Another way to resolve the Bertrand paradox,

is by lifting the assumption of product homogeneity.

Product homogeneity, is on the eye of the beholder.

Some things, some people,

see them as same.

Some other people, see them as different.

There are people that they cannot understand the difference

between even red and white wine.

And there are some other people that they will pay

thousands of dollars for a bottle of wine because

they consider that is different than anything else.

So, in this case, we can have heterogeneous product that they again,

change the Bertrand paradox.

Why? Because, they make price to not be the only thing that affects decisions to buy.

Market shares are determined by other things, by other variables,

that they make the products to be heterogeneous or as we say,

to be simply different.

What is that? Differences in perceived quality.

Some people consider that something is better than something else.

You go to the supermarket, some of the times,

you only look at the brands that you like and you do

not pay attention to the small differences in prices,

of course, if they're small enough.

You also look at the design,

which one you like more,

you're willing to pay more,

for things that you like the design more.

What is the performance,

how these products works for you?

Or what is the durability?

Maybe to products they look the same,

but one, you bought it and it broke down very quickly,

while when you had the other,

it lasted for more time and now you will prefer this one,

even if, it is a little more expensive.

So, again, in the market with differentiated goods,

it makes sense to compete with prices rather than, in quantities.

So, you can quantify the differences in quality,

better in dollars, than in quantities of products.

It makes sense to say that, "Yes,

this product is a little better than the other one,

that's why it's 10 percent more expensive."

So, it makes more sense in this case,

to compete with quality.

So, also, we hit the Bertrand paradox by making customers to not desert the firm,

over a small price increase relative to the competition.

If you make a good product,

you can charge a little more and you can get away with it.

Don't take it wrong, people do not want to pay more for anything.

You can get away from charging more,

if you first convince the consumers,

that your product it's worth it to be sold for more,

and this is something that then will change the Bertrand paradox.

Let's see now, a technical model that will explain and will

quantify how product differentiation works with the Bertrand model.

So, we have two firms that they face symmetric demand curves.

Firm one, has a demand curve that it's quantity is equal to a constant a minus d,

times p_1 so, the price of the first firm multiplied by coefficient d,

plus, the price of the other firm.

And then, there is a symmetric demand function for the other firm.

So, quantity for each firm decreases with its own price.

So, as you increase your own price,

the quantity that you expect to sell decreases,

but also increases with the price of your competitor.

So, if your competitor increases the price,

while you keep it constant,

then you should expect to receive more customers because,

some of your competitors customers,

will now probably come to you because you are cheaper.

Not all of them, but some of them.

I want you to observe this d there.

I'm not going to write that on the slides,

but because you will become an economist and

because you know how to read behind the mathematical equations,

you have to expect there that this d,

has to be more than one.

Why's that? Because, you should reasonably expect that your own price,

will make a bigger difference in your quantity,

than the price of your rival.

This is very important, keep it in mind.

So, the marginal cost is given to be again,

constant to see as simple as possible.

Firms choose prices simultaneously,

decision on the price is binding.

Once you announce it, you cannot take it back.

So, we are going to do nothing different,

than what we did previously in Cournot.

We are going to set up the profit function,

and we will calculate the reactions.

The only difference, small difference,

is that instead of using a choice variable of quantity now,

because we are talking about Bertrand,

we're going to use a control variable,

a choice variable of price.

So, the profit for each firm,

I will use their i, j, notation again.

I is us, j are the others, simple.

So, the profit is equal to a minus d_p_i plus p_j.

This is the quantity that we will sell,

times, the profit per unit.

So, quantity times profit per unit,

will give us our total profit.

Each firm will maximize profit by setting its first

derivative equal to zero and this of course,

will yield the reaction functions in this model.

So, this will be the reaction function for firm one,

for i, for us,

and then for j,

if we said j equal to two,

we're going to get the reaction function for the other firm.

For the reaction functions,

we can say that they are kind of different than in Cournot,

they look a little different.

First of all, they are both positively slopped while in Cournot,

they were always negatively slopped because,

as the other Cournot,

as the other competitor increases their quantity,

you want to decrease yours,

because if you keep increasing yours,

prices will go very low and you will both lose money.

If your competitor in Cournot decreases the quantities,

then you want to fill in the gap and increase your quantity to substitute.

So, you have some strategic substitutability in Cournot.

While here in Bertrand,

you do not have that, you have strategic complementarity.

If someone increases their price,

you say, "I can increase it too also,

maybe a little less than them but also,

I can increase it" If they decrease the prices,

then you will have to follow them.

You will have to move in the same direction and

match them or else you are going to lose a lot of customers.

So, that's why, here in Bertrand model,

we should expect that our reaction functions are always positively slopped.

They also intersect at the positive quartile,

where the price equilibrium price intersection will be above c,

above the cost because,

if you're still below cost,

it means that you make losses.

And if you solve at cost,

it means that you have not solved the Bertrand paradox.

So, let's see how it works in our model.

So, this will be the reaction function for firm one,

this will be the reaction function for firm two,

and then in the same manner,

they want to behave in a way that what they do,

is the optimal response to what the other guy does.

And by doing that simultaneously,

you get the simultaneous solution of this system, and in this case,

you get the Bertrand equilibrium,

which is nothing more than another version of Nash equilibrium again.

So, this will be done in the prices,

p_1-star and p_2-star which we denote, the equilibrium prices.

We are going to have a particular application,

another example that will give us some nice numbers and then,

we are going to use these numbers in order to understand

some very useful things about general firm interaction.

We're going to get as deep to firm competition as possible,

in the next segments. Stay with us.