In the previous episode,

we talked about co-insurance a market way to overcome the moral hazard problem.

The problem of moral hazard was first observed in the insurance industry.

And, it was in this industry when people realized that it produces quite a bit of damage.

Now, in this episode and in the next one,

we will talk about a more advanced model of analyzing moral hazard insurance.

Namely, we will see that the insurance company that does not observe the behavior of

the individuals who buy insurance still

can influence the behavior by offering them different contracts.

And, in this case without restoring observability,

we will be able to somewhat alleviate the problem of moral hazard.

Now, let's consider the following model.

We have two parties here.

A risk averse individual who maximizes her expected utility,

and the risk neutral insurance company.

That means that other things being equal,

the individual who buys insurance makes

her choices on the basis of the maximum expected utility,

while the insurance company cares only about breaking even.

So, if the insurance company pays something to the individual in the bad case, let's say,

then it should somehow compensate itself by collecting

the same amount of money from the individual in the good case.

Now, we can immediately see that the key story here is

the probability of which cases occur.

Now, first of all,

let's put some charts and numbers here.

We said that the individual maximizes expected utility.

In economics, it can be shown that the function

of expected utility as a function of income,

C is consumption level or income in this case.

This is utility, must be a convex function like this.

That comes from the well-known law of diminishing marginal utility.

Basically, that means that if you have one dollar,

then one more dollar is really available for you.

If you have a million dollars,

then one more dollar is much less

available to you compared to the amount that you already have.

And in this small episode we will approximate-.

This will be falling curve.

We will say that the expected utility as a function of C

is a square root of C. This function is,

well, it comes from here,

and that is consistent with the law of diminishing marginal utility.

Now, so we said that the individual is a risk averse,

and maximizes expected utility.

So that comes from the idea that people are in general greedy and risk

averse and here we will see that greed will be the number one criteria.

Now the setup is like this.

There is the high state and the low state,

and the high state occurs with a probability pi,

while the low state occurs with the probability one minus pi.

Now, the income or

consumption level of the individual before insurance in the high state is C_H,

and here it's C_L.

Now, the insurance contract works like this.

In the high state,

in which the individual does not need the coverage,

she pays to the insurance company some amount of X, and therefore,

after insurance we have here C_H minus X.

In the low state,

when the individual does need the insurance,

the insurance company pays

something to the individual and therefore the consumption level becomes C_L plus Y.

So this is the amount of Y.

This is the most general case.

We can see that the insurance company must break even so

that basically means that the expected amount of money that the company pays,

which is pi times X,

should be equal to the expected amount of money that the company receives,

pi times X, is the same as the amount that it pays,

which is Y times one minus pi.

So this is the criterion that allows the company to offer the contract.

And then, we will see that

the individual make her choice on the basis of the maximum expected utility.

Let's flip over the chart and put some formulas here.

And we will start with the first case.

Case one, that will be all complete insurance.