In the previous episode, we talked about complete insurance. And if the individual is completely insured clearly, he or she would not take any efforts to behave properly. Now, we will study another more interesting case of incomplete insurance, the situation in which the company by offering an individual a special contract will induce him to behave in a corresponding way. So we are talking about incomplete insurance. Well, as soon as we pronounced the word incomplete, that means that the individual is not having the same level of utility or, in our model, the same level of consumption in high and low states. So levels of consumptions differ. Now, that, as you can see, is not a great thing for the individual because he is risk-averse. But we will see that because greed also plays an important role here, that the individual will behave so as to maximize the expected utility. Now, the question goes, what if pi, the probability of the high state, changes? We will study that in the following two cases. Case one will be efforts that the individual takes so he or she does something, let's say drives more prudently, parks the car at the secured spot, and so on, and so forth. In this case, we will say that pi is back to 0.5, to the same one half. But because these efforts, they cost something, then the utility of consumption is diminished, and we have to subtract minus delta, which is equal to minus 5. So we calculate the utility for this case with these probabilities of one half in the good state and one half in the high state and one half in the low state and then subtract 5. Now, there is case two in which the individual takes no efforts. And in this case, we will say that pi drops to 0.2, but the utility here is unchanged. Well, unchanged, we mean that we do not subtract anything from that. Now, we know the answer for the no efforts from the previous episode. The utility is 51.57. But let's now analyze what's going on here. The insurance company offers the following contract. It says that, "I will pay you $400 in the low state, and I will receive from you $400 in the high state." Now, again, high or low, the same initial numbers, 8500 and 1200. But, now, the contract looks like that the incomplete insurance, and the amount is 400 so that after insurance, how much do we have? We have 8100 here, and we have 1600 here because, here, the individual pay is 400 to the insurance company. Here, he receives that amount. Well, clearly, the consumption level is way different here. It's more than five times. But let's calculate the utility of consumption in this case. By the way, this contract is feasible only if pi is 0.5 because you can see that, in this case, the insurance company breaks even. If the probabilities are like that, 0.2 and 0.8, then unfortunately, the insurance company will lose money, and the contract will not be feasible. Well, let's calculate the utilities. So utility of efforts is, in this case, this is 0.5 times the square root of 8100. We calculate it to the individual after the contract, plus 0.5 times the square root of 1600, and we subtract 5 as we agreed. Now, the numbers are made up so that you can easily take the square roots, and the result is 60. Well, if you recalculate that for the second case for no efforts, then they will have 0.2 times the square root 8100 plus 0.8 of the square root of 1600 and no subtraction. And that will be just 50. Now, see what happens. You can see that, in this case, the individual will indeed decide to take some effort because, in this case, the utility of consumption is much higher. Well, what does that mean in reality? That means that the company offers such a contract, and then the individual by him or herself decides to behave properly. Now, the key story is that the company does not observe the behavior of the individual. The company does not put any kind of special behavior as part of the contract, but the company enjoys these probabilities as a result of the fact that the individual realizes that prudent behavior is in his or her benefit. So this is the key story here. So we can see that income with insurance provides an important signal for the individual. And, therefore, by just offering that, grow to the forces or induces, better to say, the individual to behave prudently. Now, you can say, "Well, what do we observe in reality? " First of all, you can say that the numbers here are in a way off each other, and that is done only to emphasize the situation. But in reality, what happens is that the insurance companies oftentimes, they offer deductibles. Let's say, any damage to your car up to $500, you cover by yourself. So if you scratch it, if you have some minor damage, then the company does not compensate you. However, when it comes to the more significant damage or, God forbid, if your car gets stolen, then in this case, the company does cover the damage. So what signal does that provide for the individual? So if you behave properly, if you park your car at secure spots, if you drive safely, if you don't ignore some potentially dangerous places, then in this case, you will save this deductible. If anything bad happens, and if your car gets crashed or stolen, then in this case, this is sort of not your fault, and then we will cover you. So we can see that this is a very powerful tool of sort of sharing risk. And the story here is very simple. Part of the risk is being moved on the individual. And, again, like I said, the individual doesn't like this because of risk aversion but still puts up with that because that provides a much higher utility. Now, we are wrapping up our discussion of moral hazard in this way, and we're making a small shift. And in the next episode, we will build a model for the analysis of another important problem caused by private information, namely, the problem of adverse selection.
