11.2 Insurance Market

In the previous section, we saw how an individual makes decisions under uncertainty by evaluating alternatives based on expected utility. We observed that a risk-averse individual is willing to give up part of the expected payoff of their wealth in order to reduce exposure to unfavorable events. In this section, we extend the analysis to a market context: the insurance market. Link to the website of an IVASS project for the promotion of insurance awareness. We will study the functioning of a competitive market in which insurance companies offer policies to individuals who are identical in terms of wealth and risk of loss, but differ in their attitude toward risk. We will analyze the equilibrium between demand and supply and see how it allows for efficient risk coverage, maximizing the surplus generated by the exchange.

Individual Demand for Insurance

Suppose the population consists of many individuals. For simplicity, we assume that initial wealth is the same for everyone and equal to $10000$ euros, and that all face the same risk: each individual, with probability $p$, loses all their wealth, and with probability $1 - p$, suffers no loss. While the probability $p$ is identical for all, what distinguishes individuals is their attitude toward risk. More precisely, each individual has a utility function of the form $u(W) = W^\pi$, where $\pi$ measures their degree of risk preference: individuals with $\pi < 1$ are risk-averse, those with $\pi > 1$ are risk-loving, and those with $\pi = 1$ are risk-neutral.

Now consider an insurance contract that guarantees full reimbursement in case of loss. How much would an individual with a given value of $\pi$ be willing to pay to eliminate the risk? The upper panel of the following figure provides the answer to this question. Since full coverage guarantees the individual a certain amount equal to $10000$ minus the premium paid to the company, it is beneficial to insure if this certain amount is at least equal to the certainty equivalent of the initial risky situation, $CE$. Therefore, the maximum amount the individual is willing to pay, indicated as $WTP$ in the graph, is equal to $10000 - CE$. As shown in the graph, this willingness to pay can be written as the sum of two components:

• the fair premium, that is, the expected value of the loss, which we denote by $FP$;
• the risk premium, that is, the additional amount the individual is willing to pay to eliminate the risk, denoted by $RP$.

In the lower panel of the figure, we plot the willingness to pay for a policy as a function of the parameter $\pi$. Since the risk premium is higher the more risk-averse the individual is (that is, the lower $\pi$), it follows that willingness to pay decreases as $\pi$ increases.

Market Demand

In the lower panel of Figure 11.4 — identical to the left panel of Figure 11.5 below — we plotted the willingness to pay for a policy as a function of the parameter $\pi$ that measures risk preference. Starting from that relationship, in the right panel of Figure 11.5 we construct a new graph that represents the market demand curve for insurance policies — the relationship between the price of a policy and the number of individuals willing to purchase it.

We assume that there are exactly $1000$ risk-averse individuals in the population, meaning individuals with a value of $\pi$ between $0$ and $1$. As shown in the left panel of Figure 11.5, an extremely risk-averse individual ($\pi \approx 0$) has a certainty equivalent close to zero and is thus willing to pay nearly all their wealth, that is, $10000$, for a policy. A weakly risk-averse individual ($\pi \approx 1$) values the policy at just above the fair premium, that is, $10000p$. The demand curve must therefore pass through the point $(0,10000)$, which represents the most risk-averse individual, and the point $(1000,10000p)$, which corresponds to the last individual for whom purchasing the policy is worthwhile.

Assuming, for simplicity, that the demand curve is linear, the only curve compatible with these two conditions is \(P = 10000 - 10 \times (1 - p) \times Q\) The curve is downward-sloping: as the price increases, the number of individuals for whom insurance is worthwhile decreases, because only those who are more risk-averse are willing to pay more.

Supply and Equilibrium

On the supply side, we assume that the probability of loss $p$ is known to insurance companies and that, being able to diversify over a large portfolio of clients, they are risk-neutral. We also assume that the market is competitive: companies therefore offer policies at a price equal to the expected value of the indemnity, earning zero profits. Since each individual has probability $p$ of losing $10000$, the expected value of the indemnity — that is, the firm’s expected marginal cost — is $10000p$. The supply is thus represented by a horizontal line at the fair premium level: $P = 10000p$.

The market equilibrium is socially efficient. Since the price equals the expected cost, each policy sold generates a net welfare gain for the insured and no loss for the insurer. Every transaction that takes place increases the expected utility of risk-averse individuals without worsening the situation of insurance firms. On the other hand, any transaction not carried out (with risk-loving individuals, who value the policy less than the fair premium) would have generated a negative surplus.

In equilibrium, there is therefore an efficient allocation of risk: the only agents who remain exposed to risk are the insurance companies, who are risk-neutral and therefore indifferent, and the risk-loving individuals, for whom insurance is not worthwhile. The market transfers risk from those who do not tolerate it (the risk-averse) to those who do (the companies), ensuring an efficient redistribution. This result serves as a benchmark that we will use in the next chapter to evaluate the effects of asymmetric information.