Demand for insurance
Consider an individual - riskophobe, whose preferences are described by the expected utility function. Let the wealth of this agent be equal to w, however there is a possibility of losing part of this wealth due to flooding. If the flood occurs, then this individual will suffer losses equal to L, 0 The flood probability is p, where pe (0,1). The insurance company offers flood insurance: the insurance unit costs y for each unit of coverage (that is, in the event of a flood the individual is guaranteed a payment of 1). Let r units be the amount of insurance purchased by an individual. How much insurance will an individual buy? what is the demand for insurance?
If the individual buys 2 insurance units, then his wealth in case of a flood will be wL + z-yz, and in the absence of a flood - w - yz. Choosing a value of 2, the individual maximizes the expected utility function:
At the optimal point z , will be met.
Note that due to the strict concavity of the expected utility function of the risk agent, the first-order conditions described above are necessary and sufficient conditions for the global maximum.
Whether the solution of the agent is internal or angular, i.e. will he buy insurance or refuse to buy it? The answer to this question depends on the ratio of the likelihood of the occurrence of the insured event (ie the probability of flooding) and the price of insurance offered by the insurance company, and also on its attitude to risk.
Consider, for example, the case of actuarially fair insurance, i.e. such when the probability of occurrence of an insured event is equal to the cost of a unit of insurance, y = p. Why is this insurance called actuarially fair? Suppose that the insurance company does not have transaction costs and the only costs are related to payment under the insurance policy. Then the expected profit of the company is equal to yz - pz (the insurance company receives from the individual yz as payment for the policy improbability p pays insurance z). At y = p the expected profit is zero regardless of how much insurance is purchased. Note that this reasoning is correct only when the company knows exactly the probability of occurrence of the insured event, i.e., in particular, in the absence of information asymmetry.
Thus, in the case of actuarially fair insurance, the conditions of the first order of the problem of maximizing the expected utility can be rewritten in the following form:
Note that the first-order condition can not be satisfied as a strict inequality, i.e. z * can not be zero. Indeed, let for some individual . Then according to the first-order condition . On the other hand, w - L & lt; w, that in view of decreasing marginal utility (because of strict concavity) and opposite in sign to the inequality: . This contradiction means that z * & gt; 0.
Since z * & gt; 0, the first-order condition of the maximization of expected utility will be fulfilled as an equality:
Since the utility function is increasing and strictly concave, the last condition will be satisfied if and only if the arguments of the function on the left and right sides of the expression are equal, i.e.,
So, the risk agent will put forward a positive demand for insurance if the insurance conditions are actuarially fair.
The merchant Cayus from St. Petersburg bought in Amsterdam goods that he could sell for 10,000 rubles if they were in St. Petersburg. He sends them by sea, but I'm not sure if they should be insured. At the same time, he knows that out of a hundred ships that leave Amsterdam for Petersburg at this time of year, usually five perish. Nevertheless, he can not find anyone who would be less than 800 rubles for a price. would accept insurance, but to him this price seems excessively high. The question is: how great should the state of Cayus be, except for the above goods, so that his waiver of insurance could be considered reasonable? We denote this state by its state x; then it together with the hope of a happy arrival of goods will be expressed as follows:
if he refuses insurance; If he, on the contrary, agrees to it, then he has a reliable general condition Cx + 9200). If we equate both these quantities to each other, we get:
and from here approximately x = 5043. Thus, if Caius, in addition to the hope for his goods, still has a sum of more than 5043 rubles, then, having refused insurance, he acts reasonably; if he has less, then he should have agreed to it. Now we ask: how, at the very least, should a person have a status who takes out insurance for 800 rubles, so that his deed can be considered reasonable? To calculate this state y , we have the equation and from here approximately y = 14,243 is a number that could be derived from the above without new calculations. The one who has less, acts unreasonably, if he takes on insurance, whereas someone with a large fortune will do it perfectly right. This shows how profitable the introduction of such insurance was, since it can be of great benefit to both parties. If Caius managed to negotiate insurance for 600 rubles, it would be unwise of him to refuse from it, since he has less than 20,478 rubles, and, conversely, he would act excessively cautiously if he insured his goods with a condition in excess of 20,478 rubles. On the other hand, anyone who has less than 29,878 rubles would have acted unwise if he had offered Cayus insurance for 600 rubles, and vice versa, it would have been more profitable for him if he had had more. However, no one, even if he was twice richer, would not have made a profitable business, taking on himself such insurance for 500 rubles.
Bernoulli D. Experience of the new theory of lot measurement // Milestones of economic thought. T. 1. Theory of consumer behavior and demand/ed. V. M. Galperin. - St. Petersburg: Economic School, 1999. P. 20-21.
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