# THE DEMAND FUNCTION IN THE INSURANCE MARKET: COMPARISON OF MAXIMIZATION OF THE PARETO PROBABILITY OF SURVIVAL WITH THE VON NEUMANN-MORGENSTERN EU THEORY AND KAHNEMAN-TVERSKY PROSPECT THEORY

## INSURANCE IN THE MODEL OF MAXIMIZATION OF AN AGENT'S PARETO PROBABILITY OF (ECONOMIC) SURVIVAL

To address the problem of rationality of the insured, we need to quantify the economic gain/loss associated with an agent's decision whether or not to buy insurance. We regard our generalized microeconomic theory — where the agent maximizes his probability of (economic survival) — as a suitable methodological approach in principle for examining the issue of insurance.

This utility function (in contrast to the EU theory and prospect theory— see below) reflects the fact that the agent's economic situation is the key factor for his insurance decision (and consequently for the insurance demand function). Agents with very high income will reject insurance (as an unfair game). However, agents with extremely low income will not buy insurance either, because if they did they would have to for go other consumer goods which, from their perspective (and given their economic situation), they regard as more necessary.

As a consequence of their economic situation, agents may therefore adopt the risky strategy of "not insuring", since they "simultaneously" consider two risks: the risk of losing the item to which the insurance relates, and the risk of

having insufficient "residual" income after paying the premium. Even if they are not risk-attracted, their situation can force them to take risks.^{[1]}

Throughout this chapter we assume that the agent must cover any loss immediately. This means that an insurance loss represents not only a loss of wealth, but also a loss of income. This simplifying [but acceptable at the level of abstraction of this book] assumption will allow us to avoid complications linked with the relationship between the wealth and income of the agent.

Let us assume that the agent is considering insuring againsta loss of L money units and that the probability of loss is *p* and the insurance premium is *a.* The expected loss is E(L) money units. For insurance companies to be able to exist, the premium must be higher than the expected loss: *a >* E(L), The agent's income is *d.* We will denote his subsistence level (the boundary of his economic extinction zone] by *b.* We assume that the agent will not cross this boundary if he pays the premium: *a< b - d.*

We assume that demand for insurance is driven by the number of agents whose risk of economic extinction will be reduced by buying insurance. If an agent does not take out insurance, he will save *a* money units for the premium but he will face the risk of losing *L* money units with a loss probability of *p.*

The key factors in this decision include not only the premium amount *a,* the loss probability *p* and the related loss L, but also the agent's income *d,* or rather his income relative to the boundary of the economic extinction zone *b.* These are the variables that determine his survival probability. We assume that the agent is price taker as regards the premium offered. He therefore has just two options:

**A. To insure **by paying the premium of *a* money units. His risk of (economic] extinction (resulting from a decline in his income below the subsistence level *b)* is then:

**B. Not to insure. **In this case his risk of extinction (due to loss *L* with probability *p)* is:

The key factor in the agent's insurance decision is the difference *R*1*(d)* - *R*2*(d).* If it is positive the agent will not take out insurance because the risk of economic extinction associated with paying the premium exceeds the risk of loss. Such a loss might be highly unpleasant (or even ruinous for very poor people), but is relatively highly unlikely.

The illustrative plots in Figures 10, 11, and 12 provide a comparison of the model agent's strategy to insure with his strategy not to insure (for three different premiums). The agent's income is represented on the horizontal axis. If, for income *d,* the plot of function *R*t*(d)* - *R*2*(d*) lies above the horizontal axis, i.e. if R1(d) > *R*2*(d),* it is profitable for the agent to take out insurance at a premium of *a.*

**Figure 10: Graph of function R**1**(d) - R**2**(d): comparison of the probability of extinction (in %) with insurance at a premium of ****a**** = 60 money units and with no insurance ****(b = L =**** 250)**

**Figure 11: Ditto for a premium of ****a**** = 70 money units**

**Figure 12: Ditto for a premium of ***a =* **80 money units**

To be able to derive the shape of the insurance demand function *D(a),* all we need to do now is introduce an assumption about the income distribution in the relevant population. For now we will limit ourselves to the simplest uniform income distribution over the range (*b,* 5), where *8* is the highest income in the system. If the entire plot of the function *R*1*(d)* - *R*2*(d)* lies below the horizontal axis over the entire domain *(b, δ)*, it is not profitable for any agent to take out insurance, i.e. *D(a)* = 0. If this is not the case, we will denote the minimum and maximum income of the insured with premium *a* by *d*1*(a)* and *d*2*(a*) respectively. The value of the demand function *D(a)* will then correspond to the ratio of the length of the range *(d*1*(a), d*2*(a*)) to the spread of the incomes in the system:

In the illustrative cases depicted in the following two graphs (Figures 13 and 14) we assume δ *= 4b,* i.e. a uniform income distribution over the range (*b*, *4b).* The corresponding insurance demand function is plotted in Figure 13:

for

for

**Figure 13: The insurance demand function in the survival probability maximization model with uniform income distribution over the range (b****,**** 4b)**

Figure 14 shows the agents for which taking out insurance is the less risky strategy (in the sense of the probability of economic extinction) in the given situation at various premiums. If point (*d, a)* lies in the grey region *Q,* an agent with income *d* regards premium *a* as an acceptable offer. It can be seen from Figure 14 that agents with very low income (*d <* 300) will not accept even a fair game, i.e. one where the premium equals the expected insurance claim *a* = 50, whereas the rather better off will even accept a premium that is higher than the expected claim. Agents with extremely high income, however, will reject the game even when the premium slightly exceeds the expected loss. In the real insurance market, though, the premium always exceeds this level, as otherwise insurance companies would be loss-making.

**Figure 14: Income characteristics of insurance demand in the survival probability maximization model: agents for whom **(**a,d)**** Î** **Q**** will take out insurance**

In the following two sections (4.2 and 4.3) we will construct the insurance demand function for the two most common approaches to modelling an agent's relationship to risk. We will work first of all with von Neumann and Morgen-stern's assumption of expected utility maximization (EU theory) and then with Kahneman and Tversky's prospect theory. At the end of Chapter 4 we will compare the insurance demand function D1(a) constructed under the assumption of economic survival probability maximization with the demand functions constructed using EU theory and prospect theory.

- [1] See footnote 26 in section 2.3 for more on the issue of agents being forced into taking risks by their situation.