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Competition in quantities.

Now, we will get a little bit technical.

We'll go over the models and we will try to

understand a little bit of the mechanics because this will be important for the end.

We have to show and to prove some things.

Just bear with me, I will try to explain everything with very simple words.

So, assume that we have two firms.

As simple as possible, a duopoly.

The minimum amount of firms that you can have an interaction. It's two.

So, firm one and firm two,

they produce a good that is homogeneous.

That is, to the eyes of the consumers this is the same good.

Has no difference.

The cost for producing the products is related to quantities.

It's a function, c of q,

and it should be increasing in quantity.

It means that as you produce more,

you should expect to have more cost.

You cannot expect to produce more and have less cost.

This would be non reasonable to expect.

You have a market demand.

That is, the consumers have some preferences which are embodied into a demand function,

and this demand function has to show you a relationship between price and quantity.

So, it's a function,

p of q, p is the price,

q again is the quantity.

Now, this function has to be negatively slow,

meaning that you cannot expect your consumers to

want to buy more as the price is increasing.

So, therefore, we have a decreasing function p of q.

Now, by q, we mean the total quantity here that most of the firms are producing.

That is, q_1 plus q_2 from each firm.

So, firms choose quantity simultaneously.

At the same time, they decide how much quantity they're going to produce.

Once they make these decisions,

this decision is binding.

Meaning that they cannot take it back.

Once they announce the quantity and they produce it,

they cannot unproduce it or change it and in the same period produce more.

So, this is an assumption of the model and it will be given with our assumptions.

So, the interaction lasts only for one round.

That is, firms do not have

a prior history and they do not care for what is going to happen in the future.

They only care for what happens within these specific periods.

But firms know all of the above from the beginning.

Now, if you look at these assumptions,

you'll understand that what I'm giving you here is nothing

more than a start game of complete information.

So, everybody knows everything and they

compete in a simultaneous way with binding auctions, and they have to choose.

Now, they do not have to choose between two strategies.

They have to select q,

which is going to be any positive number in this model.

Alright, so we're going to stick to what we said in the previous lecture about Gates.

We're going to apply the same principles.

Nothing's going to be very different.

We just are going to see it in a little more depth and how firms are doing it in reality.

Before we get to competition,

let's examine the benchmark.

Let's have something to compare our results to.

So, we're going to talk about the case where the firms get

together and they decided to act cooperatively as one entity.

Meaning that we have a collusion situation.

So, the two firms get together and they act as one.

The profit in this case will be given by

their demand curve which will express their price of the product.

So, the price will be a function.

The first parenthesis there is a function of the sum of quantities,

p of q_1 plus q_2.

This is a function, a negative function.

I'm not going to give you a specific form for now.

This general form will work just fine.

We have to multiply the price times the total quantity that's times q_1 plus q_2.

And then, we have to subtract the cost that we need to

pay in order to produce these total quantity,

that is, q_1 plus q_2.

So, this is our profit.

And then, when you give a profit function to an economist,

automatically we are going to maximize it,

so we are going to take the derivative of that, the first derivative.

And since this problem here is a concave problem,

we do not need to take any more derivative second or another degree.

And therefore by taking this derivative,

we'll equate it to zero and we will see to what quantity we maximize the total profit.

So, this will give us this function that you can see in this slide.

Now, be careful here because we have, again,

the demand curve their plus the quantity multiplied by

this fraction dp over d of total quantity is what?

If you look at it carefully,

this is the slope of the demand curve.

So, I'm taking the derivative of the demand curve there and then in the other fraction,

minus dc over d total quantity,

this will be the first derivative of cost,

which is the slope of cost which as you probably know from your microeconomics course,

this denotes the marginal cost.

We have to equate this to zero.

And if we solve it with respect to p,

I do not need to say anymore that is p function of total quantity.

I'm just going to solve the first order condition with respect

to p and I will get these relationship.

Which tells me that the first function, again,

is a marginal cost minus quantity times the slope of the demand curve.

I told you from the beginning,

but I'm sure you knew it from before you took this class,

that the slope of demand curves,

we have the demand curve, is always negative.

So, I have a minus sign there.

And then, I have a negative slope.

So, this is going to mean that

my equilibrium price will be the marginal cost plus something.

Because in relationship number one,

that you can see here,

in the second factor we have minus and

negative thing and this will give us plus something.

So, this is not very mathematically precise,

plus something, but that's what it is in reality.

So, it's marginal cost increased by something.

This is going to be my pricing in the collusion benchmark.

Right. So, this is how the firm is going to behave in the collusion benchmark.

Let's get now to the case where we have competition in quantities.

This is a model that was developed by the French economist Corneau in 1838,

and it assumed a very simple case of a duopoly between

two sellers of the same homogeneous good

with the same assumptions that I told you in the beginning.

So, firms compete with respect to quantity.

Let stand for firm one.

Everything will be the same for firm two,

but we will do firm one first.

The profit for firm one is equal to

its revenue which is the quantity of firm one times the price.

But now the price, be careful,

because the price is not going to be a function of only q_1 because the other firm also

produces quantity into this market and

consumers have a possibility to buy also from the other firm.

So, therefore, the price will be a function of q_1 plus q_2,

the total quantity in the market.

And we have to subtract in order to get the profit from the revenues.

We have to subtract minus the cost of q_1,

q_2 is not going to be there because the production of q_2 is undertaken from

firm two which does not take part into the profit of firm one.

So, again, we're going to maximize this function with respect to q_1.

This will give us a similar function to

the one that we have before in the collusion case.

We have to set these equal to zero in order to find

the maximum and then the first order condition when it is sold with respect to price.

It will give us that price is, again,

these marginal cost of firm one minus the quantity of

firm one times the slope of the demand curve.

And then, again, that is that the price is equal to

the marginal cost of firm one plus something,

because again, minus the negative slope of the demand curve will give us a plus.

So, again we have that the marginal cost is

marked up by something and this will give us the price.

So, in reality, what we have here is that

both collusion and competition gives us that price will be marginal cost plus something.

Now, this two somethings from relationship number one

and relationship number two, will be different.

So, calculations are analogous for firm two.

Same problem symmetric to firm two can be solved and you can do that at home.

It's not going to be easy at all.

You just have to change the indexes there,

the subscripts, and you will end up with a very similar mission.

Now, let's go and compare the Cournot competition to our benchmark of the collusion case.

So, we said that in both cases,

in relationship one and two we're going to set price equal to MC plus something.

And this something in relationship one and two was different.

Now, we can prove that if you have well-behaved

demand functions and your costs are reasonably described by the function c,

the something from the collusion case is

bigger than the something of the competition case.

So, they will both price above the marginal cost but collusion price will be higher.

Now, if you look at the math there and how this can be proven,

you will see that the first and the second,

the left side and the right-hand side,

they look very similar but the left-hand side actually also include the same thing

like the other side but also times quantity to the slope of the demand curve.

So, it includes another positive quantity.

So, it's going to be much substantially higher than the other side.

So, collusion profit, collusion price, will be higher.

Thus, we can conclude that under Cournot,

price is higher than marginal cost but not as high as in collusion.

Collusion price will be the highest,

but then Cournot is also going to be more than marginal cost.

This is the result that we just saw.

And I'm going to give you an example in the next segment.