# INSURANCE DEMAND IN THE VON NEUMANN-MORGENSTERN MODEL OF MAXIMIZATION OF THE EXPECTED UTILITY OF INCOME (EU THEORY)

In the von Neumann-Morgenstern model^{[1]} a rational economic decision is one which maximizes the expected utility of the economic gain, not the expected gain itself (i.e. the gain multiplied by the probability of success). An economically rational agent experiences a diminishing marginal utility of money (income), i.e. for *j*>*k* the number of utils (i.e. subjective utility units) he derives from the j-th money unit is smaller than the number of utils he derives from the *k*-th money unit.

As in the previous section, the agent decides whether it is profitable for him to take out insurance at a premium of *a* money units against a loss of *L* money units that will occur with a probability of *p.* The expected loss is E(L) money units. We assume again that the premium *a* is higher than the expected loss: *a* > E(L).

The illustrative example used for the following graphs considers a potential loss of *L =* 250 money units and a loss probability of *p =* 0.2. The expected loss due to theft is therefore E(L) = 50 money units and the premium *a* is alternatively in the range of 50-80 money units. In this illustrative example we assume an income utility function in the for m

From the utility maximization perspective, the relationship between utility and income for insurance is determined by the money (income) utility function. As we are entering the domain of stochastic phenomena here, we naturally need to use cardinal utility in order to be able to quantitatively compare the expected utility of insurance cover with the loss associated with paying the premium. A key assumption of the model is that the income utility function is strictly concave, characterizing risk-averse agents. Given the description of insurance this assumption is certainly more than acceptable.^{[2]}

**Figure 15: The Von Neumann-Morgenstern model of the expected utility of money: the poorer the agent, the greater the loss of utility generated by a loss of 250 money units**

The question is whether an economically rational agent will pay the premium of *a* money units. If he does, he will experience a fall in income of *a* money units. If he does not, his expected income will decrease by E(L) money units. For all agents, expected income is *ceteris paribus* higher in the no-insurance case. If decision-takers were by guided by expected income, no one would take out insurance.

How will the situation change if we work not with expected income, but with the expected utility of income? We will assume a universal (identical for all agents) income utility function *u(d).*

We will assume *d ≥ a.* Otherwise the decision-taker would not even consider insurance.

A loss of *L* money units will give the agent a decrease in utility of a magnitude that depends on his level of income. This can be seen on the graph — in the event of an insurance loss an agent with an income of *d =* 1000 money units loses 69 utils, whereas an agent with an income of 750 money units loses 90 utils. An agent with an income of 500 money units would lose 134 utils.

The loss associated with paying a premium of 60 money units is also different for agents with different income levels: *u* (1000) - *u* (940) = 15 utils, *u* (750) - *u* (690) = 19 utils, and *u* (500) - *u* (440) = 27 utils.

An agent with an income of d = 1000 money units therefore has a utility of *u* (940) = 985 utils when insured and an expected utility of 0.8 • *u* (1000) + 0.2•1/ (750) = 0.8 • 1000 + 0.2 • 930 = 986 utils when uninsured. He will therefore decide not to buy insurance. By contrast, an agent with a lower income of *d =* 500 money units has a utility of *u* (440) = 814.4 utils when insured and an expected utility of 0.8 • *u* (500) + 0.2 • *u* (250) = 0.8 • 841 + 0.2 • 707 = 814.2 utils when uninsured. Consequently, it is profitable for him to be insured. The same thinking will also induce an agent with a low income of *d =* 310 money units to opt for insurance (with a greater incentive than an agent with income of *d =* 500 money units), as he has a utility of *u* (250) = 707 utils when insured and an expected utility of 0.8 • *u* (310) + 0.2 • *u* (60) = 696 utils when uninsured.

In this model of maximization of the expected utility of income it is irrational for the wealthy (from a certain income level upwards) to be insured, whereas for poorer people it is irrational not to be insured.^{[3]}

The threshold case (the threshold income level dx) is the level at which an economically rational agent ceases to insure himself given an increment in his income. The threshold d1 is the root of the equation^{[4]}:

We can interpret function *h(d)* as a measure of the strength of the incentive to insure. For a strictly concave and growing utility function u(d), equation *h(d*) = 0 has a single solution and the income threshold d, is therefore unique. In our illustrative example depicted in Figure 16 the threshold is *d*1 = 522 money units.

So, with a premium of *a =* 60 money units, agents with income *d <* 522 will take out insurance and agents with higher income will not.

For different premiums the willingness to buy insurance and the threshold income level both decrease. Figure 17 illustrates the relation between the income threshold (the income above which agents do not buy insurance) and the premium amount Insurance is bought by agents below the income threshold *d ≤ d*1, for which (d, a) lies within *Q.*

What is the income elasticity of demand in this expected utility maximization model? Demand is given by the premium multiplied by the number of agents below the income threshold *d*1 (which is constant given constant preferences).

**Figure 16: The income threshold in the model of maximization of the expected utility of income: root of equation h(d) = 0**

**Figure 17: The relation between the income threshold in the model of maximization of the expected utility of income and premium amount ****a**

We will denote the relation between insurance demand and income by *D(d).* An increase in income of 1% will cause a decrease in the number of agents below the income threshold, so the income elasticity is negative. Insurance is an inferior good:

Let us now explore the price elasticity of demand

For our example. The graph in Figure 18 below plots the insurance demand function in the case of a uniform income distribution.^{[5]}

**Figure 18: The insurance demand function D**2**(a) in the von Neumann-Morgenstern model of maximization of the expected utility of income**

Figure 19 shows the high elasticity of this demand function over its entire domain. The elasticity at point T = [a; D(a)] is given by the segment ratio TL/TK, which is greater than 1 for the entire demand curve.

**Figure 19: Elasticity of the insurance demand function: segment ratio TL/TK**

In this expected income utility maximization model, therefore, insurance demand is price elastic for all premium levels. The elasticity decreases with premium price.

In von Neumann and Morgenstern's EU model, insurance is simultaneously an inferior good, as it is demanded by low-income agents (hence an increase in income will reduce demand), and a luxury good, as agents react very sensitively to a rise in the price.

These properties of the EU model are realistic for high and medium levels of income: the better off such an agent is, the less attractive to him is insurance at an unfair price (i.e. a price exceeding the expected loss, a condition that always applies in the insurance market, as otherwise insurance companies would loss-making).

The model is unrealistic^{[6]} for agents with very low income *d,* who would be left with virtually no income after paying the premium. In reality such agents will not buy insurance, because paying the premium worsens their economic situation, thereby reducing their subjective satisfaction to a greater extent than the risk of an uninsured loss. A very poor person who inherits a property or a car will not insure it, because he would face economic collapse after paying the premium. He compares the risk of losing the item with the risks associated with experiencing a relatively significant fall in the money he needs to pay for the bare necessities, and he chooses to reject the offer of insurance.

Another problem with the expected utility of income model — as pointed out by economic psychologists^{[7]} — is the imperfect additivity of economic preferences. Contrary to economic rationality, people code their expenditure according to purpose (food, housing, insurance) and do not always sum their gains and losses in these sub-items.^{[8]} Moreover, people tend not to value an economic action in the same way when it is broken down into multiple actions with the same aggregate gain/loss — people are more willing to pay a high price when it is broken down into several smaller amounts than when they have to pay it all in one go ^{[9]} Consequently, the time distribution of payments plays a role in the insurance decision — a customer of an insurance company will more happily accept insurance paid for on an ongoing basis than insurance paid for in a lump sum "upfront". And when paying on an ongoing basis he willingly accepts a higher price than predicted by the von Neumann-Morgenstern expected utility maximization model. Hence, the imperfect additivity of preferences in the Kahneman-Tversky model affects the subjective degree of risk aversion.

Another tricky issue is the ability of agents to assess objectively the loss probability and its potential impacts on their utility. Tversky showed that people frequently displayed intransitivity of preferences when making choices under uncertainty,^{[10]} while Edwards found significant differences between subjective and objective probabilities.^{[11]}

Interesting experiments have been conducted with deferred gratification, i.e. with the inclusion of the time factor in insurance decision-making. A preference for immediate consumption over deferred consumption means a decrease in the weight of future uncertainty and can lead to subjective underweighting of the level of risk. This phenomenon was studied by, among others, Friedman.^{[12]} In his permanent income theory, consumers adjust their savings so as to keep their income constant over their lifetime. Future consumption is discounted at a subjective risk underweighting rate, which Friedman estimated at at least 30% (and substantially higher when inflation is in double figures). This causes risk to be underweighted in the insurance decision as well.

An interesting way to explain certain phenomena in the area of choice under uncertainty that are inconsistent with standard economic models (including the von Neumann-Morgenstern EU model) is asymmetric valuation of personal income. This is described in the next section.

- [1] See Neumann, J. von, Morgenstern, O.: Theory of Games and Economic Behavior. Princeton: Princeton University Press, 1953.
- [2] A risk-averse agent will reject all games with a zero expected payoff (fair games), but he will also reject some games with a positive expected payoff if the payoff is so low that it does not offset the decrease in utility associated with the negatively perceived risk [i.e. with the non-zero variance of random variable u(d)].
- [3] Only, of course, if they can afford to pay the premium of a. We assume they can.
- [4] In our illustrative example the equation is 0.2 • u(d – 250) + 0.8 • u(d) – u(d – 60) = 0.
- [5] With probability density function f(d) constant in the range I ≡ (M, 4M) and zero for d ( I.
- [6] See also Skořepa, M.: Zpochybnění deskriptivnosti teorie očekávaného užitku. WP IES No. 7. Praha: Fakulta sociálních věd UK, 2006, pp. 1–15.
- [7] See, for example, Tversky, A., Kahneman, D.: Judgment under Uncertainty: Heuristics and Biases. Science 185, 4157(1974): 1124–31, and Frank, R. H.: Microeconomics and Behavior. New York: McGraw-Hill, 2006, pp. 259–83.
- [8] Tversky, A., Kahneman, D.: The Framing of Decisions and the Psychology of Choice. Science 211, 4481(1981): 453–58.
- [9] For instance, a lunch with a split price for the main dish and side dish ($8 + $2) is psychologically more acceptable than the same lunch at an aggregate price ($10). In this context, Richard Thaler formulated two pragmatic rules for making commercial offers more attractive: segregate gains (“Don't wrap all your Christmas presents in one box”) and integrate losses (a hot tub seems cheaper if it is bundled into the price of a home than when valued in isolation). See Thaler, R. H.: Mental Accounting and Consumer Choice. Marketing Science 4, 3(1985): 199–214.
- [10] See Tversky, A.: Intransitivity of Preferences. Psychological Review 76, 1(1969): 31–48.
- [11] See Edwards, W.: The Theory of Decision Making. Psychological Bulletin 51, 4(1954): 380–417.
- [12] See Friedman, M.: Windfalls, the “Horizon” and Related Concepts in the Permanent Income Hypothesis. In C. F. Christ et al., Measurement in Economics, Stanford: Stanford University Press, 1963.