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GREG HUTKO: Welcome
back to the 14.01
problem-solving videos.
Today I'm going to be working
on Fall 2010 PSET 8,
Problem Number 2.
And we've seen the case with
the monopolist where we've
only had one producer in a
market, and we've seen how
that affects consumer surplus,
produce surplus, and
deadweight loss.
Now we're going to look
at the case where we
have two firms competing.
And specifically, for the first
part, we're going to
have the Cournot equilibrium,
where they're going to be
competing not by setting the
price of their product.
Instead they're going to compete
by setting how much
they're going to produce.
Let's go ahead and read part
A of this problem.
Consider a market in which
two firms produce
a homogeneous product.
Market demand is given by
quantity equals 200 minus p.
The cost functions for Firm A
and Firm B, the total cost for
A equals 5qA and the total cost
for b is going to equal
1/2 qB squared, respectively.
Find the Cournot equilibrium
quantity
supplied by each firm.
We're going to graph our
results using reaction
functions and we're going to
find the market price, and
then calculate the profits
for each firm.
Now in the duopoly model, both
firms in the Cournot
equilibrium are going
to set their
quantities at the same time.
But what they're going to do is
they're going to know what
each other's revenues
and costs look like.
So they can know, I know that
if I make this move and set
this quantity, I know that my
competition is going to have
this reaction.
Since they can plan for each
other's reaction, they can
decide how much quantity they're
going to produce at
the same time and they're going
to reach an equilibrium.
What this looks like on our
reaction curves, up here we're
going to have qA.
And down here is
going to be qB.
We're going to graph the
reaction function for Firm A
first. And we're now going to
graph the reaction function
for Firm B.
Now all this tells us is if
we're at this point, if we're
Firm A and we decide to produce
this much, then we
know that Firm B's best reaction
is going to produce
this quantity here, the
intersection of how much I'm
producing with their
reaction curve.
Similarly, if Firm B is going
to decide to produce this
much, then my best reaction
going over to Firm A's
reaction function straight
across, is going to be to
produce this much.
Now since they're both choosing
at the same time,
neither of the firms
can decide to
declare a larger amount.
And they have to kind of use
their intuition to determine
what they're going to produce.
So since they're choosing at
the same time, they have to
plan for each other's
reactions.
They're going to produce at
this point where the two
reaction curves meet.
And that's what we're going
to be calculating.
We're going to be calculating
the quantity that Firm A
produces and the quantity that
Firm B produces when the
reaction curves are set
equal to each other.
Now to get the reaction curves,
we're going to start
off with our revenue function
for both Firm A and Firm B.
And revenue is just going to
be the price times the
quantity that either Firm A
or Firm B is producing.
Now instead of saying that we
are just going to take the
marginal revenue according to
this P times Q, we're going to
plug-in for P from the demand
curve 200 minus a
disaggregated quantity where
we disaggregate into the
amount Firm A is producing
and the
amount Firm B is producing.
We're going to plug this in
to the revenue function.
And so just like the monopolist
did where they're
maximizing by setting marginal
revenue equal to marginal
cost, Firm A and Firm B we're
going to do the same thing.
We're going to set marginal
revenue equal to marginal
cost. And then we're going to
solve through for qA in terms
of the quantity that the other
firm is producing.
That's why it's considered a
reaction curve because it's in
terms of what the other
firm is producing.
So solving through for Firm A's
marginal revenue, we're
going to find that marginal
revenue for Firm A is equal to
200 minus 2qA minus qB.
And it makes sense that the more
that they're producing,
Firm A and Firm B, the lower the
revenue that they're going
to be taking in.
Now we're going to also
calculate the marginal cost
for Firm A by taking the
derivative with respect to the
total cost function.
We're going to find that the
marginal cost is going to be
equal to 5.
Now we're just going to set
the marginal cost and the
marginal revenue for
Firm A equal.
And we're going to solve
through for qA.
When we do that, we're going
to find that qA is equal to
97.5 minus 0.5 qB.
And we're going to repeat this
exact same process for Firm B.
But when we do it for Firm
B, instead of taking the
derivative with respect to qA,
we're going to take the
derivative with respect to qB.
So the marginal revenue for Firm
B is going to be equal to
200 minus qA minus 2 qB.
And the marginal cost is just
going to be equal to qB.
Again, we're going to set
marginal cost and marginal
revenue equal to each other.
And now we're going to
solve through for qB.
And when we do this, we're
going to have Firm B's
reaction curve.
And now what we're going to
do since we have these two
reaction curves, we have a
reaction for Firm A and a
reaction for Firm B, all we're
going to do is we're going to
plug-in for this qB, qB's
reaction curve.
And when we do that, when we
plug-in 66.67 minus 0.33qA, we
can solve through for just qA.
Doing this we're going to find
that Firm A is going to
produce approximately
77 units.
And then taking this 77 and
plugging it in to Firm B's
reaction function, we
can solve for Firm
B's production amount.
And we're going to find that
Firm B is going to produce
approximately 41 units.
Now to find the equilibrium
price, we're just going to
come back up here to our
disaggregated demand function,
and we're going to plug-in
for qA and qB.
And we can solve through for the
price being equal to 82.
Now what we've just done for
this problem is we solved on
our graph for the intersection
of the two reaction functions.
We found that qB is going to
be equal to 41 and we found
that qA is going to
be equal to 77.
So we just calculated the
intersection point for the
Cournot equilibrium.
Now the last part of this
problem asks us to calculate
the profits for both the firms.
The profits for Firm A
are just going to be price times
the quantity that A is
producing minus the cost
as a function of qA.
So we're just going to take the
total revenues minus the
total costs.
For Firm A, we're going to find
that the total profits
are going to be about $5,929.
And doing the same process for
Firm B, we can find that the
profits for Firm B are
going to be equal to
approximately $2,521.
Now part B of this problem is
going to ask us instead of
having this Cournot equilibrium
where neither firm
can go ahead and produce a
higher quantity or move first
in the market, we're going to
look at something different
than the Cournot equilibrium.
We're going to look at the case
where one of the firms
gets to decide how much they're
going to produce
before the other firm.
And if you get to decide first,
you get to produce a
higher quantity and get
more of the profits.
Part B says, now suppose that
Firm A chooses how much to
produce before firm B does.
In this case, Firm A
is a Stackelberg
leader and B a follower.
We're going to calculate the
quantities, the market price,
and the profit for each firm.
Now coming over to this side of
the board, we see that I'm
going to keep Firm B's reaction
function the same.
So Firm B is going to be
reacting in the same way to
Firm A's decision.
Only now, the only difference is
when we calculate marginal
revenue equal the marginal cost
for Firm A, instead of
just saying the qB is going
to be random, we
can't account for it.
We're going to plug-in, we're
going to take into account
Firm B's reaction when we're
maximizing or taking the
derivative with respect to qA.
So instead of having qB in here,
I'm going to plug-in
this reaction function.
So in this case, the equation
that I'm going to be
maximizing is going to be
this one right here.
I'm going to take the derivative
with respect to qA
to find the marginal
revenue for A.
And again, I'm just going to set
this equal to the marginal
cost, which we found earlier
is equal to 5.
And when we set these equal, we
can solve through for the
quantity that Firm A is
going to produce.
And we're going to find just
like we predicted that the
leader is going to
produce more.
In this case, Firm A
has increased their
production to 96.25.
And then plugging in this
quantity in to Firm B's
reaction function, we can find
that Firm B in this case, is
going to produce 34.6.
Now in this case, we can again
calculate the price by taking
the demand function that we
have. We can take the demand
function that we're given
in the problem.
Plugging in for qA and qB, we
find that the new price in the
Stackelberg problem is
going to be 69.15.
And again, we can calculate the
profits going through the
same process of doing total
revenue minus total cost. And
we're going to have that the
profit for Firm A is going to
be equal to about 6,174.
And the profit for Firm
B is going to be
equal to about 1,794.
And so what we can do here is
we can compare the profits
that we had in the Stackelberg
case to the profits that we
had at the start
of our problem.
So before we can see that Firm
A was not as profitable when
they had to choose their
quantity at the same time as
Firm B. We can see that their
profits have increased.
But we can see that Firm B,
their profits have actually
decreased because they're
a follower in
the Stackelberg model.
Now the last thing, and the
thought I want to leave you
with is, how do we actually
interpret this when we look at
the reaction functions
on our graph?
We're no longer at the point
where we're setting the two
reaction functions equal.
What's happening now is we're
way up here and qA is choosing
their production way up here.
And qB is forced to react by
choosing their production
right here.
And what happens is since they
both increased their
production or since qA has
increased their production and
qB has decreased their
production, but since
production has increased
overall, the price has dropped
compared to when they were at
the Cournot equilibrium.
So total profits have actually
dropped as well.
So really what the first two
parts of these problems were
having us look at, they were
looking at two different
situations of duopoly
where we have two
competitors in the market.
The first one they were
choosing their
outputs at the same time.
And in the second problem, one
of the firms had the advantage
of getting to choose a higher
quantity and making a credible
threat that they were
going to make that
quantity to begin with.
For the last parts of these
problems, you're going to go
ahead and you can look at what
the implications are when we
think about what the total
quantity is produced in a
competitive market aggregating
the supplies of these two
firms. And then you can compare
the output in the
three different scenarios.
But for now, I'm going
to leave you here.
Go ahead and finish the rest of
the problem, and I hope you
found this part helpful.
