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PROFESSOR: We've been talking
so far about basically
overviews of supply and demand
relationships and
understanding how
markets work.
Now we're going to step back and
get behind the supply and
demand curves and understand
where those curves
themselves come from.
So we talked about, given that
we have supply and demand
curves, how they interact.
Now we're going to get behind
that and see where these
curves actually come from.
Thank you, by the way
for coming down.
I appreciate it.
So what we're going to do is,
we're going to start with the
demand curve, and we're going to
spend the next few lectures
talking about consumers and how
consumer preferences are
ultimately what leads
to the construction
of the demand curve.
Then after that and after the
first exam-- that will cover
what's on the first exam--
after the first exam, we'll
start talking about firms and
what determines the
firm supply curve.
So today we'll talk about
consumers and we're going to
talk about where the demand
curve comes from.
And where it comes from, and
where all consumer behavior
coming from in economics is
from utility maximization.
That's where everything with
consumers starts is with
utility maximization.
That's the basic building block
of consumer behavior.
And basically, utility
maximization--
that's what this lecture will
be about, describing it.
But basically, an overview
is, we posit some type of
preferences.
We posit consumer preferences,
what consumers would like.
We posit some budget constraint,
what resources
consumers have to get
what they'd like.
And then we do a constrained
maximization problem that
says, given your preferences,
given what you'd like, subject
to the resources you have
available, what
choices will you make?
And in particular, we're going
to ask, the term we'll use is
we'll ask what bundle of goods
makes you the best off?
Given your preference, given
your constraints,
what bundle of goods?
So think about consumers
choosing
across a set of goods.
Typically, we'll think about two
goods because graphs are
easier to think about two
dimensions than more.
So we'll typically think about
trading off two goods.
So think about consumers with
preferences across two goods,
some budget they can
allocate, and how
they make those choices.
But this basic framework applies
to the multiplicity of
choices we all make along
many, many dimensions.
So doing two dimensions as one
of the simplifying assumptions
I'll talk about.
But that's just a simplifying
assumption.
So basically what we we're going
to do is we're going to
go through this in three
steps, not just in this
lecture, but over the
next few lectures.
Step one, is we're going to talk
about what assumptions we
make about preferences.
So I'll talk today about
preference assumptions.
So the axioms that underlie how
economists model consumer
preferences.
We'll then talk about how we
translate these preferences
assumptions into mathematical
tractability through the use
of the utility function, which
is basically a mathematical
representation of underlying
consumer preferences.
So we'll talk about how we
basically take these
preferences and translate them
into something that we can
work with here at MIT by making
it mathematical, by
making a utility function.
And then finally, we'll talk
about budget constraints.
And armed with these three
things, we'll then be able to
model how consumers
make decisions.
Now, importantly for today's
lecture, we are not dealing
with budget constraints.
So this is not happening
today.
So today we're not going to
worry about the budget
constraints.
Today we're in a world where
we're just going to talk about
what people want and we're going
to put out of our mind
whether or not they
can afford it.
So just talk about people want,
and we'll put out of
mind for today.
We'll come back next time to
whether they can afford it.
We're just going to think
about unconstrained
preferences for today's
lecture.
So let's talk about our
preference assumptions.
So, to model consumers'
preferences across goods,
we're going to impose three
preference assumptions.
Three preference assumptions.
Assumption one--
now once again, let me remind
you from the first lecture,
this is getting to some of
the harder material.
I'm going to write messily and
talk quickly, so stop me if
anything is unclear.
And if you don't stop me, I'll
just go faster and faster
until I explode.
So basically, feel free to
interrupt and stop me with
questions and such.
Three assumptions
on preferences.
The first assumption
is completeness.
The first assumption is the
assumption of completeness.
When comparing two bundles of
goods, you prefer one or the
other, but you don't
value them equally.
OK when comparing two bundles
of goods, you prefer one or
you prefer the other, but
you're not indifferent.
Completeness is the same
as no indifference.
So what we're saying is whenever
I offer you two
bundles of goods, you
could always tell me
what you like better.
Now it could be infinitesimally
better.
I'm not saying you have to
have strong preferences.
But you cannot say
I'm indifferent.
You can never be purely
indifferent.
There always at least some
slight preference for one
bundle of goods over another.
That's the completeness
assumption.
This is an assumption we make.
Now in reality, oftentimes
we are indifferent.
Well once again, this is one
of these simplifying
assumptions that will
make the model work.
And in fact, in reality if
forced, you can always decide
whether you like one thing
better than another, we just
often follow heuristic rules
which say we're roughly
indifferent.
We're just going to say, more
precisely, you are never
purely indifferent.
So, I'm not sure is
not an option.
You can never say I don't know,
I don't know which I
prefer, I'm indifferent.
I'm sorry, let me back up.
I'm using the wrong word.
Forget I said indifferent,
because we'll want to use that
word in a different
context later.
You can't say, I'm not sure.
You can't say, I'm not sure.
You can't say, I'm not sure,
can't say I don't know, I
don't know how I feel
about that.
Scratch what I said a few
minutes ago, because I want to
use indifference differently.
Completeness is not about not
being different, we're going
to use that.
What I'm saying is it's
about not being sure.
You've got to value every
bundle of goods.
You've got to be willing to
value every bundle of goods
that's given you.
So you can't say that I don't
know, I don't know how I feel
about that.
You've got to have some
feeling about stuff.
You can't say I'm not sure.
You've got to have a complete
set of preferences over all
bundles of goods that
are given you.
OK, that's completeness.
The second is transitivity.
Which is something we've been
learning since kindergarten
about transitivity, right?
And also, it's a different
context.
That's just if you prefer x to
y, and y to z, you've got to
prefer x to z.
OK, you guys should
do transitivity in
your sleep by now.
OK, so the standard transitivity
we always assume
in math class, we're going
to assume here as well.
OK, that should be pretty
noncontroversial.
OK, and then finally, and
probably most controversial,
is we're going to assume
non-satiation.
Or the famous economic
assumption that
more is always better.
OK.
More is always better, that is,
you never would turn down
having more.
Now we're going to talk later
today and tomorrow about why
you might not like the next unit
as much as you like the
current unit.
But you'll always like
it greater than zero.
You're always happy
to have more.
You never say, I've had enough,
I literally value at
zero the next unit.
You may value it as epsilon, but
you'll always value it as
greater than zero.
That's the non-satiation
assumption,
more is always better.
Now, this is the most
controversial.
And obviously we can think
of many contexts in
which that's not true.
But if we don't allow for this
assumption, the modeling gets
a lot trickier.
So once again, let's put
it out of our mind.
Realistically, we know once
we've eaten a certain amount,
we literally do not
want any more.
OK, so we're going to
put that aside.
Assume we're always in a space
where we can always eat a
little bit more.
OK, we'll call it the
Jewish mother space.
OK, you can always eat
a little bit more.
OK, you can always eat
a little bit more.
We're just going to assume we're
in that space for now.
OK.
And so, for large ranges,
we can see it is not an
unreasonable assumption.
Although, I think in extremes,
you could see this becomes
unreasonable.
OK, so those are assumptions.
Completeness, which once
again, I screwed up in
describing.
Come back to the second way I
described it, which means you
can't say you're not sure.
You always have preferences
over things.
That doesn't seem
unreasonable.
Transitivity which we've been
living with since we were
kindergartners.
And non-satiation, which could
be a little controversial, but
we'll live with it for now.
Now given these, we're going to
talk about the properties
of what we call indifference
curves.
This is why I screwed
up before.
Of course you can be indifferent
between things.
That's the whole point
of economics.
I don't know why I
got that wrong.
I haven't taught this course
about six years, so I lost
track of things.
Properties of indifference
curves.
So indifference curves are our
name for what you could also
think of as preference maps.
In economics, we like to be able
to describe everything,
as I said, three ways,
intuitively, graphically, and
mathematically.
Preference maps are the
graphical representation of
people's preferences which we
do through graphics that we
call indifference curves.
So now let's go to the example
I'm going to use that I'm
going to use throughout these
next couple lectures of a
decision you have to make.
Now I tried to think of a cool
way to make this example cool,
and I just couldn't.
So its going to be
a boring example.
It's going to be, imagine your
parents gave you some money
and you had to decide whether
to buy pizza or see movies.
I tried to make it at least a
little bit relevant even if I
couldn't make it cool.
You've got to decide whether
to buy pizza or see movies.
That's your decision.
That's the trade-off
you're making.
We're in a world with only two
goods, pizza and movies.
And you're deciding how to
allocate the money your
parents gave you over
pizza and movies.
Now let's say we're going to
consider three choices of
pizza and movies.
So go to figure 4-1a.
We're going to consider, you
could have two pizzas and one
movie, that's point A. You could
have one pizza and two
movies, that's point B. Or you
can have two of both, that's
point C. That's just three
choices you're facing.
Once again, we're ignoring
paying for them.
Budget constraints
is next time.
Now we're just saying I'm giving
these three choices.
Well how do you feel
about them?
Well let's assume that
you're indifferent--
and this is why you can
be indifferent.
What I said before,
just strike.
Let's say you're indifferent
between two pizzas and one
movie, and one pizza
and two movies.
Let's say, if you had two pizzas
and one movie, or one
pizza and two movies, you pretty
much feel the same
about them.
But clearly you like two pizzas
and two movies better
than either of the first
two combinations.
Then what we can do is we
can draw what we call
indifference curves.
And that's in figure 4-1b.
These are maps of your
preferences.
An indifference curve is the
curve showing all combinations
of consumption along which the
individual is indifferent.
And I'll say that again,
very important concept.
An indifference curve is a curve
showing all combinations
of consumption along which an
individual is indifferent.
So you have an indifference
curve.
I said you were indifferent
between A and B. So you have
an indifference curve that runs
between A and B. That
means that all, and I'm assuming
that all combinations
along this curve, you're
indifferent.
So you're equally happy getting
two pizzas and one
movie or one pizza
and two movies.
But point C, which is two pizzas
and two movies is on a
different indifference curve.
You're not indifferent between
point C and points A and B.
You're indifferent
between A and B--
I'm just assuming this, I'm
not saying you are.
But I'm just assuming, let's
imagine you are.
But you clearly like two pizzas
and two movies better
than one of one and
two of the other.
Yeah?
AUDIENCE: Does that break the
completeness rule for the--
PROFESSOR: Does that break it?
Why would that break it?
AUDIENCE: Do you prefer
pizza over movies
or movies over pizza?
PROFESSOR: No.
Because this is my
screw up before.
Completeness just means you
know how you feel about
everything.
So strike from the record
my initial description.
Completeness means you just
know how you feel about
everything.
You're allowed to
be indifferent.
Completeness just means you
can't say, I don't know, I
don't know how I feel
about pizza.
You've got to have feelings
for pizza.
OK.
You've got to know how
you feel about stuff.
That's what completeness is.
So armed with those assumptions,
there are four
key properties of indifference
curves that we have
to keep track of.
Four key properties of
indifference curves.
The first is that consumers
prefer higher
indifference curves.
So you prefer higher
indifference curves.
Prefer higher indifference
curves.
What I mean by that is, the
further out the indifference
curve, the more you prefer it.
And this comes naturally from
the non-satiation assumption.
Given that we've assumed
non-satiation, you must always
prefer an indifference curve
that's further from the origin
because it's more, and
more is better.
OK so given non-satiation,
you will always prefer an
indifference curves that are
further from the origin.
That follows directly
from non-satiation.
The second point is that
indifference curves are always
downward sloping.
Indifference curves are always
downward sloping.
Indifference curves are always
downward sloping.
And that, once again, comes
from non-satiation.
To see this, let's look at the
next figure, an upward sloping
indifference curve.
Why does an upward sloping
indifference curve, someone
tell me, violate
non-satiation.
Yeah?
AUDIENCE: Because you're
indifferent to getting more.
PROFESSOR: Yeah.
Because this would say
you're indifferent
between (1,1) and (2,2).
It's not quite drawn right.
We ought to just have this go
through to point (2,2).
But basically, this would say
you're indifferent between
getting one pizza and
one movie or two
pizzas and two movies.
You can't be because that
violates more is better.
So indifference curves can't be
upward sloping, they've got
to be downward sloping by the
non-satiation assumption.
OK, that's the second property
of indifference curves.
The third property of
indifference curves is
indifference curves
cannot cross.
Indifference curves
cannot cross.
Why can't indifference
curves cross?
Well here I forgot to have
Jessica do a pretty diagram,
so you'll have to deal with
my ugly handwriting here.
So why can't indifferent
curves cross?
Well imagine a situation
where you have your
pizza and your movies.
And imagine a situation where
you have one indifference
curve that looks like this,
and one indifference curve
that looks like this.
OK, two indifference curves.
And you've got, let's label
these points A, B, and C.
Now could someone give me, based
on the properties of
indifference curves that we
talked about over here, given
these three properties,
can someone tell me
why this is a violation?
Yeah?
AUDIENCE: Because A and B are
on the same curve, meaning
you're indifferent between A and
B. A and C are also on the
same curve because you're
indifferent between the two.
But that means you're also
indifferent between B and C
which can't be true because
more is better.
PROFESSOR: Exactly.
So transitivity says I must then
be indifferent between B
and C through the logic
you just laid out.
But I can't be indifferent
between B and C because B
dominates C. B has a basically
the same number of movies, but
more pizza, so I must
like B better.
So by the combination of
transitivity and non-satiation
indifference curves
can't cross.
And finally, completeness, which
is the most awkward of
these assumptions, it simply
means you can't have more than
one indifference curve
through a point.
So basically, the idea of every
possible bundle has one
indifference curve.
You can't have two indifference
curves through it
sayin, I'm not sure which
indifference curve I'm on.
I'm not sure how I
feel about this.
You know how you feel.
There's one indifference curve
through every bundle.
There's not two indifference
curves through a bundle.
So this is the way we think
about preference maps which is
the sort of core building
block of utility theory.
Now I was an undergrad here,
took this course, but I never
really understood indifference
curves until I had a year off
with a grad student who was
trying to decide where to take
a job and he did it through
just showing me an
indifference map.
He said look, I'm trying to
decide where to take a job,
and I care about two things.
I care about how good the place
is and where it is.
So he said here, he had
location and he
had academic rank.
And he said look, I'm
indifferent between Princeton
which has a shitty location but
a wonderful academic rank.
I'm from New Jersey, but it's
still a shitty location.
OK, and Santa Cruz.
And Santa Cruz which has not
such a good academic
reputation, but a pretty
awesome location.
And he said here's my
indifference map.
And where did he end up going?
He ended up going to the IMF,
the international monetary
fund in DC which had a better
location than Princeton--
worse than Santa Cruz, but a
better reputation than Santa
Cruz and worse than Princeton.
So he decided he was indifferent
along this map,
and he ended up choosing
a point in the middle.
But indifference curves are just
a way of representing two
dimensional choices.
Now very few choice in life are
really two dimensional,
but that's a nice example.
Question in the back?
AUDIENCE: I was wondering if
IMF, the point would be
actually not on the curve,
but further out?
PROFESSOR: If it were
further out.
A great question.
So imagine if IMF were here.
What should he have done?
Definitely go to IMF.
Here he was indifferent.
He could flip a coin and be
equally happy at all three.
But if IMF were out here, and
maybe it was because that's
what he chose.
That's a good point.
I don't know if IMF
was here or here.
The fact that he chose
IMF, it can reveal it
wasn't anywhere in here.
It's a very good
point actually.
It can reveal it wasn't
anywhere in here.
That we know.
But I can't tell if it
was on the curve
or outside the curve.
It could have been on the
curve because he's
indifferent, so who knows, he
could have flipped the coin.
Or it could have been outside
the curve because it's better.
We can't tell that.
That's a good point.
All right.
So that's a preference map.
That's indifference curves.
Now let's step from indifference
curves, which is
a building block of preferences,
to utility.
Now everything you need to
know about preferences is
represented in those
indifference maps.
The problem is they're pretty
awkward to work with when we
need to actually prove theorems
and solve and
understand how people
make decisions.
That's a lot easier if we have
a mathematical representation
of those preference maps.
And that's the utility
function
So the utility function is a
mathematical representation of
preferences.
That's all it is.
You're going to be hearing this
term in your nightmares
for the next semester.
Utility functions.
But remember, it's just a
mathematical representation of
people's underlying
preferences.
Don't be scared of it.
And the key thing is that we
assume individuals have these
well-defined utility functions,
and by maximizing
those utility functions we can
tell what choices they're
going to make.
So for example, suppose that
I said that your utility
function over pizza and movies
was the square root of pizza
times movies.
That's a utility function.
I'm going to say, what the
hell does that mean?
Well, it doesn't mean anything,
it's a utility function.
It's your preferences.
It's a mathematical
representation of your
preferences.
What does that mean?
What it means is--
it doesn't mean anything
inherently, but it tells us
about your preferences.
What it tells us is that
your preferences can be
represented.
If you flip back to figure 4-1b,
it tells us those are
your preferences because you're
indifferent between two
pizzas and one movie and one
pizza and two movies.
Of course you're indifferent.
They both give a utility
square root two.
But you prefer two pizzas and
two movies because that gives
a utility of two.
So this is a mathematical
representation consistent with
those utility indifference
curves.
Not the only one.
There's other mathematical
representations that could be
consistent with those
indifference curves.
But let's posit that this is
your utility function.
This is a mathematical
representation of your tastes.
Now what does utility mean?
Utility means nothing in the
sense that it is not a
cardinal concept.
It's only an ordinal concept.
So if I say to you that you get
two utils from two pizzas
and two movies, that doesn't
mean anything.
It just means that you get
more than from one
pizza and one movie.
And we can even get the ratio
that you get square root of
two more, than you get from
one pizza and two movies.
We can do ranking and
ordinality, but we can't
assign cardinality.
I can't say how happy you are
in some abstract absolute
sense from two pizzas
and one movie.
I can't give a cardinal
form preference.
But this is an ordinal ranking
of preferences.
I can tell what you like
better than what else.
That's why utility function
is a representation of
indifference maps.
They're just a mathematical tool
for comparing bundles,
they're not some inner answer
to the value of your soul or
something like that.
Don't imbue these with
too much magic.
They're just mathematical ways
of representing preferences.
The key concept, the single most
important concept, for
consumer theory for
understanding how consumers
make decisions is the concept
of marginal utility.
We'll talk a lot this semester
about marginal this and
marginal that.
And this is our first example.
Marginal utility.
That is how your utility changes
with each additional
unit of the good, or
the derivative
of the utility function.
If you want to do it in calculus
terms, marginal
utility is the derivative of
your utility function with
respect to one of the inputs.
But if you don't want to put it
in calculus terms, it's as
you add each unit of one of the
elements of the utility
function, how does
utility change
So to see this, let's do an
example of marginal utility.
Imagine for a moment that you
have two pizzas, p equals two.
You've got two pizzas,
they're there.
Your roommate's got
them or something.
OK, now I want to ask, how does
your utility change as
you see additional movies?
And to show that, let's
look at figure
4-3 which isn't here.
Whoops.
There's no figure 4-3.
Do you got that figure 4-3?
AUDIENCE: There was never
any figure 4-3.
PROFESSOR: There was never
any figure 4-3.
So let's go to 4-5.
So basically--
AUDIENCE: Figure 4-4?
PROFESSOR: No but--
actually fine.
4-4.
So basically what this is
showing, what figure 4-4 is
showing, is it showing how--
no actually, let's go to 4-5.
They're out of order.
Let's go to 4-5.
What 4-5 is showing--
no, that's not going to work.
OK, back to 4-4.
What figure 4-4 is showing, is
it's showing how your marginal
utility for movies evolves, how
your utility evolves as
you get more movies.
Given that you have two pizzas,
this is the evolution
of your utility as you
get more movies.
So each additional movie
increases your utility.
The slope is positive.
By more is better,
we know that.
Even if it's some date
movie, it still
improves your utility.
So it still improves your
utility, but at
a diminishing rate.
And that's the key
is that we assume
diminishing marginal utility.
The key assumption underlies
everything we'll do for
consumers is diminishing
marginal utility.
We assume that additional movie
increases your utility,
but at an ever diminishing
rate.
So basically, we can actually
graph your margins.
And that's what figure
4-5 is, is a graph of
your marginal utility.
So basically, when you have
two pizzas and one movie,
utility is square root
of 2, right?
Now what I'm saying is if you
get one more movie, your
utility is going to rise from
square root of 2 to 2.
So the marginal utility
of that next movie --
is that right?
Two movies.
1.4.
Yeah, it's going to rise by
the square root of 2.
You're going to multiply your
utility by the square root of
two, so your marginal
utility--
you're going to go from the
utility of square root of 2 to
utility of two.
So utility is going
to increase by the
square root of 2.
Utility is going to increase--
I'm doing this wrong, hold on.
One second.
From one movie.
I see.
I see.
So, I'm sorry.
This isn't the delta, this is
the level of marginal utility.
So I'm graphing the actual level
of marginal utility.
Back up.
OK, so I'm graphing the actual
level of marginal utility.
So when you have two pizzas and
one movie, your marginal
utility, your actual utility--
I see, that's what this is.
This is the actual utility
I'm graphing.
So I told you a minute ago, we
can't measure utility as a
cardinal concept, but actually
here I'm doing it anyway
because it's to illustrate
marginal utility.
So your utility, OK.
When you have one movie is
1.4, square root of 2.
That's your utility.
Now when you move from one movie
to a second movie, your
utility goes up from square
root of 2 to 2.
Your utility goes up by 0.6.
So the marginal utility of
that second movie is 0.6.
Utility was 1.4, was
a square root of 2.
Now it's increased to 2.
So the marginal utility of
the first movie is 0.6.
Now let's say you add another
movie, you go to three movies.
What's your utility now?
It's the square root of 6.
So it's gone from 4,
to the square root
of 6, which is 2.45.
So your marginal utility of
the third movie is 0.45.
This graph is messed up because
the first one is an
actual utility level.
So the first one I say,
for one movie, you
have a utility 1.4.
And then for the second movie,
I give the marginal utility,
the third movie marginal
utility.
So, this graph sort of-- yeah?
AUDIENCE: It shows the marginal
utility of the very
first movie.
PROFESSOR: Yeah, I guess
that's right
because you're zero.
You're zero movies.
OK, right.
You're right.
OK, so the first one is the
marginal utility of the very
first movie, you're right.
So the very first movie gives
you marginal utility of 1.4
because you go from 0
to square root of 2.
That's right.
My bad.
So you go from 0 to square root
of 2 to get a marginal
utility of 1.4 for the first
movie From square root of 2 to
2, you get 0.6 the next movie.
From 2 to square root
of 6, you get 0.45
for the third movie.
For square root of 6 to square
root of 8, you only get 0.38
from the fourth movie,
and so on.
So the key point is that these
marginal utilities are ever
decreasing.
Each additional movie gives you
less incremental utility.
And if you stop and
think about it,
it's kind of intuitive.
Just stop and think, think about
the movies you want to
see right now.
The four movies you
want to see.
Presumably whichever you ranked
first would give you
more utility to see than
whichever you ranked second.
And if you think the movies
that are out right now are
pretty crappy like I do, by the
time you get to the fourth
movie, you're not getting much
utility from it at all.
Thinking about movies that are
out now, you're getting a lot
of utility from that first
movie you see.
Marginal, extra utility from
the first movie you see.
But each additional one is
giving you less and less.
And that's the idea of
diminishing marginal utility.
Likewise with pizzas, if you
haven't eaten all day, that
first pizza can give you a very
high marginal utility.
The enjoyment you get from
eating that first pizza can be
very large.
But the second pizza,
not so much.
You're already pretty full.
Third pizza, even less.
And then fourth pizza would
probably violate
non-satiation.
So that's the basic idea.
Yeah?
AUDIENCE: I have a question.
Do we assume that the goods
are homogeneous.
Is it the same movie
watched four times?
Or different movies?
PROFESSOR: Actually, that's
a great question.
And you have to specify that
as part of the problem.
I haven't specified that here.
Obviously it can't be the same
pizza eaten four times.
It could be the same kind of
pizza eaten four times.
But do you see the same movie?
I haven't specified that here.
So there's not a general
assumption about that.
It depends on how
I define movies.
Did I define movies as--
I don't know.
God, I'm terrible.
All I know that's out now is
the Guardians of G'ahoole
because I've got a little kid
who is interested in it.
Whatever movie's out.
Do I define movies as Guardians
of G'ahoole, or do I
define movies as
seeing a movie?
And I didn't specify that.
Implicit in my examples,
I specify
movies as seeing a movie.
But you have to specify that to
be more precise if you're
actually trying to
figure out--
it depends what you're
maximizing over.
If you're maximizing over seeing
any movie or maximize
over seeing the same movie.
And I didn't specify here.
AUDIENCE: It can work
in both cases.
PROFESSOR: It would work
in both cases.
Clearly you could imagine,
actually it's a very good
point, where do you think your
marginal utility would
diminish more?
Seeing the same movie.
So what your example points out
is that different goods
will have different rates of
diminishing marginal utility.
OK, so marginal utility will
always be diminishing, but at
very different rates for
different goods.
So the general principle is
that they'll be generally
diminishing marginal utility.
But at different rates
for different goods.
So after all my mess ups,
let me just review.
Marginal utility is diminishing
because each good
is worth less to you.
It's always positive because
of non-satiation.
And this graph represents the
marginal utility you get from
each movie you see conditional
on having eaten two pizzas.
Marginal utility is
the increment from
the next unit consumed.
Now let's get back
on track here.
Now let's go to thinking
about--
now that we have this concept
of utility and marginal
utility, let's now bring
utility back
to preference maps.
Let's ask, given what we know
about utility, what can this
teach us about the shape
of preference maps?
What's the linkage between
utility and preference maps?
And that linkage comes through
something we call the marginal
rate of substitution.
The marginal rate of
substitution is the
mathematical concept that links
preference utility with
preference maps.
The marginal rate of
substitution technically is
the slope of the indifference
curve.
It's delta P over delta M. The
slope of the indifference
curve is the marginal rate
of substitution.
That's what it means
graphically, but here's what
you have to understand
at a deeper level.
What it really is, it's
the rate which you are
willing to trade off.
The rate at which you are
willing to trade off the
y-axis for the x-axis.
The rate at which you're
willing to trade
off pizza for movies.
So that's what it means
intuitively.
The slope of the curve tells you
that you're indifferent.
Remember, you're indifferent
between any points along with
this indifference curve.
You're indifferent between four
pizzas and one movie,
you're indifferent between two
pizzas and two movies, and
four movies and one pizza.
You're indifferent
along all those
combinations of figure 4-6.
The MRS is the slope of that
curve telling you the rate at
which you're willing to trade
off pizza for movies.
Now just a side note here,
you're never, of course,
actually trading.
There's not some market
where you bring a
pizza and get a movie.
So I didn't say trade, it's
not like baseball cards.
I said trade off.
What I mean is ultimately you
have some budget, and you have
to allocate that budget.
So if you decide to allocate
it on pizzas, you can't
allocate it on movies.
Or the more you allocate on
pizzas, the less you can
allocate on movies.
So there's always a trade-off.
Remember, I said, economics is
always about trade-offs.
Given your limited budget,
there's always a trade off.
And the rate at which you're
willing to trade off is your
marginal rate of substitution.
Given that you're going to have
to trade off-- and we
haven't got a bunch of
constraints yet, we'll get to
that next time--
the rate at which you're willing
to is your marginal
rate of substitution.
Yeah?
AUDIENCE: Is that rate usually
related to the price?
PROFESSOR: Ultimately no.
I'm sorry.
The marginal rate of
substitution purely comes from
your preferences.
Ultimately to decide how much
you actually consume, you'll
need to bring in the price.
So remember, I haven't talked
about prices here, we haven't
talked about that here.
But this is a preference
concept.
This has nothing to
do with prices.
But you're getting
ahead of us.
We'll see next time, to decide
how much you actually consume,
you're going to relate the
marginal rate of substitution
to the prices you face
in the market.
And that will decide how
much you consume.
This is just a utility
concept.
Yeah?
AUDIENCE: Did you say it was
the y-axis or the x-axis?
That would be negative?
PROFESSOR: It's negative.
Yes.
Of course.
Right, of course.
The point is how many movies are
you willing to give up to
get another pizza?
How many pizzas are you willing
to give up to get
another movie?
MRS, it's very hard to remember
what's on the top,
what's on the bottom.
Be very careful on this.
But that's why I said remember
it's the y-axis or the x-axis.
It's how many pizzas you're
willing to trade off to get
another movie.
Basically remember when I say
trade off, here, this is not
that you're literally trading,
it's that ultimately you're
going to have to make
that trade-off.
Ultimately when we come to the
next lecture and face a budget
constraint, you're going to have
to decide how do I want
to allocate my budget across
pizzas and movies?
The way you're going to decide
that is by the relationship of
how you feel about trading
off one for the other.
Now here's the key feature
of the MRS which
is the MRS is yeah?
Question?
Yeah.
AUDIENCE: That and exchange
rates are always changing
depending on how much you
happen to be trading.
PROFESSOR: Exactly.
The MRS is diminishing.
Technically when you go to grad
school, you realize that
marginal utility isn't actually
technically always
diminishing.
I said it is.
For this course it is.
But if you want to get
mathematically correct, really
what's always diminishing that
you prove is the marginal rate
of substitution is always
diminishing.
So we have diminishing marginal
utility for the
purpose of this course, but the
really important concept
is you have diminishing marginal
rate of substitution.
The rate at which you're willing
to trade off pizza for
movies is going to fall
as you have less
pizza and more movies.
So to see that, look at this
graph, and let's compute the
marginal rate of substitution
along each segment.
So let's localize.
Imagine the segments were
linear, imagine we had two
linear segments between
these points.
We don't.
But imagine for a
second we did.
So the marginal rate of
substitution from the first
point, four pizzas and one
movie, to the second point,
two pizzas and two movies,
the marginal rate of
substitution is -2.
You are willing to give up two
pizzas to get one movie.
This is the same graph,
figure 4-6.
This isn't on the graph, you
have to write it on.
So going from that first point
to that second point, you're
willing to give up two pizzas
to get one movie.
So that rate of marginal
substitution is -2.
However, when you're at two
movies and two pizzas, and I
say OK, how about giving up one
more pizza to see movies?
Now you say, wait a second.
To give up one more pizza,
I need to see two movies.
My marginal rate of substitution
on that second
segment is -1/2.
The marginal rate of
substitution on the first
segment is -2.
The marginal rate of
substitution on the second
segment is -1/2.
Once again, assuming they're not
linear, so it's actually
changing everywhere, but if they
were linear, that's what
it would be.
Can someone tell me why?
Why is the marginal rate of
substitution falling?
Why is the marginal rate of
substitution lower on that
second segment than
on the first?
AUDIENCE: Because marginal
utility increases the fewer of
something you have.
PROFESSOR: Exactly.
So go ahead, flesh it out, the
fewer of somthing you have, so
tell me in terms of the trade
you're willing to make.
AUDIENCE: You value it more, so
you want to trade more of
something else for it.
PROFESSOR: The point is when I
have four pizzas my marginal
utility of that last pizza
is not very high.
And I'm fine to give
up two pizzas--
and plus I'm only seeing one
movie, there's a second movie
I really want to see.
So you say to me, look,
I've got four pizzas,
I'm seeing one movie.
You say hey, there's a
second movie out I
know you want to see.
I know you don't really
value four pizzas.
At the end, you're
totally full.
Would you be willing to give
up two pizzas to see the
second movie?
And you're like, sure why not?
Well once you have two pizzas
and you've seen two movies,
you're not that interested in
a third movie and you'll be
hungry if you have less than two
pizzas, so then you say,
wait a second.
If you want me to give up
another pizza, you've got to
give me two movies.
Because my marginal utility of
pizzas is rising, my marginal
utility of movies is falling.
And that's why the marginal
rate of substitution
diminishes along the
indifference curve.
So that allows us to write
mathematically the definition
of the marginal rate of
substitution is the negative
of the marginal utility of
movies, or more generally
what's on the x-axis, over the
marginal utility of pizza, or
more generally what's
on the y-axis.
The marginal rate of
substitution, the first key
formula you need to know for
this course, the marginal rate
of substitution is equal to the
ratio of marginal utilities.
Now this is tricky.
Maybe you guys don't find it
tricky, it's the kind of thing
I find tricky.
Which is I defined it as delta
y-axis over delta x-axis.
And yet, when I defined here
the marginal utilities, I
flipped it.
I did the marginal utility of
what's on the x-axis over the
marginal utility of what's
on the y-axis.
Why is that?
Can anyone tell me
why that is?
Why is it flipped
when defined in
terms of marginal utilities?
Yeah?
AUDIENCE: It's a denominator.
So utility over movies--
PROFESSOR: Well, let me try for
slightly more, how does
marginal utility relate?
Yeah.
AUDIENCE: Marginal utility
is delta P over
P. So it gets flipped.
Because of--
PROFESSOR: OK.
Yeah, you're giving the
same answer, which
is technically right.
What I was more looking for but
it's the intuitive version
of that, marginal utility is a
negative function of quantity.
Marginal utility is a negative
function of quantity.
So the fact that it's a ratio of
the quantity of pizza over
the quantity of movies is the
same thing as the marginal
utility of movies over the
marginal utility of pizza.
Because marginal utility is a
negative function of quantity.
The more quantity you
have, the lower is
your marginal utility.
And that's the key
to understand.
So it's the slope of the
indifference curve which is
the ratio of the marginal
utilities, but it's the
marginal utility of
movies over pizza.
Because what that's saying is
that as you get more movies,
you care less about each
additional movie and ditto
with pizzas.
Let's just look at this for
a minute, think about it
intuitively for a minute.
We've seen it graphically,
we're seeing it
mathematically, let's make
sure we understand it
intuitively.
What this is saying is that
as you get more movies--
so let's relate this
to the graph.
As you get more movies and less
pizza, as you move down
that curve, more movies, less
pizza, what's happening?
What's happening to the marginal
utility of movies as
you move down that curve?
What direction is it heading?
AUDIENCE: It's decreasing.
PROFESSOR: What?
AUDIENCE: It's decreasing.
PROFESSOR: It's decreasing.
Because you're getting more
movies and marginal utility is
a negative function
of quantity.
Likewise, the marginal utility
of pizza is increasing because
you're getting less
pizza so you care
about each pizza more.
And that's why the marginal
rate of substitution
diminishes.
That's why it diminishes because
as you move down that
curve, the numerator is falling,
the denominator is
increasing.
And that's why we have
everywhere diminishing
marginal rates of
substitution.
So another way to think about
this is imagine for a moment
what life would be like if we
didn't have diminishing
marginal rates of
substitution.
And once again I'm going to try,
once again Jessica, next
year we'll let you
make this pretty.
But I'm going to try to
draw it crudely here.
Let's do pizzas and
movies again.
Let's do pizzas and
movies again.
Movies and pizza.
And that's one, two,
three, four.
One, two, three, four.
Now let's imagine that instead
of diminishing marginal
utility and instead of
indifference curves being
convex to the origin, imagine
if indifference curves were
concave to the origin, which
is what increasing marginal
rate substitution would imply.
So that would be something where
you'd be indifferent
between four pizzas and one
movie, between three pizzas
and two movies, and between one
pizza and three movies.
So your indifference curve
would look like that.
Not quite to scale,
but you get it.
It would be concave to the
origin instead of convex to
the origin.
In this case, marginal rates
of substitution would be
everywhere increasing.
That is, basically I'd be
willing to give up one pizza
to get one movie.
But to get that next movie,
I'd give up two pizzas.
But as you can see, that
doesn't make sense.
It doesn't make sense that given
that as long as you're
ranking movies, or even more in
the example of seeing the
same movie over and over
again, it's maybe more
compelling.
That basically what you can see
is that if you're willing
to give up one pizza to see that
movie a second time, why
would you possibly give up two
pizzas to see it a third time?
That makes no sense at all.
If you only like it so much you
only give up one pizza to
see it a second time, why would
you possibly give up two
pizzas to see it a third time?
You wouldn't.
It doesn't make sense.
And that's why marginal rate
of substitution has to be
everywhere decreasing, it
can't be increasing.
Yeah?
AUDIENCE: Could it
remain constant?
PROFESSOR: It could actually
remain constant.
Yes, that's right.
You can be indifferent.
My indifference curves--
how many of you guys have
seen Toy Story 3?
I think it's one of my 10
favorite movies of all time.
The greatest children's
movie ever made.
I've seen it three times.
My indifference curve
is virtually--
I've enjoyed it the third times
as much as the first--
it's virtually flat with
respect to Toy Story 3.
I could see it 10 more times and
feel pretty much the same.
So that's certainly possible
that it would be constant,
that I'd be willing to give
up whatever I pay--
$10 a shot to see it.
It's possible.
So basically, almost always,
inequalities will be greater
than or equal to, or less than
or equal to in this course.
It's more fun to talk about
the not equal to case, the
non-linear case.
But linear cases will
exist as well.
It's just a can't be can't
be opposite sign.
You can't have an increasing
marginal rate of substitution.
Another question over here?
AUDIENCE: What about
addictions?
You could want it more
the second time.
PROFESSOR: That's interesting.
So how would addiction
work? so basically--
AUDIENCE: Well it's not
really decreasing.
You need more the second
time, right?
So it has to--
PROFESSOR: That's very
interesting.
I mean in some sense.
So you give up one pizza for
the first shot of heroin.
And then, you're hooked, so then
you'd be willing to give
up two pizzas for the
next shot of heroin.
Yeah, I guess so.
I guess that's right.
I guess we're going to have to
stay away from addiction in
this course.
I guess an addictive good
could look like that.
That's a very good point.
Other questions, comments?
So what we're doing is we're
going to stop here,
understanding that we're
going to have--
leaving this example aside--
we're going to have
diminishing marginal--
yes one more question?
AUDIENCE: [UNINTELLIGIBLE]
PROFESSOR: Basically, we're
assuming by non-satiation that
ever happens.
So once again, that would
violate the non-satiation.
The problem with the
addictiveness example is the
reason it wouldn't work in
this course is eventually
you'd violate your budget
constraint because you'd want
more and more and more.
Maybe not.
But in any case, we're going to
ignore that example, assume
diminishing marginal rate of
substitution, and we'll come
back next time as I put this
together with a budget
constraint to actually
dictate your choices.
