Now, we're introducing a model of liquidity.

The key idea here is that investors maximize their expected utility.

So, we will calculate the utility given various kinds of potential offerings,

and then we'll see how these investors will prefer certain choices to others.

Then, we will also simplify the world around,

and we will use a three moment model of this world.

In what follows, we will build up on

the findings of the famous article by Diamond that is included in our handouts,

that he did it quite a few time ago.

But then, now we are just using that in a much more simplified way.

So, this is the model of stroboscopic economy if you will.

So, there is flashlight at point zero,

something happens, then there is darkness.

Then, we arrive at point one, flashlight again,

something happens and then darkness,

and then flashlight again at point two.

So, this is a three-point economy.

Nothing happens in between.

Now, we have 100 depositors to invest.

Each invests $1,

and each investor has an uncertain need for liquidity.

So, these investors can consume at point one,

or at point two but not at both.

So, we will denote π,

the probability of consumption at point one,

and clearly here that will be 1-π.

And, the decision is made on the maximization of expected utility.

And, for the purpose of this model,

we will say that the expected utility as

a function of the consumption level is the following:

Expected utility is 10 times square root of C minus one,

where C is the amount of money that the investor receives at this point.

So, clearly, overall expected utility,

and we will also say that expected utility at point one

is π times expected utility at point one,

plus one minus π expected utility of point two.

So, overall, we have the following thing.

We have utility of consumption at T equal to zero,

is equal to π square root of C_1 minus one,

plus one minus π square root of C_2 minus one and all that multiplied by 10.

So, this is what we will maximize.

And, this is the model,

and we will see exactly what can these people do because now,

we will introduce certain instruments of investments.

So, what these investors can have or can buy.

And we will start out with two very simplistic assets.

So, overall, we have three assets,

but now we'll start with two more simplest.

Again, I will always reproduce this timeline,

and first comes the asset that I will call a blue bag.

This is an asset that costs $1 at point zero,

and if it's liquidated at point one,

it brings you exactly $1,

and then point two, also exactly $1.

So, it's a perfect liquid asset basically as a bill.

So, you have a $1 bill in your pocket,

and you do nothing with it.

You get at it point zero, and you hold,

you can use that at point one or at point two,

or you can actually put this bill in the bank, and at point one,

the bank gives back to you this bill,

or if you don't take it here,

then you can take it at point two.

So, you have no interest,

you have no return, no nothing,

and clearly, this utility of this,

then is equal to 10 times square root of C whatever, it's one or two,

it's the same minus one which is exactly zero,

because it's at this point zero and

in the other point zero regardless of the probability.

Well, you can see that this is a nice instrument in terms of liquidity,

but it's not nice at all in terms of return.

Well, there exists another asset that I will label as a black box,

that also costs $1 at point zero,

but then it has the following feature.

So, if you sell it at point one,

it brings you back the same $1 with no interest.

But, if you can afford to hold it until point two,

then at point two in a magic way,

it gives you $2.

So, this is a grossly exaggerated example of a time deposit.

So, if you held them till expiration,

you can reap some interest.

And here, I, at least with the reasons,

I put a very high interest.

Now, we can calculate the utility of the square.

The utility of the square is equal to 10 times,

now, we have to deal with these probabilities.

Let me, for the future of this episode,

put that π is equal to 20 percent,

and one minus π will be 80 percent.

We will talk in some more detail what happens if these probabilities cannot be fixed,

or are likely to move,

that we will postpone for some time.

Now, if we did that,

by the way here,

we didn't care because regardless of the probability,

we would have had zeros.

Now, that is 10 times point two times the square root of one minus one,

this is zero, plus point eight times the square root of two minus one,

so this is zero,

this is one, square root gives one,

so that is eight.

Whatever it means, all units here do not play a significant role.

The key is that clearly,

this asset has a positive expected utility,

and as far as this is concerned,

we know that investors would actually prefer

this black box to this blue bag because you can see that it costs the same.

At point one, it's exactly the same,

but at point two clearly the black box is much better, way better if you will.

So far so good,

but so far there is no bank.

So far, we just said that,

because remember the discussion over the previous week,

so that was more like dealing with the borrower in a direct way.

But now, we are moving to the third most interesting asset.

Again, the same timeline and I will label that a red triangle.

Red triangle also costs one dollar at point zero,

but then it promises to pay its owner $1.25 at point one,

and $1.85 at point two.

Well, clearly the red triangle is better than the blue bag,

because it costs the same,

but it provides its owner a greater amount of money both at point one and at point two.

Well, clearly you remember that the utility of the blue bag was zero.

Well, I put these numbers specifically that way to not, well,

it really doesn't have to be exactly zero,

but for our discussion that is even easier.

Now, let's calculate what's the utility of this triangle.

Again, if we put probabilities here of 120,

am sorry, of point 20 and point 80.

Now, the utility of the triangle is,

10 is overall, and we take a brace here, it's how much?

It's point two times the square root of point 25,

then I subtract one from here,

plus point eight times the square root of point 85.

Well, I'm sorry, this is taken easily this is point five,

this is point 92,

but these calculations are quite trivial,

the result is 8.38.

So, what is important?

If we go back for a moment,

you can see that we had eight for the black box.

And now, for the triangle,

we beat that, and we have 8.38.

So, an investor who maximizes

her expected utility would stick to the red triangle.

Now, clearly the depositors are now happy.

Remember, from the end of the next week,

I said that depositors should be given some kind of a carrot.

Now, they are. Now as a bank,

I say, "Well, I offer you red triangles.

So, if you lend to businesses through me,

then your utility will be, well,

not that much but still higher,

than compared to the case when you lend to the businesses directly."

All of that is great.

Now, there is something that remains here.

We have to prove that the bank not only declares that,

but can actually deliver on this promise.

So, given our setup of 100 investors and the universe of just three assets, the blue bag,

the black square, and the red triangle,

I will show to you what exactly the bank does to create this liquidity.

This process of liquidity creation,

or if you will the autopsy of the liquidity creation process,

I will introduce starting from the next episode.