WEBVTT

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Prof: Now,
the course, just to summarize

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again,
the course is the standard

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financial theory course that was
made popular over the last ten

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years in a bunch of business
schools,

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and those guys who developed
this material basically thought

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that markets were great and
finance was almost a separate

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part--
could be walled off from much

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of economics.

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So here at Yale we've never
taught finance that way.

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We've always taught it as a
part of economics and the crisis

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recently,
I think, has made it clear that

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that's probably the way one
should really think about the

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problem.

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So it's become very fashionable
now to say that financial

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theorists had everything all
wrong and to ask how it is that

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they got everything all wrong.

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Why didn't they anticipate the
crash?

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And the two standard critiques
of standard financial economics

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are a) it didn't allow for
psychology,

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and you'll hear about that from
Shiller next semester,

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and b) it didn't take into
account collateral.

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And it was all done in a very
special case,

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a knife-edge case.

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If you looked at it a little
more broadly you would realize

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that the kind of crisis we've
had now is not such an

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unfathomable thing.

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In fact it's happened many
times before.

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So that's the perspective I'm
going to take in this class.

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So to put it a different way,
Krugman,

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very recently in the New York
Times,

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you may have read his magazine
article,

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wrote exactly that,
that there are two problems.

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The financial theory has failed
us.

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Why has it failed us?

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Well, because it didn't have
psychology and it didn't have

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collateral.

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And he didn't talk much about
the collateral which is

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obviously something he's not
thought very much about before.

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But together with the
collateral he sort of said it's

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too much--how did he put it?

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He said, "Too much
seduction by mathematics.

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The financial economists were
seduced by their own mathematics

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into believing stuff that a
sensible person who didn't pay

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so much attention to mathematics
wouldn't do."

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Well, although the critique in
this course is going to be

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partly based on collateral the
rest of what Krugman said I

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completely disagree with.

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I regard that as a kind of
Taliban approach to economics.

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The more technology and
firepower you use the more

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you're going to be misled.

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That's what the Taliban
believe, and they want to get

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rid of modern technology and
return to first principles.

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So I think, in fact,
the problem with modern finance

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was not too much mathematics,
but too little mathematics,

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and they made these very
special simplifying assumptions

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and didn't realize how important
the assumptions were to the

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conclusions.

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So we're going to reexamine all
that and that's what we're

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starting at now.

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We're going to consider,
first of all,

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the argument that free markets
work best.

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So we started with a little
example.

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Oh, by the way,
the first problem set,

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if I didn't mention it,
is due Tuesday.

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So you need to bring it to
class and there will be a box

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with each of your section
leaders' names on it.

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So supposedly you've been able
to sign up for sections by now.

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Is that right?

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Anyway, it's on the web,
so you pick a section and sign

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up for it.

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If you wait too long your
section will fill up the time.

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So there are eight sections.

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You ought to be able to find
one of them that fits your

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schedule.

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And you need to turn it in on
Tuesday in class,

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by the end of class.

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Maybe you can scribble
something during the class,

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but by the end of the class
we're going to take the problem

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sets and after that it's too
late to hand them in.

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So all of you are going to have
problems,

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you're going to have midnight
sessions,

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all night things that you're
going to have to do for some

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other course,
or grandmothers are going to

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die.

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All sorts of things are going
to happen, but we don't take the

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problem sets late.

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So there will be ten problem
sets.

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We're only going to count nine
of the grades so you'll have one

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free pass, and that's what life
is.

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And it's just too complicated
to keep track of people handing

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them late all the time.

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The answers are going to go on
the web right after the class,

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and so it's just in the past we
negotiated with every person who

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was late and it's just too
complicated.

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And when you make a simple rule
grandmothers don't die anymore.

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So anyway, that's how we're
going to work it in the class.

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You just have to turn it in.

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And there are ten of them.

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Only nine count.

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If you miss one altogether it's
really not going to change your

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grade anyway.

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If you miss half of them that's
going to have some effect on

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your grade, and so I don't
encourage you to do that,

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but I'm sure you won't do that.

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So I think that's all the
preliminaries.

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There are two midterms,
one in the middle of the class

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and one right at the end.

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Anyway, the question we want to
spend the whole of the class on

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today is whether the free market
is really such a great idea.

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And the quintessential example
in which it is a great idea is

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the one we did in class on the
very first day.

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We had a bunch of football
tickets and there were buyers

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each of whom knew his own
valuation and sellers each of

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whom knew her own valuation and
we threw everybody together and

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just very briefly explained the
rules.

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By the way, the only important
rule was--there were two

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important rules.

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You had to announce publically
and loudly what price you were

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offering.

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That's very important.

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You don't have any secret deals.

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That would have screwed
everything up.

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And secondly,
we had a rule about once you

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make a deal what happens.

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How does the thing actually get
transferred?

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So one of the TAs stood by and
wrote it down,

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and the two people exchanged
the footballs and agreed to it

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and walked off stage.

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So the actual mechanics of the
transaction to make sure that

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the person turning over the
money actually gets the football

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ticket,
that of course is incredibly

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important and that's the thing
that gets left out often in

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finance.

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That's the collateral business
that we're going to come to

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later.

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How do you know that the guy's
actually going to pay you what

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he promises?

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Well, he's got to put up
collateral so that you can trust

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him.

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So without that there would
have been a big problem,

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and we're going to come talk
about that later.

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In the old days when you bought
a stock someone on a bicycle

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would carry the certificate from
one place--

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someone would carry the check
from one guy to another guy and

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then the bicyclist would get the
stock certificates and ride the

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bike back to the buyer.

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So it was one broker to another
broker.

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Sometimes it took a couple of
hours,

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so there was a spacing in
between when the guy gave over

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the money and when the guy got
the stock back,

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and you have to allow for that.

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Maybe it would take a couple
days to process on everybody's

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books.

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So there's a thing called
ex-dividend.

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When you buy a stock the old
buyer continues to get the

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dividends for a while--
the old owner--until a

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particular date after which the
new buyer starts getting the

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dividends if there are any.

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And everybody has to understand
that, because you have to allow

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for the actual trading
technology.

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So all those things are going
to come up later,

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and they were in the background
of this example.

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But never mind that.

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The point now is that these
people,

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everybody just knew their own
valuation,

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not anybody else's valuation,
and chaos ensued,

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hardly any rules,
and miraculously almost

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instantly within less than two
minutes they figured out what to

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do and they managed to get the
football tickets into the hands

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of the people who liked them the
most.

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And I'm going to just say that
slightly more mathematically.

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You could model what everybody
did as having a utility function

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for football tickets.

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So the top person,
Mr.

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44 gets utility of 44 for
having one ticket and tickets

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beyond that don't give him any
extra utility,

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just still utility 44.

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Similarly Miss 6 there at the
bottom she had utility 6 for a

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football ticket,
but there's also money in the

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background.

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So the welfare function
depended on the football tickets

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and money and was U(X) M.

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So why does that capture what
went on?

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Because Mr.

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44, knowing that the football
ticket's worth 44,

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he would say to himself,
"Should I get a ticket or

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not get a ticket?"

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Well, if the price is under 44
the amount of money he gives up

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and loses is going to be less
than the amount of utility he

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gains for the football ticket so
therefore he'll buy the ticket.

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So this utility function U(X) M
captures the idea,

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represents the goals of the
people involved in the

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experiment.

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Each of them has a different U
of X, but all of them are of the

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form U(X) M.

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And so the conclusion was of
the experiment and of the

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theory,
supply equals demand,

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the conclusion is that the
football tickets are going to

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end up in the hands of the
people who like them the best.

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So what does that mean?

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That means that in equilibrium
the final allocation maximizes

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total welfare.

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Now what does that mean?

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Well, each person,
i, has a different utility

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function,
and so if you add up over all

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the people i you get the total
welfare of every single person,

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the economy's total welfare.

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I'm now about to prove--but it
should be obvious--that total

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welfare is maximized at that
equilibrium that you actually

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found in class,
almost.

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There was just one tiny,
tiny deviation from the theory.

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Nobody made a mistake,
by the way.

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I think I mentioned this many
times.

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I've done this experiment
before and Mr.

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12 gets so upset that he can't
buy any ticket and he's standing

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there embarrassed that everyone
else has traded and he's still

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sitting there with no football
ticket--

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he ends up bidding 30 or
something to get a football

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ticket.

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Nobody made a mistake this
time, and so it happened almost

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as the theory predicted.

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So let's just think of the
theoretical outcome.

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In the theoretical outcome
where the price is 25 and those

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top eight people have it that
final allocation maximizes total

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welfare.

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Why is that?

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Well, whatever the money
distribution was you couldn't

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change total welfare,
because if i gives up some of

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his money to her,
to j, the total amount of money

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is still the same,
and if you add up the welfares

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of all the people you're just
going to get the total amount of

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money on the right and it won't
make any difference.

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So therefore to maximize
welfare all you have to do is

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maximize the sum of--
you have to put the football

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tickets in the hands of the
people who want them the most.

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That's going to maximize
welfare, because that maximizes

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the sum of the U_i of
X's and rearranging the M's

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doesn't matter.

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So we found in
equilibrium--that equilibrium

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maximized total welfare.

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So that was the original
argument for why equilibrium was

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such a great idea.

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The greatest good to the
greatest number became

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mathematical.

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It maximizes the sum of the
welfare, so the sum utilities

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they called it.

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That was the utilitarian view
of economics,

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utilitarian view.

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And that's sort of the view
that prevailed in 1871.

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All right now,
they made one generalization

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from that--
we found before,

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which is that if you think of
not eight different buyers,

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but one buyer,
or maybe two different buyers

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where the utility functions are
this concave function,

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so they're U_I(X) M
where U_I of 1 could

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be 44 and U_I of 2
could be 44 40 which is 84.

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That consolidated person would
behave exactly like the

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individual people who'd buy as
many tickets as they would

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collectively and so nothing
would change.

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The football tickets would
still end up in the hands of the

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people who wanted them the most.

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Maybe it would be one guy,
who held three tickets,

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instead of three different
people, but the tickets would

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still be in the hands of the
highest valuation holders and so

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you would get exactly the same
conclusion.

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One last lesson from that
example is that the price turned

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out to have nothing to do with
the total value of football

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tickets.

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The price turned out to be
equal, more or less,

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to what the marginal buyer and
the marginal seller thought it

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was worth.

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So Mr.

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26 and Miss 24,
they're the ones who controlled

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the price,
what 44 thought was totally

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irrelevant,
so that was why it was called

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the marginal revolution in
economics.

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So Adam Smith who was so
puzzled because he said water is

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so valuable and has such a low
price,

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and diamonds are so useless
really when it comes down to it

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and they have such a high price
the answer to his puzzle is

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simply yes.

12:55.538 --> 12:59.628
Water, at the beginning Miss 44
would be 44,000,

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whereas if we had a same model
for diamonds,

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diamonds would also be very big
at the beginning.

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But the point is there's so
much water in the economy that

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the marginal value of an extra
gallon of water is not very

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high.

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The marginal value of water is
low even though the total value,

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which is the area under the
demand curve,

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is very high.

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So that's the lesson that we
learned in the first class.

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And now we want to generalize
this to a much more

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sophisticated model,
but one you can still compute

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very easily,
and we're going to see how

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these special assumptions don't
quite work so well.

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Yes?

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Please interrupt me at anytime
with questions.

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Student: So could you
explain again how you said that

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the welfare function and the
utility function somehow involve

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three or four people together.

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Prof: Yes.

13:58.509 --> 14:03.789
So her question is--I said very
quickly which you'll see later,

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I pressed the button too soon.

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If I press down and it's
halfway up does that just break

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the whole thing?

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Anyway, so her question is she
would like to know,

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again, why it is that I can
sort of combine people into one

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person and what do I mean by
that.

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So what I meant by that is if I
have two people,

14:29.091 --> 14:29.301
Mr.

14:29.303 --> 14:30.303
44 and Mr.

14:30.298 --> 14:35.008
40--I'm taking the buyers to be
hes and the sellers to be

14:35.009 --> 14:39.719
hers--I have two guys,
44 and 40, and I set any price.

14:39.720 --> 14:42.610
If the price is above 44
neither of them will want to

14:42.605 --> 14:42.935
buy.

14:42.940 --> 14:44.290
If it's between 44 and 40 just
Mr.

14:44.285 --> 14:45.585
40 will buy [correction:
just Mr.

14:45.591 --> 14:48.471
44],
and if it's below 40 both of

14:48.472 --> 14:51.442
them will buy,
but each of those guys is only

14:51.442 --> 14:52.772
interested in one ticket.

14:52.769 --> 14:53.089
Mr.

14:53.092 --> 14:57.622
44's utility is U_I
of 1 is 44.

14:57.620 --> 14:59.190
U_I of 2 is still 44.

14:59.190 --> 15:01.040
He doesn't get anything out of
a second ticket.

15:01.038 --> 15:06.538
Suppose now I had a third
person, a bigger person whose

15:06.543 --> 15:12.973
utility U_K of 1 is 44
and U_K of 2 is 84,44

15:12.966 --> 15:13.676
40.

15:13.678 --> 15:16.558
So that person,
now, is a bigger guy.

15:16.559 --> 15:18.379
He's interested in more tickets.

15:18.379 --> 15:19.619
That's what I mean by bigger.

15:19.620 --> 15:23.580
He's going to behave himself
exactly the same way the other

15:23.576 --> 15:26.506
two separate people behaved
collectively.

15:26.509 --> 15:30.629
So if the price is above 44 he
won't buy either ticket.

15:30.629 --> 15:34.769
If it's 50 and he buys one
ticket he'll have lost 50 and

15:34.767 --> 15:38.397
he'll have gained 44 in utils,
so he will have been worse off

15:38.399 --> 15:40.389
than when he started,
and he certainly won't buy two.

15:40.389 --> 15:41.359
He would be even worse off.

15:41.360 --> 15:44.720
If the price is between 44 and
40 he'll say to himself,

15:44.720 --> 15:48.780
this is the marginal
revolution, "If I buy the

15:48.775 --> 15:52.665
first ticket I pay 42 and I get
44 out of it,

15:52.668 --> 15:55.898
so I've gained two utils,
but now if I think about buying

15:55.898 --> 15:59.298
a second ticket for 42 and I
only get 40 out of it I'm going

15:59.298 --> 16:02.478
to start losing,
so I'll stop with one ticket.

16:02.480 --> 16:06.320
So that guy will behave exactly
the same way the two people

16:06.318 --> 16:08.568
separately behave,
so whether it's two guys

16:08.565 --> 16:10.805
separately or one guy together
it doesn't make any difference.

16:10.808 --> 16:13.108
Their total action's exactly
the same.

16:13.110 --> 16:16.350
And in the end the football
tickets have ended in the hands

16:16.346 --> 16:18.576
of the people who liked them the
most.

16:18.580 --> 16:21.230
Maybe I need a lot more water
than you do,

16:21.230 --> 16:27.150
but if I need it much more than
you do then all those gallons,

16:27.149 --> 16:31.109
maybe the first twenty gallons
I drank I needed more than your

16:31.107 --> 16:35.197
first gallon and after that you
started needing water as much as

16:35.196 --> 16:36.646
I did,
something like that.

16:36.649 --> 16:41.159
So the tickets or the water
ends up in the hands of whoever

16:41.158 --> 16:44.338
needs it most,
and it may be that there is

16:44.344 --> 16:47.924
more than one unit in each
person's hands.

16:47.918 --> 16:52.818
Any other questions before I
raise the thing again?

16:52.820 --> 16:55.160
All right, so that sounds all
very convincing,

16:55.159 --> 16:58.229
but it's not going to turn out
to be quite so convincing.

16:58.230 --> 17:01.510
So let's try and generalize
this model to a more

17:01.505 --> 17:03.035
sophisticated thing.

17:03.038 --> 17:05.608
And so I'm following an example
which is in the notes.

17:05.608 --> 17:12.688
So you find that if you read
the notes--oh,

17:12.688 --> 17:15.888
so the textbooks.

17:15.890 --> 17:19.220
Because the approach I take and
that we take a Yale is quite

17:19.224 --> 17:22.504
different than the standard
approach you're not going to be

17:22.503 --> 17:24.203
able to follow a textbook.

17:24.200 --> 17:26.130
That's why I give you a whole
list of textbooks.

17:26.130 --> 17:27.130
I encourage you to read them.

17:27.130 --> 17:27.770
They're great books.

17:27.769 --> 17:28.569
They're famous people.

17:28.568 --> 17:32.058
Most of them are quite good
friends of mine so I endorse

17:32.057 --> 17:34.307
them all,
but they're differently

17:34.310 --> 17:38.200
presented than this course and
that's why you need to rely on

17:38.202 --> 17:39.892
the notes a little bit.

17:39.890 --> 17:47.550
So let's just take this first
example.

17:47.548 --> 17:50.698
Suppose now that we have two
goods, but they're going to be

17:50.702 --> 17:51.412
continuous.

17:51.410 --> 17:54.470
You don't have to have just one
football ticket or two football

17:54.474 --> 17:54.974
tickets.

17:54.970 --> 17:58.020
We have two goods which we're
going to call X and Y,

17:58.018 --> 18:00.288
and we've got two agents,
A & B.

18:00.288 --> 18:06.148
And so W^(A) of X and Y,
that's the welfare function

18:06.146 --> 18:11.766
like we had before,
is going to be whatever I had,

18:11.773 --> 18:15.223
100X - 1 half X squared Y.

18:15.220 --> 18:19.710
And now the endowments of
goods, which I was a little bit

18:19.711 --> 18:28.231
fast and loose about before,
E^(A) of X, E^(A) of Y,

18:28.230 --> 18:31.630
that's the endowment of A,
how much he has to begin with

18:31.634 --> 18:32.444
of X and Y.

18:32.440 --> 18:37.600
I say it's 4 and 5,000.

18:37.598 --> 18:39.148
And then let's make another
person,

18:39.150 --> 18:47.140
W^(B), his welfare function or
let's say his and her welfare

18:47.141 --> 18:52.561
function is 30X - 1 half X
squared Y,

18:52.558 --> 19:09.088
and her endowment E^(B) of X
and E^(B) of Y equals 80 and

19:09.087 --> 19:11.447
1,000.

19:11.450 --> 19:15.280
So this is supposed to be
shorthand for an economy in

19:15.284 --> 19:18.164
which there are thousands of
people,

19:18.160 --> 19:20.930
millions of people,
every person characterized by

19:20.932 --> 19:24.182
the utility they get,
the goals they have over the

19:24.180 --> 19:27.120
consumption goods,
and their starting endowments,

19:27.124 --> 19:30.244
and they're all going to be
thrown together and expected to

19:30.244 --> 19:30.734
trade.

19:30.730 --> 19:35.760
Another example,
we can do another example,

19:35.761 --> 19:37.201
let's say.

19:37.200 --> 19:39.860
We're going to work out both of
these, the same kind of examples

19:39.855 --> 19:41.705
you're going to the do in the
problem set.

19:41.710 --> 19:45.120
Another example,
I'm sorry I can't remember my

19:45.116 --> 19:45.946
examples.

19:45.950 --> 19:54.740
I won't need to look at these
after I write the examples down.

19:54.740 --> 20:08.770
So another one is W^(C)(X,
Y) = 3 quarters log X 1 quarter

20:08.770 --> 20:17.630
log Y, and E^(C),
the endowment of C,

20:17.632 --> 20:22.312
equals 2 and 1.

20:22.308 --> 20:31.738
And meanwhile I have another
person D of X and Y whose

20:31.740 --> 20:37.080
endowment [correction:
utility]

20:37.078 --> 20:47.398
is 2 thirds log X 1 third log Y
and her endowment 1,2.

20:47.400 --> 20:50.150
And of course I could have had
an economy in which they were

20:50.150 --> 20:52.710
all there at the same time,
but I'm just going to do two

20:52.713 --> 20:53.323
examples.

20:53.318 --> 20:55.788
So you see the economy consists
of many people,

20:55.788 --> 20:58.588
many goods maybe,
many people and each with

20:58.585 --> 21:01.775
different endowments and
different utilities,

21:01.778 --> 21:04.338
and if you throw them all
together what's going to happen?

21:04.338 --> 21:07.688
And so we have a theory now,
a theory of equilibrium that

21:07.692 --> 21:09.132
explains what happens.

21:09.130 --> 21:12.680
And we can use a few tricks,
which I'm going to teach you

21:12.675 --> 21:14.355
now,
to actually solve concretely

21:14.362 --> 21:16.672
for what's going to happen in
each of these cases,

21:16.670 --> 21:18.000
and it's very simple.

21:18.000 --> 21:21.330
And the next step is going to
be to add finance to it and

21:21.330 --> 21:24.370
financial variables,
but at the bottom we still want

21:24.365 --> 21:26.205
to have economic variables.

21:26.210 --> 21:29.380
See here we've got the
consumption of two different

21:29.375 --> 21:33.045
goods X and Y and we want to see
what's going to happen.

21:33.048 --> 21:40.658
All right, so equilibrium is
always defined by turning things

21:40.662 --> 21:42.822
into equations.

21:42.818 --> 21:46.188
So we said the equations here
are going to be that.

21:46.190 --> 21:51.630
So what is A going to do?

21:51.630 --> 22:03.850
Well, the endogenous variables
are going to be P_X,

22:03.847 --> 22:09.347
P_Y,
X^(A), Y^(A),

22:09.346 --> 22:13.416
X^(B) and Y^(B).

22:13.420 --> 22:15.010
That's what everybody has to
decide.

22:15.009 --> 22:19.159
In the end A has to decide,
the prices have to emerge for X

22:19.156 --> 22:19.726
and Y.

22:19.730 --> 22:21.370
We're assuming,
again, that these people,

22:21.368 --> 22:23.568
by the way--I have one agent A
and one agent B,

22:23.568 --> 22:26.768
I really mean there's a million
agents just like A and a million

22:26.773 --> 22:29.783
agents just like B and they're
all shouting and screaming at

22:29.776 --> 22:32.316
each other and they're in some
kind of market.

22:32.318 --> 22:35.658
So if there's only one agent of
each type there'd be bargaining

22:35.659 --> 22:38.029
and threats and it'd be very
complicated,

22:38.029 --> 22:39.749
but with lots of people of each
type,

22:39.750 --> 22:42.790
that's what I'm talking
about--so in our football ticket

22:42.786 --> 22:45.986
example there were sixteen
people competing with each other

22:45.988 --> 22:48.858
and you don't really need much
more than three,

22:48.858 --> 22:51.818
or four, or five on a side,
at least four,

22:51.819 --> 22:54.869
to get competition.

22:54.868 --> 22:58.468
So with enough competition the
theory says the prices are going

22:58.469 --> 23:02.069
to emerge and people are going
to look at the prices and decide

23:02.067 --> 23:03.807
how much they want to buy.

23:03.808 --> 23:05.288
What do they want to end up
consuming?

23:05.288 --> 23:08.628
So A has to make his decisions
and B has to make her decisions.

23:08.630 --> 23:11.470
And so those are the endogenous
variables.

23:11.470 --> 23:17.050
The exogenous variables were
all of the 80,1,000,1 half,

23:17.047 --> 23:21.507
1 times y, 100 times x,
all those numbers are

23:21.509 --> 23:22.929
exogenous.

23:22.930 --> 23:24.950
The utility functions,
the endowments,

23:24.953 --> 23:25.833
are exogenous.

23:25.828 --> 23:31.588
So these are all the exogenous
things.

23:31.588 --> 23:35.598
So the theory is going to say,
how do you go from exogenous to

23:35.598 --> 23:39.608
endogenous and it's going to be
just a bunch of equations,

23:39.609 --> 23:41.699
so what do they each want to do?

23:41.700 --> 23:51.950
So A is going to maximize W^(A)
of X and Y such that,

23:51.950 --> 23:56.380
as we said, the critical
insight, the budget constraint--

23:56.380 --> 23:59.180
[clarification:
here, writes but does not say

23:59.175 --> 24:02.735
out loud P_X X]
P_Y Y is less than or

24:02.736 --> 24:06.486
equal to P_X times
E^(A)_X P_Y

24:06.486 --> 24:09.666
times E^(A)_Y),
but we know what these numbers

24:09.667 --> 24:09.857
are.

24:09.858 --> 24:18.928
E^(A)_X is 4 and
E^(A)_Y is 5,000.

24:18.930 --> 24:22.920
So A takes it for
granted--theory says this,

24:22.920 --> 24:26.700
it's very shocking--but it says
A takes it for granted that he

24:26.704 --> 24:29.564
can sell all his endowment if he
wanted to,

24:29.558 --> 24:33.808
4 units of X and 5,000 units of
Y and get the money from doing

24:33.805 --> 24:38.045
that and use the money to spend
on his final consumptions--

24:38.048 --> 24:42.578
let's call this X^(A) and
Y^(A)--

24:42.579 --> 24:45.379
of X^(A) and Y^(A).

24:45.380 --> 24:50.320
So let's leave out the A's here
for a minute because those are

24:50.318 --> 24:52.018
the choices he has.

24:52.019 --> 24:55.169
He's wants to max over X and Y,
so there are many

24:55.170 --> 24:56.220
possibilities.

24:56.220 --> 24:58.680
It has to satisfy this budget
constraint.

24:58.680 --> 25:06.610
And similarly Y is going to be
maximizing over X and Y,

25:06.608 --> 25:12.088
W^(B) of X, Y such that
P_X X P_Y Y

25:12.086 --> 25:16.446
less than or equal to
P_X times 80

25:16.446 --> 25:19.486
P_Y times 1,000.

25:19.490 --> 25:21.920
I think I remembered the
numbers finally.

25:21.920 --> 25:27.390
So, and now what we want to do
is we want to solve for these

25:27.392 --> 25:30.882
variables so that when A,
taking P_X and

25:30.875 --> 25:33.705
P_Y as given,
maximizes his utility function,

25:33.710 --> 25:38.430
he'll choose X^(A) and X^(B),
and B will choose Y^(A) and

25:38.428 --> 25:42.378
Y^(B) such that demand equals
supply.

25:42.380 --> 25:47.440
And so I'm going to write,
over here, maybe,

25:47.442 --> 25:50.152
demand equals supply.

25:50.150 --> 25:54.470
So we know that in the end so
whatever these choices are

25:54.472 --> 25:59.192
they're going to lead to him
choosing X^(A) and Y^(A) and her

25:59.190 --> 26:01.470
choosing X^(B) and Y^(B).

26:01.470 --> 26:08.370
And it's got to be that X^(A)
X^(B) = E^(A)_X

26:08.368 --> 26:15.268
E^(A)_X which equals 4
80 which equals 84.

26:15.269 --> 26:23.529
And it's got to be that Y^(A)
Y^(B) has to equal

26:23.526 --> 26:31.776
e^(A)_Y
e^(B)_Y which equals

26:31.782 --> 26:38.292
5,000 1,000,
which equals 6,000.

26:38.288 --> 26:39.508
I hope I've remembered
everything.

26:39.509 --> 26:42.869
So all right,
so those are two of the

26:42.868 --> 26:43.988
equations.

26:43.990 --> 26:47.300
Supply has to equal demand,
and now let's just do a little

26:47.296 --> 26:50.076
trick here to get some of the
other equations.

26:50.078 --> 26:51.748
A is going to spend all his
money.

26:51.750 --> 26:54.200
What's the point in not
spending money--because the more

26:54.202 --> 26:56.832
X he has and the more Y he has,
certainly the more Y he has,

26:56.834 --> 26:57.954
the better off he is.

26:57.950 --> 26:59.300
So he's not going to waste
money.

26:59.298 --> 27:03.178
So this is going to turn out to
be an equality here.

27:03.180 --> 27:06.700
Okay, so that's actually an
equation, not just a variable.

27:06.700 --> 27:13.120
So P_X times X^(A)
(now the actual solution)

27:13.118 --> 27:20.268
P_Y times Y^(A) has to
equal P_X times 4

27:20.266 --> 27:23.896
P_Y times 5,000.

27:23.900 --> 27:26.160
That's an equation,
and then similarly Y,

27:26.163 --> 27:28.263
she's not going to waste her
money.

27:28.259 --> 27:30.409
She's going to spend it all it
she's optimizing,

27:30.407 --> 27:32.097
so this will turn into an
equality.

27:32.098 --> 27:37.008
And so this will give
P_X times X^(B)

27:37.011 --> 27:42.251
P_Y times Y^(B) =
P_X times 80

27:42.246 --> 27:45.446
P_Y times 1,000.

27:45.450 --> 27:49.040
So we've got four equations,
and now we have to do the

27:49.037 --> 27:52.487
marginal equation,
the crucial marginal equation.

27:52.490 --> 27:56.330
So what does that say?

27:56.329 --> 27:57.819
We talked about this last time.

27:57.818 --> 27:59.638
You've all seen it before so I
can go quickly,

27:59.644 --> 28:01.714
but this was the critical
insight that took years to

28:01.712 --> 28:02.202
develop.

28:02.200 --> 28:03.710
Marx couldn't figure it out.

28:03.710 --> 28:06.470
Until his dying day he was
trying to understand what these

28:06.468 --> 28:07.918
marginalist guys were doing.

28:07.920 --> 28:15.300
So the idea is that if you've
optimized by choosing X^(A) and

28:15.301 --> 28:17.761
Y^(A),
if he's optimized choosing

28:17.762 --> 28:19.832
X^(A) and Y^(A),
it has to be that the last

28:19.827 --> 28:22.197
dollar he spent--
he was indifferent between

28:22.198 --> 28:23.358
where he spent it.

28:23.358 --> 28:25.748
Otherwise he would have moved a
dollar from one thing to the

28:25.751 --> 28:26.321
other thing.

28:26.318 --> 28:31.828
So it has to be that the
marginal utility of X at X^(A)

28:31.827 --> 28:37.437
and Y^(A) divided by the price
of X, so what is that?

28:37.440 --> 28:40.760
What is the marginal utility of
X?

28:40.759 --> 28:45.209
That's the derivative of X,
100 - 2 times 1 half,

28:45.210 --> 28:59.700
100 - X, has to equal the
marginal utility to A of Y--

28:59.700 --> 29:07.010
divided by the--sorry,
I meant to leave room here.

29:07.009 --> 29:15.119
Equals the marginal utility of
A of Y evaluated at X^(A) and

29:15.118 --> 29:19.928
Y^(A), divided by the price of
Y.

29:19.930 --> 29:28.190
So that equals 1.

29:28.190 --> 29:30.080
Marginal utility of Y is just 1.

29:30.079 --> 29:34.069
The derivative of Y is 1.

29:34.068 --> 29:36.828
And then we have to write the
same thing for B.

29:36.828 --> 29:47.138
The marginal utility of X for
X^(B) and Y^(B) divided by the

29:47.137 --> 29:49.407
price of X.

29:49.410 --> 29:51.660
So what is for B?

29:51.660 --> 29:58.940
It's (30 - X) divided by
P_X.

29:58.940 --> 30:02.650
Not very good board management.

30:02.650 --> 30:06.370
Has to equal,
and this is also going to turn

30:06.367 --> 30:11.637
out to be 1 over P_Y
equals marginal utility of B of

30:11.644 --> 30:14.484
Y,
at X^(B), Y^(B),

30:14.478 --> 30:17.818
all over P_Y.

30:17.819 --> 30:19.389
So those are the equations.

30:19.390 --> 30:22.860
Now, does that make sense to
everybody?

30:22.858 --> 30:24.828
I think I need to pause for a
minute.

30:24.828 --> 30:30.238
I'm going to do exactly the
same thing with that other

30:30.237 --> 30:35.847
system, but let's just see if we
can figure this out.

30:35.848 --> 30:38.628
So equilibrium is this very
involved thing.

30:38.630 --> 30:41.100
What everybody does depends on
what everybody else is going to

30:41.102 --> 30:43.612
do because how much should you
pay for something depends on how

30:43.614 --> 30:46.334
much you think you can get it by
offering it to some other guy.

30:46.328 --> 30:47.968
If there are a million A's and
a million B's,

30:47.972 --> 30:50.292
you're dealing with one of the
B's, maybe the other B will give

30:50.288 --> 30:51.108
you a better deal.

30:51.108 --> 30:53.248
So you have to think about what
the other people are doing

30:53.253 --> 30:54.913
before you can decide what to do
yourself.

30:54.910 --> 30:58.090
All that is captured by the
idea of the prices.

30:58.088 --> 31:01.008
Somehow people get into their
minds what the best deal they

31:01.013 --> 31:01.673
can get is.

31:01.670 --> 31:03.420
That's the prices,
P_X and

31:03.420 --> 31:04.220
P_Y.

31:04.220 --> 31:06.420
Given those prices,
A, each agent,

31:06.416 --> 31:10.406
looks as his budget set or her
budget set and decides what to

31:10.413 --> 31:10.883
do.

31:10.880 --> 31:12.070
And what should they do?

31:12.068 --> 31:16.068
They should equate marginal
utilities.

31:16.069 --> 31:17.439
That's the key insight.

31:17.440 --> 31:21.160
The marginal utility per dollar
of X has to equal the marginal

31:21.163 --> 31:22.753
utility per dollar of Y.

31:22.750 --> 31:26.240
That just says that the budget
set is tangent to the

31:26.241 --> 31:27.681
indifference curve.

31:27.680 --> 31:28.900
That's what that says.

31:28.900 --> 31:31.700
So you take the ratio of
marginal utilities--it equals

31:31.698 --> 31:32.858
the ratio of prices.

31:32.858 --> 31:35.638
And cross multiplying,
it says the marginal utility

31:35.641 --> 31:39.181
per dollar,
the slope of the indifference

31:39.176 --> 31:43.886
curve is marginal utility,
let's say, of X over marginal

31:43.887 --> 31:48.127
utility of Y and the slope of
the budget set is P_X

31:48.125 --> 31:49.675
over P_Y.

31:49.680 --> 31:52.140
So if I just put the
P_X down here and the

31:52.144 --> 31:55.044
marginal utility up there that
just says the marginal utility

31:55.044 --> 31:57.994
of X divided by P_X
equals the marginal utility of Y

31:57.990 --> 31:59.490
divided by P_Y.

31:59.490 --> 32:04.340
That's something that you could
waste a huge amount of time on.

32:04.338 --> 32:08.578
I don't have to do it because I
know that you all have seen it

32:08.576 --> 32:12.536
before, and the one guy who
hasn't seen it before is going

32:12.535 --> 32:14.545
to figure it out himself.

32:14.548 --> 32:20.288
So we have a tremendous
advantage here.

32:20.288 --> 32:22.688
I can just skip over that
immediately and make use of that

32:22.693 --> 32:22.993
fact.

32:22.990 --> 32:24.400
So that's the critical insight.

32:24.400 --> 32:27.300
You've taken this incredibly
complicated system and reduced

32:27.298 --> 32:30.398
it to a bunch of equations which
you can put on a computer,

32:30.400 --> 32:31.750
which is about--what I'm about
to do,

32:31.750 --> 32:35.830
and solve it with a flick of a
button.

32:35.828 --> 32:41.708
So are there any questions
here--let me pause again--with

32:41.710 --> 32:44.650
how I got these equations?

32:44.650 --> 32:47.650
So it's a little bit
complicated, but of course once

32:47.652 --> 32:50.422
you've understood it it's not so
complicated.

32:50.420 --> 32:52.970
Now, who first thought of all
this stuff?

32:52.970 --> 32:54.670
The amazing thing is,
incidentally,

32:54.670 --> 32:56.720
these equations always have a
solution.

32:56.720 --> 32:59.770
If you take typical equations
in any field,

32:59.772 --> 33:02.902
physics, mathematics,
just random equations,

33:02.898 --> 33:05.588
they're not going to be
solvable.

33:05.588 --> 33:08.638
X squared 1 = 0,
that's just one equation,

33:08.636 --> 33:10.566
doesn't have a solution.

33:10.568 --> 33:13.428
And if you have simultaneous
equations why should there be a

33:13.426 --> 33:13.956
solution?

33:13.960 --> 33:17.380
The economic system always has
a solution.

33:17.380 --> 33:22.590
This is an astonishing fact
first proved by Arrow,

33:22.588 --> 33:26.608
my thesis advisor,
Debreu who did it at Yale as an

33:26.605 --> 33:29.645
assistant professor,
and didn't get tenure,

33:29.646 --> 33:32.966
and later won the Nobel Prize,
which has happened several

33:32.970 --> 33:36.020
times at Yale--
Arrow, Debreu,

33:36.019 --> 33:41.759
and McKenzie all separately,
although these two guys ended

33:41.757 --> 33:45.247
up writing a joint paper,
anyway, they found that this

33:45.250 --> 33:47.630
system always has a solution.

33:47.630 --> 33:50.330
There's something special about
the economic system that has a

33:50.328 --> 33:53.028
solution that has to do with
diminishing marginal utility,

33:53.029 --> 33:54.489
which we're not going to talk
about in this class,

33:54.490 --> 33:56.130
but it's quite a fascinating
thing.

33:56.130 --> 34:00.730
And they based their argument
on an argument that Nash had

34:00.728 --> 34:02.178
given for games.

34:02.180 --> 34:05.530
And this whole thing is very
related to Nash equilibrium.

34:05.528 --> 34:08.788
And I'm sure you've heard of
Nash and many of you have maybe

34:08.789 --> 34:11.219
seen the movie,
A Beautiful Mind.

34:11.219 --> 34:15.699
Well, about five years ago,
a couple of years after the

34:15.697 --> 34:19.327
movie came out,
Nash is still very much alive

34:19.333 --> 34:22.443
and not quite as wacky as he
used to be,

34:22.440 --> 34:27.780
and so the Indian Game Theory
Society opened.

34:27.780 --> 34:31.350
It was founded believe it not,
just five years ago despite all

34:31.349 --> 34:33.339
the brilliant Indian economists.

34:33.340 --> 34:37.110
The Game Theory Society was
founded about five years ago and

34:37.110 --> 34:40.760
they had an opening conference
where six people gave talks

34:40.755 --> 34:41.965
including Nash.

34:41.969 --> 34:44.019
I was one of the people who
gave a talk,

34:44.018 --> 34:47.278
and there were thousands of
people who showed up,

34:47.280 --> 34:49.530
mostly because of the movie,
I mean,

34:49.530 --> 34:52.250
there was just thousands and
thousands of people.

34:52.250 --> 34:55.950
So afterwards we went on tour,
traveling to a bunch of

34:55.945 --> 34:58.895
different cities,
and every city we went to we'd

34:58.904 --> 35:01.724
get off the train or the
limousine or something there'd

35:01.719 --> 35:04.799
be a throng of people there
waiting to meet Nash and there'd

35:04.797 --> 35:06.567
always be a press conference.

35:06.570 --> 35:10.050
And after the press conference
there'd be a picture on the

35:10.054 --> 35:13.304
front page of whatever city,
and these were all great

35:13.304 --> 35:16.434
cities, a city we'd gone to,
and always there was Nash and

35:16.434 --> 35:18.764
everybody else was cropped out
of the picture.

35:18.760 --> 35:21.840
But anyway, in one of these
first conferences,

35:21.840 --> 35:25.370
I'm just illustrating Nash
equilibrium here,

35:25.369 --> 35:27.719
somebody said,
some reporter says,

35:27.715 --> 35:31.505
"We've seen the movie,
but can you really tell us in a

35:31.507 --> 35:34.227
word what is Nash equilibrium,
competitive equilibrium,

35:34.233 --> 35:35.883
just say in a word what does it
mean,

35:35.880 --> 35:37.530
what does it mean for us?

35:37.530 --> 35:42.650
And so each of us took a try at
trying to explain what Nash

35:42.654 --> 35:45.574
equilibrium was including Nash.

35:45.570 --> 35:48.650
It didn't go too well,
the explanations,

35:48.648 --> 35:50.778
until they got to Aumann.

35:50.780 --> 35:53.180
So he was also one of the
people who spoke,

35:53.184 --> 35:55.594
and he subsequently won the
Nobel Prize.

35:55.590 --> 35:58.290
But anyway, at the time he
hadn't won it yet and he's

35:58.289 --> 35:58.809
Israeli.

35:58.809 --> 36:00.729
He's also a great figure.

36:00.730 --> 36:04.380
And so Aumann says,
"That question reminds

36:04.378 --> 36:06.698
me,"--
I can't do his Israeli

36:06.697 --> 36:09.977
accent--"that question
reminds me of the first press

36:09.976 --> 36:13.166
conference Khrushchev"--
who you might remember was

36:13.168 --> 36:14.638
Premier of the Soviet Union.

36:14.639 --> 36:18.149
This was in the time of Kennedy
and thumping the table and the

36:18.152 --> 36:20.562
Cold War and stuff--
"the first press

36:20.559 --> 36:23.759
conference Khrushchev gave to
western reporters and somebody

36:23.760 --> 36:26.130
said,
'Can you tell me in a word,

36:26.134 --> 36:30.114
describe in a word the health
of the Russian economy,' and

36:30.112 --> 36:32.782
Khrushchev says,
'Good.'

36:32.780 --> 36:36.090
And then the reporter says,
'I didn't really mean one word.

36:36.090 --> 36:38.590
Take two words and tell us,
what is the health of the

36:38.594 --> 36:39.514
Russian economy?'

36:39.510 --> 36:45.830
And Khrushchev says,
'Not good.'"

36:45.833 --> 36:51.943
So Aumann says,
"Equilibrium in one word

36:51.942 --> 36:54.572
is interaction,
in two words,

36:54.572 --> 36:56.412
rational interaction."

36:56.409 --> 37:00.149
So his definition managed to
get into the newspapers and none

37:00.146 --> 37:01.016
of ours did.

37:01.018 --> 37:02.648
So that pretty much summarizes
it.

37:02.650 --> 37:05.450
It's interaction,
but rational interaction.

37:05.449 --> 37:08.739
So, and it's captured by the
idea that everybody anticipates

37:08.739 --> 37:11.969
the prices and those prices are
going to really lead to the

37:11.972 --> 37:13.202
market's clearing.

37:13.199 --> 37:17.789
So they're all anticipating the
right prices and behaving as

37:17.786 --> 37:21.516
optimally as they can,
choosing the best thing in

37:21.518 --> 37:23.228
their budget sets.

37:23.230 --> 37:28.010
I put this on a computer and
solved it,

37:28.010 --> 37:29.640
which we're going to do in a
second,

37:29.639 --> 37:32.199
but there's a trick to solving
this by hand,

37:32.199 --> 37:35.139
so I might as well just do the
tricks by hand because on an

37:35.135 --> 37:36.245
exam,
for example,

37:36.248 --> 37:38.318
I'm not going to be able to
give--

37:38.320 --> 37:40.360
you're going to use the
computer, it's very simple.

37:40.360 --> 37:43.230
You'll see in one minute you
can solve this on a computer,

37:43.230 --> 37:48.230
but by hand it's worth knowing
how to do and you probably know

37:48.233 --> 37:51.263
how to do this,
but let me describe it.

37:51.260 --> 37:59.800
So the first thing to observe
is that the prices don't really

37:59.795 --> 38:04.345
matter up to scalar multiples.

38:04.349 --> 38:06.959
Walras, by the way,
was the first who made this

38:06.963 --> 38:07.593
argument.

38:07.590 --> 38:13.460
So Walras was one of the
marginalists in 1871 from

38:13.460 --> 38:14.780
Lausanne.

38:14.780 --> 38:16.960
So he says, "Look,
doubling the prices isn't going

38:16.961 --> 38:17.651
to do anything.

38:17.650 --> 38:19.550
It's just like changing dollars
into cents."

38:19.550 --> 38:21.540
If you look at everybody's
budget set and double

38:21.539 --> 38:24.079
P_X and P_Y
you're doubling both sides of

38:24.079 --> 38:24.799
the equation.

38:24.800 --> 38:25.860
You're not doing anything.

38:25.860 --> 38:27.970
So if P_X and
P_Y are part of an

38:27.969 --> 38:29.419
equilibrium,
2P_X and

38:29.422 --> 38:32.292
2P_Y will also be part
of the equilibrium because the

38:32.288 --> 38:35.148
prices only appear here in the
budget set and doubling them all

38:35.154 --> 38:36.314
doesn't do anything.

38:36.309 --> 38:45.099
So really you might as well
assume that P_X equals

38:45.103 --> 38:45.703
1.

38:45.699 --> 38:47.679
So he says, "Well,
that gets rid of one variable.

38:47.679 --> 38:50.769
You've got six variables and
six equations so you can all

38:50.771 --> 38:54.031
solve them, but there's so many
it seems too complicated.

38:54.030 --> 38:57.380
But now you got rid of one
variable, well you can also get

38:57.382 --> 38:59.562
rid of one equation,"
he says.

38:59.559 --> 39:01.169
So how can you get rid of one
equation?

39:01.170 --> 39:04.910
Well, suppose we clear the X
market.

39:04.909 --> 39:10.159
We find X^(A), X^(B),
Y^(A) and Y^(B) and

39:10.163 --> 39:13.033
P_X and P_Y
and all the equations are

39:13.032 --> 39:15.242
satisfied,
one through five.

39:15.239 --> 39:18.599
All these equations are
satisfied, one,

39:18.596 --> 39:22.656
three, four,
five and six are all satisfied.

39:22.659 --> 39:25.659
We haven't checked equation two
though, whether that market's

39:25.659 --> 39:26.509
going to clear.

39:26.510 --> 39:28.660
And Walras said,
"Well, it has to clear.

39:28.659 --> 39:33.259
The last market we don't need
to worry about."

39:33.260 --> 39:34.700
Why is that?

39:34.699 --> 39:39.969
Because if X^(A) X^(B) = E^(A)
E^(B) that means collectively

39:39.969 --> 39:45.149
all the agents are spending on
good X exactly all the money

39:45.148 --> 39:50.148
that they're collectively
getting by selling good X.

39:50.150 --> 39:52.750
That's what the top equation
says because when you multiply

39:52.753 --> 39:55.273
through the whole thing by
P_X the total amount

39:55.268 --> 39:58.138
people are spending on good X is
equal to the total amount agents

39:58.143 --> 40:00.123
are getting by selling all the
good X.

40:00.119 --> 40:03.349
So since everybody's spending
all their money that must mean

40:03.349 --> 40:06.249
the rest of their money
collectively is just all their

40:06.251 --> 40:08.771
money they're getting from
selling good Y.

40:08.768 --> 40:13.528
They must be spending it all
collectively on buying good Y.

40:13.530 --> 40:16.100
That means the next equation
automatically has to hold

40:16.099 --> 40:19.149
because everybody spent all his
money so therefore all the money

40:19.152 --> 40:21.722
collectively that was spent on
good X equals to what's

40:21.721 --> 40:24.051
purchased [correction:
revenue received in sales]

40:24.047 --> 40:26.857
of good X because supply equals
demand for good X.

40:26.860 --> 40:29.550
So good Y it has to be that all
the people, the income that

40:29.552 --> 40:31.782
they're getting on spending
[correction; selling]

40:31.782 --> 40:34.572
good Y, all of that was spent
on buying Y collectively,

40:34.570 --> 40:36.680
not any person,
each person that's selling Y

40:36.684 --> 40:39.294
and buying X or something,
but collectively all the money

40:39.289 --> 40:41.299
we've just deduced spent on
[correction: received by

40:41.295 --> 40:42.745
selling]
Y had to go to buying Y,

40:42.750 --> 40:45.030
so therefore the Y market is
clearing too.

40:45.030 --> 40:48.050
So once you've cleared all the
other markets you know that the

40:48.052 --> 40:49.392
last market has to clear.

40:49.389 --> 40:57.289
So without loss of generality
don't worry about last market.

40:57.289 --> 41:00.739
So that reduced it to five
equations and five unknowns,

41:00.735 --> 41:01.815
so that helped.

41:01.820 --> 41:06.370
We got rid of one equation and
we got rid of one unknown.

41:06.369 --> 41:08.949
So we got rid of the top
equation, let's say,

41:08.954 --> 41:10.134
and P_X.

41:10.130 --> 41:13.990
One of those two equations,
the market clearing equations,

41:13.994 --> 41:17.114
doesn't matter as long as we do
all the others,

41:17.112 --> 41:19.962
and one of the prices we can
fix at 1.

41:19.960 --> 41:23.930
So as we can fix P_Y,
let's say, at 1,

41:23.931 --> 41:27.481
we might as well fix
P_Y at 1.

41:27.480 --> 41:29.560
This becomes a much simpler
equation.

41:29.559 --> 41:33.739
This now I can replace with 1,
and this I can replace with 1.

41:33.739 --> 41:38.019
We already know what the price
is of Y, it's 1.

41:38.018 --> 41:43.818
But now things get very,
very simple because you have

41:43.818 --> 41:50.288
(100-X) over P_X = 1,
so I just write that again,

41:50.286 --> 41:54.186
(100-X) over P_X = 1.

41:54.190 --> 42:01.240
So I bring the P_X to
the other side and I have 100-X

42:01.244 --> 42:05.914
= this is X^(A)-- equals
P_X.

42:05.909 --> 42:13.549
Another way of writing that is
X^(A) = 100-P_X.

42:13.550 --> 42:23.150
Then from this bottom equation
I've got 30-P_X--30

42:23.150 --> 42:26.730
minus--X^(B),
sorry.

42:26.730 --> 42:29.230
These are A's and this was B.

42:29.230 --> 42:31.360
I forgot the superscript.

42:31.360 --> 42:35.760
So (30-X^(B)) over
P_X) has to

42:35.764 --> 42:42.094
equal--well, the marginal
utility is 1 and the price is 1

42:42.088 --> 42:44.458
so that equals 1.

42:44.460 --> 42:48.200
So I just have 30-X^(B) =
P_X,

42:48.199 --> 42:52.169
or, in other words I have X^(B)
(just writing this--

42:52.170 --> 43:00.310
bring it to the other side) =
30-P_X.

43:00.309 --> 43:01.449
So you look at the demand.

43:01.449 --> 43:02.639
This is what Walras did.

43:02.639 --> 43:04.859
He said, "Forget about all
these equations just look at

43:04.858 --> 43:06.738
demand and see where demand
equals supply."

43:06.739 --> 43:09.109
So here, given the price
P_X and P_Y

43:09.108 --> 43:11.518
we know without loss of
generality P_Y is 1.

43:11.518 --> 43:16.838
So given P_X this is
how much A is going to demand of

43:16.835 --> 43:17.175
X.

43:17.179 --> 43:20.249
And given P_X this is
how much B is going to demand.

43:20.250 --> 43:29.740
And we know in equilibrium by
that top equation that plus that

43:29.739 --> 43:32.539
has to equal 84.

43:32.539 --> 43:33.999
So now I can solve it.

43:34.000 --> 43:39.450
So I know that 100-P--well,
I'm just going to solve it

43:39.451 --> 43:40.481
quickly.

43:40.480 --> 43:50.760
So it's 130-2P_X =
84, which implies that

43:50.755 --> 44:02.465
46,2P_X = 46 implies
that P_X = 23.

44:02.469 --> 44:08.049
Once you have P_X =
23 then you can figure out what

44:08.045 --> 44:13.185
X^(A) is,
because 100-X^(A) has to be

44:13.193 --> 44:22.163
P_X so that implies
that X^(A) is 23 and it implies

44:22.159 --> 44:28.369
that X^(B) is--
no, X^(A) is 77,

44:28.373 --> 44:33.443
right, where is X^(A)?

44:33.440 --> 44:37.380
X^(A) is 100-P_X,
so if P_X is 23 X^(A)

44:37.380 --> 44:38.400
has to be 77.

44:38.400 --> 44:41.830
It implies that X^(B),
I can do X^(B) from over here.

44:41.829 --> 44:48.029
30-P_X is 7,
and sure enough 77 7 really do

44:48.027 --> 44:49.337
equal 84.

44:49.340 --> 44:52.010
So we've cleared the top market.

44:52.010 --> 44:54.010
And now we don't have to worry
about the other market.

44:54.010 --> 44:55.560
We can figure out what X^(B) is.

44:55.559 --> 44:56.889
How do we figure out what X^(B)
is?

44:56.889 --> 44:58.559
We go into this budget set.

44:58.559 --> 45:02.989
At a price of 23 he's going to
consume 77.

45:02.989 --> 45:04.209
That's going to cost a bunch of
money,

45:04.210 --> 45:05.690
and this is how much income he
has,

45:05.690 --> 45:07.530
and we subtract if off,
and P_Y is 1,

45:07.530 --> 45:09.240
we can figure out what Y is.

45:09.239 --> 45:15.009
So we can figure out from this
Y^(A) and Y^(B),

45:15.009 --> 45:21.159
and we know that that's going
to clear the market,

45:21.155 --> 45:25.165
so we've solved the problem.

45:25.170 --> 45:29.980
But we can do this on a
computer.

45:29.980 --> 45:31.530
All right, so are there any
questions how I did this?

45:31.530 --> 45:34.350
I'm going to do it one more
time with this model and then

45:34.347 --> 45:36.107
I'm going to do it on a
computer.

45:36.110 --> 45:41.110
And so this is the kind of
problem that hopefully will be

45:41.108 --> 45:45.748
second nature to you after you
do the problem set.

45:45.750 --> 45:47.380
It's a very elementary thing.

45:47.380 --> 45:48.370
Of course the first time you do
it,

45:48.369 --> 45:50.359
it seems very complicated,
but it's a very mechanical

45:50.362 --> 45:52.412
elementary thing,
but it's going to give us a lot

45:52.409 --> 45:54.529
of insight into the economy,
so any questions?

45:54.530 --> 45:55.480
Yes?

45:55.480 --> 45:57.110
Student: I was just
wondering what those two lines

45:57.114 --> 45:57.324
said.

45:57.320 --> 45:58.750
I think the first word says
assume.

45:58.750 --> 45:59.590
I just can't read it.

45:59.590 --> 46:01.770
Prof: Assume,
this says, without loss of

46:01.771 --> 46:03.481
generality that P_X =
1.

46:03.480 --> 46:04.830
Student: And the second
line?

46:04.829 --> 46:06.179
Prof: Except that I took
P_Y.

46:06.179 --> 46:07.489
This is P_Y not
P_X.

46:07.489 --> 46:11.569
P_Y = 1.

46:11.570 --> 46:14.220
And the second line,
this one says,

46:14.219 --> 46:16.909
so without loss of generality
P_Y is 1,

46:16.909 --> 46:20.499
so having put P_Y = 1
here I then looked at these

46:20.503 --> 46:23.843
equations,
(100-X^(A)) over P_X

46:23.840 --> 46:27.630
= 1 over P_Y,
but I took P_Y to be

46:27.625 --> 46:29.575
one so that's 1 over 1 which is
1.

46:29.579 --> 46:33.609
And I took this equation which
is (30-X^(B)) over P_X

46:33.610 --> 46:36.730
= 1 over P_Y,
so (30-X^(B)) over

46:36.733 --> 46:39.843
P_X = 1 over
P_Y,

46:39.840 --> 46:42.520
but that's 1/1 which is 1,
so I wrote that.

46:42.518 --> 46:47.388
So this is how I got my two
critical equations.

46:47.389 --> 46:50.919
These two equations here,
this and that,

46:50.918 --> 46:52.728
went down to this.

46:52.730 --> 46:55.760
And then I just rewrote this
one as X^(A) is

46:55.764 --> 46:59.654
100-P_X and this one
you can write as X^(B) is

46:59.646 --> 47:01.126
30-P_X.

47:01.130 --> 47:04.450
So 100-P_X,
30-P_X and then I

47:04.445 --> 47:08.515
added X^(A) to X^(B) and I got
130-2P_X = 84 and I

47:08.519 --> 47:10.039
got P_X.

47:10.039 --> 47:10.969
Yeah?

47:10.969 --> 47:13.509
Student: So we can set
P_Y to any number?

47:13.510 --> 47:14.300
Prof: Any number.

47:14.300 --> 47:15.550
Student: And get the
same results?

47:15.550 --> 47:16.080
Prof: Yes.

47:16.079 --> 47:17.169
You'll just multiply the pries.

47:17.170 --> 47:19.930
If you set P_Y to be
2 you'd have gotten

47:19.927 --> 47:22.907
P_X to be 46 and you
get the same answer.

47:22.909 --> 47:24.849
Student: So they're like
relative prices?

47:24.849 --> 47:27.539
Prof: So the only thing
that matters is relative prices.

47:27.539 --> 47:29.379
So this is what Walras pointed
out.

47:29.380 --> 47:34.000
If you change dollars to cents
you're going to multiply every

47:33.998 --> 47:36.768
price by 100,
but the relative price of

47:36.766 --> 47:40.636
oranges and tomatoes is going to
be the same as it was before.

47:40.639 --> 47:44.689
So the theory only produces
relative prices.

47:44.690 --> 47:45.570
Any other questions?

47:45.570 --> 47:47.410
There was someone else raising
their hand.

47:47.409 --> 47:48.709
Nope?

47:48.710 --> 47:51.390
All right, let's just do it one
more time so you see you get the

47:51.385 --> 47:53.675
hang of this and then we're
going to talk about why the

47:53.679 --> 47:56.179
market's so good and we're going
to see things are getting a

47:56.184 --> 47:57.634
little bit more complicated.

47:57.630 --> 48:01.160
So let's do this one.

48:01.159 --> 48:11.769
This one is going to work--oh,
so I wasn't very clever here.

48:11.768 --> 48:17.508
Aha, maybe I could be clever,
more clever.

48:17.510 --> 48:19.110
So how do we do this one?

48:19.110 --> 48:20.770
Well, we have to write down all
the equations.

48:20.769 --> 48:22.469
So what are they going to be?

48:22.469 --> 48:25.169
They're going to be the same as
before, X^(A) X^(B) =

48:25.172 --> 48:26.992
E^(A)_X
E^(B)_X.

48:26.992 --> 48:28.712
[correction:
as pointed out later,

48:28.706 --> 48:30.786
should be X^(C) X^(D) =
E^(C)_X

48:30.786 --> 48:32.186
E^(D)_X].

48:32.190 --> 48:36.690
That's supply and demand but
this is just 2 1 = 3,

48:36.693 --> 48:41.383
and over here we have--it's not
A and B any more.

48:41.380 --> 48:46.160
It's C and D,
I guess I called them,

48:46.163 --> 48:47.533
C and D.

48:47.530 --> 48:58.930
And now we have for the second
one, we have Y^(C) Y^(D) =

48:58.929 --> 49:09.109
E^(C)_Y endowment of D
of Y = 1 2 = 3.

49:09.110 --> 49:11.460
All right, so that's supply and
demand.

49:11.460 --> 49:12.890
Then we have to do the budget
sets.

49:12.889 --> 49:14.079
They're going to be simple.

49:14.079 --> 49:23.369
P_X times X^(C)
P_Y times Y^(C) has to

49:23.371 --> 49:31.851
equal P_X times 2
P_Y times 1.

49:31.849 --> 49:32.309
All right?

49:32.309 --> 49:38.039
And then budget set for D is
P_X times X^(D)

49:38.039 --> 49:43.229
P_Y times Y^(D) =
P_X times 1

49:43.228 --> 49:46.038
P_Y times 2.

49:46.039 --> 49:49.609
And finally we have to do the
marginal business.

49:49.610 --> 49:51.070
So what's the marginal business?

49:51.070 --> 49:52.190
So somebody tell me this.

49:52.190 --> 50:03.850
So we need the marginal utility
of A over the price of X.

50:03.849 --> 50:04.469
What's that?

50:04.469 --> 50:09.849
What's the marginal utility of
X to Mr. A?

50:09.849 --> 50:11.159
Student: 3 fourths X.

50:11.159 --> 50:12.679
Prof: 3 fourths what?

50:12.679 --> 50:14.659
Student: X.

50:14.659 --> 50:17.449
Prof: That's what you
said 3 fourths X,

50:17.454 --> 50:19.694
exactly, divided by
P_X.

50:19.690 --> 50:20.530
So how did I do that?

50:20.530 --> 50:22.840
I took the derivative of 3
fourths times log X.

50:22.840 --> 50:24.030
This is the only thing you have
to know.

50:24.030 --> 50:26.400
The derivative of log X is 1
over X,

50:26.400 --> 50:30.990
and that's going to be equal to
the marginal utility of

50:30.987 --> 50:33.357
[clarification:
at the point]

50:33.364 --> 50:37.804
X_A,
Y_A with respect to Y

50:37.802 --> 50:41.902
over P_Y and that's
equal to what?

50:41.900 --> 50:43.530
Student: 1 fourth Y.

50:43.530 --> 50:48.120
Prof: 1 fourth Y.

50:48.119 --> 50:50.219
Student: Are your A's
C's?

50:50.219 --> 50:51.589
Prof: Yes,
my A's are C's.

50:51.590 --> 50:52.120
Thank you.

50:52.119 --> 50:57.079
I'm glad you pointed that out.

50:57.079 --> 50:58.419
Thanks.

50:58.420 --> 51:00.560
So it's embarrassing to make
all these mistakes,

51:00.556 --> 51:03.056
but you'll find in 30 years
you'll start making mistakes

51:03.056 --> 51:03.416
too.

51:03.420 --> 51:09.290
So that's that equation and
then we have to do the same

51:09.288 --> 51:10.808
thing for Y.

51:10.809 --> 51:12.829
And I should have been more
clever and left more room,

51:12.827 --> 51:13.397
but I didn't.

51:13.400 --> 51:16.490
But anyway, the last equation
is going to be marginal utility.

51:16.489 --> 51:17.709
So what is the last equation?

51:17.710 --> 51:30.270
Marginal utility of D,
of X, over P_X =

51:30.268 --> 51:32.098
what?

51:32.099 --> 51:37.229
2 thirds times 1 over Y
[correction: X]

51:37.230 --> 51:41.820
divided by P_X =
what?

51:41.820 --> 51:45.640
I'm not going to write out
marginal utility of -Y over

51:45.639 --> 51:47.369
P_Y is what?

51:47.369 --> 51:50.059
Student:
>

51:50.059 --> 51:52.399
Prof: [1 third times 1
over]

51:52.402 --> 51:55.162
Y^(D), [all]
divided by P_Y.

51:55.159 --> 51:56.919
So those are the equations.

51:56.920 --> 52:01.100
So now we're going to put these
on a computer,

52:01.096 --> 52:06.476
but we can solve these by hand
again, and were going to see

52:06.478 --> 52:10.468
it's very useful to be able to
do this.

52:10.469 --> 52:17.499
Almost, so there's another
trick to doing this.

52:17.500 --> 52:21.150
So the trick is we can take one
of them to be 1,

52:21.150 --> 52:23.170
whichever we want to do.

52:23.170 --> 52:27.700
Take P_Y to be 1 or
P_X to be 1.

52:27.699 --> 52:30.939
So take P_X = 1.

52:30.940 --> 52:33.650
Here you see things are a
little bit more symmetric.

52:33.650 --> 52:37.140
There, there was the special Y
that had constant margin utility

52:37.141 --> 52:39.621
just like in our football
tickets example.

52:39.619 --> 52:41.679
Here there's nothing that is
constant marginal utility.

52:41.679 --> 52:44.029
X and Y are much more symmetric.

52:44.030 --> 52:47.830
So this move to more symmetry
without the special X if very

52:47.827 --> 52:50.157
important,
and the guy who first did that

52:50.161 --> 52:53.121
was actually Irving Fisher at
Yale who you're going to hear a

52:53.123 --> 52:54.263
lot about very soon.

52:54.260 --> 52:59.460
So it's a little bit more
complicated this time.

52:59.460 --> 53:06.170
So here's the critical
equation, this one let's say.

53:06.170 --> 53:10.560
So now I'm going to solve this.

53:10.559 --> 53:12.709
So how can I do this?

53:12.710 --> 53:15.470
I want to do the same trick as
before.

53:15.469 --> 53:19.349
I now want to solve--so let's
take P_Y to be 1 like

53:19.349 --> 53:22.129
I did before,
take P_Y to be 1.

53:22.130 --> 53:28.430
So let's just solve this
equation, solve for X.

53:28.429 --> 53:36.479
It's not going to be so easy to
do this.

53:36.480 --> 53:39.530
So now there's a tremendous
trick here.

53:39.530 --> 53:48.920
What does this say if I rewrite
this?

53:48.920 --> 53:53.590
I can bring the X down here and
I get P_X times X.

53:53.590 --> 53:58.540
And I can bring the Y down here
and I get P_Y times Y.

53:58.539 --> 54:02.389
So it says that the amount you
spend on X relative to 3

54:02.391 --> 54:06.531
quarters is equal to the
amount--this is by the way Mr.

54:06.530 --> 54:11.250
C--the amount C spends on X
relative to 3 quarters is equal

54:11.246 --> 54:15.636
to the amount he spends on Y
relative to 1 quarter,

54:15.639 --> 54:20.659
but the total spending on X and
Y has got to be all his income.

54:20.659 --> 54:25.009
So basically it says that he
spends 3 quarters of his income

54:25.005 --> 54:26.915
on X and 1 quarter on Y.

54:26.920 --> 54:28.980
That's the crucial insight.

54:28.980 --> 54:33.360
So you can solve these
logarithmic examples very easily

54:33.360 --> 54:34.660
by that trick.

54:34.659 --> 54:48.509
So it's evident from marginal
utility equation that C will

54:48.509 --> 55:03.329
spend 3 quarters of his income
on X, and D will spend 2 thirds

55:03.331 --> 55:08.921
of her income on X.

55:08.920 --> 55:12.320
By the same argument she's
going to have 2 thirds--

55:12.320 --> 55:17.060
I can bring X^(D) down and her
spending on X relative to 2

55:17.059 --> 55:22.049
thirds is equal to her spending
on Y relative to 1 third,

55:22.050 --> 55:24.380
but she's spending all her
income on X and Y.

55:24.380 --> 55:29.240
So clearly 2 thirds of it is
being spent on X and 1 third of

55:29.242 --> 55:29.822
it Y.

55:29.820 --> 55:35.900
So that property of the log
utilities is no accident.

55:35.900 --> 55:38.350
They were invented for exactly
that purpose.

55:38.349 --> 55:39.959
So this is a story you probably
heard,

55:39.960 --> 55:42.540
but there is a famous--I'm from
Illinois--

55:42.539 --> 55:45.949
there was a famous Senator
Douglas from Illinois,

55:45.949 --> 55:47.949
there have been several
Douglas' from Illinois.

55:47.949 --> 55:52.209
One of them debated Lincoln.

55:52.210 --> 55:54.670
Maybe he wasn't from Illinois.

55:54.670 --> 55:55.940
Lincoln was from Illinois.

55:55.940 --> 55:59.480
There was a famous senator
named Douglas from Illinois

55:59.483 --> 56:03.033
after the Civil War and he
noticed that farmers' labor

56:03.028 --> 56:06.838
tended to get 2 thirds or 3
quarters of all the income and

56:06.840 --> 56:08.380
capital the rest.

56:08.380 --> 56:11.480
So he said, "What kind of
utility function would make me

56:11.480 --> 56:14.430
always spend the same fraction
of my money on a particular

56:14.425 --> 56:15.195
good."

56:15.199 --> 56:18.439
And so he went to his college
math teacher,

56:18.440 --> 56:22.170
Cobb, and asked him if he could
invent a utility function which

56:22.172 --> 56:26.032
had the property that you always
spent a fixed proportion of your

56:26.025 --> 56:28.635
money on each good,
and so Cobb invented the

56:28.644 --> 56:31.344
Cobb-Douglas utility function
and this is where it is.

56:31.340 --> 56:34.680
This is just the sum of logs.

56:34.679 --> 56:37.919
So it's called Cobb-Douglas
utility.

56:37.920 --> 56:39.240
So it has that property,
this property,

56:39.239 --> 56:42.419
that each person in this
Cobb-Douglas utility spends a

56:42.423 --> 56:45.673
fixed proportion of her money on
each of the goods,

56:45.670 --> 56:47.910
a different proportion on each
of the goods and different

56:47.909 --> 56:49.389
people have different
proportions,

56:49.389 --> 56:52.549
but any single person can
always spend a given proportion

56:52.552 --> 56:54.702
3 quarters on X and 1 quarter on
Y,

56:54.699 --> 56:56.729
2 thirds on X, 1 quarter on Y.

56:56.730 --> 56:59.990
And because of that it's very
easy to solve for equilibrium.

56:59.989 --> 57:03.189
So we go, X^(A) is going to be
3 quarters.

57:03.190 --> 57:05.380
What is her income?

57:05.380 --> 57:17.310
Her income is P_X
times 2 P_Y times 1.

57:17.309 --> 57:18.729
So her income has to be that.

57:18.730 --> 57:23.270
She's going to spend 3 quarters
of her income on X,

57:23.271 --> 57:26.271
so I could write X^(A) as that.

57:26.269 --> 57:27.529
So this is C.

57:27.530 --> 57:35.600
And X^(D) is going to be,
she's spending 2 thirds of her

57:35.599 --> 57:44.409
income on X and 1 third on Y,
so her endowments are 1 unit of

57:44.405 --> 57:47.775
X and 2 units of Y.

57:47.780 --> 57:50.180
So this is her income,
P_X times 1

57:50.175 --> 57:53.445
P_Y times 2,
and so she's spending 2 thirds

57:53.445 --> 57:56.695
of it on X so therefore
P_X times X^(D),

57:56.699 --> 57:58.759
that's the amount of money she
spends on X,

57:58.760 --> 58:00.990
that's what she spends on X,
has to be that,

58:00.989 --> 58:02.879
so X^(D) is that.

58:02.880 --> 58:09.660
And if I add these two,
when I add them up I have to

58:09.659 --> 58:11.519
get 2 1 = 3.

58:11.519 --> 58:14.179
So I can just now solve this.

58:14.179 --> 58:18.079
Well, I can do a trick and pick
one of my--either P_X

58:18.083 --> 58:20.733
or P_Y to be 1,
so either one.

58:20.730 --> 58:21.810
I keep going back and forth.

58:21.809 --> 58:22.749
It doesn't make any difference.

58:22.750 --> 58:27.430
Let's try P_Y as 1,
can take that to be 1.

58:27.429 --> 58:28.659
And now I can solve it.

58:28.659 --> 58:42.899
So this is just 3 quarters,
times (P_X times 2 1

58:42.896 --> 58:49.636
divided by P_X.

58:49.639 --> 58:56.919
Then the other one is 2 thirds
(P_X times 1 2)

58:56.916 --> 59:00.486
divided by P_X.

59:00.489 --> 59:05.809
So I can add those,
and when I add those it equals

59:05.806 --> 59:06.236
3.

59:06.239 --> 59:11.369
So I know that 3P_X,
if I multiply through by

59:11.367 --> 59:16.017
P_X I get
3P_X = 3 halves--

59:16.018 --> 59:17.858
hopefully I did this right--3
halves P_X--

59:17.860 --> 59:20.340
this'll be very embarrassing if
I didn't--

59:20.340 --> 59:30.610
3 halves P_X 3
quarters 2 thirds P_X

59:30.610 --> 59:35.840
4 thirds = 3P_X.

59:35.840 --> 59:40.320
Oh, is this right?

59:40.320 --> 59:42.120
So who can do this in their
heads?

59:42.119 --> 59:47.999
3 halves P_X from
that is 3 halves P_X,

59:47.996 --> 59:53.466
so 3 halves - 2 thirds,
3 halves is 9 sixths - 4 sixths

59:53.467 --> 59:55.087
is 5 sixths.

59:55.090 --> 59:59.090
It looks like 5 sixths
P_X.

59:59.090 --> 1:00:00.890
Does anyone believe that?

1:00:00.889 --> 1:00:06.519
If I do this in terms of 6
that's 9 sixths and that's 4

1:00:06.523 --> 1:00:11.223
over 6, that's 13 over 6 and
that's 18 over 6,

1:00:11.217 --> 1:00:14.867
so it's 5 over 6 P_X.

1:00:14.869 --> 1:00:21.679
That looks right,
and 3 quarters 4 thirds if I go

1:00:21.681 --> 1:00:29.771
to 12ths that's 9 twelfths and
16 twelfths is 25 twelfths,

1:00:29.768 --> 1:00:34.308
so this looks like 5 halves.

1:00:34.309 --> 1:00:38.449
So that means P_X = 5
halves.

1:00:38.449 --> 1:00:42.889
So there I've got the answer.

1:00:42.889 --> 1:00:46.839
Does that look right to you?

1:00:46.840 --> 1:00:48.650
Is this clear what I did here?

1:00:48.650 --> 1:00:57.220
I just took for this trick all
I did was I solved for--so let's

1:00:57.222 --> 1:01:00.682
just repeat what I did.

1:01:00.679 --> 1:01:03.949
Just like over there I reduced
it to simultaneous equations in

1:01:03.954 --> 1:01:05.764
a mechanical way,
in a very simple

1:01:05.757 --> 1:01:08.387
straightforward mechanical way
which the first time you see

1:01:08.391 --> 1:01:11.481
looks very complicated,
but it's very simple,

1:01:11.476 --> 1:01:14.656
in fact, after you've done it
once.

1:01:14.659 --> 1:01:17.189
Then it allows you to take
these very complicated models

1:01:17.186 --> 1:01:18.516
and say something concrete.

1:01:18.518 --> 1:01:20.928
So I've got all the
peoples'--their welfare

1:01:20.927 --> 1:01:22.817
functions and their endowments.

1:01:22.820 --> 1:01:25.410
So I say in equilibrium what
has to happen.

1:01:25.409 --> 1:01:28.719
Whatever they decide to eat C
and D what he eats plus what she

1:01:28.724 --> 1:01:30.414
eats has to be the endowment.

1:01:30.409 --> 1:01:32.039
The total endowment is 3.

1:01:32.039 --> 1:01:34.829
So the total consumption of X
has to be 3.

1:01:34.829 --> 1:01:38.019
The total consumption of Y
between what he eats and what

1:01:38.021 --> 1:01:39.881
she eats also has to be three.

1:01:39.880 --> 1:01:42.250
Now each of them is going to
spend all their money.

1:01:42.250 --> 1:01:43.490
He's going to spend all his
money.

1:01:43.489 --> 1:01:45.769
She's going to spend all her
money.

1:01:45.768 --> 1:01:49.228
Because it's Cobb-Douglas,
because it's logarithmic,

1:01:49.230 --> 1:01:52.310
and you do this marginal
utility stuff you find out,

1:01:52.309 --> 1:01:56.279
and this was the only trick,
so this is a non-obvious trick

1:01:56.277 --> 1:01:59.827
which some senator and
professional mathematician had

1:01:59.833 --> 1:02:02.763
to invent,
Cobb-Douglas is designed so

1:02:02.757 --> 1:02:06.967
that you can say right away with
those utility functions D is

1:02:06.974 --> 1:02:11.054
clearly going to spend 2 thirds
of her money buying X,

1:02:11.050 --> 1:02:15.730
and C is going to spend 3
quarters of his money buying X.

1:02:15.730 --> 1:02:18.020
It's just obvious from the
first order conditions,

1:02:18.018 --> 1:02:19.698
from this marginal utility
conditions,

1:02:19.699 --> 1:02:20.999
they're called first order
conditions,

1:02:21.000 --> 1:02:23.060
from this equating marginal
utilities.

1:02:23.059 --> 1:02:24.849
That was the crucial trick.

1:02:24.849 --> 1:02:28.869
So that's a trick that you have
to internalize and from now on

1:02:28.871 --> 1:02:33.091
that's all you have to know that
C's going to spend 3 quarters of

1:02:33.090 --> 1:02:36.090
his money on X,
1 quarter of his money on Y and

1:02:36.094 --> 1:02:39.634
D's going to spend 2 thirds of
her money on X and 1 third on Y,

1:02:39.630 --> 1:02:41.130
but supply has to equal demand.

1:02:41.130 --> 1:02:42.400
So what is C?

1:02:42.400 --> 1:02:43.720
What is he actually buying?

1:02:43.719 --> 1:02:45.469
Here's his total money.

1:02:45.469 --> 1:02:47.839
He has two units of X and one
unit of Y.

1:02:47.840 --> 1:02:50.800
So he's selling his units of X
at the price P_X,

1:02:50.795 --> 1:02:53.125
and his units of Y at the price
P_Y,

1:02:53.128 --> 1:02:55.408
and he's spending 3 quarters of
it on X.

1:02:55.409 --> 1:02:57.779
So how much X is he actually
buying?

1:02:57.780 --> 1:03:00.020
This is the amount of money
he's spending on X.

1:03:00.018 --> 1:03:02.858
Divide by the price of
P_X,

1:03:02.855 --> 1:03:05.765
that's how much money he buys
of X.

1:03:05.768 --> 1:03:08.908
She, D, she's going to spend,
here's her income which is not

1:03:08.907 --> 1:03:11.347
quite the same as his,
because her endowment is

1:03:11.353 --> 1:03:13.113
different, that's her income.

1:03:13.110 --> 1:03:16.940
She spends 2 thirds of it on X,
so the amount of X she wants to

1:03:16.940 --> 1:03:20.460
buy is the amount of money she
spends on it divided by the

1:03:20.461 --> 1:03:21.081
price.

1:03:21.079 --> 1:03:22.809
That's how much she wants to
buy.

1:03:22.809 --> 1:03:27.079
Now I just have to add X^(C)
X^(D) and it's very hard for me

1:03:27.079 --> 1:03:30.409
to do at the board and you to
follow there,

1:03:30.409 --> 1:03:34.039
but of course if you stare at
the page for a minute at home

1:03:34.041 --> 1:03:36.111
it'll be very simple to follow.

1:03:36.110 --> 1:03:38.450
I do Walras' trick.

1:03:38.449 --> 1:03:41.189
I said I can always take
P_Y to be 1,

1:03:41.190 --> 1:03:43.940
and if I take P_Y to
be 1 I'm going to get this

1:03:43.938 --> 1:03:46.438
income is P_X [times]
2 times [correction:

1:03:46.443 --> 1:03:48.263
plus]
1 times 1 which is just 1,

1:03:48.260 --> 1:03:50.010
so 3 quarters of this divided
by P_X,

1:03:50.010 --> 1:03:51.390
that's what he's buying.

1:03:51.389 --> 1:03:55.119
She's buying 2 thirds of her
income which is P_X

1:03:55.123 --> 1:03:57.683
times 1 1 times 2 which is plus
two,

1:03:57.679 --> 1:03:59.989
divided by P_X,
that's what she's buying,

1:03:59.989 --> 1:04:04.199
and I just add this to this and
do a little algebra.

1:04:04.199 --> 1:04:07.229
So I just add and do a little
algebra and lo and behold

1:04:07.231 --> 1:04:09.311
P_X is equal to 5
halves.

1:04:09.309 --> 1:04:11.459
So I happened to remember
that's the right number so I

1:04:11.463 --> 1:04:12.523
actually did this right.

1:04:12.518 --> 1:04:15.478
So P_X is equal to 5
halves, and we've solved the

1:04:15.481 --> 1:04:16.301
whole problem.

1:04:16.300 --> 1:04:24.320
So if P_X is equal to
5 halves how much is she

1:04:24.324 --> 1:04:27.684
actually buying of X?

1:04:27.679 --> 1:04:29.759
Well, I could always plug this
back in,

1:04:29.760 --> 1:04:35.180
plug in P_X is equal
to 5 halves and find out that's

1:04:35.182 --> 1:04:37.082
5 1 is 6,
times 3 quarters,

1:04:37.081 --> 1:04:38.111
divided by 5 halves.

1:04:38.110 --> 1:04:42.010
That would tell me how much
X_C she was buying.

1:04:42.010 --> 1:04:46.730
So and I could plug in 5 halves
for P_X and I'd get

1:04:46.728 --> 1:04:51.678
how much D was buying of X and I
could also plug P_X =

1:04:51.682 --> 1:04:56.182
5 halves and PY = 1,
and find out what they were

1:04:56.177 --> 1:04:57.767
doing of good D.

1:04:57.768 --> 1:05:00.798
So you can solve it by hand
very easily, but let's just

1:05:00.795 --> 1:05:03.985
solve it by computer instead
unless there's a question.

1:05:03.989 --> 1:05:06.359
Ha, I stopped it.

1:05:06.360 --> 1:05:07.870
Any questions about what I did
here?

1:05:07.869 --> 1:05:08.709
Yes?

1:05:08.710 --> 1:05:12.540
Student: So we just
maximized utility so there's no

1:05:12.536 --> 1:05:15.486
other allocation of utilities
any greater?

1:05:15.489 --> 1:05:19.739
Prof: Well,
now we haven't gotten here yet.

1:05:19.739 --> 1:05:23.029
I've run over a little bit,
so I'm going to finish the

1:05:23.025 --> 1:05:26.305
class by repeating this
calculation on a computer just

1:05:26.311 --> 1:05:29.971
by pressing a button and you'll
see what the answer is.

1:05:29.969 --> 1:05:33.189
But then we have to examine the
question, have we really

1:05:33.193 --> 1:05:36.073
maximized utility here,
and to give away the punch

1:05:36.065 --> 1:05:38.405
line, that utility was very
special.

1:05:38.409 --> 1:05:41.819
It was constant marginal
utility of 1 in a particular

1:05:41.818 --> 1:05:42.408
good Y.

1:05:42.409 --> 1:05:45.209
That's what made this example
as almost identical to the

1:05:45.206 --> 1:05:46.526
football ticket example.

1:05:46.530 --> 1:05:49.790
The final equilibrium is going
to maximize the sum of

1:05:49.788 --> 1:05:50.538
utilities.

1:05:50.539 --> 1:05:53.649
Here, this equilibrium is not
going to maximize the sum of

1:05:53.646 --> 1:05:54.296
utilities.

1:05:54.300 --> 1:05:57.780
There's no reason it should
maximize the sum of utilities.

1:05:57.780 --> 1:06:00.770
And so you need a different
definition of why the free

1:06:00.773 --> 1:06:02.473
market is such a good thing.

1:06:02.469 --> 1:06:04.489
So economists made a tremendous
mistake.

1:06:04.489 --> 1:06:07.859
They thought that the original
criterion for a good market is

1:06:07.864 --> 1:06:09.894
you maximize the sum of
utilities.

1:06:09.889 --> 1:06:13.519
That's not even true in an
example like this one,

1:06:13.518 --> 1:06:16.438
so we need a different
definition which we call Pareto

1:06:16.436 --> 1:06:19.296
efficiency that illustrates why
the market's good.

1:06:19.300 --> 1:06:22.370
But if they made a mistake once
it stands to reason they could

1:06:22.369 --> 1:06:23.879
make a mistake another time.

1:06:23.880 --> 1:06:26.240
So there's something special
even about this example.

1:06:26.239 --> 1:06:29.379
When we put in financial
variables I'm going to argue you

1:06:29.382 --> 1:06:32.532
shouldn't expect to get the
optimal outcome all the time,

1:06:32.525 --> 1:06:34.205
but that'll be next class.

1:06:34.210 --> 1:06:35.140
Yes?

1:06:35.139 --> 1:06:38.659
Student: Beyond like
arithmetic use is there any

1:06:38.661 --> 1:06:41.861
reason you would choose to
assume P_Y or

1:06:41.856 --> 1:06:44.526
P_X is 1 or it's
arbitrary?

1:06:44.530 --> 1:06:44.930
Prof: No.

1:06:44.929 --> 1:06:47.199
There's no reason to pick
P_X or P_Y

1:06:47.195 --> 1:06:49.415
to be 1, whichever one you want
you can choose to be 1,

1:06:49.418 --> 1:06:50.488
and I keep going back.

1:06:50.489 --> 1:06:53.039
I can never make up my mind
which one to do,

1:06:53.043 --> 1:06:55.303
so yeah, just whatever it works
out.

1:06:55.300 --> 1:06:58.170
This one it clearly worked out
arithmetically easy to take

1:06:58.168 --> 1:07:01.188
P_Y = 1 because the
marginal utility of Y was 1 and

1:07:01.188 --> 1:07:02.848
that canceled everything out.

1:07:02.849 --> 1:07:05.709
Here I could have taken either
one price to be 1 and it

1:07:05.710 --> 1:07:06.930
wouldn't have helped.

1:07:06.929 --> 1:07:11.099
So I picked P_Y to be
1 again.

1:07:11.099 --> 1:07:16.529
In the last five minutes let's
just show how to solve this by

1:07:16.525 --> 1:07:17.515
computer.

1:07:17.518 --> 1:07:19.978
So this is something you also
are going to be able to do.

1:07:19.980 --> 1:07:21.540
And it sounds like,
"Oh, there's so many

1:07:21.536 --> 1:07:22.276
complicated things.

1:07:22.280 --> 1:07:24.240
There's these new
equations,"

1:07:24.240 --> 1:07:26.500
if you do this for the problem
set,

1:07:26.500 --> 1:07:28.600
after you've done it once for
the problem set--

1:07:28.599 --> 1:07:30.129
you may have a little trouble
with the problem set,

1:07:30.130 --> 1:07:33.220
the TAs will help you,
but after you do it once this

1:07:33.217 --> 1:07:34.547
will be very simple.

1:07:34.550 --> 1:07:38.140
Now, doing it by computer is
also very simple.

1:07:38.139 --> 1:07:39.609
And it's going to sound
complicated,

1:07:39.610 --> 1:07:43.430
but as all you young people
know if any old guy can figure

1:07:43.425 --> 1:07:47.305
out how to work a computer you
can do it vastly quicker.

1:07:47.309 --> 1:07:55.519
So let's just take the second
example here.

1:07:55.519 --> 1:07:56.649
And we have five minutes left.

1:07:56.650 --> 1:07:58.590
That's all it'll take.

1:07:58.590 --> 1:08:00.000
So this is Excel.

1:08:00.000 --> 1:08:03.710
Now, Excel is this program
that's made zillions of dollars.

1:08:03.710 --> 1:08:09.890
The inventor of Excel,
by the way, was the inventor of

1:08:09.887 --> 1:08:10.817
Lotus.

1:08:10.820 --> 1:08:13.110
Oh, what was the guy's name?

1:08:13.110 --> 1:08:14.940
His sister was in my class at
Yale.

1:08:14.940 --> 1:08:17.200
He was two years ahead of me.

1:08:17.198 --> 1:08:19.758
Not Gabor, Mitch Gabor,
[correction:

1:08:19.762 --> 1:08:22.622
Mitch Kapor]
something like that was his

1:08:22.619 --> 1:08:23.279
name.

1:08:23.279 --> 1:08:25.839
Anyway, she was in my class,
and he was two years ahead of

1:08:25.835 --> 1:08:26.055
me.

1:08:26.060 --> 1:08:27.720
And he invented this thing
called Lotus,

1:08:27.724 --> 1:08:28.924
which made a lot of money.

1:08:28.920 --> 1:08:32.280
And then it got bought out by a
few people.

1:08:32.279 --> 1:08:35.779
And then Excel just basically
copied the entire thing,

1:08:35.778 --> 1:08:39.408
Microsoft, and made a fortune
and had to pay him off for

1:08:39.408 --> 1:08:41.188
plagiarizing the thing.

1:08:41.189 --> 1:08:44.399
But anyway, it's basically
Mitch Gabor was the inventor,

1:08:44.402 --> 1:08:47.032
a Yale undergraduate two years
ahead of me.

1:08:47.029 --> 1:08:49.309
So he's a billionaire now.

1:08:49.310 --> 1:08:51.980
So let's just solve the problem.

1:08:51.979 --> 1:08:53.889
Let's do the second one because
I may not have time for the

1:08:53.885 --> 1:08:54.275
first one.

1:08:54.279 --> 1:08:55.509
So what did I do?

1:08:55.510 --> 1:08:58.650
I said let's write down the
exogenous variables first,

1:08:58.646 --> 1:09:00.656
sorry let's just go up a
little.

1:09:00.658 --> 1:09:04.528
So the exogenous variables are
the endowment of X,

1:09:04.530 --> 1:09:07.770
of the two goods,
A and B, that's 2 and 1,

1:09:07.770 --> 1:09:09.430
and B is 1 and 2.

1:09:09.430 --> 1:09:10.910
Now what are the variables?

1:09:10.908 --> 1:09:12.228
P_X,
P_Y,

1:09:12.226 --> 1:09:13.586
X^(A), Y^(A),
X^(B) and Y^(B),

1:09:13.591 --> 1:09:14.911
we don't what those are.

1:09:14.908 --> 1:09:17.688
So I've plugged in
P_X and P_Y.

1:09:17.689 --> 1:09:20.429
I'll guess both of them are 1,
which is obviously going to be

1:09:20.429 --> 1:09:22.239
wrong,
and I'll guess that people just

1:09:22.243 --> 1:09:25.763
end up with their endowments,
which is obviously not right.

1:09:25.760 --> 1:09:27.590
So then I look at the budget
set.

1:09:27.590 --> 1:09:29.740
So those are my guesses.

1:09:29.738 --> 1:09:32.498
These are the endogenous
variables and wild guesses about

1:09:32.500 --> 1:09:33.240
the solution.

1:09:33.239 --> 1:09:34.719
Now, what are the equations?

1:09:34.720 --> 1:09:35.930
Well, we wrote them down.

1:09:35.930 --> 1:09:39.220
There's the budget set of A,
so that's just the budget set

1:09:39.216 --> 1:09:39.616
of A.

1:09:39.618 --> 1:09:41.648
So how do you write these
equations down?

1:09:41.649 --> 1:09:46.139
You simply name the--it's up
here if you haven't used Excel

1:09:46.136 --> 1:09:47.526
before, up here.

1:09:47.529 --> 1:09:53.129
You write down the letter,
say B35 that's P_X,

1:09:53.126 --> 1:09:55.056
so B35 times B31.

1:09:55.060 --> 1:10:00.850
That's P_X,
is B35 times endowment X^(A).

1:10:00.850 --> 1:10:03.310
That's the income.

1:10:03.310 --> 1:10:04.500
I wrote the income first.

1:10:04.500 --> 1:10:07.400
That's the income A has minus
how much she spends or he

1:10:07.404 --> 1:10:07.894
spends.

1:10:07.890 --> 1:10:12.710
B35 times B37,
B35 remember is the price of X,

1:10:12.706 --> 1:10:15.486
B37 is how much he buys.

1:10:15.489 --> 1:10:16.669
So that's just the budget set.

1:10:16.670 --> 1:10:18.960
So for each of these equations,
the marginal utility,

1:10:18.957 --> 1:10:20.187
I just did the same thing.

1:10:20.189 --> 1:10:23.729
Remember the 3 quarters,
over P_X times X = 1

1:10:23.729 --> 1:10:27.329
quarter over P_Y times
Y, so this [difference]

1:10:27.333 --> 1:10:29.173
should be equal to zero.

1:10:29.170 --> 1:10:32.540
Instead of saying this equals
that I subtract the right hand

1:10:32.538 --> 1:10:34.308
side from the left hand side.

1:10:34.310 --> 1:10:38.090
So you want all these equations
to be equal to zero.

1:10:38.090 --> 1:10:41.060
I just wrote down the six
equations.

1:10:41.060 --> 1:10:45.140
And so Excel now tells me that
of course the budget set is

1:10:45.137 --> 1:10:48.927
going to be satisfied
automatically because people are

1:10:48.930 --> 1:10:51.150
consuming their endowments.

1:10:51.149 --> 1:10:53.769
And the budget set of B is
automatically satisfied because

1:10:53.768 --> 1:10:55.788
I just had them choosing their
endowments.

1:10:55.788 --> 1:10:58.098
And markets are going to clear,
of course, because everybody's

1:10:58.099 --> 1:11:00.219
choosing their endowments,
but they're not optimizing.

1:11:00.220 --> 1:11:03.770
So this marginal utility stuff
is all screwed up.

1:11:03.770 --> 1:11:05.120
So what do I do on the right?

1:11:05.118 --> 1:11:08.068
For every error in the equation
I square it.

1:11:08.069 --> 1:11:10.469
So I've squared all the errors.

1:11:10.470 --> 1:11:12.720
So these are my equations I
need to satisfy,

1:11:12.721 --> 1:11:15.081
one, two, three,
four, five, six equations.

1:11:15.078 --> 1:11:17.528
One, two, three,
four, five, six,

1:11:17.530 --> 1:11:19.600
and I summed the squares.

1:11:19.600 --> 1:11:22.690
So if I make the sum of the
squares zeros each of those has

1:11:22.686 --> 1:11:23.376
to be zero.

1:11:23.380 --> 1:11:25.820
So Excel, now,
can minimize the sum of squared

1:11:25.820 --> 1:11:26.310
errors.

1:11:26.310 --> 1:11:29.980
Excel is going to search over
all endogenous variables,

1:11:29.979 --> 1:11:33.979
P_X through Y^(B) to
find the things that makes this

1:11:33.980 --> 1:11:35.950
number as small as possible.

1:11:35.948 --> 1:11:38.578
Once this number becomes zero
it means all the ones above it

1:11:38.582 --> 1:11:41.392
have to be zero because they're
all squared numbers adding up to

1:11:41.393 --> 1:11:44.263
that,
and so I will find the solution.

1:11:44.260 --> 1:11:47.230
So you see that all you have to
do--if you've done this before

1:11:47.233 --> 1:11:49.673
of course it's obvious,
if you haven't it's just so

1:11:49.671 --> 1:11:51.281
simple to write the equation.

1:11:51.279 --> 1:11:54.589
Supply and demand I just name
the box,

1:11:54.590 --> 1:12:00.630
B32 that's the endowment of
Y^(A) the endowment of Y^(B)

1:12:00.625 --> 1:12:06.765
equals the consumption of A plus
the consumption of B.

1:12:06.770 --> 1:12:09.650
That's the difference we want
to make zero.

1:12:09.649 --> 1:12:11.509
So here's how you solve it.

1:12:11.510 --> 1:12:14.370
There's a thing called solver.

1:12:14.368 --> 1:12:19.018
So you go to tools and you hit
solver, and now solver says you

1:12:19.019 --> 1:12:21.229
want to take a target cell.

1:12:21.229 --> 1:12:21.719
I cheated.

1:12:21.720 --> 1:12:24.230
I already knew what it was, C47.

1:12:24.229 --> 1:12:25.509
So it's the target cell.

1:12:25.510 --> 1:12:28.280
I hit minimize,
so I want to minimize that.

1:12:28.279 --> 1:12:30.659
And now what cells do I change?

1:12:30.658 --> 1:12:34.758
Well, I have to tell Excel what
to search over.

1:12:34.760 --> 1:12:37.970
So now Excel,
what are the cells?

1:12:37.970 --> 1:12:39.940
I could say P_X,
P_Y,

1:12:39.939 --> 1:12:41.199
you know, all the endogenous
variables,

1:12:41.198 --> 1:12:44.408
but I know I can fix
P_X to be 1 so I'm

1:12:44.413 --> 1:12:48.503
going to forget that one and I'm
going to say just these five,

1:12:48.500 --> 1:12:49.170
right?

1:12:49.170 --> 1:12:53.110
I don't need all six of them,
just five because I can always

1:12:53.112 --> 1:12:55.052
take P_X to be 1.

1:12:55.050 --> 1:12:58.580
So the solver now knows it
wants to minimize this number,

1:12:58.578 --> 1:13:02.628
which is the squared errors of
all the equations I want to hold

1:13:02.631 --> 1:13:04.751
equal,
it's going to minimize that by

1:13:04.752 --> 1:13:06.492
searching over all those
numbers.

1:13:06.488 --> 1:13:08.368
It's not very smart about
searching for it,

1:13:08.372 --> 1:13:10.212
and sometimes it never finds an
answer.

1:13:10.210 --> 1:13:14.220
We know there always is an
answer, and so how do you solve

1:13:14.217 --> 1:13:14.567
it?

1:13:14.569 --> 1:13:17.649
You just hit solve and it's
going to search and do it.

1:13:17.649 --> 1:13:19.579
And what should the answer be?

1:13:19.578 --> 1:13:23.648
If I fix P_X to be 1,
remember the answer was when

1:13:23.649 --> 1:13:27.029
P_Y is 1,
X turns out to be 5 halves.

1:13:27.029 --> 1:13:32.039
If I fix P_X to be
one, what should P_Y

1:13:32.037 --> 1:13:32.467
be?

1:13:32.470 --> 1:13:36.260
The solution we got before was
P_X = 5 halves and

1:13:36.257 --> 1:13:37.627
P_Y is 1.

1:13:37.630 --> 1:13:40.900
Now I'm going to fix
P_X to be 1,

1:13:40.898 --> 1:13:42.608
so what should Y be?

1:13:42.609 --> 1:13:45.459
X was 5 halves times Y,
so Y should be .4,

1:13:45.463 --> 1:13:49.363
so if this solves right we
should get P_Y to be

1:13:49.362 --> 1:13:49.852
.4.

1:13:49.850 --> 1:13:55.250
So I just hit solve and voila I
get P_Y to be .4.

1:13:55.250 --> 1:13:57.410
I find X^(A) is 1.8.

1:13:57.409 --> 1:13:58.509
I find all the numbers.

1:13:58.510 --> 1:14:00.890
I just solved it just like that
instantly.

1:14:00.890 --> 1:14:04.940
So you can see how useful it's
going to be to use solver and do

1:14:04.939 --> 1:14:06.049
these problems.

1:14:06.050 --> 1:14:08.630
Student: If you change
the endowments does that change

1:14:08.634 --> 1:14:08.854
it?

1:14:08.850 --> 1:14:09.310
Prof: Of course.

1:14:09.310 --> 1:14:12.310
If I change the endowments I'll
get a different answer,

1:14:12.306 --> 1:14:14.916
and if I increase the
endowments yes it does and

1:14:14.916 --> 1:14:16.356
that's very important.

1:14:16.359 --> 1:14:18.319
Student:
>

1:14:18.324 --> 1:14:20.344
increasing it
>

1:14:20.340 --> 1:14:22.110
Prof: If I double
everybody's endowment?

1:14:22.109 --> 1:14:24.159
Student: If you double
one endowment.

1:14:24.158 --> 1:14:26.388
Prof: If I double one
endowment that's going to change

1:14:26.386 --> 1:14:26.976
things around.

1:14:26.979 --> 1:14:29.869
If I double everybody's
endowment it won't change

1:14:29.873 --> 1:14:30.903
anything, yeah.

1:14:30.899 --> 1:14:35.999