Now, let's build a model of adverse selection in credit markets.
In what follows, we will use the approach of that
was first developed in the 1981 article by the late Nobel Prize winner Joe Stiglitz.
And the more simplified approach that we'll use
here was offered by Dr. Bradley Chadwick from UCLA,
some 20 something years ago.
Now, we will see the situation like this.
Now, there is again the high state and
the low state that occur with probability one here,
that can not be changed.
And now, there are three kinds of companies that come at a the bank.
Now, all companies need to borrow $8 million.
Now, the bank is greedy and it's a monopolist.
So, the bank can charge as high of its value as it likes.
But the bank maximizes its expected cash flow.
So, we will see that if a bank charges too much,
then not all borrowers will be willing to take a loan.
We will think that there are three kinds of companies.
Companies number one of that are riskless and
nice with cash flows of 12 and 12 in both cases,
black companies, then blue companies number two,
that have cash flows of 18 and 6.
So, you can see that the expected cash flow is probably is the same for them.
But this is a riskier company.
And then, there are also the riskiest companies number three,
that have cash flows of 20 and 0.
Well, this company is also,
in terms of expected cash flow is a little bit worse,
here it's 12, here it's just 10.
But that's what we observe as a model.
Now, lets take the view of the bank.
What the bank observes is,
that if F is less or equal than 12,
then we can expect that all three kinds of borrowers come.
Well, to be better, I will put F with black here,
that supports the idea that all of them have.
Well, I'm a greedy bank,
so I can go over 12,
but if I do so, let's say if F is greater then but less or equal than 18,
then all the borrowers of kinds two and three stay, and number one,
the best borrowers, they drop out,
best I mean, the borrowers with the lowest risk.
And, if I persist with my greed,
if I say that F is greater then 18,
then only the red guys stay.
Now, that should be less or equal then 20,
because if I for example charge 25,
then no one borrows,
and i have no business.
Well, but what happens on these threshold points 12 and 18?
When a company comes to me, to the bank,
to take a loan,
I cannot distinguish what kind of company it is.
So, I have to inscribe a certain probability to this company.
And now I see that if F is less or equal then 12,
then all companies will take the loan.
I will inscribe the probability of one-third.
If I go over 12,
then my probability goes to one half because these guys drop out.
And if I go over 18,
then both these guys and these drop out and my probability
goes up to one but I have business only with companies like three.
Well, lets see what is my expected cash flow.
If I put F a little bit less than 12,
I put like 12 minus Epsilon ,
where Epsilon is a small amount,
very close to zero.
Then what is my expected cash flow to the bank?
In this case, all three kinds of borrowers come,
and what is the contribution of borrowers of number one?
Its probability is one-third,
and in the high state,
I have my 12 minus Epsilon,
and in the low state,
I have the same.
So, this first brace is the contribution of black borrowers number one.
Now, at the same time,
we know that with the same probability one-third,
I have business with blue borrowers that are different.
Now, what do I have then? In high state,
I have the same 12 minus Epsilon ,
again this is my F,
but in a low state unfortunately,
they have only six to pay me back.
So, I do not really get all my money back.
Well, this is not it.
And then I also have a contribution of the red riskiest borrowers.
In this case, I have in the high state,
they have enough money to pay me back,
but in the low state unfortunately, they have nothing.
Now, If I summarized all that,
I will get that the result is 9 minus two-thirds of an Epsilon.
So, this is the amount of money that the bank gets.
So, I gave almost 12 but I collected just a little bit less than nine.
And that is because in this case,
I have less than my face value,
and in this case, I have nothing.
Well, as we stated the bank is quite greedy,
so it thinks, what if I went over 12?
So, if I charged 12 plus Epsilon,
let's see what happens.
Well, it seems to be a very clear calculation the same as it was, but it changes,
if F is 12 plus Epsilon,
we see that black borrowers drop out.
So, now I have to put it in blue,
and then erase what happens,
my expected cash flow.
What's the contribution of borrowers number two?
Now, the probability is half because first guys dropped out.
Here, I have one half times 12 plus Epsilon plus one half times 6,
and then there's also a contribution of the other guys which is again,
half here, one half 12 plus Epsilon, because Epsilon is small,
so it cannot go over 20,
and then here I have times zero,
and all that adds up to 7.5 plus Epsilon over two.
So, we see what happened.
I just went over 12 and before I had almost nine,
but now I dropped to 7.5.
Well, if we proceeded,
then I can say,
well, if I dropped, what if I keep raising my F up to 18 minus Epsilon?
Now, nothing changes.
Well, the borrowers pay a higher interest,
but both two and three still borrow.
And if you redid all this calculation,
you would see that the expected cash flow to the bank
would rise up to 10.5 minus Epsilon over two.
So, the bank says, fine,
if I went ahead and jumped over 12,
I will have to go up to 18 minus Epsilon.
Well, finally, if I go let's say jump over 18,
in this case, the contribution is very simple,
Pie is equal to one, I have all the red borrowers and then,
expected cash flow to the bank for 18 plus Epsilon,
is 9 plus Epsilon over two.
So again, from this point to that, I do drop.
Well, all these things I will summarize on the chart on the next page.
That goes like this.
So, this is the face value,
and this is expected cash flow to the bank.
Now, five, 10, 15, and here I'll put also, five, 10, 15, 20.
There are special points here.
One special point is,
this is 12, and this is 18.
So, see what happens.
We know that if I go to 12 from here,
then I get nine minus two thirds of an Epsilon.
So, we have the following picture.
This is nine minus two thirds of an Epsilon.
Well, clearly this is here but this is a linear thing,
so Epsilon may be quite large too.
Well, this is, again,
