WEBVTT

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[SQUEAKING]

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[RUSTLING]

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[CLICKING]

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ROBERT TOWNSEND: So let
me catch us all back up.

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We have one more week of classes
consisting of two lectures.

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They're pretty exciting
because in effect, everything

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comes together at the end.

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The one today is the failure
of the welfare theorems,

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but not just that a
theorem doesn't work,

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but why it doesn't
work and also remedies

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that are going to make its way
into proposals for new market

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structures that get rid
of the externalities,

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existing and new.

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And then the one on Thursday
is on monetary theory

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where we will actually have
money and cryptocurrency

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in the model.

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And we can address things like
the price of Bitcoin, which

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yesterday hit an
all-time record,

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but I haven't
looked at it today.

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And that's kind of the point.

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So on the reading list, today
is Failure of the Well Welfare

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Theorems.

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There's two starred
articles here.

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It's a little bit unconventional
relative to other lectures

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in the sense that I'm not going
to cover either one of these

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in enormous amount of detail,
and yet they are a starred.

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So it's fair game
in terms of things

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that I expect you to know.

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I only have four or
five slides at the end

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on the Jain-Townsend paper, but
it relates to cryptocurrency

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and platforms for exchange.

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So it is very consistent
with some of the themes

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we keep hitting
on in the course.

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And this Arrow article, very
famous one, I have only today

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extracted two or three--

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really two slides from a
much larger and much more

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interesting paper.

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So take a look at those.

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Are there-- before
I ask you all,

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are there any questions
from last time,

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about the lecture
from last time?

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So last time was identification
and falsification with data.

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There is a question that's
actually not written down here,

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but I feel I want to ask anyway.

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Namely, what was the
point of the lecture?

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Why do we care about being able
to falsify a model with data?

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And I'll take volunteers.

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Why is it important to be
able to reject a model?

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STUDENT: I mean, I guess
it makes it more testable.

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ROBERT TOWNSEND: Well--

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STUDENT: If we have
empirical data,

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we can tell whether it's
even feasible for it

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to be that model.

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ROBERT TOWNSEND: Yeah, yeah.

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That's good.

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Or a more brutal way to put it
is, if it's not falsifiable,

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then it can't be rejected.

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And if it can't
be rejected, what

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is the point of taking it to
the data in the first place?

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That's a little bit
harsh in the sense

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that you could accept the
model as an abstract version

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of reality, use it to
estimate parameters,

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conduct policy
experiments-- and we

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did that several times in the
lectures on trade and so on.

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But this lecture embraces the
full empirical experimental

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science point of
view that we want

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to make some progress
by ruling things out.

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So you really want to if you're
in the wrong class of models.

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And the way to do
that is to at least

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have the possibility
of data that

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would be inconsistent with the
model so that you would know.

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OK.

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So that wasn't asked.

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Thank you for
volunteering an answer.

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So apropos along
that line, how do we

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test whether an observed
demand function came

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from a consumer-maximizing
utility?

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Or I guess more
specifically, does anyone

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remember one, two, or all
three of the implications that

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come from the premise of
consumer optimization for data?

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STUDENT: It should have
homogeneity of degree 0.

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It has to satisfy while
reaching equilibrium.

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And the Slutsky matrix has
to be symmetric and negative

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[INAUDIBLE].

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ROBERT TOWNSEND: I'm not
actually hearing you very well.

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Can you repeat that?

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STUDENT: Oh yeah.

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The demand function has to
be homogeneous of degree 0.

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It has to satisfy while
reaching equilibrium.

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And the Slutsky matrix has to be
negative, [? semi-definite. ?]

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ROBERT TOWNSEND: Yeah, perfect.

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So satisfy-- I guess
we call it Walras' law,

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but essentially means
that they're spending

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all their money and they're not
in the interior of the budget

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set.

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And yeah, homogeneity of degree
0, no inflation illusions.

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And the Slutsky
thing is the thing

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we derived pretty
carefully the last time.

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Does anyone remember what--

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and can state the weak axiom
of revealed preference?

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STUDENT: Let's say we have
two pairs of observations

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containing prices and
consumption bundles.

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And for example, for like p,
p2, the x2 is preferred over x1.

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And then for p1, the x2
should be outside of budget.

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Otherwise, x1 would not
be preferred under p1.

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ROBERT TOWNSEND: Yep.

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Yeah, you said it right I
think except for the very end.

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We start with two pairs,
p2x2 and p1x1, and--

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so maybe I have it here.

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I'm not finding
the slide I wanted

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that I thought was going to
be right in front of me, but--

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STUDENT: It's in a
previous page, the--

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yeah, yeah.

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It was just a statement, yeah.

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ROBERT TOWNSEND: So we
have these two pairs

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and they're not the same, so
there's a real experiment here.

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And we start with
the second pair.

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Under prices p2, they choose x2.

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They could have chosen
x1, but did not.

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They could have chosen
because x1 is in the budget

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and they did not.

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So they revealed to
prefer x2, quote-unquote.

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But if that's true,
then when at price is p1

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when they're choosing x1,
x2 should not be available.

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Because if it were
available at prices p1,

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having already deduced
that x2 is preferred to x1,

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they would likewise choose
x2 in that situation and that

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would be a contradiction.

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So thank you, yeah,
you said it fine.

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OK.

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So again, those
are just sample--

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somewhat not-so-randomly-sampled
questions,

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but there are others
there I could have asked.

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I was tempted to go over
the thing about the Slutsky

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matrix again and the concavity
of the expenditure function

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and so on, but I--

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and how we end up being able
to deduce restrictions on data

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from the Hicksian demand
which we don't observe.

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But anyway, I didn't ask.

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So OK.

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So that's-- oh.

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And then the one
other thing, which

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is actually the reason
I had this open--

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wow.

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I'm having trouble
manipulating the slide deck.

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This Echenique.

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So one of you asked me last
time, I guess in my office hour

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on Thursday, what's
going on here?

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It's actually all on the slide.

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I took the time this morning
to look up the paper on the--

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I don't have the book.

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The book is in my
MIT econ office

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and I have been in
there for nine months.

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But portions of the
paper are online.

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And it's algorithmic
and just saying,

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if you have the data, like
the kind of data we have in

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revealed preference--

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so we see prices and quantities
and we're trying to--

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in integrability, we're
trying to deduce the utility

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function or a utility function
that would generate the data.

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So the issue is, if
you had a utility

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function the household
would be maximizing

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it to generate
the data, how hard

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is it to maximize
the utility function?

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And the spirit of
this was, there

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may be a whole set
of utility functions.

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There may be some
that would generate

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the data for which
it would be actually

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hard to compute the solution.

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Like Lagrangians are
iterating over possibilities

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and taking local derivatives
and so on and so forth.

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When you only have a couple
of commodities and so on,

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it's obvious graphically
what the maximum is.

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But in higher dimensional
spaces, it's not so obvious.

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And the point of
this article was

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only to say that if you
choose a utility function that

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is consistent with the data
that is hard to compute,

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there's another one that would
generate the same data that

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for which the
algorithm increases

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in complexity as a polynomial
with the number of goods.

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So it's definitely what you
would have thought all along.

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It's a computer science
paper about algorithms.

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And to get in into it
in any more detail,

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one would actually have
to look at the algorithm

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and deduce why it's
only increasing

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in polynomial complexity.

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Anyway, I just wanted to
clarify that since one of you

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asked me this last time.

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OK.

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So for today, the failure
of the welfare theorems.

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Sounds ominous.

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OK.

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So here's an outline
of the lecture.

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We'll do failure of the
second welfare theorem

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first, because ironically,
it's kind of easier.

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Then we'll go to failure of the
first welfare theorem, which

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is harder in some sense
because it's so easy for it

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to be satisfied,
but will feature

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local satiation
which violates one

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of the sufficient assumptions.

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And then when other reasons for
failure of the welfare theorem

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have to do with
incomplete markets

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where there is not a
market in every good.

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Pollution is an example of
that, and you may or may not

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be aware of markets
for pollution rights

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as in cap and trade, and
I'll go through that.

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The generalization of
that is when households

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are altruistic about each
other or jealous of each other,

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they really care about what
other households are consuming

00:13:07.200 --> 00:13:09.330 align:middle line:84%
in a positive or
negative way, that

00:13:09.330 --> 00:13:13.440 align:middle line:84%
generates another externality,
but in that Arrow article,

00:13:13.440 --> 00:13:15.030 align:middle line:84%
there is a way to
solve the problem,

00:13:15.030 --> 00:13:18.690 align:middle line:84%
by creating markets
for the missing goods.

00:13:18.690 --> 00:13:22.680 align:middle line:84%
Then I'll get into another
failure of the welfare theorem.

00:13:22.680 --> 00:13:25.710 align:middle line:84%
Go through the
proof that we used

00:13:25.710 --> 00:13:27.960 align:middle line:84%
and show you where it
could fail if there's

00:13:27.960 --> 00:13:31.590 align:middle line:84%
an infinite number of goods,
which brings us to overlapping

00:13:31.590 --> 00:13:34.320 align:middle line:84%
generations, nobody
lives forever,

00:13:34.320 --> 00:13:37.760 align:middle line:84%
but the economy hopefully
goes on forever.

00:13:37.760 --> 00:13:40.290 align:middle line:84%
And we have money
in the economy.

00:13:40.290 --> 00:13:44.850 align:middle line:84%
So that will take us to
the lecture next Thursday.

00:13:44.850 --> 00:13:47.900 align:middle line:84%
And finally, I want
to touch bases again

00:13:47.900 --> 00:13:52.160 align:middle line:84%
on these platforms for trading
cryptocurrency or anything

00:13:52.160 --> 00:13:55.970 align:middle line:84%
else for that matter,
which seem to be very far

00:13:55.970 --> 00:13:59.660 align:middle line:84%
away from the welfare theorems
and the frameworks we're using.

00:13:59.660 --> 00:14:02.600 align:middle line:84%
It looks like there's
an externality again.

00:14:02.600 --> 00:14:04.640 align:middle line:84%
And the theorem ought
to fail, but I'll

00:14:04.640 --> 00:14:08.630 align:middle line:84%
show you how to design
those markets in such a way

00:14:08.630 --> 00:14:11.060 align:middle line:90%
as to recover optimality.

00:14:11.060 --> 00:14:13.040 align:middle line:84%
So at that point, we
kind of crossed the line

00:14:13.040 --> 00:14:15.350 align:middle line:84%
from positive to
normative economics.

00:14:15.350 --> 00:14:19.820 align:middle line:84%
We're using economic science
to advise policymakers

00:14:19.820 --> 00:14:21.300 align:middle line:90%
on what to do.

00:14:21.300 --> 00:14:24.720 align:middle line:84%
So here's a statement of
the second welfare theorem,

00:14:24.720 --> 00:14:27.620 align:middle line:84%
which is literally pasted
in from earlier lectures.

00:14:27.620 --> 00:14:30.200 align:middle line:84%
So you know everything
on this slide.

00:14:30.200 --> 00:14:32.810 align:middle line:84%
An economy consists
of consumption sets

00:14:32.810 --> 00:14:36.260 align:middle line:84%
and preferences and
endowments, production assets,

00:14:36.260 --> 00:14:38.060 align:middle line:90%
ownership shares.

00:14:38.060 --> 00:14:39.530 align:middle line:84%
And for the second
welfare theorem,

00:14:39.530 --> 00:14:41.090 align:middle line:90%
we kind of needed a lot.

00:14:41.090 --> 00:14:44.270 align:middle line:84%
We needed convexity of
the consumption sets,

00:14:44.270 --> 00:14:48.530 align:middle line:84%
we needed the production
sets to be convex and admit

00:14:48.530 --> 00:14:50.960 align:middle line:90%
this concave transformation.

00:14:50.960 --> 00:14:56.150 align:middle line:84%
Preferences, also concave,
locally non-satiated.

00:14:56.150 --> 00:15:00.378 align:middle line:84%
And these kinds of interior
points, so to speak.

00:15:00.378 --> 00:15:02.170 align:middle line:84%
And then we have a
statement of the theorem

00:15:02.170 --> 00:15:05.500 align:middle line:84%
that for any lambda, hence,
for any Pareto optimum,

00:15:05.500 --> 00:15:09.310 align:middle line:84%
there exists a price vector and
an assignment of wealth that

00:15:09.310 --> 00:15:12.790 align:middle line:84%
would constitute a Walrasian
equilibrium with transfers

00:15:12.790 --> 00:15:14.930 align:middle line:90%
with those wealth assignments.

00:15:14.930 --> 00:15:19.630 align:middle line:84%
So the lecture where we went
through this featured two ways

00:15:19.630 --> 00:15:21.040 align:middle line:90%
to attack the problem.

00:15:21.040 --> 00:15:25.150 align:middle line:84%
One, the Lagrangian problem
both for the Pareto problem

00:15:25.150 --> 00:15:29.200 align:middle line:84%
with the lambda weights and for
each of the firm and household

00:15:29.200 --> 00:15:31.330 align:middle line:84%
maximization
problems, then we just

00:15:31.330 --> 00:15:33.730 align:middle line:84%
matched up all the
Lagrange multipliers,

00:15:33.730 --> 00:15:36.130 align:middle line:90%
but that's that only work--

00:15:36.130 --> 00:15:40.540 align:middle line:84%
typically only works for sure
if you have all this convexity

00:15:40.540 --> 00:15:42.010 align:middle line:90%
and concavity.

00:15:42.010 --> 00:15:46.180 align:middle line:84%
And the second way we did it was
just separating and supporting

00:15:46.180 --> 00:15:51.130 align:middle line:84%
hyperplane theorem which,
in turn, relied on concavity

00:15:51.130 --> 00:15:53.430 align:middle line:90%
or convexity of the steps.

00:15:53.430 --> 00:15:59.920 align:middle line:84%
So on that note, here's an
illustration when it works,

00:15:59.920 --> 00:16:03.450 align:middle line:84%
suppose you had a single
representative consumer,

00:16:03.450 --> 00:16:08.550 align:middle line:84%
this could be our Gorman man,
maximizing utility subject

00:16:08.550 --> 00:16:11.970 align:middle line:84%
to production possibility
set-- and this is obviously

00:16:11.970 --> 00:16:14.040 align:middle line:90%
the overall maximum.

00:16:14.040 --> 00:16:17.910 align:middle line:84%
So then we can create
this budget line

00:16:17.910 --> 00:16:21.210 align:middle line:84%
for the household
and a isoprofit line

00:16:21.210 --> 00:16:25.530 align:middle line:84%
for the producer which
separates these two convex

00:16:25.530 --> 00:16:29.190 align:middle line:84%
sets, the production set
from the weak upper contour

00:16:29.190 --> 00:16:32.850 align:middle line:84%
set of bundles that give
you at least as much utility

00:16:32.850 --> 00:16:35.490 align:middle line:90%
as that this red dot.

00:16:35.490 --> 00:16:39.430 align:middle line:84%
What if preferences
were not convex?

00:16:39.430 --> 00:16:45.000 align:middle line:84%
So we have a nice pink convex
production possibilities set,

00:16:45.000 --> 00:16:47.730 align:middle line:84%
but this is an
illustrative indifference

00:16:47.730 --> 00:16:50.290 align:middle line:90%
curve with all these waves.

00:16:50.290 --> 00:16:55.350 align:middle line:84%
So it has concave,
but convex portions.

00:16:55.350 --> 00:16:58.680 align:middle line:84%
Or alternatively,
the upper contour set

00:16:58.680 --> 00:17:01.020 align:middle line:84%
of utility points
which are greater

00:17:01.020 --> 00:17:03.930 align:middle line:84%
than or equal to the utility
achieved at the red dot, that's

00:17:03.930 --> 00:17:07.800 align:middle line:84%
not a convex set as
you can see by taking

00:17:07.800 --> 00:17:12.510 align:middle line:84%
say these two points in the blue
set in a weighted combination

00:17:12.510 --> 00:17:15.569 align:middle line:84%
of those two points would
not be in the blue set.

00:17:15.569 --> 00:17:19.079 align:middle line:84%
Another way to say that is
at this budget line, which

00:17:19.079 --> 00:17:23.040 align:middle line:84%
separated on the previous
slide, would not separate here,

00:17:23.040 --> 00:17:25.349 align:middle line:84%
because although we have
profit maximization,

00:17:25.349 --> 00:17:28.380 align:middle line:84%
we do not have
utility maximization,

00:17:28.380 --> 00:17:31.680 align:middle line:84%
and you can imagine
doing better here,

00:17:31.680 --> 00:17:35.130 align:middle line:84%
say, a tangency that would
achieve a higher level

00:17:35.130 --> 00:17:38.350 align:middle line:90%
of utility than a red dot.

00:17:38.350 --> 00:17:41.370 align:middle line:84%
So the second welfare
theorem would fail.

00:17:41.370 --> 00:17:43.410 align:middle line:84%
This is a competitive
equilibrium,

00:17:43.410 --> 00:17:46.920 align:middle line:90%
but it's not Pareto optimum.

00:17:46.920 --> 00:17:52.160 align:middle line:84%
Over here, we have
non-convexity of the technology.

00:17:52.160 --> 00:17:55.300 align:middle line:84%
So the upper contour
set is convex,

00:17:55.300 --> 00:17:59.170 align:middle line:84%
but you see these
non-concave portions.

00:17:59.170 --> 00:18:01.360 align:middle line:84%
So there's not a
concave transformation

00:18:01.360 --> 00:18:04.090 align:middle line:84%
representing this production
possibility set of-- production

00:18:04.090 --> 00:18:06.670 align:middle line:90%
possibility set is not convex.

00:18:06.670 --> 00:18:08.740 align:middle line:84%
So again, the
candidate here would

00:18:08.740 --> 00:18:12.820 align:middle line:84%
be the budget line for
the household for which we

00:18:12.820 --> 00:18:15.010 align:middle line:90%
have utility maximization.

00:18:15.010 --> 00:18:19.510 align:middle line:84%
At that budget line, we would
not have profit maximization

00:18:19.510 --> 00:18:23.230 align:middle line:84%
because the producer
could be doing better.

00:18:23.230 --> 00:18:26.380 align:middle line:84%
I think I misstated
the result over here.

00:18:26.380 --> 00:18:28.720 align:middle line:90%
We have a Pareto optimum.

00:18:28.720 --> 00:18:30.550 align:middle line:84%
In the second welfare
theorem we start

00:18:30.550 --> 00:18:32.870 align:middle line:90%
with Pareto optimal points.

00:18:32.870 --> 00:18:34.990 align:middle line:84%
And then we're trying
to decentralize it.

00:18:34.990 --> 00:18:38.080 align:middle line:84%
In this case, we would
have profit maximization,

00:18:38.080 --> 00:18:40.420 align:middle line:84%
but we would not have
utility maximization.

00:18:40.420 --> 00:18:44.050 align:middle line:84%
In this case, we would have
utility maximization but not

00:18:44.050 --> 00:18:46.100 align:middle line:90%
profit maximization.

00:18:46.100 --> 00:18:48.910 align:middle line:84%
So these are perfectly
fine Pareto-optimal points,

00:18:48.910 --> 00:18:52.840 align:middle line:84%
they're just not decentralizable
and the second welfare theorem

00:18:52.840 --> 00:18:54.600 align:middle line:90%
fails.

00:18:54.600 --> 00:18:57.240 align:middle line:84%
Now it doesn't just because
we don't have the convexity

00:18:57.240 --> 00:19:00.330 align:middle line:90%
doesn't mean it always fails.

00:19:00.330 --> 00:19:02.070 align:middle line:90%
This is illustrative.

00:19:02.070 --> 00:19:04.440 align:middle line:84%
I've shown you this
picture before.

00:19:04.440 --> 00:19:08.670 align:middle line:84%
We have this upper contour
set with various wiggles in it

00:19:08.670 --> 00:19:12.630 align:middle line:84%
and the production possibility
set with portions of increasing

00:19:12.630 --> 00:19:17.760 align:middle line:84%
returns, but nevertheless, at
this optimum and the associated

00:19:17.760 --> 00:19:21.540 align:middle line:84%
supporting hyperplane, we
would have utility maximization

00:19:21.540 --> 00:19:23.530 align:middle line:90%
and profit maximization.

00:19:23.530 --> 00:19:26.910 align:middle line:84%
So we might get lucky even
with the non-convexities,

00:19:26.910 --> 00:19:29.130 align:middle line:90%
but it's just not guaranteed.

00:19:29.130 --> 00:19:31.290 align:middle line:84%
Those assumptions in the
second welfare theorem

00:19:31.290 --> 00:19:35.210 align:middle line:84%
are sufficient
assumptions, not necessary.

00:19:35.210 --> 00:19:36.450 align:middle line:90%
First welfare theorem.

00:19:36.450 --> 00:19:38.610 align:middle line:84%
And again, I've kind
of, quote-unquote,

00:19:38.610 --> 00:19:41.250 align:middle line:84%
just pasted in this
slide, let's start

00:19:41.250 --> 00:19:43.900 align:middle line:84%
with a price equilibrium
with transfers,

00:19:43.900 --> 00:19:46.140 align:middle line:84%
including, say,
Walrasian equilibrium

00:19:46.140 --> 00:19:49.230 align:middle line:90%
with x star, y star, p, and w.

00:19:49.230 --> 00:19:51.960 align:middle line:84%
And then if preferences
are rational

00:19:51.960 --> 00:19:55.830 align:middle line:84%
and locally non-satiated,
x star is a Pareto optimum.

00:19:55.830 --> 00:19:56.940 align:middle line:90%
That's the theorem.

00:19:56.940 --> 00:19:58.770 align:middle line:84%
Start with a
competitive equilibrium,

00:19:58.770 --> 00:20:02.100 align:middle line:84%
it will be Pareto optimum
if it's sufficient--

00:20:02.100 --> 00:20:06.000 align:middle line:84%
that is to say, to have
rational locally non-satiated

00:20:06.000 --> 00:20:06.600 align:middle line:90%
preferences.

00:20:06.600 --> 00:20:09.360 align:middle line:84%
I remember in presenting
this, I emphasized

00:20:09.360 --> 00:20:12.960 align:middle line:84%
how weak the assumptions were,
because you might think they're

00:20:12.960 --> 00:20:15.900 align:middle line:84%
always satisfied, hence,
every competitive equilibrium

00:20:15.900 --> 00:20:20.230 align:middle line:84%
or equilibrium with transfers
would be Pareto-optimal.

00:20:20.230 --> 00:20:23.670 align:middle line:84%
However, I'll show you a
picture of where it's violated

00:20:23.670 --> 00:20:26.160 align:middle line:84%
and we're going to
pick on this local--

00:20:26.160 --> 00:20:29.400 align:middle line:84%
I'm going to show you an
example of locally-satiated

00:20:29.400 --> 00:20:32.640 align:middle line:84%
preferences, and hence, failure
the first welfare theorem,

00:20:32.640 --> 00:20:33.870 align:middle line:90%
and here we are.

00:20:33.870 --> 00:20:36.420 align:middle line:90%
So this is the Edgeworth box.

00:20:36.420 --> 00:20:38.430 align:middle line:84%
One household's over
here in the southwest

00:20:38.430 --> 00:20:40.590 align:middle line:90%
another here in the northeast.

00:20:40.590 --> 00:20:43.180 align:middle line:84%
Southwest one's easy
to see what's going on.

00:20:43.180 --> 00:20:45.240 align:middle line:84%
There's an endowment
point, the budget line

00:20:45.240 --> 00:20:47.160 align:middle line:84%
runs through the
endowment point.

00:20:47.160 --> 00:20:50.550 align:middle line:84%
This household southwest
is maximizing utility

00:20:50.550 --> 00:20:53.370 align:middle line:84%
according to the red
indifference curves

00:20:53.370 --> 00:20:57.630 align:middle line:84%
and has achieved the highest
possible utility level

00:20:57.630 --> 00:21:00.330 align:middle line:90%
within his or her budget set.

00:21:00.330 --> 00:21:04.620 align:middle line:84%
Household over here, northeast,
has indifference curves

00:21:04.620 --> 00:21:07.380 align:middle line:84%
that are, in general,
monotone-increasing

00:21:07.380 --> 00:21:09.510 align:middle line:90%
as you move southwest.

00:21:09.510 --> 00:21:12.570 align:middle line:84%
However this
shaded-in blue region

00:21:12.570 --> 00:21:15.600 align:middle line:84%
is the region of
local satiation.

00:21:15.600 --> 00:21:19.560 align:middle line:84%
So the idea is that that
household has an indifference

00:21:19.560 --> 00:21:22.920 align:middle line:84%
band, indifference curve
band, not just curves,

00:21:22.920 --> 00:21:25.230 align:middle line:84%
but this whole
interval of points

00:21:25.230 --> 00:21:27.600 align:middle line:84%
are such that the
utility level is

00:21:27.600 --> 00:21:31.530 align:middle line:84%
the same for this
northeastern household.

00:21:31.530 --> 00:21:35.810 align:middle line:84%
And maybe just in that region,
maybe the rest doesn't matter.

00:21:35.810 --> 00:21:38.400 align:middle line:84%
OK, so what's the
first welfare theorem?

00:21:38.400 --> 00:21:41.280 align:middle line:84%
It says that the competitive
equilibrium is optimal.

00:21:41.280 --> 00:21:43.560 align:middle line:84%
Here's the competitive
equilibrium.

00:21:43.560 --> 00:21:47.190 align:middle line:84%
Household southwest is
certainly maximizing utility.

00:21:47.190 --> 00:21:50.370 align:middle line:84%
Household northeast
is maximizing utility

00:21:50.370 --> 00:21:53.580 align:middle line:84%
because this point gives
at least as much utility

00:21:53.580 --> 00:21:56.910 align:middle line:84%
and maybe more than any other
point in household's northeast

00:21:56.910 --> 00:21:59.650 align:middle line:90%
budget set in this whole region.

00:21:59.650 --> 00:22:01.410 align:middle line:84%
So it is a competitive
equilibrium.

00:22:01.410 --> 00:22:03.630 align:middle line:90%
But is it Pareto-optimal?

00:22:03.630 --> 00:22:05.100 align:middle line:90%
No.

00:22:05.100 --> 00:22:06.360 align:middle line:90%
Why?

00:22:06.360 --> 00:22:09.360 align:middle line:84%
Because we could move in
this northeast direction

00:22:09.360 --> 00:22:12.870 align:middle line:84%
and increase the utility
of the southwest household

00:22:12.870 --> 00:22:17.220 align:middle line:84%
without hurting the utility
of the northeast household.

00:22:17.220 --> 00:22:18.720 align:middle line:90%
So there you go.

00:22:18.720 --> 00:22:20.680 align:middle line:90%
That's an illustration.

00:22:20.680 --> 00:22:24.510 align:middle line:84%
So we definitely use
implicitly this assumption

00:22:24.510 --> 00:22:27.060 align:middle line:90%
about local non-satiation.

00:22:27.060 --> 00:22:29.520 align:middle line:84%
And I could have
drawn pictures that

00:22:29.520 --> 00:22:32.940 align:middle line:84%
had to do with discrete points
and consumption sets which

00:22:32.940 --> 00:22:36.870 align:middle line:84%
also violates local satiation
because there's nothing nearby.

00:22:36.870 --> 00:22:39.510 align:middle line:90%
OK, so let me go back.

00:22:39.510 --> 00:22:41.370 align:middle line:84%
Another thing that's
easy to forget

00:22:41.370 --> 00:22:45.270 align:middle line:84%
in this definition of an economy
and the associated welfare

00:22:45.270 --> 00:22:50.160 align:middle line:84%
theorems is we have L goods,
say, finite dimensional,

00:22:50.160 --> 00:22:53.910 align:middle line:84%
and households have
endowment and preferences

00:22:53.910 --> 00:22:58.320 align:middle line:84%
over consumption for each
and every item in that L

00:22:58.320 --> 00:22:59.710 align:middle line:90%
dimensional space.

00:22:59.710 --> 00:23:02.850 align:middle line:84%
So when we're coming
up with these prices p,

00:23:02.850 --> 00:23:05.520 align:middle line:84%
we're pricing each
and every commodity

00:23:05.520 --> 00:23:08.130 align:middle line:90%
in the entire commodity space.

00:23:08.130 --> 00:23:12.210 align:middle line:84%
Another language we could use is
incomplete markets rather than

00:23:12.210 --> 00:23:13.320 align:middle line:90%
complete markets.

00:23:13.320 --> 00:23:16.110 align:middle line:84%
There could be items that
households care about

00:23:16.110 --> 00:23:19.920 align:middle line:84%
and that firms can produce for
which there is no price system.

00:23:19.920 --> 00:23:21.420 align:middle line:90%
You can't trade it.

00:23:21.420 --> 00:23:25.980 align:middle line:84%
And that will also cause the
first welfare theorem to fail.

00:23:25.980 --> 00:23:28.530 align:middle line:84%
So I'm going to give you a
well-known example, which

00:23:28.530 --> 00:23:30.160 align:middle line:90%
is pollution.

00:23:30.160 --> 00:23:34.470 align:middle line:84%
So in the example there are
two goods, one representative

00:23:34.470 --> 00:23:37.500 align:middle line:90%
consumer, and one firm.

00:23:37.500 --> 00:23:41.220 align:middle line:84%
The firm can transform
one of the goods, good 1,

00:23:41.220 --> 00:23:45.600 align:middle line:84%
into the second good, good 2,
according to this production

00:23:45.600 --> 00:23:46.930 align:middle line:90%
function.

00:23:46.930 --> 00:23:50.890 align:middle line:84%
And remember, for firms,
inputs are negative.

00:23:50.890 --> 00:23:53.920 align:middle line:84%
So minus y1 is the
positive quantity

00:23:53.920 --> 00:23:57.730 align:middle line:84%
of the input of the first good
mapped through the production

00:23:57.730 --> 00:24:02.990 align:middle line:84%
function F to the output
of the second good.

00:24:02.990 --> 00:24:06.570 align:middle line:84%
Now the idea here is that
the firm generates pollution

00:24:06.570 --> 00:24:09.180 align:middle line:90%
when it produces a second good.

00:24:09.180 --> 00:24:13.170 align:middle line:84%
The act of transforming
good 1 to good 2

00:24:13.170 --> 00:24:15.750 align:middle line:84%
produces this level of
pollution, but for the minute

00:24:15.750 --> 00:24:18.240 align:middle line:84%
we're not going to have
a market in pollution.

00:24:18.240 --> 00:24:19.710 align:middle line:90%
It's a genuine good--

00:24:19.710 --> 00:24:24.600 align:middle line:84%
actually, a bad, but for
the moment no prices on it.

00:24:24.600 --> 00:24:30.720 align:middle line:84%
Preferences and production
both specify this bad pollution

00:24:30.720 --> 00:24:31.350 align:middle line:90%
good.

00:24:31.350 --> 00:24:34.590 align:middle line:84%
Households care about their
consumption of good 1, good 2,

00:24:34.590 --> 00:24:38.910 align:middle line:84%
but also this pollution
P. And I should have said,

00:24:38.910 --> 00:24:44.700 align:middle line:84%
down here in red, consumer 1
begins with 1 unit of good 1

00:24:44.700 --> 00:24:46.290 align:middle line:90%
and no units of good 2.

00:24:46.290 --> 00:24:49.830 align:middle line:84%
So if the consumer is going
to get a hold of good 2,

00:24:49.830 --> 00:24:53.070 align:middle line:84%
it has to come from
firms producing it

00:24:53.070 --> 00:24:55.650 align:middle line:84%
with the input
surrendered by good--

00:24:55.650 --> 00:24:58.350 align:middle line:84%
of the household of
good 1 as an input.

00:24:58.350 --> 00:24:59.770 align:middle line:90%
OK.

00:24:59.770 --> 00:25:01.980 align:middle line:84%
So the utility
maximization problem

00:25:01.980 --> 00:25:05.070 align:middle line:84%
is just maximized
consumption of good 1

00:25:05.070 --> 00:25:10.530 align:middle line:84%
and good 2 with the fallout
that there's pollution, P,

00:25:10.530 --> 00:25:12.315 align:middle line:90%
as a consequence of y2.

00:25:12.315 --> 00:25:16.980 align:middle line:84%
y2, the second good is exactly
what the consumer is eating,

00:25:16.980 --> 00:25:19.920 align:middle line:84%
but y2 comes from
production using

00:25:19.920 --> 00:25:21.750 align:middle line:90%
the input of the first good.

00:25:21.750 --> 00:25:25.890 align:middle line:84%
And the input of the first
good, being negative,

00:25:25.890 --> 00:25:28.680 align:middle line:84%
subtracts off from
the unit endowment

00:25:28.680 --> 00:25:31.410 align:middle line:84%
that the household
has of that good,

00:25:31.410 --> 00:25:35.070 align:middle line:84%
the residual being household's
consumption of good 1.

00:25:35.070 --> 00:25:39.060 align:middle line:84%
OK, so this is like
our Pareto problem

00:25:39.060 --> 00:25:42.240 align:middle line:84%
with a couple of simplifications
and one complication.

00:25:42.240 --> 00:25:46.080 align:middle line:84%
There's only one household, no
need for the lambda weights.

00:25:46.080 --> 00:25:50.050 align:middle line:84%
You can think of this as a
Gorman person if you want.

00:25:50.050 --> 00:25:52.840 align:middle line:84%
And there's only one
firm, although we've

00:25:52.840 --> 00:25:54.840 align:middle line:90%
done that before.

00:25:54.840 --> 00:25:57.810 align:middle line:84%
And now we've had we have
this non-traded good,

00:25:57.810 --> 00:26:04.120 align:middle line:84%
this pollution aspect jointly
produced along with y2.

00:26:04.120 --> 00:26:04.720 align:middle line:90%
OK.

00:26:04.720 --> 00:26:07.880 align:middle line:90%
So this is a Pareto problem.

00:26:07.880 --> 00:26:10.750 align:middle line:90%
So I'm about to determine--

00:26:10.750 --> 00:26:15.860 align:middle line:84%
characterize the set of all
Pareto-optimal allocations.

00:26:15.860 --> 00:26:20.630 align:middle line:84%
We do that by maximizing
utility subject to effectively

00:26:20.630 --> 00:26:23.960 align:middle line:84%
the production set and
the resource constraints.

00:26:23.960 --> 00:26:28.010 align:middle line:84%
I've lined up the Lagrange
multipliers on equations 5, 6,

00:26:28.010 --> 00:26:31.940 align:middle line:90%
and 7 and ordered them 1, 2, 3.

00:26:31.940 --> 00:26:35.540 align:middle line:84%
So those are the gammas,
those are the shadow prices.

00:26:35.540 --> 00:26:40.080 align:middle line:84%
OK so everything's
convex in this world.

00:26:40.080 --> 00:26:42.560 align:middle line:84%
So we can do the
first-order conditions

00:26:42.560 --> 00:26:45.530 align:middle line:84%
for the Lagrangian
differentiating with respect

00:26:45.530 --> 00:26:50.180 align:middle line:84%
to x1, x2; consumption
for the households,

00:26:50.180 --> 00:26:54.965 align:middle line:84%
y1 and y2, inputs and
outputs for the firm.

00:26:54.965 --> 00:26:56.900 align:middle line:90%
You can check me.

00:26:56.900 --> 00:27:01.220 align:middle line:84%
At this point, you should be
familiar with the Lagrangians.

00:27:01.220 --> 00:27:03.260 align:middle line:84%
You may want to go
back and review it.

00:27:03.260 --> 00:27:06.170 align:middle line:84%
The only subtlety here is
to get the signs right.

00:27:06.170 --> 00:27:08.360 align:middle line:84%
When we write down
these constraints,

00:27:08.360 --> 00:27:12.300 align:middle line:84%
we should write them as greater
than or equal to inequalities

00:27:12.300 --> 00:27:16.610 align:middle line:84%
so that we get the interpretable
sign on the shadow prices.

00:27:16.610 --> 00:27:18.380 align:middle line:84%
Well, they should
look kind of familiar.

00:27:18.380 --> 00:27:22.430 align:middle line:84%
This says the marginal utility
of good 1 is equal to lambda 1.

00:27:22.430 --> 00:27:25.160 align:middle line:84%
The marginal utility of
good 2 equal to lambda 2.

00:27:25.160 --> 00:27:27.560 align:middle line:84%
And the household's
utility function.

00:27:27.560 --> 00:27:32.420 align:middle line:84%
This is a statement about the
marginal product of good 1

00:27:32.420 --> 00:27:33.930 align:middle line:90%
in production.

00:27:33.930 --> 00:27:37.010 align:middle line:84%
And this is a statement
about how the Lagrangian is

00:27:37.010 --> 00:27:39.560 align:middle line:90%
changing as we change y2.

00:27:39.560 --> 00:27:44.510 align:middle line:84%
Here and only here you can
see that the Pareto optimum

00:27:44.510 --> 00:27:47.090 align:middle line:84%
is taking into account
the extent to which

00:27:47.090 --> 00:27:49.650 align:middle line:84%
pollution is created
by production.

00:27:49.650 --> 00:27:53.840 align:middle line:84%
And if you go back and look
over here, the y2 is written--

00:27:53.840 --> 00:27:57.950 align:middle line:84%
P is pollution written
as a function of y2.

00:27:57.950 --> 00:28:01.190 align:middle line:84%
So when we differentiate
with respect to y2,

00:28:01.190 --> 00:28:05.480 align:middle line:84%
we pick up that partial P,
partial y2 in the household's

00:28:05.480 --> 00:28:07.070 align:middle line:90%
utility function.

00:28:07.070 --> 00:28:09.830 align:middle line:84%
And of course, we pick up y2
and the two other spots where

00:28:09.830 --> 00:28:12.290 align:middle line:90%
it enters in equation 6 and 7.

00:28:12.290 --> 00:28:15.170 align:middle line:84%
So that's where this
thing is coming from.

00:28:15.170 --> 00:28:17.630 align:middle line:90%
OK, so the rest is algebra.

00:28:17.630 --> 00:28:21.920 align:middle line:84%
You can see that we could
solve for the inverse.

00:28:21.920 --> 00:28:27.710 align:middle line:84%
If we took df to dy1 and set it
equal to lambda 1 over lambda

00:28:27.710 --> 00:28:32.870 align:middle line:84%
3, we could then go over here
and find lambda 3 and lambda 1

00:28:32.870 --> 00:28:34.490 align:middle line:90%
and make some substitutions.

00:28:34.490 --> 00:28:37.170 align:middle line:84%
Actually, we flipped
it upside-down.

00:28:37.170 --> 00:28:40.310 align:middle line:84%
Instead of df dy1,
we take 1 over that,

00:28:40.310 --> 00:28:43.430 align:middle line:84%
but we do go and look for
those substitutions, which

00:28:43.430 --> 00:28:48.230 align:middle line:84%
would be gamma 3, which
picks up this mess over here,

00:28:48.230 --> 00:28:51.270 align:middle line:84%
and gamma 1, which
picks up this guy.

00:28:51.270 --> 00:28:54.986 align:middle line:84%
So that's the algebra
behind this expression.

00:28:54.986 --> 00:28:57.440 align:middle line:90%
OK, so let's interpret it.

00:28:57.440 --> 00:29:00.410 align:middle line:84%
The marginal rate of
substitution for the household

00:29:00.410 --> 00:29:03.350 align:middle line:90%
had a Pareto-optimal allocation.

00:29:03.350 --> 00:29:06.740 align:middle line:84%
And here, we kind of do
something unconventional.

00:29:06.740 --> 00:29:09.500 align:middle line:84%
This is the marginal rate
of substitution of good 2

00:29:09.500 --> 00:29:12.920 align:middle line:84%
for good 1 and not
the other way around.

00:29:12.920 --> 00:29:16.550 align:middle line:84%
Normally if you go back and
look at the consumer lecture 2,

00:29:16.550 --> 00:29:20.360 align:middle line:84%
I guess it was, we had the
marginal rate of substitution

00:29:20.360 --> 00:29:23.990 align:middle line:84%
of good 1 for good
2 being u1 over u2.

00:29:23.990 --> 00:29:28.190 align:middle line:84%
And also the price ratio
we'll get to being P1 over P2.

00:29:28.190 --> 00:29:30.690 align:middle line:84%
So here, it's kind of
flipped the other way around.

00:29:30.690 --> 00:29:33.200 align:middle line:84%
It's not wrong, you
just gotta remember,

00:29:33.200 --> 00:29:36.140 align:middle line:84%
we're redefining the
marginal rate of substitution

00:29:36.140 --> 00:29:38.930 align:middle line:84%
to be the opposite of
what we did before.

00:29:38.930 --> 00:29:39.710 align:middle line:90%
OK.

00:29:39.710 --> 00:29:42.800 align:middle line:84%
So the marginal rate of
substitution for the household

00:29:42.800 --> 00:29:45.920 align:middle line:84%
is the marginal
utility of good 2,

00:29:45.920 --> 00:29:49.010 align:middle line:84%
marginal utility divided by
the marginal utility of good 1,

00:29:49.010 --> 00:29:53.480 align:middle line:84%
but this is equal to, from this
expression, the first part,

00:29:53.480 --> 00:29:56.820 align:middle line:84%
numerator-denominator plus
numerator-denominator.

00:29:56.820 --> 00:30:01.490 align:middle line:84%
So this plus part
numerator-denominator is here,

00:30:01.490 --> 00:30:03.260 align:middle line:90%
it's just rewritten.

00:30:03.260 --> 00:30:05.720 align:middle line:84%
So this is the full
characterization

00:30:05.720 --> 00:30:09.420 align:middle line:84%
of the Pareto-optimal
allocation.

00:30:09.420 --> 00:30:13.190 align:middle line:84%
And it kind of looks like the
marginal rate of substitution

00:30:13.190 --> 00:30:16.070 align:middle line:84%
for the consumer is
equal to something

00:30:16.070 --> 00:30:17.930 align:middle line:84%
having to do with
the marginal product

00:30:17.930 --> 00:30:20.720 align:middle line:84%
or marginal rate of
transformation in production,

00:30:20.720 --> 00:30:23.240 align:middle line:84%
but we have this extra
term and this extra term

00:30:23.240 --> 00:30:26.210 align:middle line:84%
reflects the fact
that to produce y2,

00:30:26.210 --> 00:30:28.040 align:middle line:84%
in order for the
household to get it,

00:30:28.040 --> 00:30:31.940 align:middle line:84%
x1 gets surrendered, put
in the production function,

00:30:31.940 --> 00:30:34.070 align:middle line:90%
but it also produces pollution.

00:30:34.070 --> 00:30:38.240 align:middle line:84%
So we're taking into account--
subtracting, as it were,

00:30:38.240 --> 00:30:43.220 align:middle line:84%
the bad thing that happens, the
extra pollution that happens

00:30:43.220 --> 00:30:46.490 align:middle line:84%
as a consequence of producing
good 2 which generates

00:30:46.490 --> 00:30:50.490 align:middle line:84%
pollution P. I'm not taking
you through all the--

00:30:50.490 --> 00:30:53.810 align:middle line:84%
I'm probably taking you through
too much of the algebra.

00:30:53.810 --> 00:30:56.060 align:middle line:84%
You've got to be careful
about these signs, negative

00:30:56.060 --> 00:30:56.990 align:middle line:90%
versus positive.

00:30:56.990 --> 00:30:59.780 align:middle line:84%
I was just pondering
that, but you were just

00:30:59.780 --> 00:31:03.230 align:middle line:84%
subtracting the P's
from the left-hand side.

00:31:03.230 --> 00:31:04.810 align:middle line:90%
OK.

00:31:04.810 --> 00:31:07.180 align:middle line:90%
You can check the algebra later.

00:31:07.180 --> 00:31:10.240 align:middle line:84%
Now we want to
decentralize this optimum

00:31:10.240 --> 00:31:14.740 align:middle line:84%
like we've been doing before
in the first welfare theorem.

00:31:14.740 --> 00:31:17.060 align:middle line:84%
But we're going to
do it in two steps.

00:31:17.060 --> 00:31:20.710 align:middle line:84%
In the first step, we have
a market in the two goods,

00:31:20.710 --> 00:31:22.610 align:middle line:90%
but not in pollution.

00:31:22.610 --> 00:31:25.310 align:middle line:84%
So the firm would
maximize profits.

00:31:25.310 --> 00:31:31.060 align:middle line:84%
Let's assume that the price
of the first good, good 1,

00:31:31.060 --> 00:31:32.710 align:middle line:90%
is the numeraire.

00:31:32.710 --> 00:31:34.840 align:middle line:84%
So I'm not going to
be writing p1 anymore,

00:31:34.840 --> 00:31:37.030 align:middle line:90%
but I will be writing p2.

00:31:37.030 --> 00:31:39.760 align:middle line:84%
So this is the price of
good 2 relative to good 1

00:31:39.760 --> 00:31:41.930 align:middle line:84%
because a good one
is the numeraire.

00:31:41.930 --> 00:31:45.050 align:middle line:84%
This is the output, this
is the revenue of the firm.

00:31:45.050 --> 00:31:47.860 align:middle line:90%
This is the input cost.

00:31:47.860 --> 00:31:49.510 align:middle line:90%
p1 to y1.

00:31:49.510 --> 00:31:53.550 align:middle line:84%
y1 is negative, that's
why it's being subtracted.

00:31:53.550 --> 00:31:56.730 align:middle line:84%
Subject to this
transformation set which

00:31:56.730 --> 00:31:58.930 align:middle line:90%
we've talked about before.

00:31:58.930 --> 00:32:01.080 align:middle line:84%
Now I'm going to
make a substitution.

00:32:01.080 --> 00:32:06.210 align:middle line:84%
If y2 is F of minus y1, then
take the inverse function,

00:32:06.210 --> 00:32:10.890 align:middle line:84%
F inverse on both sides,
and we get F inverse of y2

00:32:10.890 --> 00:32:13.320 align:middle line:90%
equals minus y1.

00:32:13.320 --> 00:32:18.390 align:middle line:84%
Multiply it by a minus sign
and we get y1 equal minus F1

00:32:18.390 --> 00:32:19.830 align:middle line:90%
inverse of y2.

00:32:19.830 --> 00:32:24.090 align:middle line:84%
So I'm going to substitute
this expression into this term

00:32:24.090 --> 00:32:28.050 align:middle line:84%
y1 in the objective-- in the
profit objective function.

00:32:28.050 --> 00:32:30.060 align:middle line:84%
Then when we
differentiate with respect

00:32:30.060 --> 00:32:32.490 align:middle line:84%
to how much y2 to
produce, we've already

00:32:32.490 --> 00:32:35.970 align:middle line:84%
taken into account the amount
of x1 that has to be higher

00:32:35.970 --> 00:32:37.150 align:middle line:90%
to get there.

00:32:37.150 --> 00:32:39.090 align:middle line:84%
The derivative of
this whole expression

00:32:39.090 --> 00:32:43.300 align:middle line:84%
is going to be p for the first
term, that's the revenue,

00:32:43.300 --> 00:32:46.860 align:middle line:84%
and then the derivative
of the second term, which

00:32:46.860 --> 00:32:49.440 align:middle line:84%
is how this inverse production
function is changing

00:32:49.440 --> 00:32:51.850 align:middle line:90%
as we change y2.

00:32:51.850 --> 00:32:54.700 align:middle line:84%
It is the inverse function,
but there's an inverse function

00:32:54.700 --> 00:32:57.550 align:middle line:84%
theorem which you
may not remember,

00:32:57.550 --> 00:33:00.550 align:middle line:84%
which is the derivative
of the inverse function is

00:33:00.550 --> 00:33:03.730 align:middle line:84%
1 over the derivative
of the direct function.

00:33:03.730 --> 00:33:06.750 align:middle line:84%
So this is the derivative of
the direct function mapping

00:33:06.750 --> 00:33:09.240 align:middle line:90%
input y1 into y2.

00:33:09.240 --> 00:33:12.540 align:middle line:84%
So in a competitive
equilibrium, we

00:33:12.540 --> 00:33:16.290 align:middle line:84%
get exactly what I was
alluding to before,

00:33:16.290 --> 00:33:17.700 align:middle line:90%
that the marginal rate of--

00:33:17.700 --> 00:33:19.680 align:middle line:90%
well, I haven't said it yet.

00:33:19.680 --> 00:33:25.320 align:middle line:84%
The marginal revenue product
of a good 1 into production

00:33:25.320 --> 00:33:27.150 align:middle line:90%
is equal to its price.

00:33:27.150 --> 00:33:30.540 align:middle line:84%
And it's also going to be
true that this price is going

00:33:30.540 --> 00:33:34.830 align:middle line:84%
to be equal, from the consumer
side you'll see momentarily

00:33:34.830 --> 00:33:38.820 align:middle line:84%
right here, equal to the
marginal rate of substitution

00:33:38.820 --> 00:33:42.720 align:middle line:84%
in utility terms
for the household.

00:33:42.720 --> 00:33:45.120 align:middle line:84%
So it looks like you
get the right thing

00:33:45.120 --> 00:33:48.180 align:middle line:84%
when you decentralize it,
that the marginal rate

00:33:48.180 --> 00:33:52.050 align:middle line:84%
of substitution is equal to the
rate of product transformation,

00:33:52.050 --> 00:33:54.600 align:middle line:84%
but it won't be the right
thing because it's neglecting

00:33:54.600 --> 00:33:57.960 align:middle line:84%
the indirect impact on
pollution from using

00:33:57.960 --> 00:33:59.860 align:middle line:90%
good 1 in production.

00:33:59.860 --> 00:34:04.080 align:middle line:84%
Here's the other half of
the Walrasian definition.

00:34:04.080 --> 00:34:07.110 align:middle line:84%
This is the household's
maximization problem.

00:34:07.110 --> 00:34:10.260 align:middle line:84%
To maximize utility,
including acknowledging

00:34:10.260 --> 00:34:14.070 align:middle line:84%
the disutility of
pollution subject to having

00:34:14.070 --> 00:34:20.010 align:middle line:84%
1 unit of good 1 valued at p1
equal to 1 plus the profits--

00:34:20.010 --> 00:34:23.730 align:middle line:84%
this profit-- that the firm
produces that's handed back

00:34:23.730 --> 00:34:25.290 align:middle line:90%
lump sum to the household.

00:34:25.290 --> 00:34:27.179 align:middle line:84%
We only have one
household, one firm.

00:34:27.179 --> 00:34:30.270 align:middle line:84%
This household has
complete ownership share

00:34:30.270 --> 00:34:32.010 align:middle line:90%
of profits of the firm.

00:34:32.010 --> 00:34:34.300 align:middle line:84%
And we get this
first-order conditions,

00:34:34.300 --> 00:34:36.610 align:middle line:90%
which is what I've said.

00:34:36.610 --> 00:34:38.310 align:middle line:84%
The marginal rate
of substitution

00:34:38.310 --> 00:34:40.540 align:middle line:90%
is equal to the price ratio.

00:34:40.540 --> 00:34:45.030 align:middle line:84%
So then if we take 11 and
go back and remember 10,

00:34:45.030 --> 00:34:49.139 align:middle line:84%
10 was this thing from
production, and substitute in,

00:34:49.139 --> 00:34:52.199 align:middle line:84%
we'll say the marginal rate of
substitution in an equilibrium

00:34:52.199 --> 00:34:55.230 align:middle line:84%
is equal to the marginal rate
of product transformation,

00:34:55.230 --> 00:34:57.570 align:middle line:84%
but that rate of
product transformation

00:34:57.570 --> 00:35:00.930 align:middle line:84%
is less than what you
would get if you take

00:35:00.930 --> 00:35:02.580 align:middle line:84%
the marginal rate
of transformation

00:35:02.580 --> 00:35:05.070 align:middle line:90%
and add on a positive term.

00:35:05.070 --> 00:35:06.810 align:middle line:84%
It may not be obvious
in the algebra

00:35:06.810 --> 00:35:08.910 align:middle line:84%
right away that this
term is positive,

00:35:08.910 --> 00:35:15.270 align:middle line:84%
but it is, because the marginal
utility is decreasing with P.

00:35:15.270 --> 00:35:17.760 align:middle line:90%
So this is a negative object.

00:35:17.760 --> 00:35:19.210 align:middle line:84%
And you've got a
minus sign here.

00:35:19.210 --> 00:35:21.570 align:middle line:84%
So negative-negative
equals positive.

00:35:21.570 --> 00:35:23.290 align:middle line:84%
This whole thing
is a positive term,

00:35:23.290 --> 00:35:26.370 align:middle line:84%
so that's why we have
this strict inequality.

00:35:26.370 --> 00:35:30.210 align:middle line:84%
This thing on the right-hand
side, though, is, as I said,

00:35:30.210 --> 00:35:34.560 align:middle line:84%
from 10, the marginal rate of
substitution under the optimum,

00:35:34.560 --> 00:35:36.090 align:middle line:90%
this guy here.

00:35:36.090 --> 00:35:38.040 align:middle line:90%
Actually, it was 8.

00:35:38.040 --> 00:35:40.080 align:middle line:90%
We used 10 to get this.

00:35:40.080 --> 00:35:43.170 align:middle line:84%
So the conclusion is the
competitive equilibrium

00:35:43.170 --> 00:35:45.330 align:middle line:84%
is not Pareto-optimal,
because we

00:35:45.330 --> 00:35:47.940 align:middle line:84%
want this expression
to hold as an equality,

00:35:47.940 --> 00:35:50.095 align:middle line:90%
it's holding as in inequality.

00:35:50.095 --> 00:35:51.250 align:middle line:90%
OK.

00:35:51.250 --> 00:35:54.600 align:middle line:84%
So by the way, here's
the economic intuition

00:35:54.600 --> 00:35:56.010 align:middle line:90%
for what's going on.

00:35:56.010 --> 00:35:59.610 align:middle line:84%
This marginal rate of
substitution is too small.

00:35:59.610 --> 00:36:01.590 align:middle line:90%
How do we get it to be bigger?

00:36:01.590 --> 00:36:04.600 align:middle line:84%
We'd like to increase
the numerator.

00:36:04.600 --> 00:36:06.600 align:middle line:84%
How do you get the marginal
utility with respect

00:36:06.600 --> 00:36:08.100 align:middle line:90%
to good 2 be higher?

00:36:08.100 --> 00:36:10.110 align:middle line:90%
Less of good 2.

00:36:10.110 --> 00:36:13.290 align:middle line:84%
And/or get the
denominator to be smaller,

00:36:13.290 --> 00:36:15.660 align:middle line:84%
that would also make the
marginal rate of substitution

00:36:15.660 --> 00:36:16.620 align:middle line:90%
higher.

00:36:16.620 --> 00:36:20.130 align:middle line:84%
How do you get marginal utility
of good 1 to be smaller?

00:36:20.130 --> 00:36:21.600 align:middle line:90%
Increase x1.

00:36:21.600 --> 00:36:23.970 align:middle line:84%
So the economics
here is telling us we

00:36:23.970 --> 00:36:26.640 align:middle line:84%
want to take the
competitive equilibrium

00:36:26.640 --> 00:36:29.580 align:middle line:84%
and move to the direction
of increasing x1,

00:36:29.580 --> 00:36:32.010 align:middle line:84%
eat it rather than
use it as an input

00:36:32.010 --> 00:36:35.130 align:middle line:84%
to produce x2 of which
we have too much.

00:36:35.130 --> 00:36:37.800 align:middle line:84%
So it's intuitive
in the economics

00:36:37.800 --> 00:36:40.980 align:middle line:84%
that if you don't have a
market in the pollution rights,

00:36:40.980 --> 00:36:44.250 align:middle line:84%
you're going to have
too much pollution.

00:36:44.250 --> 00:36:47.590 align:middle line:84%
But we can fix it by having
the pollution rights.

00:36:47.590 --> 00:36:50.050 align:middle line:90%
So what do I mean by that?

00:36:50.050 --> 00:36:54.240 align:middle line:84%
Suppose the household can sell
rights to pollute to the firm

00:36:54.240 --> 00:36:57.360 align:middle line:84%
and that we can enforce
those rights in the sense

00:36:57.360 --> 00:37:00.030 align:middle line:84%
that if the firm doesn't
have the rights, then

00:37:00.030 --> 00:37:01.440 align:middle line:90%
it can't pollute.

00:37:01.440 --> 00:37:05.820 align:middle line:84%
The rights will specify the
number of units of P. Firms

00:37:05.820 --> 00:37:07.770 align:middle line:84%
are going to have
to buy those rights.

00:37:07.770 --> 00:37:09.300 align:middle line:90%
It comes with y2.

00:37:09.300 --> 00:37:11.430 align:middle line:84%
So the level of P
that comes from y2

00:37:11.430 --> 00:37:13.590 align:middle line:84%
has to be consistent
with the rights

00:37:13.590 --> 00:37:19.200 align:middle line:84%
that the firm purchases, and
likewise, consistent at a price

00:37:19.200 --> 00:37:19.980 align:middle line:90%
of rights--

00:37:19.980 --> 00:37:21.780 align:middle line:84%
with the rights
that the household

00:37:21.780 --> 00:37:23.628 align:middle line:90%
is willing to surrender.

00:37:23.628 --> 00:37:25.170 align:middle line:84%
The household is
going to get revenue

00:37:25.170 --> 00:37:27.730 align:middle line:84%
from selling the rights
to pollute to the firms.

00:37:27.730 --> 00:37:29.490 align:middle line:90%
So that's a positive thing.

00:37:29.490 --> 00:37:30.990 align:middle line:84%
But on the other
hand, the household

00:37:30.990 --> 00:37:33.690 align:middle line:84%
knows that the firm will
produce with those rights

00:37:33.690 --> 00:37:35.830 align:middle line:84%
and it will suffer
that pollution.

00:37:35.830 --> 00:37:37.630 align:middle line:84%
So there's a trade-off
for the household,

00:37:37.630 --> 00:37:39.780 align:middle line:84%
but this is the
right trade-off now,

00:37:39.780 --> 00:37:42.870 align:middle line:84%
because we have the market
for the pollution rights.

00:37:42.870 --> 00:37:44.310 align:middle line:90%
OK.

00:37:44.310 --> 00:37:46.920 align:middle line:84%
In notation, we
maximize the utility

00:37:46.920 --> 00:37:51.060 align:middle line:84%
of the household over the two
goods, x1, x2, and pollution

00:37:51.060 --> 00:37:55.740 align:middle line:84%
level P bar subject to the
budget, which is get revenue

00:37:55.740 --> 00:37:57.570 align:middle line:84%
from selling your
endowment of good 1,

00:37:57.570 --> 00:37:58.950 align:middle line:90%
you get profits from the firm.

00:37:58.950 --> 00:38:02.250 align:middle line:84%
You also get revenue from
selling pollution rights

00:38:02.250 --> 00:38:06.810 align:middle line:84%
to the firm, the price
being q, the per-unit price.

00:38:06.810 --> 00:38:09.060 align:middle line:90%
Per unit of pollution is q.

00:38:09.060 --> 00:38:10.990 align:middle line:90%
First time we've used that.

00:38:10.990 --> 00:38:13.980 align:middle line:84%
And if you look at the
solution to this problem

00:38:13.980 --> 00:38:17.130 align:middle line:84%
through Lagrangian's,
we now have three goods,

00:38:17.130 --> 00:38:20.490 align:middle line:90%
good 1, good 2, and p, p bar.

00:38:20.490 --> 00:38:24.540 align:middle line:84%
If you thought about mu
as the Lagrange multiplier

00:38:24.540 --> 00:38:28.170 align:middle line:84%
on this budget constraint
12, you get the obvious

00:38:28.170 --> 00:38:30.180 align:middle line:84%
that the marginal
utilities of the goods

00:38:30.180 --> 00:38:33.030 align:middle line:84%
are equal to the marginal
utility of income

00:38:33.030 --> 00:38:34.890 align:middle line:90%
times the price.

00:38:34.890 --> 00:38:36.840 align:middle line:84%
Again, p1 is equal
to 1, so there's

00:38:36.840 --> 00:38:38.400 align:middle line:90%
no price in the first term.

00:38:38.400 --> 00:38:40.380 align:middle line:84%
Third term is kind
of interesting,

00:38:40.380 --> 00:38:42.430 align:middle line:84%
because it's got
a negative sign.

00:38:42.430 --> 00:38:45.510 align:middle line:84%
But again, pollution
is a bad from the point

00:38:45.510 --> 00:38:46.600 align:middle line:90%
of view of the household.

00:38:46.600 --> 00:38:48.480 align:middle line:84%
So this marginal
utility is negative,

00:38:48.480 --> 00:38:50.700 align:middle line:84%
the negative sign
makes it positive.

00:38:50.700 --> 00:38:52.680 align:middle line:90%
Positive equal positive.

00:38:52.680 --> 00:38:55.230 align:middle line:84%
The marginal utility
of reducing pollution

00:38:55.230 --> 00:38:58.410 align:middle line:84%
is equal to the marginal
utility of income times

00:38:58.410 --> 00:39:00.880 align:middle line:90%
the price of pollution rights.

00:39:00.880 --> 00:39:05.010 align:middle line:84%
So now in an equilibrium,
we've got the marginal rate

00:39:05.010 --> 00:39:06.930 align:middle line:84%
of substitution
over the two goods

00:39:06.930 --> 00:39:09.450 align:middle line:84%
from the household point
of view equal to p.

00:39:09.450 --> 00:39:11.760 align:middle line:84%
We could also look at the
marginal rate of substitution

00:39:11.760 --> 00:39:14.280 align:middle line:90%
of good 1 for pollution P--

00:39:14.280 --> 00:39:17.100 align:middle line:84%
P bar, actually, in
the household problem,

00:39:17.100 --> 00:39:19.050 align:middle line:90%
and that's equal to q.

00:39:19.050 --> 00:39:22.290 align:middle line:84%
So we have three goods
and we have two ratios.

00:39:22.290 --> 00:39:26.050 align:middle line:84%
For the firm's problem, they're
going to maximize profits.

00:39:26.050 --> 00:39:26.880 align:middle line:90%
This is revenue.

00:39:26.880 --> 00:39:28.440 align:middle line:90%
This is costs.

00:39:28.440 --> 00:39:31.980 align:middle line:84%
They have to buy
those rights, same q.

00:39:31.980 --> 00:39:33.870 align:middle line:90%
And rights are an input.

00:39:33.870 --> 00:39:37.140 align:middle line:84%
So we put a negative sign
here, because they've

00:39:37.140 --> 00:39:39.570 align:middle line:84%
got to buy them as an
input, the way we've

00:39:39.570 --> 00:39:41.730 align:middle line:90%
been signing inputs all along.

00:39:41.730 --> 00:39:45.190 align:middle line:84%
And anyway, it's a cost,
so it makes a lot of sense.

00:39:45.190 --> 00:39:47.790 align:middle line:84%
So we rewrite this
problem, substituting

00:39:47.790 --> 00:39:53.460 align:middle line:84%
in this inverse function for
y1, and take derivatives,

00:39:53.460 --> 00:39:57.600 align:middle line:84%
and we get, with respect
to y1, where does it enter?

00:39:57.600 --> 00:40:02.100 align:middle line:90%
It enters here and here.

00:40:02.100 --> 00:40:04.700 align:middle line:90%
y1-- I should have said y2.

00:40:04.700 --> 00:40:08.270 align:middle line:84%
By making the substitution, we
don't have y1 in the expression

00:40:08.270 --> 00:40:09.270 align:middle line:90%
anymore.

00:40:09.270 --> 00:40:12.620 align:middle line:84%
We have y2 in terms
of revenue in y2

00:40:12.620 --> 00:40:16.050 align:middle line:84%
in terms of production and
the associated pollution.

00:40:16.050 --> 00:40:20.600 align:middle line:84%
So then just manipulating
this equation--

00:40:20.600 --> 00:40:23.030 align:middle line:84%
well, sorry, looking at
the first-order condition

00:40:23.030 --> 00:40:27.050 align:middle line:84%
we get this thing, which
is the marginal revenue

00:40:27.050 --> 00:40:29.660 align:middle line:84%
on the left-hand side and
the derivative of these two

00:40:29.660 --> 00:40:31.710 align:middle line:90%
terms on the right-hand side.

00:40:31.710 --> 00:40:34.910 align:middle line:84%
And now we can go
back and do our checks

00:40:34.910 --> 00:40:36.770 align:middle line:90%
of the first-order condition.

00:40:36.770 --> 00:40:39.260 align:middle line:84%
The marginal rate of
substitution for households

00:40:39.260 --> 00:40:41.030 align:middle line:90%
of the two goods is p.

00:40:41.030 --> 00:40:45.390 align:middle line:84%
The p is equal to
this thing from 14.

00:40:45.390 --> 00:40:49.110 align:middle line:84%
So substituted in, leave
the first term the same,

00:40:49.110 --> 00:40:52.520 align:middle line:84%
and then find the
expression for q

00:40:52.520 --> 00:40:55.460 align:middle line:84%
up here from the
consumer's problem,

00:40:55.460 --> 00:40:57.980 align:middle line:90%
and keep writing dP dy2.

00:40:57.980 --> 00:40:59.810 align:middle line:90%
So that's the algebra.

00:40:59.810 --> 00:41:01.550 align:middle line:90%
And now, guess what?

00:41:01.550 --> 00:41:03.320 align:middle line:90%
We've got a full optimum.

00:41:03.320 --> 00:41:06.650 align:middle line:84%
This is exactly the
expression we wanted over

00:41:06.650 --> 00:41:10.220 align:middle line:84%
here in the full
optimum problem, not

00:41:10.220 --> 00:41:13.250 align:middle line:84%
the decentralize problem,
but in the optimum problem.

00:41:13.250 --> 00:41:15.290 align:middle line:84%
The marginal rate
of substitution

00:41:15.290 --> 00:41:17.480 align:middle line:84%
is equal to the
rates of production

00:41:17.480 --> 00:41:21.050 align:middle line:84%
taking into account pollution
and the adverse utility

00:41:21.050 --> 00:41:22.920 align:middle line:90%
consequences of pollution.

00:41:22.920 --> 00:41:27.110 align:middle line:84%
This equation 8 is exactly
what we just derived here

00:41:27.110 --> 00:41:29.490 align:middle line:90%
at the bottom of this slide.

00:41:29.490 --> 00:41:31.370 align:middle line:84%
So we've achieved it,
but we achieved it

00:41:31.370 --> 00:41:34.880 align:middle line:84%
by creating the missing market,
which was, in this case,

00:41:34.880 --> 00:41:36.570 align:middle line:90%
was pollution.

00:41:36.570 --> 00:41:39.810 align:middle line:84%
So OK, so there's a
general lesson here.

00:41:39.810 --> 00:41:41.750 align:middle line:84%
Now you may have heard
of pollution rights

00:41:41.750 --> 00:41:46.100 align:middle line:84%
before under the
terminology cap and trade.

00:41:46.100 --> 00:41:47.960 align:middle line:90%
So that's kind of related.

00:41:47.960 --> 00:41:51.150 align:middle line:84%
Particularly you let firms
pollute all they want,

00:41:51.150 --> 00:41:53.800 align:middle line:84%
but they have to buy
the rights to do it.

00:41:53.800 --> 00:41:58.840 align:middle line:84%
So you-- maybe endow firms
with arbitrary assignments

00:41:58.840 --> 00:42:03.910 align:middle line:84%
of rights, and amazingly,
it's not going to matter who--

00:42:03.910 --> 00:42:06.430 align:middle line:84%
it's not going to matter for
production who is initially

00:42:06.430 --> 00:42:08.290 align:middle line:84%
assigned a high right
to pollute and who's

00:42:08.290 --> 00:42:11.770 align:middle line:84%
assigned a low right
to produce and pollute.

00:42:11.770 --> 00:42:13.930 align:middle line:90%
And then you'll see that now.

00:42:13.930 --> 00:42:17.650 align:middle line:84%
So we have the two
firms, call them a and b.

00:42:17.650 --> 00:42:21.790 align:middle line:84%
They're both endowed with
pollution rights, Pa bar and Pb

00:42:21.790 --> 00:42:23.050 align:middle line:90%
bar.

00:42:23.050 --> 00:42:26.110 align:middle line:84%
And then we look at
the profit maximization

00:42:26.110 --> 00:42:29.070 align:middle line:90%
problem of each firm j.

00:42:29.070 --> 00:42:35.160 align:middle line:84%
To produce revenue from
output less the input required

00:42:35.160 --> 00:42:37.480 align:middle line:84%
of good 1, that's
this inverse function,

00:42:37.480 --> 00:42:40.680 align:middle line:84%
which is what we've been
doing before, except now we

00:42:40.680 --> 00:42:43.950 align:middle line:84%
have a j everywhere--
this is firm type j.

00:42:43.950 --> 00:42:48.630 align:middle line:84%
And then we have
this revenue or cost.

00:42:48.630 --> 00:42:51.030 align:middle line:84%
Q is the price of
pollution rights,

00:42:51.030 --> 00:42:54.250 align:middle line:84%
they're endowed with
pollution rights, P bar.

00:42:54.250 --> 00:42:57.940 align:middle line:84%
So we could stop there and
just say they get that revenue.

00:42:57.940 --> 00:42:59.730 align:middle line:84%
But not if they're
producing, because they

00:42:59.730 --> 00:43:02.040 align:middle line:84%
need to have the rights
corresponding to the amount

00:43:02.040 --> 00:43:04.480 align:middle line:90%
of y2 that they're producing.

00:43:04.480 --> 00:43:07.200 align:middle line:84%
So we subtract off the
corresponding rights

00:43:07.200 --> 00:43:11.340 align:middle line:84%
that they need for a
chosen level of output y2.

00:43:11.340 --> 00:43:12.990 align:middle line:90%
This term could be--

00:43:12.990 --> 00:43:16.120 align:middle line:84%
the whole term could be
positive or negative.

00:43:16.120 --> 00:43:19.440 align:middle line:84%
If they're overly endowed with
rights, they'll be selling some

00:43:19.440 --> 00:43:22.320 align:middle line:84%
and be left with a
positive residual term.

00:43:22.320 --> 00:43:24.300 align:middle line:84%
If they're underendowed
with rights,

00:43:24.300 --> 00:43:28.260 align:middle line:84%
they will be buying more rights
than they were endowed with.

00:43:28.260 --> 00:43:30.720 align:middle line:90%
But the economics is the same.

00:43:30.720 --> 00:43:33.060 align:middle line:84%
Regardless of what
the P bar is, you

00:43:33.060 --> 00:43:37.290 align:middle line:84%
want to evaluate the
price of the permits

00:43:37.290 --> 00:43:39.300 align:middle line:90%
as an opportunity cost.

00:43:39.300 --> 00:43:42.270 align:middle line:84%
That is to say, take the
case where they're overly

00:43:42.270 --> 00:43:45.120 align:middle line:84%
endowed with rights,
they could hoard them,

00:43:45.120 --> 00:43:48.660 align:middle line:84%
and then they could effectively
sell them in the market.

00:43:48.660 --> 00:43:50.740 align:middle line:90%
That would be fine.

00:43:50.740 --> 00:43:51.900 align:middle line:90%
Or they could use them up.

00:43:51.900 --> 00:43:54.390 align:middle line:84%
So the opportunity
cost of using it up

00:43:54.390 --> 00:43:56.130 align:middle line:84%
is what they could
have done with it,

00:43:56.130 --> 00:43:58.080 align:middle line:90%
which is just sell it outright.

00:43:58.080 --> 00:44:00.510 align:middle line:84%
So this opportunity
cost argument

00:44:00.510 --> 00:44:02.520 align:middle line:90%
works for both types of firms.

00:44:02.520 --> 00:44:06.760 align:middle line:84%
And it is the logic of why
the endowments don't matter.

00:44:06.760 --> 00:44:09.060 align:middle line:84%
What matters is what
they do on the margin.

00:44:09.060 --> 00:44:12.720 align:middle line:84%
That's what they take under
consideration when they're

00:44:12.720 --> 00:44:14.340 align:middle line:90%
thinking about production.

00:44:14.340 --> 00:44:16.770 align:middle line:84%
A way to see that in
the algebra is just

00:44:16.770 --> 00:44:20.070 align:middle line:84%
to take the derivative of these
profit statements for each firm

00:44:20.070 --> 00:44:24.060 align:middle line:84%
j with respect to the
choice variable y2,

00:44:24.060 --> 00:44:26.640 align:middle line:84%
taking into account
that pollution is also

00:44:26.640 --> 00:44:30.310 align:middle line:84%
a function of y2, and
we would get, again,

00:44:30.310 --> 00:44:35.400 align:middle line:84%
the derivative with respect
to the revenue with p minus--

00:44:35.400 --> 00:44:37.000 align:middle line:84%
or put it on the
right-hand side,

00:44:37.000 --> 00:44:41.010 align:middle line:84%
the derivative of production
through the 1 over the direct

00:44:41.010 --> 00:44:41.520 align:middle line:90%
inverse--

00:44:41.520 --> 00:44:44.760 align:middle line:84%
the derivative of the
direct production function,

00:44:44.760 --> 00:44:49.230 align:middle line:84%
making that substitution plus q
times the way pollution rights

00:44:49.230 --> 00:44:50.550 align:middle line:90%
are changing--

00:44:50.550 --> 00:44:52.950 align:middle line:84%
pollution permit
requirements are changing

00:44:52.950 --> 00:44:55.490 align:middle line:90%
as we're changing production.

00:44:55.490 --> 00:44:55.990 align:middle line:90%
OK.

00:44:55.990 --> 00:45:03.820 align:middle line:84%
So this equation 16 holds for
both firms j of type a or b.

00:45:03.820 --> 00:45:06.490 align:middle line:90%
Can we see who's polluting more?

00:45:06.490 --> 00:45:10.870 align:middle line:84%
Well, let's suppose that this
coal mine in West Virginia

00:45:10.870 --> 00:45:15.280 align:middle line:84%
has really bad
consequences for pollution

00:45:15.280 --> 00:45:19.270 align:middle line:84%
as opposed to something
using clean energy.

00:45:19.270 --> 00:45:22.660 align:middle line:84%
The marginal effect of
pollution is higher for type

00:45:22.660 --> 00:45:25.550 align:middle line:84%
a firms relative
to type b firms.

00:45:25.550 --> 00:45:28.930 align:middle line:84%
But if this term is
higher for a than for b,

00:45:28.930 --> 00:45:31.810 align:middle line:84%
and for both a and
b, whatever else

00:45:31.810 --> 00:45:35.290 align:middle line:84%
is going on the right-hand side,
the two terms have to sum up

00:45:35.290 --> 00:45:39.220 align:middle line:84%
to p, then for a, if this
is higher, q is the same,

00:45:39.220 --> 00:45:40.930 align:middle line:90%
this term has to be lower.

00:45:40.930 --> 00:45:44.510 align:middle line:84%
How do we get for a firms
this term to be lower?

00:45:44.510 --> 00:45:47.680 align:middle line:84%
We make the marginal
product of the input higher.

00:45:47.680 --> 00:45:48.830 align:middle line:90%
How do we do that?

00:45:48.830 --> 00:45:51.500 align:middle line:84%
We do that by
reducing the input.

00:45:51.500 --> 00:45:53.290 align:middle line:84%
So again, the
economics make sense.

00:45:53.290 --> 00:45:56.290 align:middle line:84%
The firm that-- type
that pollutes the most

00:45:56.290 --> 00:46:01.820 align:middle line:84%
will use less of the input
and therefore pollute less.

00:46:01.820 --> 00:46:04.900 align:middle line:84%
So this is another example
of pollution rights

00:46:04.900 --> 00:46:08.230 align:middle line:84%
and maybe a more familiar
one, which is cap and trade.

00:46:08.230 --> 00:46:12.310 align:middle line:84%
Which, I think you know,
is implemented in the US.

00:46:12.310 --> 00:46:16.630 align:middle line:84%
California has various kinds
of permits for polluting

00:46:16.630 --> 00:46:17.590 align:middle line:90%
and so on.

00:46:17.590 --> 00:46:21.460 align:middle line:84%
They were increasingly
popular and then things

00:46:21.460 --> 00:46:23.050 align:middle line:84%
happened over the
last four years,

00:46:23.050 --> 00:46:24.870 align:middle line:90%
but we won't go into that.

00:46:24.870 --> 00:46:25.990 align:middle line:90%
OK.

00:46:25.990 --> 00:46:28.160 align:middle line:84%
Walrasian equilibrium
with pollution rights.

00:46:28.160 --> 00:46:31.220 align:middle line:84%
So there's a more
general idea here,

00:46:31.220 --> 00:46:35.170 align:middle line:84%
which is that be externalities
all over the place.

00:46:35.170 --> 00:46:38.920 align:middle line:84%
Households can care
positively or negatively

00:46:38.920 --> 00:46:41.140 align:middle line:84%
about what other
households are eating.

00:46:41.140 --> 00:46:45.220 align:middle line:84%
Households could care about what
firms are producing-- you just

00:46:45.220 --> 00:46:46.790 align:middle line:90%
saw an example of that.

00:46:46.790 --> 00:46:51.610 align:middle line:84%
So the consumption set would
be the standard one over the L

00:46:51.610 --> 00:46:54.790 align:middle line:84%
dimensional commodity
space, but in addition, it

00:46:54.790 --> 00:46:58.150 align:middle line:84%
would specify what other
households are eating

00:46:58.150 --> 00:47:02.650 align:middle line:84%
of each of the goods over
all the other households,

00:47:02.650 --> 00:47:04.480 align:middle line:90%
i minus 1 households.

00:47:04.480 --> 00:47:06.640 align:middle line:84%
And again, what
the firm is doing.

00:47:06.640 --> 00:47:09.250 align:middle line:84%
Or if they're j firms, what
each and every one of them

00:47:09.250 --> 00:47:10.690 align:middle line:90%
is doing potentially.

00:47:10.690 --> 00:47:12.620 align:middle line:84%
Not everything
needs to be in here,

00:47:12.620 --> 00:47:14.710 align:middle line:84%
but this is--
externalities written down

00:47:14.710 --> 00:47:16.340 align:middle line:90%
in its most general form.

00:47:16.340 --> 00:47:16.870 align:middle line:90%
OK.

00:47:16.870 --> 00:47:19.720 align:middle line:84%
So we did a version of this
just now with pollution,

00:47:19.720 --> 00:47:23.630 align:middle line:84%
and we can do a version
with consumption.

00:47:23.630 --> 00:47:26.800 align:middle line:84%
So this is that part of
Arrow I was mentioning when

00:47:26.800 --> 00:47:28.840 align:middle line:90%
we went over the reading list.

00:47:28.840 --> 00:47:32.860 align:middle line:84%
We have a pure exchange economy
to make it a bit easier.

00:47:32.860 --> 00:47:35.170 align:middle line:90%
We have the usual.

00:47:35.170 --> 00:47:39.820 align:middle line:84%
Household cares about
consumption good k.

00:47:39.820 --> 00:47:42.040 align:middle line:90%
There are n households.

00:47:42.040 --> 00:47:44.170 align:middle line:90%
So i goes from 1 to n.

00:47:44.170 --> 00:47:49.270 align:middle line:84%
There are m consumer goods,
so k goes from 1 to m.

00:47:49.270 --> 00:47:52.730 align:middle line:84%
So this part so far
is pretty standard.

00:47:52.730 --> 00:47:55.450 align:middle line:84%
However, suppose that
the utility function

00:47:55.450 --> 00:47:59.720 align:middle line:84%
of the household depends on
the consumption of the others.

00:47:59.720 --> 00:48:01.930 align:middle line:84%
So if we started
with household j,

00:48:01.930 --> 00:48:04.660 align:middle line:84%
for example, if they
could choose it,

00:48:04.660 --> 00:48:09.700 align:middle line:84%
this would be the rights
to eat that j would assign

00:48:09.700 --> 00:48:12.460 align:middle line:90%
to household i for commodity k.

00:48:12.460 --> 00:48:16.180 align:middle line:84%
In other words, household j
cares about what household

00:48:16.180 --> 00:48:17.320 align:middle line:90%
i is doing.

00:48:17.320 --> 00:48:19.550 align:middle line:84%
We're going to try
to decentralize this.

00:48:19.550 --> 00:48:22.300 align:middle line:84%
So we want to create a
market in those rights.

00:48:22.300 --> 00:48:24.850 align:middle line:84%
Mathematically we're just
adding another subscript

00:48:24.850 --> 00:48:28.720 align:middle line:84%
and you have to memorize
that it goes what j wants

00:48:28.720 --> 00:48:32.980 align:middle line:90%
to assign to i for commodity k.

00:48:32.980 --> 00:48:35.560 align:middle line:84%
Well, that's household
j, here's household i.

00:48:35.560 --> 00:48:37.090 align:middle line:90%
It's the same thing.

00:48:37.090 --> 00:48:40.120 align:middle line:84%
Household has
utility function Ui

00:48:40.120 --> 00:48:43.750 align:middle line:84%
over what i wants to
assign of the first good

00:48:43.750 --> 00:48:45.730 align:middle line:84%
for the first household
all the way up

00:48:45.730 --> 00:48:49.000 align:middle line:84%
to what i wants to assign
for the n-th household

00:48:49.000 --> 00:48:50.140 align:middle line:90%
of the m-th good.

00:48:50.140 --> 00:48:51.730 align:middle line:84%
So this is a great,
big, long vector

00:48:51.730 --> 00:48:54.490 align:middle line:84%
running over all goods
and all households,

00:48:54.490 --> 00:48:59.020 align:middle line:84%
including, actually, the
own terms, i i terms,

00:48:59.020 --> 00:49:02.020 align:middle line:84%
as if assigning to your
goods to themselves.

00:49:02.020 --> 00:49:02.800 align:middle line:90%
Anyway.

00:49:02.800 --> 00:49:05.860 align:middle line:84%
Then we have-- if we're going
to solve the Pareto problem,

00:49:05.860 --> 00:49:09.940 align:middle line:84%
we have the typical
constraint over here

00:49:09.940 --> 00:49:13.270 align:middle line:84%
that if household i
is consuming good k,

00:49:13.270 --> 00:49:15.670 align:middle line:84%
and we sum up
overall households,

00:49:15.670 --> 00:49:19.750 align:middle line:84%
we'll get the total amount
of good k being allocated

00:49:19.750 --> 00:49:21.820 align:middle line:90%
like a resource constraint.

00:49:21.820 --> 00:49:25.210 align:middle line:84%
But in addition,
we would have that

00:49:25.210 --> 00:49:28.960 align:middle line:84%
what household j
experiences in terms of what

00:49:28.960 --> 00:49:31.390 align:middle line:90%
i is eating of household--

00:49:31.390 --> 00:49:36.160 align:middle line:84%
of good k, whatever household
i is doing of good k,

00:49:36.160 --> 00:49:40.880 align:middle line:84%
it's impacting household j
and all the other households.

00:49:40.880 --> 00:49:46.070 align:middle line:84%
So xik, this guy here,
what i is really doing of k

00:49:46.070 --> 00:49:50.270 align:middle line:84%
is what j experiences that
i is doing of k over all

00:49:50.270 --> 00:49:52.220 align:middle line:90%
the other households j.

00:49:52.220 --> 00:49:55.700 align:middle line:84%
So this is an externality,
you could think, positively

00:49:55.700 --> 00:49:58.370 align:middle line:84%
or negatively inflicted on
all the other households.

00:49:58.370 --> 00:50:01.220 align:middle line:84%
Whatever household i is
doing, it as consequences

00:50:01.220 --> 00:50:02.840 align:middle line:90%
for all the other households.

00:50:02.840 --> 00:50:05.870 align:middle line:84%
Well anyway, it looks like
a standard Pareto problem.

00:50:05.870 --> 00:50:08.030 align:middle line:84%
We could maximize
lambda-weighted sums

00:50:08.030 --> 00:50:11.780 align:middle line:84%
of utilities and we would have
these two resource constraints,

00:50:11.780 --> 00:50:15.920 align:middle line:84%
17 being conventional,
but 18 being

00:50:15.920 --> 00:50:17.760 align:middle line:90%
the new thing that we need.

00:50:17.760 --> 00:50:20.630 align:middle line:84%
So let's think
about shadow prices.

00:50:20.630 --> 00:50:25.250 align:middle line:84%
This guy here, 17, is going to
produce a Lagrange multiplier

00:50:25.250 --> 00:50:26.540 align:middle line:90%
for every good k.

00:50:26.540 --> 00:50:28.580 align:middle line:90%
So we can call that qk.

00:50:28.580 --> 00:50:33.530 align:middle line:84%
This guy here, 18, is going to
produce a Lagrange multiplier

00:50:33.530 --> 00:50:36.260 align:middle line:90%
for every j because there--

00:50:36.260 --> 00:50:40.370 align:middle line:84%
j goes from 1 to n
for every household i

00:50:40.370 --> 00:50:42.200 align:middle line:90%
and for every good k.

00:50:42.200 --> 00:50:44.210 align:middle line:84%
So there's lots of
Lagrange multipliers

00:50:44.210 --> 00:50:48.710 align:middle line:84%
and we can call them Pijk
for all these constraints.

00:50:48.710 --> 00:50:50.270 align:middle line:90%
These are shadow prices.

00:50:50.270 --> 00:50:53.570 align:middle line:84%
These are going to be the
shadow prices of the rights.

00:50:53.570 --> 00:50:55.280 align:middle line:84%
And with these
shadow prices, we're

00:50:55.280 --> 00:50:58.190 align:middle line:84%
going to be able to
decentralize to get

00:50:58.190 --> 00:51:01.200 align:middle line:84%
to Pareto-optimal
competitive equilibrium.

00:51:01.200 --> 00:51:04.080 align:middle line:84%
If you looked at the
first-order conditions,

00:51:04.080 --> 00:51:06.930 align:middle line:90%
where does jik enter?

00:51:06.930 --> 00:51:10.410 align:middle line:84%
It enters into the
utility function of j.

00:51:10.410 --> 00:51:15.510 align:middle line:84%
There's a lambda j for the
Pareto weight of household j,

00:51:15.510 --> 00:51:17.730 align:middle line:84%
because we'd be maximizing
lambda-weighted sums

00:51:17.730 --> 00:51:20.040 align:middle line:84%
of utilities in
the Pareto problem.

00:51:20.040 --> 00:51:23.160 align:middle line:90%
And where else does xjik enter?

00:51:23.160 --> 00:51:25.690 align:middle line:90%
It enters here.

00:51:25.690 --> 00:51:28.590 align:middle line:84%
So then we pick up
that Pijk Lagrange

00:51:28.590 --> 00:51:32.400 align:middle line:84%
multiplier because
everything else is linear.

00:51:32.400 --> 00:51:35.450 align:middle line:84%
So that's where
this is coming from.

00:51:35.450 --> 00:51:40.390 align:middle line:84%
And then this second first-order
comes from going back

00:51:40.390 --> 00:51:42.820 align:middle line:90%
to look at the xik.

00:51:42.820 --> 00:51:46.720 align:middle line:84%
Now the xik appears here
in 17, so we pick up

00:51:46.720 --> 00:51:49.600 align:middle line:84%
the conventional Lagrange
multiplier there.

00:51:49.600 --> 00:51:54.310 align:middle line:84%
But that xik also enters into
a multitude of constraints here

00:51:54.310 --> 00:51:59.000 align:middle line:84%
on 18, namely one
for each of the j's.

00:51:59.000 --> 00:52:03.460 align:middle line:84%
So we have a Pijk down
here, summing over all those

00:52:03.460 --> 00:52:06.290 align:middle line:84%
because it appears in each
and every one of them,

00:52:06.290 --> 00:52:09.740 align:middle line:90%
and that's this expression.

00:52:09.740 --> 00:52:11.140 align:middle line:90%
So there's our price system.

00:52:11.140 --> 00:52:13.410 align:middle line:84%
We have a conventional
price system

00:52:13.410 --> 00:52:17.990 align:middle line:84%
qk for each of the underlying
goods, but in addition to that,

00:52:17.990 --> 00:52:21.370 align:middle line:84%
we have these price of
rights to assign consumption

00:52:21.370 --> 00:52:23.590 align:middle line:90%
for other households.

00:52:23.590 --> 00:52:26.380 align:middle line:84%
And I deleted the
rest of the slides

00:52:26.380 --> 00:52:28.240 align:middle line:90%
because this is the intuition.

00:52:28.240 --> 00:52:29.890 align:middle line:90%
It's enough.

00:52:29.890 --> 00:52:34.480 align:middle line:84%
You've already seen us take
prices as Lagrange multipliers

00:52:34.480 --> 00:52:35.890 align:middle line:90%
and decentralize.

00:52:35.890 --> 00:52:38.200 align:middle line:84%
You can write down the
problem of the household

00:52:38.200 --> 00:52:41.005 align:middle line:84%
and the decentralized
problem with these prices

00:52:41.005 --> 00:52:42.880 align:middle line:84%
through the Lagrangian,
take the derivatives,

00:52:42.880 --> 00:52:45.010 align:middle line:84%
and you will get these
expressions back.

00:52:45.010 --> 00:52:47.950 align:middle line:84%
So that's the way we
proved the second welfare

00:52:47.950 --> 00:52:51.380 align:middle line:84%
theorem, that any Pareto-optimal
allocation can be achieved.

00:52:51.380 --> 00:52:53.750 align:middle line:84%
Now there's a lot of
externalities here.

00:52:53.750 --> 00:52:55.060 align:middle line:90%
Where did they all go?

00:52:55.060 --> 00:52:58.640 align:middle line:84%
The answer is we created
a market in all the goods.

00:52:58.640 --> 00:53:00.910 align:middle line:84%
Another way to view this
on the production side

00:53:00.910 --> 00:53:05.320 align:middle line:84%
is that when a household
is deciding what to eat,

00:53:05.320 --> 00:53:08.410 align:middle line:84%
it is aware of the utility
consequences and also

00:53:08.410 --> 00:53:09.910 align:middle line:90%
the expense in the budget.

00:53:09.910 --> 00:53:14.200 align:middle line:84%
But it's also getting
revenue, because it's

00:53:14.200 --> 00:53:18.880 align:middle line:84%
producing the jik goods that the
other households j care about.

00:53:18.880 --> 00:53:22.640 align:middle line:84%
So it's like as a firm, it's
selling those for revenue,

00:53:22.640 --> 00:53:25.450 align:middle line:84%
which is kind of like combining
the examples that we've had

00:53:25.450 --> 00:53:26.700 align:middle line:90%
so far.

00:53:26.700 --> 00:53:27.490 align:middle line:90%
OK.

00:53:27.490 --> 00:53:31.870 align:middle line:84%
So let me get to the
failure of the first welfare

00:53:31.870 --> 00:53:34.190 align:middle line:90%
theorem in a different way.

00:53:34.190 --> 00:53:36.580 align:middle line:84%
Instead of having
missing markets,

00:53:36.580 --> 00:53:39.630 align:middle line:84%
I'm going to have
too many markets.

00:53:39.630 --> 00:53:41.790 align:middle line:90%
An infinite number of goods.

00:53:41.790 --> 00:53:44.460 align:middle line:84%
So you may remember how the
proof went of the first welfare

00:53:44.460 --> 00:53:48.250 align:middle line:84%
theorem, but just in case,
I'll review it for you.

00:53:48.250 --> 00:53:49.950 align:middle line:84%
So the first welfare
theorem was that

00:53:49.950 --> 00:53:52.980 align:middle line:84%
any competitive equilibrium
would be Pareto-optimal.

00:53:52.980 --> 00:53:54.870 align:middle line:90%
We did it by contradiction.

00:53:54.870 --> 00:53:55.950 align:middle line:90%
We said no.

00:53:55.950 --> 00:53:58.560 align:middle line:84%
Suppose it were
not Pareto-optimal,

00:53:58.560 --> 00:54:03.780 align:middle line:84%
then this baseline x allocation
would be weakly dominated

00:54:03.780 --> 00:54:07.500 align:middle line:84%
for some households, the
baseline would be the star.

00:54:07.500 --> 00:54:09.760 align:middle line:84%
The alternative is
without the star.

00:54:09.760 --> 00:54:13.200 align:middle line:84%
So the dominating one weakly
dominates for some households

00:54:13.200 --> 00:54:15.180 align:middle line:84%
and strictly
dominates for others

00:54:15.180 --> 00:54:17.460 align:middle line:90%
relative to the baseline.

00:54:17.460 --> 00:54:19.410 align:middle line:84%
And then we kind
of did this algebra

00:54:19.410 --> 00:54:22.890 align:middle line:84%
and decided that clearly for
the households for which this

00:54:22.890 --> 00:54:25.440 align:middle line:84%
dominates, it would
not be affordable.

00:54:25.440 --> 00:54:28.440 align:middle line:84%
We used a lemma that even for
the households for which they

00:54:28.440 --> 00:54:31.680 align:middle line:84%
would be indifferent, it
costs at least as much.

00:54:31.680 --> 00:54:34.620 align:middle line:84%
We then add it up over
all the households

00:54:34.620 --> 00:54:36.660 align:middle line:84%
and got that the
value of expenses

00:54:36.660 --> 00:54:40.860 align:middle line:84%
in the alternative, suppose
Pareto-dominating allocation

00:54:40.860 --> 00:54:43.920 align:middle line:84%
is strictly greater than
the valuation of expenses

00:54:43.920 --> 00:54:46.890 align:middle line:84%
in the baseline of the
competitive equilibrium.

00:54:46.890 --> 00:54:50.400 align:middle line:84%
And then using that fact along
with the other conditions,

00:54:50.400 --> 00:54:51.510 align:middle line:90%
we ended up--

00:54:51.510 --> 00:54:53.760 align:middle line:84%
the bottom line
of the proof was,

00:54:53.760 --> 00:54:56.730 align:middle line:84%
this alternative allocation
was not feasible,

00:54:56.730 --> 00:54:59.760 align:middle line:84%
because we had this inequality
strictly greater than 0

00:54:59.760 --> 00:55:00.750 align:middle line:90%
going on here.

00:55:00.750 --> 00:55:03.270 align:middle line:84%
But the alternative
Pareto-dominating allocation

00:55:03.270 --> 00:55:05.460 align:middle line:84%
is supposed to be
feasible, so this

00:55:05.460 --> 00:55:06.810 align:middle line:90%
would be the contradiction.

00:55:06.810 --> 00:55:10.120 align:middle line:84%
Now it all seems
pretty straightforward,

00:55:10.120 --> 00:55:14.310 align:middle line:84%
but what happens if there's an
infinite number of households?

00:55:14.310 --> 00:55:17.160 align:middle line:84%
We start summing up
over infinity here.

00:55:17.160 --> 00:55:18.680 align:middle line:90%
It has no meaning that--

00:55:18.680 --> 00:55:21.960 align:middle line:84%
not necessarily
having a meaning.

00:55:21.960 --> 00:55:25.130 align:middle line:84%
If it's infinity,
greater than infinity,

00:55:25.130 --> 00:55:27.780 align:middle line:84%
well that's where the logic
of the proof breaks down.

00:55:27.780 --> 00:55:30.910 align:middle line:84%
You can't have one infinity
being bigger than another one.

00:55:30.910 --> 00:55:33.360 align:middle line:84%
So this can happen in
overlapping generations

00:55:33.360 --> 00:55:36.420 align:middle line:84%
models where we have
a first generation,

00:55:36.420 --> 00:55:38.040 align:middle line:84%
a second generation,
a third one,

00:55:38.040 --> 00:55:40.290 align:middle line:90%
and it goes on without end.

00:55:40.290 --> 00:55:44.200 align:middle line:84%
Everyone has a finite life, but
the economy goes on forever.

00:55:44.200 --> 00:55:46.230 align:middle line:90%
So how many goods are there?

00:55:46.230 --> 00:55:48.180 align:middle line:90%
An infinite number of goods?

00:55:48.180 --> 00:55:49.590 align:middle line:90%
How many households are there?

00:55:49.590 --> 00:55:51.780 align:middle line:84%
An infinite number
of households.

00:55:51.780 --> 00:55:54.660 align:middle line:84%
As an example, but
this isn't exactly

00:55:54.660 --> 00:55:58.230 align:middle line:84%
overlapping generations,
it's very much in that spirit

00:55:58.230 --> 00:56:00.210 align:middle line:84%
and it's easier to
understand what goes

00:56:00.210 --> 00:56:03.180 align:middle line:90%
wrong with the welfare theorem.

00:56:03.180 --> 00:56:07.020 align:middle line:84%
Let's just outright have an
infinite number of agents

00:56:07.020 --> 00:56:10.200 align:middle line:90%
and only two goods, 1 and 2.

00:56:10.200 --> 00:56:14.700 align:middle line:84%
Each household has an endowment
of each of the two goods,

00:56:14.700 --> 00:56:17.620 align:middle line:84%
1 unit of the first and
1 unit of the second.

00:56:17.620 --> 00:56:21.760 align:middle line:84%
Each household i also has
a common utility function,

00:56:21.760 --> 00:56:23.760 align:middle line:90%
which is this log utility.

00:56:23.760 --> 00:56:26.130 align:middle line:90%
Additive Cobb-Douglas.

00:56:26.130 --> 00:56:29.130 align:middle line:84%
So let's call the price
vector of the two goods

00:56:29.130 --> 00:56:33.900 align:middle line:84%
p1, p2, let the first would be
the numeraire, second one have

00:56:33.900 --> 00:56:36.210 align:middle line:90%
a relative price of p.

00:56:36.210 --> 00:56:39.300 align:middle line:84%
You may remember that these
Cobb-Douglas-like utility

00:56:39.300 --> 00:56:42.640 align:middle line:84%
functions generate
constant expenditures.

00:56:42.640 --> 00:56:45.390 align:middle line:84%
So p, the price of
the second good,

00:56:45.390 --> 00:56:47.700 align:middle line:84%
times the quantity
of the second good

00:56:47.700 --> 00:56:51.360 align:middle line:84%
is equal to 1/2
of the endowment,

00:56:51.360 --> 00:56:55.050 align:middle line:84%
because the weights here
are equal on the two goods.

00:56:55.050 --> 00:56:59.730 align:middle line:84%
Endowment, they have the
first good at a price of 1, 1

00:56:59.730 --> 00:57:02.520 align:middle line:84%
unit of the second
good at a price of p.

00:57:02.520 --> 00:57:05.070 align:middle line:84%
So one half of the
valuation of their endowment

00:57:05.070 --> 00:57:09.220 align:middle line:84%
is spent on good 2,
1/2 is spent on good 1.

00:57:09.220 --> 00:57:12.510 align:middle line:84%
These are the demand
equations at prices p.

00:57:12.510 --> 00:57:14.830 align:middle line:90%
What's a possible solution here?

00:57:14.830 --> 00:57:18.630 align:middle line:90%
Well, let's try p equal to 1.

00:57:18.630 --> 00:57:21.420 align:middle line:84%
If the price of the second
good was equal to 1,

00:57:21.420 --> 00:57:25.050 align:middle line:84%
then the solution to these
equations are x1 and x2

00:57:25.050 --> 00:57:27.520 align:middle line:90%
is equal to 1.

00:57:27.520 --> 00:57:29.950 align:middle line:90%
1/2 of 1 plus 1, et cetera.

00:57:29.950 --> 00:57:33.850 align:middle line:84%
So that is a candidate--
autarky, actually--

00:57:33.850 --> 00:57:37.030 align:middle line:84%
equilibrium and price is p,
they choose not to trade.

00:57:37.030 --> 00:57:39.290 align:middle line:84%
If the welfare
theorem were true,

00:57:39.290 --> 00:57:41.200 align:middle line:84%
then that competitive
equilibrium

00:57:41.200 --> 00:57:44.020 align:middle line:84%
should be Pareto-optimal,
but it's not.

00:57:44.020 --> 00:57:45.340 align:middle line:90%
Why not?

00:57:45.340 --> 00:57:48.460 align:middle line:84%
OK, now I'm going to pull
this rabbit out of my hat.

00:57:48.460 --> 00:57:50.170 align:middle line:90%
It's kind of amazing.

00:57:50.170 --> 00:57:55.420 align:middle line:84%
Let's have the second household
transfer all of its endowment

00:57:55.420 --> 00:57:56.800 align:middle line:90%
to the first one.

00:57:56.800 --> 00:58:00.835 align:middle line:84%
The first one started out with
1, 1, now he gets to 2, 2.

00:58:00.835 --> 00:58:02.930 align:middle line:84%
And you say, oh,
this guy's 0, 0.

00:58:02.930 --> 00:58:04.420 align:middle line:90%
No, no, no, no.

00:58:04.420 --> 00:58:08.830 align:middle line:84%
So household 2 gets
compensated from household 3.

00:58:08.830 --> 00:58:12.760 align:middle line:84%
Transfers his or her
endowment to household 2.

00:58:12.760 --> 00:58:15.220 align:middle line:84%
Household 2 is no
worse than before.

00:58:15.220 --> 00:58:17.170 align:middle line:90%
Household 1 is better off.

00:58:17.170 --> 00:58:19.810 align:middle line:84%
If we had a finite
number of households,

00:58:19.810 --> 00:58:21.700 align:middle line:90%
the last guy would get screwed.

00:58:21.700 --> 00:58:25.210 align:middle line:84%
But this goes on forever,
so you can dominate.

00:58:25.210 --> 00:58:28.550 align:middle line:84%
You're like pulling this
good out of infinity.

00:58:28.550 --> 00:58:30.580 align:middle line:84%
Another way to
say this is what's

00:58:30.580 --> 00:58:33.020 align:middle line:84%
happening to the
welfare theorem?

00:58:33.020 --> 00:58:36.850 align:middle line:84%
As I said, what we need is,
even with an infinite number

00:58:36.850 --> 00:58:39.610 align:middle line:84%
of goods, that the
valuation over all

00:58:39.610 --> 00:58:43.990 align:middle line:84%
of them, over all the
households is a real number.

00:58:43.990 --> 00:58:46.000 align:middle line:84%
For the proof to
go through, it has

00:58:46.000 --> 00:58:48.490 align:middle line:84%
to dominate the valuation
of the endowment.

00:58:48.490 --> 00:58:51.400 align:middle line:84%
We need to be able to
evaluate the endowments

00:58:51.400 --> 00:58:55.270 align:middle line:84%
in an infinite good economy,
and in that equilibrium

00:58:55.270 --> 00:58:59.320 align:middle line:84%
and autarky, the valuation
of wealth is infinite.

00:58:59.320 --> 00:59:00.190 align:middle line:90%
Why?

00:59:00.190 --> 00:59:05.920 align:middle line:84%
Because we have the endowment
of household 1 i equal 1, 1.

00:59:05.920 --> 00:59:08.140 align:middle line:90%
The price equal 1, 1.

00:59:08.140 --> 00:59:11.020 align:middle line:90%
So this dot-product is 1, 2--

00:59:11.020 --> 00:59:11.950 align:middle line:90%
sorry.

00:59:11.950 --> 00:59:15.320 align:middle line:84%
And we add up to overall the
households and we get infinity.

00:59:15.320 --> 00:59:19.450 align:middle line:84%
So the valuation of wealth
in that very special economy

00:59:19.450 --> 00:59:22.730 align:middle line:84%
was infinite, and that's
why the proof broke down.

00:59:22.730 --> 00:59:25.300 align:middle line:84%
You could see that it broke
down because we dominated

00:59:25.300 --> 00:59:27.640 align:middle line:84%
the supposed
Pareto-optimal autarky

00:59:27.640 --> 00:59:29.800 align:middle line:90%
allocation by something better.

00:59:29.800 --> 00:59:33.100 align:middle line:84%
Now when we go to an
infinite horizon model,

00:59:33.100 --> 00:59:35.470 align:middle line:84%
something like
this could happen,

00:59:35.470 --> 00:59:38.230 align:middle line:84%
because we've got an
infinite number of goods.

00:59:38.230 --> 00:59:40.180 align:middle line:84%
Even without
overlapping generations,

00:59:40.180 --> 00:59:42.460 align:middle line:84%
if we imagine
households live forever,

00:59:42.460 --> 00:59:45.670 align:middle line:84%
like in a medieval storage
economy or whatever-- well,

00:59:45.670 --> 00:59:46.370 align:middle line:90%
that's not--

00:59:46.370 --> 00:59:47.650 align:middle line:90%
they died every 13 years.

00:59:47.650 --> 00:59:51.340 align:middle line:84%
But with an infinite
dimensional commodity space,

00:59:51.340 --> 00:59:54.430 align:middle line:84%
it could be that the valuation
of wealth is infinite

00:59:54.430 --> 00:59:58.070 align:middle line:84%
and competitive equilibrium
would not be Pareto-optimal.

00:59:58.070 --> 01:00:02.470 align:middle line:84%
However, typically those prices
are declining at rate beta,

01:00:02.470 --> 01:00:05.180 align:middle line:84%
and beta's the discount
rate and utility.

01:00:05.180 --> 01:00:07.810 align:middle line:84%
So you have an infinite
sum, but that infinite sum

01:00:07.810 --> 01:00:12.020 align:middle line:84%
converges to something finite
as we add more and more terms.

01:00:12.020 --> 01:00:14.360 align:middle line:84%
So that's why this
doesn't always happen.

01:00:14.360 --> 01:00:18.790 align:middle line:84%
The necessary and sufficient
condition is this one, 23.

01:00:18.790 --> 01:00:20.990 align:middle line:90%
So decision time.

01:00:20.990 --> 01:00:23.290 align:middle line:84%
I'm going to skip
the last five slides.

01:00:23.290 --> 01:00:25.615 align:middle line:84%
I will tell you what
the content of it

01:00:25.615 --> 01:00:28.210 align:middle line:84%
is, which is a bit frustrating,
because I don't have time

01:00:28.210 --> 01:00:29.500 align:middle line:90%
to go over it.

01:00:29.500 --> 01:00:32.530 align:middle line:84%
I anticipated this thinking
about the lecture this morning,

01:00:32.530 --> 01:00:34.640 align:middle line:90%
there's a lot of material here.

01:00:34.640 --> 01:00:37.420 align:middle line:84%
But this is the starred
paper on the reading list.

01:00:37.420 --> 01:00:40.750 align:middle line:84%
So between these five
slides and that paper,

01:00:40.750 --> 01:00:43.780 align:middle line:84%
you should be able
to understand.

01:00:43.780 --> 01:00:46.570 align:middle line:84%
The economics of this
problem is, what do I

01:00:46.570 --> 01:00:48.370 align:middle line:90%
get from owning Bitcoin?

01:00:48.370 --> 01:00:51.890 align:middle line:84%
Hopefully I have the right to
buy things from someone else,

01:00:51.890 --> 01:00:53.920 align:middle line:90%
but only if they want the coin.

01:00:53.920 --> 01:00:57.070 align:middle line:84%
So my problem is
defined as in a network.

01:00:57.070 --> 01:00:59.020 align:middle line:90%
If I'm a buyer, I want sellers.

01:00:59.020 --> 01:01:01.120 align:middle line:90%
If I'm a seller, I want buyers.

01:01:01.120 --> 01:01:05.330 align:middle line:84%
My utility is a consequence of
who else is on the platform,

01:01:05.330 --> 01:01:08.930 align:middle line:84%
whether it's a cryptocurrency
platform or something else.

01:01:08.930 --> 01:01:11.830 align:middle line:84%
So there's a natural
externality here

01:01:11.830 --> 01:01:13.930 align:middle line:84%
that these households
care about who else

01:01:13.930 --> 01:01:16.240 align:middle line:90%
is populating the platform.

01:01:16.240 --> 01:01:18.220 align:middle line:84%
You would think
with an externality,

01:01:18.220 --> 01:01:20.630 align:middle line:84%
competitive equilibrium
is going to go bad

01:01:20.630 --> 01:01:23.290 align:middle line:84%
and we're not going to
get to a Pareto optimum,

01:01:23.290 --> 01:01:25.870 align:middle line:84%
but there is a
remedy, and that is

01:01:25.870 --> 01:01:29.930 align:middle line:84%
to price the right to
participate in the platform

01:01:29.930 --> 01:01:33.040 align:middle line:84%
according to your
agent utility type.

01:01:33.040 --> 01:01:36.190 align:middle line:84%
So different types
pay different amounts.

01:01:36.190 --> 01:01:39.760 align:middle line:84%
They buy the rights to
participate on the platform,

01:01:39.760 --> 01:01:43.270 align:middle line:84%
and that's going to
internalize the externality,

01:01:43.270 --> 01:01:45.010 align:middle line:84%
much in the same
way that we were

01:01:45.010 --> 01:01:49.040 align:middle line:84%
talking about with
pollution rights and so on.

01:01:49.040 --> 01:01:51.100 align:middle line:84%
But anyway, I'm
already a minute over.

01:01:51.100 --> 01:01:53.170 align:middle line:84%
So I'm going to
skip these slides.

01:01:53.170 --> 01:01:54.847 align:middle line:90%
You can take a look at them.

01:01:54.847 --> 01:01:57.430 align:middle line:84%
And if you have questions, maybe
we can talk a little bit more

01:01:57.430 --> 01:01:58.815 align:middle line:90%
about it next time.

01:01:58.815 --> 01:02:01.170 align:middle line:90%
And that's all for today.

01:02:01.170 --> 01:02:08.000 align:middle line:90%