Almost one century after Cournot,
Germany economist Stackelberg who was living here in Moscow,
he came up with an alteration of the Cournot Model.
He said, "Firms do not always move sequentially.
Sometimes they can move in turns.
And what if one firm has the chance to move first before the other one.
Is this going to give an advantage to this firm?"
So let's assume now that we have the same model like
before but now firm one is the leader and from two becomes the follower.
Being able to lead,
being able to produce first,
may give you an advantage.
I'm telling you this before we do the model because every economist,
every person who wants to understand how business work,
should before they solve anything,
they should have expectations about this and how this is going to work,
how this is going to affect reality.
So, our expectations here should be the following.
Should be that once I produce first I'm able to show you,
to show my opponent, how much I have produced.
If I produce a lot,
my opponent will see that,
he will understand that I cannot take it back,
and then will adjust their behavior accordingly.
So perhaps, I will have an advantage if I produce first.
Let's see how this works.
So firm two as a follower does nothing different than Cournot.
They still have the same optimal response.
They see they still have to see
the quantity of the other person in order to know what is the best quantity to produce.
So we shouldn't expect the optimization of firm
two to be any different than what they did before.
So firm one on the other hand is now the leader and can take it into account that firm
two can have a reaction function and
optimal response way that firm one can take into advantage.
I told you in the first lecture that the information
is the most important thing in business.
Here this materializes.
Firm one knows that it's the leader.
Also can predict because it knows the full model,
we said in the assumption that the model that everyone
knows everything from the beginning.
So firm one is able to do the calculations for firm
two and derive the optimal response function of firm two.
And then, since they can commit with the quantity that they produce,
they can take advantage of these commitment that they have.
So the profit of firm one is the standard one,
one minus C1 minus Q2 times Q1 minus Q1 squared, and then,
once we have these standard profit,
we can observe this Q2 that now can be replaced
with the way that firm two is expected to react to this problem.
So I will replace Q2 with the reaction function of
the second firm that is R2 in the period from the previous segment.
And once I replace that I see that the profit function of
firm one does not depend any more on the quantity of firm two.
Quantity of firm two two is replaced with the optimal response of firm two.
So I will optimize that, I will maximize it,
I will take the derivative and I will set the derivative equal to zero.
And this is what will give me the optimal quantity.
If I solve that, I will see that the first mover has an advantage.
The quantity for the leader will be equal to this equation.
The quantity for the follower taken from
the reaction function 2 that we derived in the previous segment we'll
be taking from this equation if we take into
account the quantity that the first firm has produced.
And then, if the cost of the first firm,
the cost of the leader,
is not much larger than the firm's two cost,
then we will see that quantity that the leader will produce is much higher.
So the market share of the leader will be much
higher if ,the firm one is allowed to become a leader,
in other words to move first.
Now, the leader will sell higher quantity,
the follower will sell lower quantity.
The question that you can ask is me as a follower,
if I was the follower,
why would I want to comply with this?
Why would I want to produce less if the leader produces more?
If I have someone who is a leader and produce a lot of quantity.
Why should I buy this?
Why should I conform to this model?
The reason is that it's your optimal response.
If you do anything different than what the model predicts here,
you are going to be in a worse state.
This is what we said in the previous lecture about the Nash Equilibrium.
Nash Equilibrium is your best response given what the other person is doing.
Hear, given that the first firm has produced a larger quantity,
you have to go on and produce a lower quantity or else it will be worse for you,
also for the other person,
but also for you and you don't want to do that.
So we have a first mover advantage
if you compete with quantities and you are allowed to go first.
Now, let's compare that to Cournot.
In terms of quantity,
the leader produces more than Cournot.
The follower produces less than Cournot and total quantity is more than Cournot.
In terms of profits,
the leader has more profit than would have anti-Cournot.
The follower has less profit than would have
anti-Cournot and price finally is lower than it would be in Cournot.
This is because there is a higher quantity into the market.
Now, let's take all these observations.
All these derivations that we have so far,
and put them into one single graph in order to understand what is going on,
what is happening in these models because we saw collusion,
we have seen perfect competition,
we have seen Cournot are no and we have since Stachelberg but four different models.
What is going on. Let's try to clear these by putting them,
all of them into one graph.
So first of all let's put Q2 there and Q1 in the horizontal axis.
Now it's time to explain to you why I always
solve my reaction function with respect to the quantity of the second firm.
This is because I want to put them on the vertical axis.
I want to graph them on the vertical axis and it helps
a lot if you solve everything with respect to Q2.
You always put Q2 on the vertical axis and you always know what happens in your graphs.
Because even some graphs you put Q2 in the vertical axis in some other graphs you change
the axis and you put the Q1 into the vertical one on then
your understanding will not be that easy anymore.
So this is the reaction function for firm one If your plot it.
This is the reaction function for firm two.
So they will look like that.
So the darker blue line is the optimal response of firm one to actions of firm two,
and the light blue is the optimal response of firm two to actions of firm one.
Now you will see that if your problem is well behaved,
you have normal demand and normal cores functions.
You will see that these two reaction curves,
they will intersect at
the positive quartile and the intersection is the Cournot Equilibrium,
which is actually nothing different than a Nash Equilibrium.
Every firm does the best it can given that the other firm is doing also the best it can.
And these two coincide only into
the intersection of the curves and there is
the Nash Equilibrium or the Cournot Equilibrium.
So this is the Cournot Equilibrium and these are the prices that we
get here because our problem is symmetric,
we will get to prices that are the same.
Now you can see there in the graph that they are almost the same.
And then, there is this curve that we call it a
contract curve and it starts from the lowest- from
the point that R2 intersects with
the vertical axis it ends to the point that R1 intersects to the horizontal axis.
So this curve there, is the curve that the collusive equilibrium will be.
And the collusive equilibrium if we assume that the firms have equal bargaining power,
and therefore they will share the market into the middle.
One firm will not dominate over the other in their private agreement.
So they will decide to cut the pie into the middle and share equal pieces.
So, therefore, we assume that the collusive equilibrium will be exactly in the middle of
this curve and you can see that the quantities will be smaller for both firms.
All right. So now,
the competitive equilibrium will be outside of this curve because it's not on
the best response of any of the two firms.
All right. So none of the other two firms wants for some reason
to produce such a high quantity that will drop the prices.
Now, these will give us quantities that they will
coincide with the beginning and the end of the contract curve.
You can prove that,
if you want, from seeing the model.
Think a little bit what the Stachelberg Equilibrium will be if
firm one is the leader. It will be right there.
Stachelberg Equilibrium if firm one is the leader.
It will be on the reaction function of firm two
because we said that the firm two will behave in the same way like in Cournot.
But now it will be outside of the best response of firm one.
And this is because firm one can commit that by producing a higher quantity,
will make the other firm to produce a lower quantity.
Why? Because it can go ahead before the other firm, produce it,
commit into that production production something
like a threat that already has materialized,
and then firm two has no other choice than doing its best given these high quantity,
and it will go and produce a low quantity QM in this case.
So you have four different equilibria in the same graph in order to make you to
understand what happens in each model and
how each of these models compares to each other.
Kosmas MarinakisAssistant Professor
