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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.

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What does that mean? Complete insurance means

that the consumption level of the individual,

in both cases after insurance in the high state and in the low state is the same.

So that means that all risk has been covered by the insurance company.

Well, and there are two important things that we have to observe here.

One, is the pi times X is equal to one minus pi times Y.

This is that the insurance company breaks even.

Now, what is the utility of individual?

We have to maximize the following thing.

This is pi times square root of C_H minus X,

plus one minus pi square root of C_L plus Y.

Well, pi here we for now hold constant.

Now we see two equations.

Well, this is not the equation that we have to maximize that.

But then, we have an equation here that connects X and Y,

so that becomes a function of just one thing,

and then differentiating that and setting the derivative to

zero we can get a solution if you will.

It says that Y should be equal to pi times C_H minus C_L.

So, if you put that,

and if you calculate X and then you will see that actually given this,

the insurance company just breaks even.

Now, and for example,

this gives us the condition in which the complete insurance is visible,

and this is the condition of the maximum of the expected utility of the individual.

Well, as always we need some examples of this.

And, let's say, example one.

Let's set that pi is equal to one minus pi and it is equal to one half.

Our good old friend, one half.

And let's put some numbers.

Let's say C_H before insurance is 8,500 and C_L is 1,200.

By the way, let me observe that when we take

square roots you don't have to be really frightened by that.

We don't care much about dimensions here.

We can always normalize that by dividing that to some, let's say, easier level.

So don't worry about the fact that we take square root of dollars.

Now, in this case,

we can say that it turns out to be that X will be equal to Y and will be equal to 3,650.

And after insurance,

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we will see that C_H minus X will be equal to C_L plus Y,

and will be equal to 4,850.

And the utility will

be equal to the square root of 4,850,

which is approximately 69.64.

So that's example one.

But, let's also analyze the situation in which probabilities are different.

Example two.

Let's say that pi is equal to just 20 percent while one minus pi is 80 percent.

Well, you can always say that in reality in insurance companies,

the probability of a low state or insurance case is low.

But that is because there are a lot of people who buy insurance.

And the probability of cases happening to them at the same time is low.

Here, we have only one company and one individual.

For that reason, we can say that actually a situation like that is quite probable.

So in this world, unfortunately,

good things happen more rarely than bad things.

And if we did some calculation,

we'll find that Y will be equal to 1,460,

C after that will be equal to 2,660,

and utility as the square root of C will be approximately 51.57.

Now, what do we see here?

So far we can see that, well,

if the probability is one half then utility is much higher,

and if probability of a high state or a good outcome is lower .2,

the utility is much lower.

So clearly, the individual would,

other things be equal,

prefer to have the probability of the high state higher.

And then, we set a question like this.

Here we talked about complete insurance.

The situation of which the individual has

its consumption level in both cases after insurance contract is the same.

So clearly, she is not

interested in making any efforts to try to change this probabilities.

And we said that we let them constant.

But what if the individual can indeed change these probabilities?

Well, this is not such an exotic case.

Let's say, you insure your car,

and you insure not only the car but also everything inside that car.

And, for example, if someone steals a valuable thing from your car,

you can go to the insurance company and say, "Well,

I became a victim of a theft so you pay me something for that."

And it can be shown that in this case if you're fully covered, then you don't care.

You stop locking up your car,

you park it everywhere,

you do leave valuable things inside.

But, in reality people don't behave like that.

And we will see in just a few minutes why this happens.

So we pose a question like that.

What if pi can be changed

by the behavior

of the individual?

We will try to answer this question in our next episode.