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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.