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 KahnemanTversky model logically displays the lowest demand at lower prices, as it excludes all agents of belowaverage wealth from the insurance demand (with its assumption of a positive attitude to risk in the convex lefthand part of the domain). At prices above the threshold there is zero demand.
By contrast, high prices are the least offputting 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 belowaverage income, since in this case even the poorest buy insurance. For those with aboveaverage income the predicted reaction in the expected utility of income maximization model is similar as in the KahnemanTversky 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): KahnemanTversky value function model
The KahnemanTversky model therefore shows the lowest price elasticity [over the entire domain), while insurance demand is most elastic in the von NeumannMorgenstern 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 NeumannMorgenstern)
high, insurance is a luxury good
negative, insurance is an inferior good
B
(KahnemanTversky)
low, insurance is a difficulttosubstitute 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, lowermiddle class, uppermiddle 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 uppermiddle classes, i.e. all except the wealthiest lying above threshold d1(a) illustrated in Figure 17. In the KahnemanTversky value function model, insurance is purchased only by uppermiddleclass agents, specifically those with income in the range of d Î (dinfl + a; d1(a)), where dm is the point of inflection of the KahnemanTversky 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 uppermiddle 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 uppermiddle class in the KahnemanTversky 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 uppermiddle class and the wealthy in the KahnemanTversky 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 NeumannMorgenstern maximum utility of income model
B – KahnemanTversky value function model
C – Pareto survival probability maximization model
Classes
Premium
Low
High
very poor
A
A
poor
A, C
A
lowermiddle
A, C

uppermiddle
A, B, C
B
very wealthy
A, B, C

The von NeumannMorgenstern 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 KahnemanTversky model reflect the reality that the very poor are rarely insured. In the latter model, though, all economic agents with belowaverage income will decide against insurance (unlike in reality). The main difference between the KahnemanTversky model and the survival probability maximization model lies in the assumed motive for the choice "not to insure" for agents with belowaverage income. In the former case, the reason (plainly unrealistic in insurancerelated 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 reallife 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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