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4.4 COMPARISON OF THE DEMAND FUNCTIONS OF MODELS A, B, AND C (FROM THE PREVIOUS THREE SECTIONS)

The following figure compares the insurance demand functions discussed in the previous three sections.

The Kahneman-Tversky model logically displays the lowest demand at lower prices, as it excludes all agents of below-average wealth from the insurance demand (with its assumption of a positive attitude to risk in the convex left-hand part of the domain). At prices above the threshold there is zero demand.

By contrast, high prices are the least off-putting in the model of maximization of the expected utility of income, since a large proportion of agents here have a positive attitude to risk.

With an increasing premium, the survival probability maximization model will reduce insurance demand most sensitively. The expected utility maximization model of course exhibits the highest demand for agents with below-average income, since in this case even the poorest buy insurance. For those with above-average income the predicted reaction in the expected utility of income maximization model is similar as in the Kahneman-Tversky model, while there are more agents excluded than in the survival probability maximization model.

Figure 25: Comparison of insurance demand functions: D1(a): Survival probability maximization model D2(a): Expected utility of income maximization model D3(a): Kahneman-Tversky value function model

Comparison of insurance demand functions: D1(a): Survival probability maximization model D2(a): Expected utility of income maximization model D3(a): Kahneman-Tversky value function model

The Kahneman-Tversky model therefore shows the lowest price elasticity [over the entire domain), while insurance demand is most elastic in the von Neumann-Morgenstern model of maximization of the expected utility of income.

The following table presents a clear comparison of the analysed models with regard to the price and income elasticity of demand:

Table 4: Comparison of models according to the elasticity of demand for insurance

Model

Elasticity of insurance demand

Price

Income

A

(von Neumann-Morgenstern)

high, insurance is a luxury good

negative, insurance is an inferior good

B

(Kahneman-Tversky)

low, insurance is a difficult-to-substitute good

high, insurance is a luxury good

C

(survival probability maximization)

moderate

moderate

The breakdown of insured agents by income level differs considerably across the three models under consideration. If we divide the population by income level into five classes (very poor, poor, lower-middle class, upper-middle class, very wealthy) we can say that, given a moderate ("sensible”) premium a, in the model of maximization of the (declining] utility of income, insurance will be taken out by the very poor, the poor and the lower and upper-middle classes, i.e. all except the wealthiest lying above threshold d1(a) illustrated in Figure 17. In the Kahneman-Tversky value function model, insurance is purchased only by upper-middle-class agents, specifically those with income in the range of d Î (dinfl + a; d1(a)), where dm is the point of inflection of the Kahneman-Tversky value function. If dinfl + a > d1(a), no one will buy insurance. In the Pareto survival probability maximization model, the poor and the lower and upper-middle classes will take out insurance.

An increasing premium will progressively deter all agents in all models, last of all the very poor in the income utility maximization model and the upper-middle class in the Kahneman-Tversky value function model. A low premium close to the expected loss E(L) would be accepted by all agents in the income utility maximization model, by only the upper-middle class and the wealthy in the Kahneman-Tversky value function model, and by everyone except the extremely poor in the Pareto survival probability maximization model. This is illustrated clearly in the following table, which indicates which classes buy insurance in which models.

Table 5: Who will take out insurance in the three models under comparison?

A – von Neumann-Morgenstern maximum utility of income model

B – Kahneman-Tversky value function model

C – Pareto survival probability maximization model

Classes

Premium

Low

High

very poor

A

A

poor

A, C

A

lower-middle

A, C

-

upper-middle

A, B, C

B

very wealthy

A, B, C

-

The von Neumann-Morgenstern model of maximization of the expected utility of income displays an increasing incentive to insure with falling income even for extremely badly off agents. This is unrealistic. In reality, the middle classes buy insurance. Anyone who is neither extremely wealthy nor extremely poor can (unlike the very poor) afford to pay for insurance without making any great sacrifices and is motivated to do so because (unlike the very wealthy) they would suffer a large (or even ruinous) loss in the event of accident or theft. By contrast, the very poor do not buy insurance in reality.

Both the economic survival probability maximization model and the Kahneman-Tversky model reflect the reality that the very poor are rarely insured. In the latter model, though, all economic agents with below-average income will decide against insurance (unlike in reality). The main difference between the Kahneman-Tversky model and the survival probability maximization model lies in the assumed motive for the choice "not to insure" for agents with below-average income. In the former case, the reason (plainly unrealistic in insurance-related matters) is that these agents are attracted to risk (i.e. they exhibit the negative risk aversion typical of gamblers), whereas in the latter case the reason is a lack of money, meaning either that the economic agent cannot afford the premium at all or that paying it would cause an excessive fall in his standard of living.

For the reasons given above, we regard our survival probability maximization model as being more consistent with the real-life behaviour of economic agents in the insurance market than both von Neumann and Morgenstern's income utility maximization model and Kahneman and Tversky's prospect theory.

 
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