Let's see now, one model that is a little more general,

than the one we just saw.

So, we're going to talk about dynamic deterrents.

And here's how we're going to proceed.

In this segment, we're going to set up a model,

we are going to make a model that will be ready for the incumbent to deter.

And then, in the next segment,

we will see how this deterrence in this model actually work. So, let's start.

So, we have two firms,

L and F. The initials start for

leader and follower because this gives us information who moves first.

So, L and F produce a homogeneous good,

a good that consumers cannot really

understand the difference between the good produced by the two firms.

So, they see the good as homogeneous.

Their production costs are zero so,

they do not have to face a particular cost assuming that it is a spring of

natural water and they just get water from the spring at zero cost to them,

zero marginal cost to them.

And then, the total demand is a simple demand curve that we have

seen p equals one minus the total quantity.

So, in this market,

the maximum amount of product that can be absorbed is one because,

it's one minus the quantity that the firms

will produce and because price can never be negative so,

the maximum is one here.

So, we have a timeline that is at three stages.

It's an interesting timeline because it resembles reality very closely.

First of all, at stage one,

the leader chooses irrevocably,

a level of capacity.

This is usual, the firm who first enter the business,

decides how big it wants to be,

how big production capacity,

capability wants to have.

Then at stage two,

an entrant will look at the market and will decide how much investment K_F,

in this case, will want to do.

So, we'll see how much investment the leader has

done and then the follower will decide about the level of

investment that wants to do afterwards.

So, then, at stage three,

the leader and the follower,

they will compete simultaneously by setting prices.

This is a rather realistic model.

This is what happens in reality, one firm enters,

decides about its size,

about it's production capability,

then starts producing as a monopoly until and in turn decides to

enter this business and they also decide afterwards about their size,

and then once they enter,

they start competing with respect to prices.

It's something that happens very often in the markets.

So, at stage three, we will consider for now,

we will assume it, but it will turn out that is, it will be true.

It will be given endogenously from the model.

So, we'll assume that the firms sell to capacity.

They build capacity because they want to use it in this case.

So, after we figure out stage three,

which means that they sell to capacity so,

we solve the production problem of the firm.

Then we will focus on the sequential choice of capacity as what happens in stage two,

and then in stage one,

because as you probably understood we,

in these dynamics, games we always follow backward induction.

We start from the last stage and we advance to the previous stages one by one,

till we get in the beginning of the game.

So, strategic choice of your initial investment of

K. The followers profit is equal to one,

has no cost, so it equal to their revenue.

What does the revenue? Quantity which we said it is going to be K_F,

sells to capacity, times the demand.

What is the demand? One minus the total quantity.

Both of them they will sell to their capacity,

so the total quantity will be one minus K_L minus K_F.

So, K_F times one minus

K_L minus K_F altogether will give us the profit for the follower.

Now, what's the follower wants to do?

To maximize the profit.

So, they will take the derivative of that,

they will set it equal to zero and they will calculate their reaction function.

This reaction function tells me,

"Give me the leader's capacity and I will tell you what is the best capacity for me."

This is how I optimally react to the choice of the leader.

So, let's go now to the leader.

The leader in the first stage,

will maximize profit also.

So, the leader's profit is also K_L times one minus K_L minus K_F,

but the leader now,

because knows how the follower will react in the next stage to the leader's choices,

can take R_F and plug it,

in their own profit function.

So, once they do this,

they will take R_F and they will replace

these blue K_F right there at the profit function and then,

they will calculate that their profit is equal

to the one that we have calculated in our slide.

So, they will maximize this and they will end up with the best capacity,

the optimal capacity for them,

which is K_L equal one-half.

So, this is by taking into account how

their follower is predicted to react to their actions.

Now, we'll go back to the reaction function of the follower to R_F we will

plug one-half for the capacity of the leader,

and then we will calculate that the capacity of the follower,

will be equal to half of the capacity of the leader,

that is one over four.

This means that, I can now calculate the profits from

the profit function of each of the two firms.

The profit for the leader,

will be equal to one-eighth and the profit of the follower,

will be equal to one-sixteenth.

So, we see that the leader has the first mover advantage moves first,

makes double the profit of the follower.

Now, before we get to what happens with deterrence,

I want to show you what will happen if we did not have first mover advantage.

If both of the firms, they moved simultaneously.

In this case, there's no leader and follower,

but I will keep the L and F notation to make it easy for you.

So, we would have symmetrical reaction functions.

There's no difference for someone to observe

the reaction function of the other and then plug

it in their function because they cannot commit.

So, therefore, we'll have two symmetrical reactions,

functions R_F and now are R_L,

and this reaction function will give me a two-by-two system.

I can solve this linear system and then

you will see if you solve it, it is not very hard.

It's actually will take you like three lines of algebra.

And then, once you solve it, you will see that the capacities

should be equal to one over three.

So, this is more than the capacity of the follower that we saw before,

but less than the capacity of the leader.

So, they will agree somewhere in the middle, we can say.

So, the profits will be equal to one over nine for both of them individually.

And then, in the sequential model,

we can see that the leader makes more profit because,

of what we said in the very very beginning that Thomas Schelling told us because,

can commit in investing one-half.

So, when the follower comes to the game,

has already observed that the leader has the capacity of one-half can see that.

This is an irrevocable movement.

Now, this plays a very big role because it influences the decision of the follower.

If investment in capacity was not sunk,

in other words was not irreversible,

you could just invest and then take it back right away.

Then in this case,

L would start by choosing a capacity of one over two.

The leaders capacity would play the leader but then the follower, would not buy that.

He'll be okay, you'll play the leader but you're not a leader.

Instead of replying you with one over four,

I'm replying to you with one over three,

and this now, will cause a problem to the leader because it's not optimizing anymore.

So, therefore, the leader would want to take their decision back and have

the one that is predicted by the simultaneous model, equal to one-third.

This means that it plays a very important role that the investment,

when it's done, it's sunk.

You cannot take it back,

if you take it back, you will have a very big cost.

So, once the follower sees that okay,

you committed into that investment,

you are not going to take it back,

then I will have to follow my reaction function and

answer optimally to this committed decision that you have made.

But if your decision is not committed,

then I don't want to answer according to my reaction function,

I want to answer according to what would maximize my profit,

given that we do it simultaneously.

So, in the next segment,

we're going to consider what happens with entry and how we can

prevent the follower to become an entrant.

We will see it, in a moment.