[SQUEAKING]
[RUSTLING]
[CLICKING]
JONATHAN GRUBER: All right,
why don't we get started?
Today we're going to move
on to, finally, the most
realistic market structure.
We talked about perfectly
competitive markets.
Now, that was a very
useful, extreme example
to help us think about
economic efficiency.
We then flipped over to
talk about the somewhat
more real estate
case of monopoly,
but still, very few markets
have only one participant.
A true monopoly is rare
in the private market.
What most markets are
marked by are probably
more features of
oligopoly, which
is a market with a small
group of firms competing
with each other, but with
barriers to entry that keep out
an unlimited number of firms.
Think about these
as markets where
there are some
barriers to entry,
so firms just can't
consciously enter and exit
like they could in
our IBM/Dell example,
but where there's small
enough barriers to entry
that a few firms have
gotten in, not just one.
So it's not a natural monopoly.
It's not like only one
firm can be in there.
Multiple firms are in
there, but they only
know they have to
compete with each other,
not with the big, wide world.
So for example,
the classic example
of an oligopoly industry
would be the auto industry.
Auto manufacturers
clearly compete.
Clearly, if you
watch any sporting
event and watch how much
advertising that goes,
they're clearly competing
with each other.
They're comparing to
each other all the time.
But most of the
cars in the world
are produced by fewer than
10 auto manufacturers.
The notion that we have a
perfectly competitive market
of thousands of sellers
selling identical goods
is clearly not right when
it comes to buying a car.
So that's the model we're
going to want to focus on
for the next few lectures.
Now, within an oligopoly
market, whenever
we think about this
market, we want
to start by noting that
within this market,
these limited sets
of competitors
can behave in two ways.
They can behave cooperatively
or non-cooperatively.
Cooperatively
means that they can
form what's called a cartel.
So when there's an
oligopoly market
and the firms cooperatively get
together and make decisions,
that's called the cartel, the
most famous example of which
is OPEC, the Organization of
Petroleum Exporting Countries,
which are the set of countries
that control about 2/3
of the world's oil,
led by Saudi Arabia,
is the major player in OPEC.
It's a cartel of
about a dozen nations.
And what they do is they
control the vast majority
of the world's oil reserves.
And by behaving cooperatively,
they essentially
turn themselves into a monopoly.
OPEC acts as if they've
got the monopoly in oil.
Certainly they used to.
Now it's getting harder.
Other non-OPEC countries are
starting to produce more oil
and it's breaking down.
But for a long time,
they were essentially
the cooperative
producer of oil, and act
essential like a monopoly,
and they made lots of profits
like a monopoly.
That kept prices high, they kept
production inefficiently low,
and they made lots of money.
However, that's a great
outcome for producers,
but as we'll talk
about next time,
it's actually a hard
outcome to enforce.
Turns out to be hard to
keep cartels together.
And so typically,
oligopoly markets
behave in a non-cooperative
way, with the participants
competing with each other, not
cooperating with each other.
In this case, you can actually
get them driving their profits
down far below the
monopoly level,
and indeed, perhaps even all the
way to the competitive level.
So you can think about markets
as a competitive as one
extreme and the monopoly
as the other extreme,
oligopoly in between.
A cooperative oligopoly
market, like cartel,
will end up close to
the monopoly outcome.
A non-cooperative market will
end up somewhere in between,
and we're going to model
today's where in between
do they end up.
Now, to think about
this, we're going
to have to turn to a
tool, which has really
become a dominant tool in
economics over the last 40
years, which is the
tool of game theory.
Game theory.
So basically, we're going
to think of oligopoly firms
as engaging in a game.
And as with any game, you
need to know two things.
One is you need to
what's the strategy,
and the second is you need to
know when is the game over.
What's the equilibrium?
And that's, essentially,
what you do with any game.
And so basically, the
key with game theory
is that we are going to
find the equilibrium,
and that's going to
yield for us the strategy
that players are going to use.
However, equilibrium in a
game is not well-defined.
It's not like a
set of rules that
are printed out, like monopoly.
In a non-cooperative oligopoly
market, the equilibrium,
you have to actually come up
with different concepts of what
equilibrium is.
There's not a hard and
fast scientific rule.
And the typical
one that's used is
called the Nash equilibrium,
the Nash equilibrium,
named for John Nash, the famous
mathematician, who economists
have claimed as their own,
even though he was really
a mathematician.
But we gave him the
Nobel Prize anyway.
And if you think of
economists, probably
one of the most famous, you
all know about the movie
and book Beautiful Mind.
He is based on the
father of game theory.
So basically, what is
the Nash equilibrium?
The Nash equilibrium is defined
as the point at which no player
wants to change their
strategy, given what
the other players are doing.
So the point at which no player
wants to change its strategy,
given what the other
players are doing.
So in other words, every player
is happy with where they are.
Given what every other
player's decided,
I'm happy to do
what I've decided.
So I've got a strategy,
and given the strategy
other players are using, if I'm
happy with my strategy, then
that's in equilibrium.
So this is a super
abstract concept,
so let's illustrate
it with an example.
And the classic
example of game theory
is the prisoner's dilemma,
which many of you, I'm sure,
know about, maybe
the most of you,
but let's just go through it.
This is the thing from the
old cop movies you see,
where they arrest two guys and
they put them in separate rooms
and they basically
interrogate them separately.
They're put in separate rooms,
and let's say that these guys
get told the following.
They each get told separately
the following thing.
They get told that right
now, if nothing else happens,
there's enough evidence to send
them each away for one year.
However, they're told, if
they turned on their friend
and say their friend's
guilty, then they go free
and their friend
gets five years.
If their friend turns on them,
then the friend goes free
and they get five years.
But if they both turn on each
other, they both get two years.
Set up as if they both stay
silent, they both get one year.
If one turns, then that
person gets to leave
and the person gets five years.
But if they both turn, then
they each get two years.
So how do we think
about decision-making
in that context?
The way we do that is we write
down, we call, a payoff matrix.
We write down in matrix
form this decision.
So let's think about what a
playoff matrix looks like.
Up here is prisoner
B, and here you
have prisoner A. Prisoner A.
And prisoner A can remain silent
or they can talk, and
prisoner B can remain silent
or they can talk.
And then we just
write down, what
are the outcomes,
or the payoffs,
from these different strategies?
So prisoner A says nothing
and prisoner B says nothing,
then A gets one year
and B gets one year.
If prisoner A says nothing
and prisoner B says, oh yeah,
prisoner A is
definitely guilty, then
prisoner A gets five years
and prisoner gets zero years.
If the opposite happens,
if prison A says,
yeah, B's guilty, and B
doesn't say anything about A,
then A gets zero years
and B gets five years.
But if they both say
the other one's guilty,
then they each get two years.
OK, that's the payoff matrix.
And now we want to ask,
given this payoff matrix,
what is the right strategy
for each prisoner to pursue?
And the way we do this in the
Nash equilibrium concept is we
look for a dominant strategy.
Is there a strategy that
I would pursue regardless
of what the other person does?
And if there is,
I'll pursue that.
Because remember, the
Nash equilibrium concept
is, what do I want to do, given
what the other person is doing?
If I have a strategy
I want to do no matter
what the other person is
doing, then I'll do it.
So when asked, is there
a dominant strategy?
Is there a strategy that is
the best thing to do, no matter
what the other guy does?
Well, clearly, if
they're cooperating,
if these were stupid police and
they sat them in the same room,
told them and then left, the
two guys could cooperate.
Well, clearly, the dominant
cooperative strategy
is for both of us
to remain silent.
That's the dominant
cooperate strategy.
And as a team, we only
get two total years
in jail, where everything
gets many more years in jail.
So if they're buddies
and they trust each other
and they cooperate, then that's
clearly the right strategy.
But let's say the police
are smart and put them
in separate rooms.
Well, what's the dominant
non-cooperative strategy?
What is the strategy that A
or B, say A, should produce?
Yeah?
Why?
AUDIENCE: Either way, you're
going to get less years.
Like if you're the only
person silent and you talk,
you get zero, and if
they talk and you talk,
you only get two versus five.
JONATHAN GRUBER: Exactly.
For prisoner A in
this first row--
compare the first column.
We'll say prisoner B is silent.
Then clearly, you're
better off talking than not
talking, zero rather than one.
Let's say prisoner B talks.
Then you're still better off
talking than not talking.
So no matter what B
does, you should talk.
Likewise, prisoner B, no matter
what A does, B should talk.
So the non-cooperative
equilibrium
is actually this outcome.
They both end up talking.
You get sort of a
race to the bottom.
The non-cooperative
outcome is much worse
than if they could
have cooperated.
So basically, what
you get is that
the non-cooperative
equilibrium is always
worse for the players than
the cooperative equilibrium.
And this was like an
unbelievable insight of Nash.
Before Nash, we always
thought competition
was always and everywhere good.
We always thought more
competition is better,
for the reasons we talked
in the first 10 or 12
lectures of this class.
Nash was the first one
to say, no, actually,
competition can be bad.
Cooperation can be better.
I don't know if you remember
the scene in a Beautiful Mind
where they're picking
up girls in the bar.
And he described
basically a Nash strategy,
how competition will lead
to the worst outcome.
And basically, that's
what you see here,
that competition can
actually lead to a worse
outcome than cooperation,
and that was really
Nash's brilliant insight.
Now, this is a cute example
with prisoners, but actually--
well first, two points.
First of all, this
generally shows you
how you do gain favor
with Nash equilibrium.
Basically, you look
at the payoff matrix,
you find the dominant
strategy, and then
you find where those dominant
strategies intersect.
And here, the
dominant strategies
intersect at this
cell, therefore
that's the equilibrium.
So that's basically how you do
game theory in a game theory
kindergarten level.
You look at the matrix.
You find each player's
dominant strategy.
And you find the point at
which those dominant strategies
intersect, and at that point,
that's the equilibrium.
Now, that's all well and good
for a simple example like this,
but let's actually apply
to an economics example.
Let's think about advertising.
So think about Coke and Pepsi.
Right now, let's think about
their decision to advertise.
Now, obviously it's
a simple problem.
Obviously Pepsi
should just be illegal
because Coke is way better.
But sadly, it's not, and
sometimes I have to drink Pepsi
and I'm very sad.
But nonetheless, in the real
world, we have Coke and Pepsi
and they have to decide
how much to advertise.
Now, the dominant
cooperative strategy
would be to say,
look, advertising
costs us a ton of money.
Let's just split the market.
Let's have a monopoly
market and just split it.
We're close to
splitting it anyway.
Coke's got some more of it.
We're close to splitting it.
Let's just split it.
Yeah?
AUDIENCE: Can you
actually do that?
JONATHAN GRUBER: What?
AUDIENCE: Can you
actually do that?
Because I remember,
there were places
that you get where
you aren't allowed
to sell in the same place.
JONATHAN GRUBER: OK, but that's
different than the cooperative.
That's imposed not by
Coke and Pepsi jointly.
That's imposed by Pepsi
saying to a university campus,
for example, we will
cut you a better deal
if you'll agree
not to sell Coke.
That's not cooperation.
That's competition.
So there's cooperative strategy.
What if they don't cooperate?
Well, let's imagine we have
the following payoff matrix.
You've got Pepsi up here,
and they can advertise or not
advertise.
And you've got Coke here,
and they can advertise or not
advertise.
And let's say the payoff
matrix is the following.
Let's say the total amount
of profit to be made
is 16 whatever,
billion, whatever
units you want to
make it, $16 billion.
And let's say if there's
no advertising, Coke gets 8
and Pepsi gets 8.
But let's say
advertising costs money.
It costs 5, $5 billion.
So let's say if
they both advertise,
then they still end up
splitting the market,
but they only make 3.
C equals 3, P equals 3.
I'm sorry, advertise.
yes, you're right.
C equals 3, P equals 3.
And here C equals 8, P equals 8.
So basically, you
have a situation
where they both end of
splitting the market either way,
but they just split a smaller
net profit if they advertise.
So clearly, they'd
rather be here than here.
But what happens in the
off diagonal elements?
Well, let's say also that if
Coke advertises but Pepsi does
not, then let's say Coke
ends up making $13 billion
and Pepsi ends up making minus--
I'm sorry, if Coke advertises
and Pepsi does not,
they split money.
And Pepsi makes negative 2.
They actually lose money
because they have fixed costs
and they don't sell anything.
Nobody buys Pepsi.
It'll lose money.
And let's say if
Pepsi advertises
and Coke doesn't, then
Coke makes negative 2
and Pepsi makes 13.
So actually, if you
don't advertise,
you're really screwed, and the
other guy is really screwed.
Yeah?
AUDIENCE: Does this include
the cost of advertising?
JONATHAN GRUBER:
This does include
the cost of advertising.
But it's just Coke gets a huge
market, expands its market.
So now let's play the game.
Well, now let's say you're Coke.
You say, well, if I advertise
and Pepsi advertises, I make 3.
But if I don't advertise
and Pepsi advertises,
I make negative 2.
So I should advertise.
If I advertise and Pepsi
doesn't advertise, I make 13.
If I don't advertise and Pepsi
doesn't advertise, I make 8.
So either way, my dominant
strategy is to advertise.
And likewise, Pepsi
does the same thing.
I screwed up writing this
compared to my notes,
but it's good because
it shows you--
I flipped the matrix, but
the logic is the same.
It helps you not just
memorize cells of the matrix
but learn the logic.
The point is either way,
the dominant strategy
is to advertise, so
they both advertise.
So real world example
of how you can end up.
Now, so much of Pepsi
and Coke do this.
Actually, there was an
industry that did this.
So when I was a kid, you never,
ever saw ads for liquor on TV.
There were beer ads and wine
ads, but no hard alcohol ad.
No bourbon, no whiskey,
no nothing, gin.
All these Captain
Morgan's ads we see now,
they didn't exist
when I was a kid.
But it wasn't because the law.
It was because the hard liquor
industry cooperatively agreed
none of them would advertise.
So they actually imposed
the cooperative equilibrium,
and then that broke down.
I don't know the story
of how it broke down.
But it broke down.
Now they all advertise,
and they're probably
all worse off than they were
when they didn't advertise.
We'll talk next time about
why it probably broke down.
I don't know the stories.
I have a rough sense, and we'll
talk about that next time.
But this is the point of how a
non-cooperative equilibrium can
drive you to a bad outcome.
Now, basically,
this doesn't just
apply to prisoners
or businesses.
It applies to people, too.
So let's say poor Hector
back there has had
a fight with his girlfriend.
And they've had a big fight.
They've going a little while.
They've had a big fight.
And Hector has got to decide,
do I apologize or do I wait
for her to apologize?
Well, the last
thing Hector wants
is to go up there and apologize
and have her say, forget it.
I'm breaking up with you.
That'd be the worst.
If he knows she's going to
be like, oh, I'm sorry, too,
then he'd be happy to do it.
But what if he goes, no
I'm breaking up with you,
and she's thinking
the same thing.
So what happens, they break up.
We've been through this
many times in our lives.
This is the
non-cooperative strategy.
Basically, if you know what the
other person is going to do,
your dominant strategies to be
an asshole, and basically that
happens a lot in the
context of the real world.
So now we have this
sad-sounding outcome,
that basically game theory leads
to bad outcomes for producers,
at least.
But this is what's
exciting about game theory.
So when I went to
grad school, back
when dinosaurs roamed
the earth, game theory
was taught barely
in the sequence.
It was like an extra
course, taught a little bit.
Now it dominates the teaching
of microeconomics, in economics.
And it doesn't dominate, but
it's a whole like component
of our core
microeconomics education,
because it's given such
a cool set of tools
to think about these decisions.
Now, I can't give you even 1%
of the flavor of game theory.
If you want to learn more, I
highly suggest you take 1412,
which our game theory class,
and you can learn a ton.
But let me show you
one interesting wrinkle
of the things game theory
can do, to go beyond this.
And that's to imagine that Coke
and Pepsi are not playing a one
shot game, but a repeated game.
Repeated game.
So now imagine that Coke
says to Pepsi the following,
I promise to not advertise as
long as you don't advertise.
But if you ever advertise,
I will advertise forever.
Coke says to Pepsi,
I promise not
to advertise as long
as you don't advertise,
but if I ever catch
you advertising,
I'm going to advertise forever.
So think about Pepsi
choice in period 1.
Pepsi's choice in period 1.
In period 1, they could
say, ha, stupid Coke.
I'm going to jump
on and advertise.
They promised not to advertise.
So if Pepsi advertises, they're
going to make 13 in period 1
because Coke's taken
themselves off to the side.
But after period 1, they're
going to make 8 forever.
No, I'm sorry.
They're going to make 3 forever.
Because Coke's
going to advertise.
They're going to advertise.
They break down to the
non-cooperative equilibrium,
if Pepsi advertises.
Now, what if Pepsi
doesn't advertise?
As long as it doesn't advertise,
then it gets to deal with Coke,
so it makes 8 forever.
We'll talk later in the course
about how you combine numbers
that happen at different
times, but trust me,
8 forever is a way better
deal than 13, than 3 forever.
So actually, by having
this be a repeated game,
Coke has solved the
prisoner's dilemma.
It's essentially imposed
a cooperative equilibrium
on the problem.
So that's how repeated
game can fix this.
But-- this is where the game
gets really exciting-- that
only works if this
game never ends,
because once Coke
or Pepsi thinks
there's an end to the game,
the entire thing breaks down.
So imagine, for example, that
Coke makes the offer to Pepsi,
but Pepsi is worried
that in 10 years,
the government is
going to outlaw soda.
The government said, look,
we're heading that direction.
Soda is going to be
illegal in 10 years,
so I don't want to do this.
I'm sorry, I have
that in my mind.
So Coke offers the deal
now, what do I think?
Well, let's think about Pepsi's
decision in the ninth year.
They've made 8, 8, 8, 8,
8, and they get to year 9.
Now in year 9, they know that
next year there's no more game.
So what should they do?
Advertise.
Grab the 13 in the last period,
because Coke can't punish them
because the game's over.
But Coke knows this.
So what's Coke going to
do in the ninth year?
Advertise.
It's going to advertise, so
they're both going to make 3.
Well, if Pepsi
knows Coke's going
to advertise in the ninth
year no matter what,
what should Pepsi
in the eighth year?
Advertise.
And if Coke knows Pepsi
is going to advertise
in the eighth year,
what should Coke do?
And so on, and it ends up that
they both advertise all the way
through.
So the game breaks
down if it's an end.
This is really kind
of neat, and this
is what game theory
is all about,
is how do you think
through these more
complicated scenarios that
are much more complicated
than the prisoner's
dilemma, and actually
think about how firms and
individuals might actually
behave?
Yeah?
AUDIENCE: Wouldn't it also
be advantageous if they just
advertised the first year
instead of these contracts,
kind of what we were
talking about earlier?
JONATHAN GRUBER: Sure.
And once again, that's
what you cover in a field
course like game theory.
What about alternative
forms of contracting,
with exclusionary
contracting, what
we call tying in contracting?
That's great, and they would.
But that's why you
got to take 1412, OK?
Yeah?
AUDIENCE: Would it
be a better outcome
if they cooperated and switched
periods of advertising?
Like for the first period,
they get 13, they get minus 2.
JONATHAN GRUBER:
Yeah, the way I've
set this problem up,
if they could commit
to that, that would be right.
But you'd have to commit to it.
Because then the period that
Pepsi promised to take off,
if they actually advertised that
period, then Coke's screwed.
So that would work as a
repeated game solution,
but it wouldn't work as
a non-repeating game.
It would work as an infinite
repeated game but not
a non-infinite repeated game.
Good question.
OK, other questions?
All right, so that's the
basis of game theory.
That's just a taste
for the excitement
that you can learn
with game theory.
But in fact, in
economics, we like
to write those as fun
examples, but we really
prefer to do math.
So let's actually
think about the math
of how we take game
theory concepts
and put them in practice.
And the way we do that
is through the concept
of the Cournot model.
The Cournot model of
non-cooperative oligopolies.
So the Cournot model of
non-cooperative oligopoly
is the standard workhorse model.
It takes this
intuition and puts it
into the optimizing
math we've been
doing so far in this class.
Now let's imagine
non-cooperative case,
but now let's imagine
not just two choices,
but realistically, there's
a whole set of choices.
Then how would you
behave in that case?
So let's imagine that
there's two airlines, United
and American.
So we have an oligopolistic
two-firm airline industry.
Obviously, the math can
extend to more firms,
but just to start, and I'll
talk about that next lecture.
But for now, imagine a two-firm
industry, United and American.
And because the hub
and spoke system
we discussed last
time, let's imagine
that they're the only two folks
that go from Boston to Chicago.
Because it's hub
and spoke system.
The only folks that go
from Boston to Chicago
are United and American,
and they do, in fact,
dominate that line.
So let's imagine
they're the only folks,
and say no other firms
can compete on this route
because they can't get
slots at the airport.
So the question is,
how do these firms
decide how many flights to run?
It's not just advertise, don't.
It's literally a
continuous decision
of how many flights to run every
day and how much to charge.
They've got to
make that decision.
And the Nash equilibrium
here, the subset of Nash,
for this example, is called
the Cournot equilibrium.
And the Cournot
equilibrium exists
when a firm chooses
a quantity such
that, given the quantity
chosen by the other firm,
they don't want to change.
So a firm chooses, essentially,
a profit-maximizing quantity,
given the quantity
chosen by the other firm.
And that profit-maximizing
quantity, then you're
in Cournot equilibrium,
if you have
chosen a quantity that is
profit-maximizing, given
what the other firm is doing.
So basically, how do we
actually carry this out?
Let's talk about the steps.
So the first step, I'm
going to talk intuitively
about the math, what
we're going to do,
and then I'm going to
talk mathematically
and graph what we actually do.
There's essentially three steps
in solving for the Cournot
equilibrium.
The first is ask how your
demand changes when some of it's
absorbed by other firms.
So the first is solve for
your residual demand function.
What does your demand
curve look like,
given what the other firm does?
That's step 1.
Step 2 is then you develop
a marginal revenue, which
is a function of the
other firm's quantity.
Little q.
It's multiple firms.
The other firm's-- that's
really bad, hard to read--
the other firm's quality.
So your marginal
revenue is a function.
Typically, it's a function--
I'm sorry-- of both your
quantity and the other firm's
quantity.
A function of both your quantity
and the other firm's quantity.
We develop marginal revenue as
a function of your own quantity.
We know how to do that.
Now we develop a margin as
a function of your quantity
and the other firm's quantity.
Then you simply set
this marginal revenue
equal to marginal
cost, and that delivers
you a conditional answer.
That delivers you your
optimal quantity as a function
of the other firm's quantity.
Well, that doesn't do
us a whole lot of good,
except there's two firms.
So the fourth step is we do the
same thing for the other firm
and get the same
kind of equation.
Then what do we have?
Two equations and two
unknowns, so we solve.
So what we do here
is essentially
the same thing we did before,
but now your marginal revenue
is not just a function
of your own quantity,
it's a function of the
other guy's quantity.
Same with the other guy.
That gives you two
equations, two unknowns.
We solve.
And the point at which
both firms are happy
is the Cournot equilibrium.
That's confusing, so let's
actually look at that.
We'll do this both graphically
and mathematically.
Let's start with figure 13.1.
To make things easy, let's
start by imagining that American
Airlines is a monopoly.
Let's start with the world with
an American Airlines monopoly.
And let's say that the demand
function is P equals 339
minus Q. That's the demand for
flights from Boston to Chicago.
And let's say that the marginal
cost, to make life easier--
it could have a cost function
and make your life difficult,
and maybe someday I'll do that.
But for now, to
make your life easy,
let's just say it's a flat
marginal cost of $147.
I'm not going to make life
difficult with solving
for marginal cost functions.
For now, it's just a flat
marginal cost of $147.
No matter how many flights they
do, it's $147 per passenger.
So if you're a monopolist,
how do you solve this problem?
Well, first you derive your
marginal revenue function.
Well, what's marginal
revenue function?
Well, revenues is P times Q,
which is 339 minus q squared.
So your marginal revenue
function is 339 minus 2Q.
That's your marginal
revenue function,
if you're the monopolist.
What's your marginal cost?
Well, I just said it's $147.
And then you just solve.
And when you solve that, you get
that Q, the optimal quantity,
is 96 flights.
And then how do
you get the price?
How do you get the price
of monopoly problem?
How do we know
what the price is?
Yeah?
AUDIENCE: Where the quantity
intersects the demand curve.
JONATHAN GRUBER: You've
got to plug it back
into the demand curve.
Take that quantity, plug it
back into the demand curve.
So the price is 339
minus 96, or 243.
So I just solved the
monopoly problem quickly,
but that's what we've done
already in this class.
And you could see that
in the graph here.
In figure 13.1, you've
got demand curve,
which is P equals
339 minus Q. You've
got a supply curve, which is
the flat marginal cost of $147.
You develop a marginal
revenue function,
which is 339 minus 2Q.
As in our previous
example, that's
just basically an inward
shift of the demand function.
That intersects marginal
costs at 96 flights.
We have 1,000
passengers per quarter.
It doesn't really matter.
It's just all standardization.
And then to get the price, you
read it off the demand curve.
You say 96 flights means the
price of $243 per flight.
OK, that's what we do if
American was a monopolist.
Now, however, American
is not a monopolist.
American deals with United,
and American doesn't know
what United is going to do.
So what does American do?
Well, let's say American
has to deal with the fact--
it now has to recognize
that it's got its own demand
function, qa, which
is the total quantity
in the market minus qu.
So it has a residual
demand function,
which is the total demand in the
market minus what United sells.
So suppose, for example,
American thinks--
American's got a
spy inside United--
and American says,
ha, I think United
is going to fly 64 flights.
So imagine American
thinks United's
going to fly 64 flights.
Well, in that case, if they're
going to fly 64 flights,
then my demand function is
p sub a equals 339 minus q
sub a minus 64, because
the big quantity
is little qa plus little qu.
So my demand function is
339 minus q sub a minus 64.
Or in other words, my
residual demand function
is that p sub a equals
339 equals 275 minus qa.
So if I think United's
going to fly 64 flights,
then my effective demand
function is 275 minus q.
And then I'm done.
Then I just solve for, what
would I do as a monopolist,
given the other guy's
flying 64 flights?
So you can see that
in figure 13.2.
So I have a demand function.
I say, well, if United's going
to fly 64 flights, that demand
function gets shifted in by 64.
And then I'm going to do
the same thing I did before.
I solve for marginal revenue.
I'm going to solve
for marginal revenue
and I intersect that
with marginal cost.
That's going to happen at 64
flights and a price of $211.
So basically, it's
the same exercise.
It's not that hard.
You just first take out
what United is going to do.
The problem is American
doesn't have a spy.
They don't really know
what United's going to do.
They have to essentially
develop a strategy,
given the possibilities
of what United might do.
They have to say, look, I
don't know what q sub u is,
so I have to devise my optimal
strategy given q sub u.
In other words, I have
to simultaneously solve
for what I would do at every
possible quantity United
would sell.
I have to solve what I would
do for every possible quantity
United would sell.
And we call this
developing your reaction,
or best response curve.
Your reaction curve or
your best response curve,
which is, what is the
best thing to do, given
what the other guy's doing?
What is the best
thing to do, given
what the other guys doing?
You could see that in figure
13.3, we show how that works.
That shows best response curves.
So for example, look at the
intersection on the y-axis,
where the red line
hits the y-axis.
That was our
monopoly equilibrium.
I'm sorry, where the blue
line hits the x-axis, my bad.
We're doing American.
Look at where the blue
line hits the x-axis.
That is assuming
zero United flights.
Where the blue
line is the x-axis
is where there's
zero United flights.
Well, we know what
American would do there.
They would fly 96 flights.
We already solved that.
Now look at the point where
United is flying 64 flights.
Well, we also know what
American would do then.
We know that we solved,
in the previous figure,
they would then do 64 flights.
And in general, what that blue
line is is for every quantity
that United flies, what
does American want to fly?
So meanwhile, United is
doing the same mathematics.
Imagine, to make life easier--
we'll almost always do this
to make life easy--
imagine United has the same
marginal costs as American,
and obviously faces the
same market demand curve.
Well then, literally, their
math is totally symmetric.
If American wasn't
in the market,
you'd have where the red line
intersects the vertical axis.
If American was
flying zero, United
would flight 96 flights,
because their problem
is identical to American's
monopoly problem.
So the red line is United's
best response curve.
So we've graphed, for
every possible amount
of flights that
United does, what's
American's optimal
amount of flights.
We've solved for every
amount of flights
that American does with
United's also amount of flights.
Where those lines intersect
is the Cournot equilibrium.
Why is that the
Cournot equilibrium?
Because at that
point, both firms
are doing the best
they can, given
what the other firm's chosen.
Or in other words,
to say this is
given what the
other firm's doing,
neither firm wants to deviate.
The profit-maximizing choice
is to be where they are, given
the other firm's behavior.
So basically, the
Cournot equilibrium
is the only equilibrium that's
possible in this market.
And why is that?
So for example, imagine that
American came in and said,
look, I like doing 96 flights.
I love being a monopolist.
I'm just going to do 96 flights.
I'm going to do 96 flights,
I'm going to charge $243.
Well, in that case, American--
United, I'm sorry-- would
happily come in at $242
and undercut them and
sell lots of flights,
because that's still
well above marginal cost.
So that's not an equilibrium
because United and American are
choosing different outcomes.
It's only equilibrium if
they're both to the point
where the same outcome
makes them both happy.
So that's the graphics.
Let's do the math here.
Let's do the Cournot math.
In general, the residual
demand for American
is that p equals 339
minus qa minus qu.
Remember, big Q is qa plus qu.
Since the demand function
is 339 minus big Q,
I simply broke big
Q into qa and qu.
Stop me if this is all unclear.
Simply broke the big Q
into those two components.
So that means that
American's revenue function--
it's called revenue A,
revenue for American--
is 339 times qa minus
qa squared minus qaqu.
This is a new term.
This was the old
revenue function
we had when they
were monopolists.
Now we've got this new term
that didn't exist before.
So that means the marginal
revenue for American
is now 339 minus 2qa minus qu.
So now their marginal
revenue is actually
a function now of
their own behavior,
but their competitor's behavior.
That's the new margin
revenue function.
But the profit maximization
rule is the same.
We just set that equal
to marginal cost.
We set it equal to
147, and you solve.
And what you end up getting
is that q sub a star--
the outcome of q sub a is
96 minus 1/2qu star, or qu.
qa star is 96 minus 1/2qu.
If you solve this equation,
that's what you get.
That's 1/2, 1/2 qu.
So now we have the
optimal quantity,
but it's a function of
what the other guy does.
That's a problem, except that
we use the same math for United.
Now, if the problem's
symmetric, you
don't have to do the math again.
You know the best response
function will be symmetric,
but that won't
always be the case.
So I'm going to
shortcut here of saying
the best response
function for qu
is q star u equals
96 minus 1/2qa.
So I've just written down
the best response function.
This corresponds to the graph.
So q star a, that's
the blue line.
It's 96 minus 1/2u.
q star u, that's the red
line, 96 minus 1/2qa.
That's their best
response function.
Now once again, to remind you, I
could simply skip to this step,
but normally you'd have to
solve through for both firms.
They might not have identical
best response functions,
or symmetrical best
response function.
Well now we're golden.
We have two equations
and two unknowns.
We know how to deal with that.
And you solve them
and you get the qa
star equals qu star equals 64.
You solve those two
equations and two unknowns.
So 64 is the solution
of that system.
What's the price?
Someone raise their
hand and tell me.
What's the price?
Without looking at the graph.
Yeah?
AUDIENCE: [INAUDIBLE]
JONATHAN GRUBER: And
how did you get that?
AUDIENCE: [INAUDIBLE]
JONATHAN GRUBER: You
got to plug in 64 twice.
A lot of people get this wrong.
They'll say, oh, 339 minus 64.
But no, it's 339 minus 128,
because they're each flying 64
and the price comes from the
total demand in the market.
So the price is $211.
That's an important
mistake to avoid.
A lot of people get here.
They'll be super excited.
They're tired.
They throw the 64 back
at the demand equation.
But remember, demand's a
function of the total market.
If symmetrically
they're each doing 64,
then the price is
going to be $211.
And that is the Nash
or Cournot equilibrium.
Both firms are happy to fly
64 flights at a price of $211.
Neither firm wants to deviate.
And you know that because
you've maximized their profits.
When United is flying 64,
the profits of American
are maximized at flying 64.
When American's flying
64, the profits for United
are maximized at flying 64.
Therefore, that is the Nash
or Cournot equilibrium.
Now, when we get to reality,
things might not always
work out so neatly.
Things might not be symmetric.
You might also not
have an equilibrium.
How could you not have
an equilibrium here?
How could that
happen graphically?
What would that mean?
Yeah?
AUDIENCE: The curves
don't intersect.
JONATHAN GRUBER: Yeah.
The best response curves
might not intersect.
You might not get
an equilibrium.
We don't know what
the hell to do then.
All chaos breaks loose.
But you might not get an
equilibrium in this market
because the best response
curves might not intersect.
In reality, in life, you could
have funky best response curves
that are non-linear or you could
have multiple intersections.
We call it multiple equilibria.
And then it becomes an
indeterminate problem
and you have to figure out which
equilibrium they settle at,
and that involves higher order
mathematics that you talk
about in more advanced classes.
So this is the
simplest, easiest cases.
Symmetric case where linear best
response functions intersect
is your easiest case.
But in general, the
general way to solve this
is the same, which is use
the principle of game theory.
Look, go back to the
prisoner's dilemma.
All we're doing here was
creating best response
functions.
It's just there wasn't a line.
It was just a point.
The best response function
was what we laid out here.
All we did with these
United and American examples
was go to a continuum and
develop best response functions
around the best response point.
Yeah?
AUDIENCE: If the Nash
equilibrium is always
worse than when
they're cooperating,
why is it so hard to
maintain a [INAUDIBLE]??
JONATHAN GRUBER: We'll
talk about that next time.
Other questions?
OK, let's stop there.
We'll come back.
Next time we'll talk about,
why don't we all just get along
with Mr. Rogers once?
