50.259
So person 1 is going to spend a
third of his money [correction:

51:50.257 --> 51:52.577
will be two thirds],
and how much money does he

51:52.583 --> 51:53.043
have?

51:53.039 --> 52:00.639
He has (1 P_2 times
3), that's A,

52:00.641 --> 52:02.091
right?

52:02.090 --> 52:03.340
His endowment is (1,3).

52:03.340 --> 52:06.490
This is his income,
and he's spending a third of it

52:06.492 --> 52:07.252
on good 1.

52:07.250 --> 52:10.210
And the price of good 1,
P_1,

52:10.213 --> 52:11.543
here is just 1.

52:11.539 --> 52:16.289
And then B is going to spend,
he's a half,

52:16.288 --> 52:22.308
half Cobb-Douglas guy,
so this is 1 and this is 1.

52:22.309 --> 52:27.419
He's spending half of his money
and his income is [1

52:27.416 --> 52:30.916
P_2 times 1]
divided by 1,

52:30.920 --> 52:37.030
and that has to equal the total
endowment which is 2,1 1.

52:37.030 --> 52:39.570
So did I go too fast?

52:39.570 --> 52:40.180
Yes?

52:40.179 --> 52:41.069
Student: Why is it a
third?

52:41.070 --> 52:41.930
Prof: Well,
it's probably wrong.

52:41.929 --> 52:43.619
So let's try it again.

52:43.619 --> 52:47.129
Maybe it's 2 thirds.

52:47.130 --> 52:48.530
Let's see what I was doing.

52:48.530 --> 52:52.890
I've taken the financial
economy, which was very

52:52.887 --> 52:55.707
complicated,
looks very hard to solve and

52:55.706 --> 52:58.756
Fisher says of course when we
add uncertainty and things like

52:58.764 --> 53:01.164
that we're going to have to do
other tricks.

53:01.159 --> 53:04.459
But without uncertainty,
with perfect foresight and so

53:04.458 --> 53:07.568
on, and no uncertainty,
Fisher says this is an easy

53:07.572 --> 53:08.882
problem to solve.

53:08.880 --> 53:11.480
You take the financial
equilibrium with all its extra

53:11.483 --> 53:14.243
variables and you realize if
people are rational they're

53:14.235 --> 53:16.835
going to see through all that
complicated stuff.

53:16.840 --> 53:19.590
They're going to realize that
alpha is just half as good as

53:19.590 --> 53:21.720
beta,
and so they're going to realize

53:21.715 --> 53:24.755
that by holding stock they're
making a certain tradeoff

53:24.755 --> 53:26.215
between alpha and beta.

53:26.219 --> 53:35.709
And we calculated the tradeoff.

53:35.710 --> 53:38.030
What was P_2?

53:38.030 --> 53:39.210
I forgot what P_2 was.

53:39.210 --> 53:41.360
Anyway, how much did you have
to pay?

53:41.360 --> 53:45.570
If you pay pi_alpha
divided by

53:45.570 --> 53:50.100
D^(alpha)_2,
something like that,

53:50.103 --> 53:52.483
was P_2.

53:52.480 --> 53:55.310
So if it costs you a certain
amount of money,

53:55.309 --> 54:02.519
if it costs you a quarter we
said, so this is P_2,

54:02.518 --> 54:05.028
so through either stock,
like beta's the one I solved it

54:05.034 --> 54:05.314
for.

54:05.309 --> 54:07.719
I said suppose beta,
that's also equal to

54:07.722 --> 54:10.682
pi_alpha over
D^(alpha)_2,

54:10.679 --> 54:13.869
we said if the price of beta
turns out to be a quarter and

54:13.871 --> 54:17.231
you're getting two dividends
then by paying a quarter you get

54:17.231 --> 54:18.241
two dividends.

54:18.239 --> 54:21.929
So it means to get one apple it
only costs you an eighth,

54:21.929 --> 54:23.509
an eighth of a dollar.

54:23.510 --> 54:25.680
So P_2 we can figure
out.

54:25.679 --> 54:29.359
So once we've got our financial
equilibrium, it basically is

54:29.362 --> 54:31.612
determining a general
equilibrium.

54:31.610 --> 54:33.390
So instead so let's go
backwards.

54:33.389 --> 54:36.119
Instead of solving for the
financial equilibrium that looks

54:36.123 --> 54:38.673
complicated let's solve for the
general equilibrium.

54:38.670 --> 54:40.510
What is the effective general
equilibrium?

54:40.510 --> 54:43.450
It's the same utilities as
before, but we've augmented the

54:43.449 --> 54:44.119
endowments.

54:44.119 --> 54:48.169
By looking through the veil of
the stocks we realized that A

54:48.166 --> 54:50.906
actually owns 3 apples in the
future,

54:50.909 --> 54:54.159
1 because he owns all of stock
alpha and another one because he

54:54.157 --> 54:55.517
owns half of stock beta.

54:55.518 --> 54:57.888
So we've got this simple
economy that we're used to

54:57.885 --> 55:00.005
solving that you did on the
first problem set,

55:00.014 --> 55:01.864
so we can do it again and solve
it.

55:01.860 --> 55:04.680
So I'm going to now solve for
general equilibrium.

55:04.679 --> 55:06.509
I have to solve for
P_1 and P_2

55:06.510 --> 55:08.480
and all the X^(A)_1,
X^(A)_2,

55:08.483 --> 55:10.013
X^(B)_1,
X^(B)_2,

55:10.010 --> 55:11.920
but I can fix one of the prices
to be one.

55:11.920 --> 55:13.840
So I'll fix P_1 to be
1.

55:13.840 --> 55:15.750
Then what does A do?

55:15.750 --> 55:18.160
So I made a mistake which is
why you weren't following me.

55:18.159 --> 55:22.159
A, his Cobb-Douglas,
2 thirds of the weight is on

55:22.157 --> 55:24.737
good 1 and 1 third on good 2.

55:24.739 --> 55:28.369
So he's going to spend 2 thirds
of his money on the first good.

55:28.369 --> 55:31.109
So that's why this should have
been a 2 thirds as she pointed

55:31.112 --> 55:31.892
out, thank you.

55:31.889 --> 55:34.519
So 2 thirds of his money,
what's his money?

55:34.518 --> 55:37.508
His endowment is (1,3),
so it's [(P_1 times

55:37.510 --> 55:40.220
1), which is 1 times 1,
(3 times P_2)]

55:40.215 --> 55:42.455
divided by the price
P_1.

55:42.460 --> 55:45.440
2 thirds of the income divided
by the price of the first good,

55:45.438 --> 55:47.978
that's how many of the first
good he wants to eat.

55:47.980 --> 55:49.460
What does she want to do?

55:49.460 --> 55:51.470
She's patient.

55:51.469 --> 55:53.999
She's going to spend half her
income on both goods,

55:54.000 --> 55:58.460
so half of her income which is
[(1 P_2 times 1)

55:58.456 --> 56:01.766
divided by 1],
that's how many apples today

56:01.771 --> 56:05.981
she wants and that's what we
have to clear to clear the apple

56:05.978 --> 56:07.378
market at time 1.

56:07.380 --> 56:11.850
So does this make sense now?

56:11.849 --> 56:13.859
I'm looking at you in the front.

56:13.860 --> 56:23.110
Do you agree with this or is
this confusing now?

56:23.110 --> 56:24.360
Do you follow this or is this
confusing?

56:24.360 --> 56:25.330
I can say it again if it's
confusing.

56:25.329 --> 56:26.029
Student: I've got a
question.

56:26.030 --> 56:26.570
Prof: Yeah?

56:26.570 --> 56:30.910
Student: Our denominator
represents what they want to

56:30.905 --> 56:32.445
have in the future?

56:32.449 --> 56:34.989
Prof: Remember how the
Cobb-Douglas worked?

56:34.989 --> 56:38.629
This trick I'm going to use
over and over again.

56:38.630 --> 56:42.790
With log utilities everybody
will spend depending on the

56:42.789 --> 56:43.849
coefficient.

56:43.849 --> 56:46.299
So remember,
this utility in the problem

56:46.297 --> 56:50.127
set, you know that this utility
is just the same as if I put 2

56:50.128 --> 56:52.638
thirds here and 1 third here,
right?

56:52.639 --> 56:57.949
Because I'm just dividing this
by 3, so instead of 1 and a

56:57.949 --> 57:02.329
half--so the original utility is
this, right?

57:02.329 --> 57:07.359
It's (log X_1) (1
half log X_2),

57:07.358 --> 57:11.598
so the sum of this plus this is
3 halves.

57:11.599 --> 57:14.109
I can multiply by 2 thirds.

57:14.110 --> 57:17.980
So if I multiply this by 2
thirds I get 2 thirds here and 1

57:17.983 --> 57:19.323
third here, right?

57:19.320 --> 57:21.330
And I haven't changed the
utility function.

57:21.329 --> 57:23.589
And now I know this is a
familiar pattern.

57:23.590 --> 57:28.590
A is always going to spend 2
thirds of his money on the first

57:28.586 --> 57:29.166
good.

57:29.170 --> 57:34.540
And B I can multiple this whole
utility by a half and a half and

57:34.543 --> 57:37.693
B is going spend--
and now we recognize it as the

57:37.686 --> 57:40.706
common Cobb-Douglas thing and we
could say that A's [correction:

57:40.706 --> 57:42.286
B's]
going to spend half his money

57:42.286 --> 57:45.206
on the first good and half his
money on the second good.

57:45.210 --> 57:46.300
So what have I done?

57:46.300 --> 57:49.990
Fisher said look,
this model is so complicated.

57:49.989 --> 57:53.439
You're thinking in your heads,
people are deciding in period 1

57:53.443 --> 57:56.503
how much stock should I buy,
how many bonds should I buy,

57:56.500 --> 57:59.190
how many apples should I eat,
but really if they're smart

57:59.192 --> 58:00.912
they're not going to think that
way.

58:00.909 --> 58:03.679
They're going to say to
themselves how many apples

58:03.681 --> 58:04.871
should I buy today?

58:04.869 --> 58:07.219
How many apples do I want to
consume tomorrow?

58:07.219 --> 58:10.949
All these financial assets are
just methods for me getting

58:10.945 --> 58:14.015
apples tomorrow in exchange for
apples today.

58:14.018 --> 58:16.458
And what's the tradeoff between
the apples was

58:16.463 --> 58:19.293
this--P_1 and
P_2 is the tradeoff

58:19.288 --> 58:20.318
between apples.

58:20.320 --> 58:22.470
You can look through the stocks
and all that,

58:22.469 --> 58:25.849
but no matter which stock you
think of buying there's going to

58:25.846 --> 58:29.166
be the same tradeoff between
apples today and apples tomorrow

58:29.168 --> 58:30.938
because of the no arbitrage.

58:30.940 --> 58:34.270
The price of alpha is going to
have to be exactly half the

58:34.273 --> 58:35.213
price of beta.

58:35.210 --> 58:38.730
So once I solve for this
economy and get the price of

58:38.726 --> 58:42.646
alpha I'll know how many apples
today I have to tradeoff in

58:42.648 --> 58:44.878
order to get apples tomorrow.

58:44.880 --> 58:47.990
So I might as well forget about
all the stocks and just try to

58:47.989 --> 58:50.799
figure out what must that
tradeoff between P_1

58:50.795 --> 58:52.065
and P_2 be.

58:52.070 --> 58:55.070
So that's why you can forget
about the stocks,

58:55.067 --> 58:58.197
forget about the bonds,
everybody's thinking I'm

58:58.199 --> 59:01.529
trading off apples today for
apples next year.

59:01.530 --> 59:03.940
I'm making all the trades
today, because I'm trading

59:03.936 --> 59:06.906
apples today in exchange for
promises for apples next period.

59:06.909 --> 59:08.889
So it's as if everything
happens today.

59:08.889 --> 59:11.719
It's as if they're present
value prices today.

59:11.719 --> 59:14.809
We trade today at prices
P_1 and P_2

59:14.811 --> 59:17.061
for apples today and apples next
year.

59:17.059 --> 59:19.269
Of course the apples won't
appear until next year,

59:19.268 --> 59:23.278
but I can sell an apple today
at price P_1 and buy

59:23.282 --> 59:27.162
promises for apple next year at
a price P_2,

59:27.159 --> 59:28.729
and that's the tradeoff I'm
facing.

59:28.730 --> 59:32.390
If I face that tradeoff how
much of my money am I going to

59:32.389 --> 59:33.929
spend on apples today?

59:33.929 --> 59:37.919
I'm going to spend 2 thirds of
my money on apples today and the

59:37.918 --> 59:41.778
other third I'll spend on
promises for apples next period.

59:41.780 --> 59:43.500
So this is a big insight Fisher
had.

59:43.500 --> 59:45.320
It's not surprising it's a
little puzzling.

59:45.320 --> 59:47.970
I'm so used to it that I've
forgotten how puzzling it is.

59:47.969 --> 59:49.239
So ask me some more questions.

59:49.239 --> 59:51.509
This was not an obvious thought
Fisher had.

59:51.510 --> 59:52.110
Yeah?

59:52.110 --> 59:55.770
Student: Do we typically
expect the price in the 2nd

59:55.771 --> 59:59.181
period to be lower than the
price in the 1st period?

59:59.179 --> 1:00:01.999
Prof: Often there'll be
inflation,

1:00:02.000 --> 1:00:04.500
so q_2--the
contemporaneous price next

1:00:04.503 --> 1:00:07.543
period might be higher than the
contemporaneous price this

1:00:07.539 --> 1:00:09.679
period,
but we don't care about that.

1:00:09.679 --> 1:00:14.009
What we care about is how many
apples you have to give up today

1:00:14.007 --> 1:00:16.377
in order to get apples tomorrow.

1:00:16.380 --> 1:00:20.450
So P_2 is the present
value price.

1:00:20.449 --> 1:00:23.279
What do you have to give up
today to get the apple next

1:00:23.277 --> 1:00:23.747
period?

1:00:23.750 --> 1:00:28.390
So we expect P_2 to
be less than P_1.

1:00:28.389 --> 1:00:30.309
Precisely because,
well, we're going to come to

1:00:30.306 --> 1:00:31.796
that,
that's the next thing I was

1:00:31.800 --> 1:00:34.300
going to talk about because
everybody's putting more weight

1:00:34.297 --> 1:00:36.657
on consumption today than they
are on consumption in the

1:00:36.663 --> 1:00:37.183
future.

1:00:37.179 --> 1:00:39.569
That's why the price
P_1 is going to be

1:00:39.574 --> 1:00:41.474
bigger than the price
P_2.

1:00:41.469 --> 1:00:42.159
Yep?

1:00:42.159 --> 1:00:44.849
Student: When we're
solving it we're solving in real

1:00:44.851 --> 1:00:45.271
prices?

1:00:45.268 --> 1:00:47.928
Prof: So we're solving
for P_1 and

1:00:47.931 --> 1:00:50.061
P_2 in present value
prices.

1:00:50.059 --> 1:00:53.359
So the crucial thing is,
he invented this term present

1:00:53.364 --> 1:00:55.864
value prices:
the prices you pay today no

1:00:55.860 --> 1:00:58.730
matter when you're going to get
the stuff.

1:00:58.730 --> 1:01:00.060
That's his big insight.

1:01:00.059 --> 1:01:02.039
You should look at present
value prices.

1:01:02.039 --> 1:01:06.009
Holding stocks and all that
complicated stuff is just giving

1:01:06.010 --> 1:01:07.760
you goods in the future.

1:01:07.760 --> 1:01:10.160
So when you buy the stocks
today you should think,

1:01:10.157 --> 1:01:12.847
how much am I having to pay
today to get an apple in the

1:01:12.851 --> 1:01:13.391
future?

1:01:13.389 --> 1:01:15.779
You can deduce that from the
price of stocks and how many

1:01:15.777 --> 1:01:16.927
dividends they're paying.

1:01:16.929 --> 1:01:19.609
So everybody must have figured
out a P_2.

1:01:19.610 --> 1:01:20.570
What does it cost today?

1:01:20.570 --> 1:01:23.510
How much money do I have to
give up today to get an apple in

1:01:23.510 --> 1:01:24.160
the future?

1:01:24.159 --> 1:01:26.759
Well, I have to buy a stock and
then sell the dividends and all

1:01:26.755 --> 1:01:28.425
that,
but really what I should be

1:01:28.425 --> 1:01:31.285
thinking about is what's the
price today I'm paying for one

1:01:31.293 --> 1:01:33.833
apple in the future,
and that's P_2.

1:01:33.829 --> 1:01:35.439
And so when you think about it
that way,

1:01:35.440 --> 1:01:37.350
although it's an intertemporal
problem,

1:01:37.349 --> 1:01:39.249
it looks like a new model with
time,

1:01:39.250 --> 1:01:42.570
Fisher said you can reduce it,
think of it as if they're just

1:01:42.574 --> 1:01:46.014
the same problem we did before
with two goods you're trading at

1:01:46.012 --> 1:01:47.012
the same time.

1:01:47.010 --> 1:01:48.750
That's not an obvious thing to
have thought of.

1:01:48.750 --> 1:01:50.440
No one thought of it before him.

1:01:50.440 --> 1:01:51.190
Yeah?

1:01:51.190 --> 1:01:54.450
Student: The budget set,
the second equation from the

1:01:54.445 --> 1:01:55.435
right-hand side.

1:01:55.440 --> 1:02:00.040
The second item shouldn't you
have a q_2 as well,

1:02:00.036 --> 1:02:03.836
q_2 times
>?

1:02:03.840 --> 1:02:05.700
Prof: Oh, absolutely.

1:02:05.699 --> 1:02:08.349
So this should have been a
q_2 here because you'd

1:02:08.353 --> 1:02:11.193
sell the dividend and you get
money by selling the dividend.

1:02:11.190 --> 1:02:12.530
Thank you.

1:02:12.530 --> 1:02:14.830
Ho, ho, ho, very good.

1:02:14.829 --> 1:02:17.859
Who said that?

1:02:17.860 --> 1:02:19.090
Who just asked that question?

1:02:19.090 --> 1:02:20.300
Where are you?

1:02:20.300 --> 1:02:21.050
I'll remember you.

1:02:21.050 --> 1:02:22.600
That was very good, exactly.

1:02:22.599 --> 1:02:24.699
So in the future you're getting
the money.

1:02:24.699 --> 1:02:28.049
So what he pointed out is I
made another mistake here.

1:02:28.050 --> 1:02:31.920
In the future the money you're
spending on goods in the future

1:02:31.916 --> 1:02:34.576
you're going to get the
dividends paid,

1:02:34.579 --> 1:02:36.559
of course you can sell the
dividends for money and the

1:02:36.556 --> 1:02:37.486
price is q_2.

1:02:37.489 --> 1:02:39.549
So a q_2 has to
appear over here just like

1:02:39.552 --> 1:02:41.052
there's a q_2 over
there.

1:02:41.050 --> 1:02:43.510
So it's the goods times the
price.

1:02:43.510 --> 1:02:46.860
That's the money you're getting
in the future,

1:02:46.856 --> 1:02:50.346
and that's the money you're
spending on the good

1:02:50.351 --> 1:02:51.691
X_2.

1:02:51.690 --> 1:02:55.560
Very good, too bad you didn't
ask me that a while ago,

1:02:55.559 --> 1:02:56.509
but anyway.

1:02:56.510 --> 1:03:00.970
Any other questions?

1:03:00.969 --> 1:03:04.429
So we're back to this standard
general equilibrium problem.

1:03:04.429 --> 1:03:07.139
We can take a financial
equilibrium and turn it into a

1:03:07.135 --> 1:03:08.255
general equilibrium.

1:03:08.260 --> 1:03:15.440
And so when we solve this we're
going to have (2 thirds

1:03:15.436 --> 1:03:21.946
P_2 1 half 1 half
P_2) = 2.

1:03:21.949 --> 1:03:32.829
So it looks like 3 halves (I
hope I haven't done this wrong)

1:03:32.827 --> 1:03:40.567
P_2 = 2 thirds
2P_2.

1:03:40.570 --> 1:03:41.040
Student:
>

1:03:41.039 --> 1:03:42.849
Prof: Thank you,
yeah, plus 2P_2.

1:03:42.849 --> 1:03:47.119
So we have 2P_2 a
half a half P_2 so we

1:03:47.119 --> 1:03:49.329
have 5 halves P_2.

1:03:49.329 --> 1:03:52.029
That was lucky you caught that,
5 halves P_2.

1:03:52.030 --> 1:03:57.000
So 2 thirds is 4 sixths.

1:03:57.000 --> 1:03:58.760
And 3 sixths is 9 sixths.

1:03:58.760 --> 1:04:04.940
And 12 sixths - 9 sixths is...

1:04:04.940 --> 1:04:07.240
What is this?

1:04:07.239 --> 1:04:09.929
So what's 2 - 2 thirds - a half?

1:04:09.929 --> 1:04:10.429
Student: 5 sixths.

1:04:10.429 --> 1:04:11.199
Prof: 5 sixths.

1:04:11.199 --> 1:04:12.639
That's correct.

1:04:12.639 --> 1:04:16.689
So P_2 therefore
equals 1 third.

1:04:16.690 --> 1:04:19.820
All right, so we've now solved
for equilibrium.

1:04:19.820 --> 1:04:25.710
We know that P_1 has
got to be 1.

1:04:25.710 --> 1:04:29.230
P_2 has got to be 1
third.

1:04:29.230 --> 1:04:34.890
We know that we can figure out
what consumption's going to be,

1:04:34.889 --> 1:04:38.299
I mean X^(A)_1,
for example,

1:04:38.300 --> 1:04:40.910
if we wanted to solve for that
we just plug in a third here.

1:04:40.909 --> 1:04:47.209
So we'd have (2 thirds) times
(1 3), which is 2 thirds times a

1:04:47.206 --> 1:04:50.506
4, which is 8 thirds,
I guess.

1:04:50.510 --> 1:04:53.440
And X^(B)_1,
we could have solved for that

1:04:53.443 --> 1:04:54.713
too if we wanted to.

1:04:54.710 --> 1:04:58.990
X^(B)_1 is going to
be a half times 4 thirds which

1:04:58.992 --> 1:05:00.012
is 2 thirds.

1:05:00.010 --> 1:05:04.660
No, that doesn't--a half plus
what was this 1 third?

1:05:04.659 --> 1:05:09.559
No, it's 1 1 third which is 4
thirds times a half which is 2

1:05:09.561 --> 1:05:10.311
thirds.

1:05:10.309 --> 1:05:15.819
Is that right?

1:05:15.820 --> 1:05:18.950
That doesn't look right,
so maybe I did this wrong.

1:05:18.949 --> 1:05:24.489
1 1 is 2 so this is 4 thirds.

1:05:24.489 --> 1:05:25.439
That looks better.

1:05:25.440 --> 1:05:27.450
X^(A)_1 is 4 thirds.

1:05:27.449 --> 1:05:29.029
X^(B)_1 is 2 thirds.

1:05:29.030 --> 1:05:32.010
So we have 4 thirds and 2
thirds, and so we could solve

1:05:32.012 --> 1:05:35.162
similarly for X^(A)_2
and X^(B)_2,

1:05:35.159 --> 1:05:36.759
which I won't bother to do.

1:05:36.760 --> 1:05:39.790
So we can figure out what the
prices are, the present value

1:05:39.786 --> 1:05:42.026
prices and the present value
consumption.

1:05:42.030 --> 1:05:43.740
But having done that,
Fisher says,

1:05:43.737 --> 1:05:45.857
we took a hard problem we make
it easy.

1:05:45.860 --> 1:05:48.050
Let's go back to the hard
problem.

1:05:48.050 --> 1:05:53.030
So Fisher says the tradeoff
between good 1 and good 2 is 1

1:05:53.034 --> 1:05:57.324
to a third, so he
defined--here's the nominal rate

1:05:57.320 --> 1:05:58.720
of interest.

1:05:58.719 --> 1:06:04.789
Fisher defined something called
the real rate of interest.

1:06:04.789 --> 1:06:07.609
And he said that was a variable
that you should pay a lot of

1:06:07.612 --> 1:06:08.332
attention to.

1:06:08.329 --> 1:06:17.199
So the real rate of interest he
said is P_1 divided by

1:06:17.204 --> 1:06:23.404
P_2,
so this is equal to 3 and so r

1:06:23.402 --> 1:06:26.082
is 200 percent.

1:06:26.079 --> 1:06:27.569
So how did I get that?

1:06:27.570 --> 1:06:31.980
Just as someone in the front
said the good 2 is much less

1:06:31.983 --> 1:06:32.933
expensive.

1:06:32.929 --> 1:06:36.109
The present value of good 2 is
much less than the present value

1:06:36.105 --> 1:06:36.715
of good 1.

1:06:36.719 --> 1:06:41.579
People think an apple today is
much more valuable than an apple

1:06:41.577 --> 1:06:42.437
tomorrow.

1:06:42.440 --> 1:06:45.420
So if you give up an apple
today you can get 3 apples next

1:06:45.422 --> 1:06:45.792
year.

1:06:45.789 --> 1:06:49.089
So if you put an apple in the
bank it's like getting 200

1:06:49.090 --> 1:06:50.830
percent interest on apples.

1:06:50.829 --> 1:06:53.049
So he called that the real
rate--the apple rate of

1:06:53.050 --> 1:06:53.550
interest.

1:06:53.550 --> 1:06:56.570
You put an apple in the bank,
you give up an apple today,

1:06:56.570 --> 1:06:59.300
buy stocks and when it comes
out in the end you've got 200

1:06:59.295 --> 1:07:01.395
percent more apples than you
started with.

1:07:01.400 --> 1:07:04.350
So it's the real rate of
interest.

1:07:04.349 --> 1:07:06.319
So that's his crucial variable.

1:07:06.320 --> 1:07:09.210
Now, let's go back to the
original equilibrium.

1:07:09.210 --> 1:07:11.890
What is the stock price?

1:07:11.889 --> 1:07:17.799
Assume q_1 = 1.

1:07:17.800 --> 1:07:26.650
What is the stock price
pi_alpha?

1:07:26.650 --> 1:07:30.910
Well, we can figure it out.

1:07:30.909 --> 1:07:33.979
How can we figure it out?

1:07:33.980 --> 1:07:36.530
What is pi_alpha?

1:07:36.530 --> 1:07:42.530
Well, stock alpha pays 1 good
tomorrow so what is the price of

1:07:42.525 --> 1:07:44.585
pi_alpha?

1:07:44.590 --> 1:07:49.000
Student:
>

1:07:49.000 --> 1:07:50.160
Prof: What?

1:07:50.159 --> 1:07:50.869
Somebody said it.

1:07:50.869 --> 1:07:53.939
I couldn't hear it--a third.

1:07:53.940 --> 1:07:55.680
How did I get a third?

1:07:55.679 --> 1:07:58.869
Because we figured out that
once everybody looks through the

1:07:58.867 --> 1:08:00.067
veil,
assuming the price

1:08:00.074 --> 1:08:02.404
P_1 is 1 and the price
q_1 is 1,

1:08:02.400 --> 1:08:06.330
if they look through the veil
they're going to say to

1:08:06.327 --> 1:08:09.227
themselves ah-ha,
how much do I have to pay today

1:08:09.230 --> 1:08:10.620
to get an apple in the future?

1:08:10.619 --> 1:08:14.009
I have to pay a third to get
one apple in the future.

1:08:14.010 --> 1:08:16.190
P_1 is 3 times
P_2.

1:08:16.189 --> 1:08:18.759
So to get 1 apple in the
future, it's only a third of an

1:08:18.764 --> 1:08:19.424
apple today.

1:08:19.420 --> 1:08:23.480
So the stock pays 1 apple in
the future so therefore how much

1:08:23.484 --> 1:08:25.184
do I have to pay today?

1:08:25.180 --> 1:08:28.230
I have to pay 1 third of an
apple today, and since I took

1:08:28.231 --> 1:08:31.391
the price of apples to be 1 it's
going to be the price of 1

1:08:31.394 --> 1:08:31.944
third.

1:08:31.939 --> 1:08:33.409
So what's the price beta?

1:08:33.409 --> 1:08:34.639
Student: 2 thirds.

1:08:34.640 --> 1:08:36.150
Prof: 2 thirds.

1:08:36.149 --> 1:08:40.659
So Fisher said look,
we've solved now for all these

1:08:40.657 --> 1:08:42.367
financial things.

1:08:42.368 --> 1:08:54.788
So what you can't do,
Fisher's theory does not

1:08:54.792 --> 1:09:08.872
explain how much of each stock,
theta^(A), etcetera,

1:09:08.869 --> 1:09:15.219
the investors hold.

1:09:15.220 --> 1:09:16.420
Why is that?

1:09:16.420 --> 1:09:18.760
Well, because it doesn't matter.

1:09:18.760 --> 1:09:21.350
Not enough is happening in the
economy yet.

1:09:21.350 --> 1:09:23.400
Alpha and beta are exactly the
same.

1:09:23.399 --> 1:09:26.359
If you own twice as much of the
alpha tree you get exactly the

1:09:26.356 --> 1:09:27.856
same as having the beta tree.

1:09:27.859 --> 1:09:31.329
So how can you possibly tell
whether somebody's going to hold

1:09:31.333 --> 1:09:33.943
twice the alpha tree or just one
beta tree?

1:09:33.939 --> 1:09:35.849
Either way he's going to get
the same thing.

1:09:35.850 --> 1:09:38.530
So the theory can't possibly
explain which one they're going

1:09:38.527 --> 1:09:38.887
to do.

1:09:38.890 --> 1:09:41.740
Somehow they'll work it out and
divide up the tree so that

1:09:41.743 --> 1:09:44.903
everybody ends up with the right
number of apples in the end.

1:09:44.899 --> 1:09:55.429
And it also does not explain
inflation because you can't tell

1:09:55.425 --> 1:10:01.735
what q_2 is going to
be.

1:10:01.738 --> 1:10:04.448
All right, because you see in
this budget set,

1:10:04.454 --> 1:10:07.234
thanks to that inspired
question, if you double

1:10:07.230 --> 1:10:09.340
q_2,
q_2 appears

1:10:09.344 --> 1:10:11.884
everywhere;
you're not going to change the

1:10:11.881 --> 1:10:12.741
second equation.

1:10:12.738 --> 1:10:14.538
So q_2 could be
anything.

1:10:14.538 --> 1:10:18.328
You can double it or triple it,
it won't matter,

1:10:18.328 --> 1:10:21.068
and the same with
q_1.

1:10:21.069 --> 1:10:24.179
It's just like Walras said
before, you can always normalize

1:10:24.180 --> 1:10:25.200
prices to be one.

1:10:25.198 --> 1:10:29.208
He had to add another theory of
money and how many dollars were

1:10:29.213 --> 1:10:32.973
floating around in the economy
to explain q_2.

1:10:32.970 --> 1:10:34.480
This theory won't explain it.

1:10:34.479 --> 1:10:38.109
So it does not explain
inflation, and it does not

1:10:38.109 --> 1:10:42.649
explain who holds which stock,
and so it does not explain the

1:10:42.646 --> 1:10:44.836
nominal rate of interest.

1:10:44.840 --> 1:10:48.080
It does not explain i,
the nominal rate of interest,

1:10:48.081 --> 1:10:51.131
because 1 dollar,
who knows what 1 dollar's going

1:10:51.131 --> 1:10:52.151
to be worth.

1:10:52.149 --> 1:10:54.339
It depends on how much
inflation we have.

1:10:54.340 --> 1:10:57.150
But it does explain the real
rate of interest.

1:10:57.149 --> 1:11:01.499
It does explain r,
and that's the variable that

1:11:01.500 --> 1:11:06.230
Fisher said is the one
economists should always pay

1:11:06.228 --> 1:11:10.388
attention to,
the real rate of interest.

1:11:10.390 --> 1:11:13.080
So that's the crucial variable.

1:11:13.078 --> 1:11:16.508
So if you want to figure out
what's the price today of a

1:11:16.507 --> 1:11:20.667
stock,
so Fisher's famous equation's

1:11:20.666 --> 1:11:26.376
the price of the stock today,
so pi_alpha divided

1:11:26.376 --> 1:11:29.116
by q_1,
so the real price,

1:11:29.117 --> 1:11:33.147
as somebody said,
the price in terms of goods of

1:11:33.154 --> 1:11:37.944
the stock today is always going
to equal the dividend in the

1:11:37.940 --> 1:11:39.970
future divided by 1 r.

1:11:39.970 --> 1:11:40.740
Why is that?

1:11:40.738 --> 1:11:42.998
That's exactly what we already
used.

1:11:43.000 --> 1:11:46.960
This is just a rewriting of the
trick we did before.

1:11:46.960 --> 1:11:51.620
You take the dividend tomorrow,
you multiply it by

1:11:51.622 --> 1:11:56.382
P_2 and then you
realize that 1 r is just

1:11:56.381 --> 1:12:00.571
P_1 divided by
P_2,

1:12:00.569 --> 1:12:03.299
so replacing the P_1s
and P_2s by

1:12:03.297 --> 1:12:06.917
q_1 and 1 r,
today's real stock price is

1:12:06.918 --> 1:12:10.338
just the dividends tomorrow
discounted.

1:12:10.340 --> 1:12:19.270
This is what he called the
Fundamental Theorem of Asset

1:12:19.265 --> 1:12:20.915
Pricing.

1:12:20.920 --> 1:12:24.650
If you knew the real rate of
interest you'd be able to figure

1:12:24.645 --> 1:12:27.995
out what all the stocks were
worth just like we did.

1:12:28.000 --> 1:12:29.650
Once we knew P_1 and
P_2,

1:12:29.649 --> 1:12:32.649
the present value--the present
value prices determine the

1:12:32.650 --> 1:12:35.300
interest rate because they're--
just as we said.

1:12:35.300 --> 1:12:36.900
P_1 over
P_2,

1:12:36.899 --> 1:12:37.849
remember, is 1 r.

1:12:37.850 --> 1:12:40.860
So knowing P_1 and
P_2 you're always

1:12:40.855 --> 1:12:44.295
normalizing P_1 to be
1, P_2's the same as 1

1:12:44.298 --> 1:12:45.008
over 1 r.

1:12:45.010 --> 1:12:48.060
So if you know P_2 or
you know 1 over 1 r,

1:12:48.064 --> 1:12:50.704
you know what the value of the
stocks are.

1:12:50.699 --> 1:12:52.769
That's his critical insight.

1:12:52.770 --> 1:12:54.440
Now, just to finish...

1:12:54.439 --> 1:12:55.499
Yes?

1:12:55.500 --> 1:12:58.210
Student: That's
q_2 or q_1?

1:12:58.210 --> 1:13:02.030
Prof: q_1,
q_1 which is the same

1:13:02.029 --> 1:13:05.059
as P_1 because it's
today's price.

1:13:05.060 --> 1:13:10.420
The contemporaneous price today
is the present value of the

1:13:10.416 --> 1:13:11.706
price today.

1:13:11.710 --> 1:13:15.940
So let me just end on this one
note.

1:13:15.939 --> 1:13:19.179
Fisher said we can take
financial equilibrium without

1:13:19.179 --> 1:13:22.169
uncertainty, reduce it to
general equilibrium.

1:13:22.170 --> 1:13:24.310
We know everything about
general equilibrium,

1:13:24.310 --> 1:13:26.770
therefore we know everything
about financial equilibrium,

1:13:26.770 --> 1:13:29.960
and we realize that the crucial
variable in general equilibrium

1:13:29.958 --> 1:13:31.038
is relative prices.

1:13:31.039 --> 1:13:33.509
There is no just interest rate.

1:13:33.510 --> 1:13:35.230
The nominal interest rate,
who the hell cares?

1:13:35.229 --> 1:13:37.789
The real interest rate is what
we care about,

1:13:37.792 --> 1:13:41.232
and just in normal economics
there's no just relative price,

1:13:41.228 --> 1:13:43.498
there's no just real interest
rate.

1:13:43.500 --> 1:13:46.320
It depends on people's
utilities.

1:13:46.319 --> 1:13:49.689
You make them more patient and
that's going to affect the real

1:13:49.694 --> 1:13:50.584
interest rate.

1:13:50.578 --> 1:13:52.178
You make them less patient it's
going to affect the real

1:13:52.175 --> 1:13:52.635
interest rate.

1:13:52.640 --> 1:13:54.560
You give them more endowments
today versus tomorrow,

1:13:54.555 --> 1:13:56.355
that's going to affect the real
interest rate.

1:13:56.359 --> 1:13:58.669
The relative price between
today and tomorrow--that's the

1:13:58.666 --> 1:14:00.186
way you should think about
finance.

1:14:00.189 --> 1:14:02.319
That's the way you should
explain what's going on in the

1:14:02.323 --> 1:14:03.103
financial markets.

1:14:03.100 --> 1:14:06.520
So in the problem set you're
just going to do a problem like

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that, and then I'm going to give
more interpretations of this

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that Fisher gave.

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So I guess I'm out of time,
so we'll stop here.

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