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So what we're going to do now is come up with a procedure that
will let us find an equilibrium in this market.
And remember, this is a market with network effects
so the equilibrium we're looking for is an equilibrium
fraction using the product or the service.
It's complicated because of the fact that what anyone's
willing to pay for this product depends on their guess about who else
is using the product.
So this is a fairly complicated concept.
We're looking for an equilibrium really in expectations with the idea
being that people have an expectation about the fraction of the population,
the thing we called z, using the product.
And in equilibrium we want that guess to be correct.
Let's first think about why we'd want that to be our equilibrium concept.
It's in some sense a self-fulfilling expectations equilibrium concept.
We want expectations that when they operate through the market,
cause the expectations themselves to be correct.
And the basic idea is to look, if the expectations people
have are not correct then someone who's either buying the product
or not buying the product is unhappy with their decision,
they're going to want to change.
If people were too optimistic, they guessed
too many people would buy their product, and they're
going to see that actually not that many people buy their product
and that means that someone who bought it is unhappy they bought it and would
rather not have bought it.
We're looking for an equilibrium situation in which everybody's
making a decision that actually is the correct decision with expectations
that are actually fulfilled.
So let's think about what that means in the context of the model we've
been running down.
Remember, r of x is person x's reservation price without the network
effect that gets multiplied by f of z, where
z is the fraction using the product, to pick up the network effect.
So person x's reservation price, if they expect fraction z to use the product,
is f of z times r of x?
What's an equilibrium going to look like?
Well, it's got to be the case that in equilibrium the people who are actually
buying the product add up to fraction z.
Let's think about the last person buying the product,
and since the person who's just barely willing to buy the product, that
would be a person whose reservation price is exactly
equal to the price in the market for the product, the thing that we've called p.
That means that in an equilibrium, if people guess fractions
that you're using the product and person x is the last person using the product,
f of z times r of x would have to be equal to p.
And that person, x, let's figure out who that person really is.
Well, if the guess z was correct, that person x has to be person z,
have to be the same.
Because, remember, every one whose name is to the left of x
is buying the product, if x is the critical person there,
everyone to the right is not buying it.
So what's the fraction buying their product when people guess fraction z?
It's this guy, x, right at the critical point, but if it's an equilibrium
then x has to be equal to z.
And that actually gives us the procedure to use
to find the equilibrium fraction z using the product.
It's just a fraction z using their product that solves f of z times r of z
equals p.
So all we've done is plug z into r in place
of x because we know in equilibrium if you
guess z, z has to be the critical person buying the product.
Let's take that intuition now that what an equilibrium has to be
and return to our example to actually figure out how to compute one.
So in our example, remember, we had r of x equal to 1 minus x, and f of z
equal to 4 times z.
And we did p equal to 1/2, but that's not relevant quite yet.
Let's first figure out what f of z times r of z looks like.
That in this example is going to be 4z times 1 minus z.
Remember, we're plugging z into r with the idea
being that the critical person using the product
has to be person z if expectations z are correct.
So let's graph 4 z times 1 minus z.
That's a nice non-monotonic function.
It's little hill pictured here in blue.
It hits the z-axis at 0 and 1 because that's what happens to f of z, f of 0
equal to 0.
And what happens to r, that is r of 1 is equal to 0.
And otherwise it's a hill.
In fact, this hill has a maximum at z equal to 1/2,
but that too isn't really relevant.
What counts is that it's a hill.
How do we find an equilibrium?
Well, we take our price p of 1/2 and we just
find all the points where that cuts f of z times r of z.
And if you notice here in the picture that price of 1/2 drawn with this red
cuts the curve twice.
It cuts it at z prime and at z double prime.
What's that mean?
Well, it means that in this market there are actually two equilibria,
both at the low expectation of z prime and at the much higher one of z
double prime.
Last time when we were doing that iteration over
guesses we were trying to find z double prime.
We weren't finding z prime because we didn't make guesses that were low.
Had we made low guesses we would have converged to z prime.
So it looks like in this market for this particular example with p of 1/2,
we actually have two equilibria that we can pick out there.
A low when z prime with a small fraction using the product and a high when
z double prime.
Remember, again, what this means.
It means that if people expect a small fraction of the population
to use the product, that is z prime, then
that's actually what's going to happen in equilibrium.
And if they expect a large fraction, z double prime, to use the product
that's what's going to happen in equilibrium.
That's really the critical aspect of this.
How many people use the product depends on what people expect to happen.
There are two self-fulfilling expectations equilibria here
in which people are actually using the product.
And it's also important to remember that there's another one.
There's another equilibrium where everyone expects no one
to use the product, that is z equal to 0.
That one is actually easy.
We don't need all the math to find that.
If everyone thinks that no one's going to use the product
then no one wants to buy it, and therefore, no one uses it.
That's always going to be an equilibrium.
So this market has three equilibria-- no one
using the product, a small fraction, z prime, using it, and a big fraction,
z double prime, using it.
The zero case is the special one.
We found z prime and z double prime by solving 4z times the quantity 1 minus z
equal to 1/2.
It's important to remember that we did this for a particular price p
equal to 1/2.
As you move that price around, the equilibria are going to change.
An interesting case is to notice what happens
if you move the price, p, up high enough.
Well, you move it above the top of the hill
then it doesn't cut the blue curve at all,
and the only equilibrium is going to be at z equal to 0.
So at a very high price, no one uses the product.
As the price comes down, the point where p hits the blue curve first
is just at one point right at the top of the hill.
That actually occurs at p equal to 1 .
And as the price comes down further, you get two equilibria
that have people using the product.
One in which the fraction using it is large and higher and higher
as you move the price lower, and another one in which
the fraction using the product is small and gets
smaller and smaller as the price comes down.
So you typically have three equilibria-- zero, the z prime,
and the z double prime.
What we'll do next time is take this analysis we've done
and begin to use it to analyze what happens
in the market for products like this.