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So now we want to think about what happens in a market when there actually
is a network effect.
So remember that previously, we thought about markets without network effects,
where people just had reservation prices describing
how much they value the product.
Now we want to put that network effect into the model.
So to motivate this, let's be concrete about an example.
Let's think about Facebook.
For most people, or at least anyone who's interested in using Facebook,
Facebook is more valuable if more of the people that they know actually
use Facebook.
And that's true generally for social networking sites
and for lots of other products, particularly technology and software.
So I need to somehow get into our model of what happens in the market the idea
that what a person is willing to pay or the costs they're
willing to bear to use a product or service actually
depends on how many other people they think
are going to use this product or service.
So let's add that into our model now.
Let's suppose that z is the fraction of the population that's
actually using our product or service.
I want each person's reservation price to depend on z as well as, of course,
on who they are, because people have different reservation prices.
And we're going to model this very simply.
We're simply going to say that now person x's reservation price is
his old reservation price, r of x, times some function, f of z,
where f is an increasing function.
f is meant to convey the idea that your reservation price gets shifted
up or down based on how many other people are using the product.
So I want f to be increasing so that you reservation price, the actual one,
will be higher if you think more people are going to use the product.
Now, this may the market a lot more complicated,
because now I don't know anyone's actual reservation
price until I know the fraction z of the population
that they expect to use the product.
So in order to figure out what goes in the market,
we're going to have to put together individual's reservation
prices with their expectation of who's going to use the product.
And we want their expectations to be correct.
Let's build this up piece by piece first by thinking
through some fairly simple examples of how this might work.
So let's start with an example in which we say that person x's reservation
price, ignoring for the moment the network effect, is just 1 minus x.
So person 0 has a reservation price without network effect of 1.
Person one, of course, has a reservation price of 0.
But I want their reservation prices actually to also depend
on what fraction of the population is using the product.
So let's suppose that function f of z is 4 times z.
So for example, if people guess that everyone's going to use the product,
then that just multiplies everybody's reservation price by 4.
4 times equals 1 of course is 4.
That means that person 0, the person with the highest actual reservation
price, now has a reservation price of 4.
Person one, of course, still has a reservation price of 0.
That person's not going to use this product no matter
how many other people are using it.
That would now then give me the actual reservation prices
that we can see in this blue line on our curve.
But remember that's the reservation prices
if everybody thinks that everyone's going to use the product.
The question is, are they right?
How many people are actually going to use the product when everyone
expects everyone else to actually use it?
Well, that's going to depend on the price.
So let's pick a price in the market and see if their guess is correct.
Suppose for example, that the price in the market is actually 1/2.
Let's find out who's going to use the product.
We know how to do that.
We simply take 1/2 over to that reservation price curve.
And we find the value of z where the person at that point
is just willing to pay 1/2 for the product.
Doesn't really matter so much what that value is.
The key is to notice that not everybody is
going to use this product, because everybody to the left of that point z
less than 1 uses the product, of course.
Everybody to the right does not.
Those people out there to the right of the point mark z less than 1
aren't going to use the product even if they thought, of course incorrectly,
that everyone would use the product.
That means that a guess that is equal to 1 is a bad guess.
In fact, as you can quickly see from that graph,
no matter what the price is, if it's anything bigger than 0,
there's no way that everybody's going to use the product.
That can't be an equilibrium.
Now, we could try other guesses and see if we can do better.
Suppose instead we guess z is equal to 1/2.
Let us now suppose everyone guesses that only half the population is
going to use the product and the other half is not.
Now, at this point, we have to think a little bit harder,
because we have to figure out which half are we talking about using the product.
Well, that's actually not all that hard.
It's going to be the people whose names x are closest to 0,
because they're the people who always value the product highest.
So this guess of z equal to 1/2 would be a guess
that everyone whose name is less than 1/2
uses the product and everyone whose name is greater than 1/2
does not use the product.
Let's put that into our graph and see what it does the reservation prices
and check to see if it's right.
Well, what it does now, remember f of z is 4 times z,
is it takes everybody's reservation price, which is 1 minus x for person x,
and it multiplies it by by 2-- 4 times 1/2.
That's the blue line in this next graph.
We can ask now if those are the reservation
prices that we have with the guess of half the population using the product
and with the price still equal to 1/2, will, in fact, half the population
use the product?
So we take the price of 1/2 over to our reservation price curve, and we check.
Does it hit at half the population?
And as you can see in this graph, no, it hits
at some value z that's bigger than 1/2.
So the guess of 1/2 was too pessimistic.
At a price of 1/2, if everybody guessed that half the people would use it,
in fact, more than half the people want to use it.
Whereas the guess of everyone using it was too optimistic.
This is beginning to give us some idea of what
a possible candidate for a good guess is.
We could imagine picking a different guess.
Suppose we all guess that 3/4 of the people are going to use the product,
and we could run through exactly the same exercise.
If we do that, the reservation price curve
is going to be 1 minus x multiplied by 3-- 4 times 3/4.
We can take that blue curve now, which has
been shifted to have an intercept on the vertical axis of 3
and still an intercept on the horizontal axix of 1,
compare it to the price of 1/2, and see what the right fraction is.
Here you can't quite see from the picture without doing a little math,
but in fact, the fraction who's going to use the product, if everyone thinks
that 3/4 of people will use it is actually 5/6.
So 3/4 was too pessimistic.
In fact, more than that are going to use it.
We could imagine continuing to do this over and over
and try to get a guess that works.
But that's a painful process to go through,
and it's actually not a good way to do this.
What we really want to do is come up with a technique that
will let us find all of the possible correct guesses, that
is all the possible expectations people could have about who's
going to use the product that will in fact be correct.
And that's what we'll look at next.