English Appendix
Interdependent Risks
How to create allocative efficiency
Institute for Risk and Insurance
University of Hamburg
ANNETTE HOFMANN
Risk management decisions undertaken by individuals or firms often affect the risks faced by others.The management of individual risks can take different forms. An investment in prevention may reduce the probability or the severity of the potential loss. In the literature, the term loss prevention is often used to mean a reduction in the probability of loss without affecting the magnitude of loss, while loss reduction means a reduction in the magnitude of loss without affecting the probability of loss.This distinction was originally introduced by Ehrlich and Becker (1972). Our focus here is on loss prevention, which seems to be more important. Hence, we will discuss individual risk management decisions that may reduce a certain risk, but influence risks of others as well.
THE CONCEPT OF INTERDEPENDENT RISKS
In many cases of individual loss prevention decisions,
the risk of an individual (or firm) depends in some way on
actions of other individuals in the economy. The risks are then called interdependent.The concept of interdependent
risks was first introduced in an economic framework by
Kunreuther and Heal (2003).
An individual risk can result due to direct and indirect sources.The direct risk source lies in an individual´s own activities. It can be reduced or even eliminated by an investment in loss prevention.The indirect risk source is due to the behavior of others, and can in general not be influenced. It seems thus intuitive that only the direct risk can be eliminated by individual loss prevention.
A great variety of problems share this structure of risk interdependencies. For instance, one may think of the network structure of (inter-)national airports.The baggage security measures taken at one airport may influence the security at other airports.The risk that an aircraft of an airline is harmed by some explosive device depends not only on its own security measures, but also on the measures undertaken by other airlines.Therefore, an investment in a safety system can only eliminate the risk of damage resulting from directly boarding passengers.This is the direct risk.The risk of contaminated luggage being transferred from other airlines cannot be excluded.This so called indirect risk depends on the security measures at other airports.
Further examples refer to common catastrophe, health or liability risks.A company´s investment in a sprinkler system decreases the risk of fire on the neighbors´ property. Also, many health risks are imposed by others. One may easily think of smokers or animal livestock diseases. Spoken in economic theory,we are talking about prevention externalities.
The structure of the problem we are discussing is often similar. In many individual decision problems,
- the possible loss cannot be completely avoided by a preventive investment. A residual (indirect) risk remains. This risk depends on the behavior of other individuals or firms in the economy.
- those who invest in loss prevention incur some kind of cost (this can be discomfort, time or money) and, in return, receive some individual benefit through the reduced individual expected loss.
- a part of the benefit is public: this is the reduced indirect risk in the economy from which everybody else benefits. Hence, there is a negative externality associated with not investing in loss prevention.This is the increased risk to others.
A well-known result in public economics suggests that when externalities exist, equilibrium behavior will be inefficient. In a world with interdependent risks, this means the total level of preventive activities in the economy will be «too low» relative to the socially optimal level. As a result, the resulting risk allocation will not be efficient. The challenge is to find a solution to the allocation problem. A possible solution is that individuals facing interdependent risks may be encouraged to individually invest in prevention through economic incentives. An example is insurance which alters economic incentives, and for that reason this allocation problem might be solved via an insurance market. Externalities might be internalized within a compulsory insurance monopoly.
However, compulsory insurance monopolies present an intervention into the freedom of choice of individuals and firms.Therefore, a particular justification should be given that a compulsory insurance monopoly may lead to a higher social level of loss prevention and, in this way, may increase allocative efficiency. The rationale is that prevention externalities lead to a market failure, and this market failure may be corrected by a compulsory insurance monopoly.
Empirical studies find that insurance monopolies tend to be more efficient than private providers (i.e. claims levels are lower), and suggest that a market with a statemonopoly for insurance leads to a remarkably higher level of social loss prevention than private markets (see Ungern-Sternberg, 2003, 2004).Why can insurance monopolies in practice lead to a remarkably higher social level of loss prevention? It is possible to attribute the efficiency of monopolies to their potential to internalize prevention externalities.
The purpose of the present article is to implement the concept of interdependent risks to insurance markets and demonstrate that a monopolistic insurer with imperfect information may induce the socially optimal prevention level by engaging in premium discrimination.
We will explain from a theoretical point of view why the social prevention level in markets with (compulsory) insurance monopolies tends to be substantially higher than in competitive insurance markets.
SOCIALWELFARE
Assume loss prevention is a discrete choice.An
individual can freely decide whether he or she wishes to
invest in prevention or not. If he or she insvests in loss
prevention at some cost we denote by «c», the direct part
of the risk will be eliminated, but he or she will still be
exposed to the indirect risk that results from the actions
of others.We make the reasonable assumption that
individuals differ in their prevention cost «c». This cost may
be anything: discomfort, time or money.
Given externalities (of loss prevention) in this market, social welfare is not at its maximum in the market equilibrium.This is illustrated in figure 2 by c* < c**, which means the market equilibrium c* is smaller than the social optimum c**.
In other words, the proportion of individuals who do not invest in prevention is higher in equilibrium than it should actually be (in the welfare optimum). As a result, the indirect risk in the economy is «too high» from a social planner´s viewpoint, and the risk allocation is inefficient.
This result comes from rational investment decisions of each individual who does not take into account in expected utility maximization the externalities caused by his or her investment. Consider figure 3 which represents the excess of expected utility from investing in prevention over not investing. It can be seen that up to the equilibrium cost c*, it is worthwhile for an agent to invest in prevention (i.e. expected utility with an investment in prevention is indeed higher than without such investment). Hence, the task to internalize externalities reduces simply to shift the excess of expected utility function over all individuals up to the optimum c**.
Summing up, we have seen why the social level of
loss prevention in a situation without insurance is «too
low» from a social welfare point of view. This result is
well-known in public economics: in an economy with
externalities, the equilibrium outcome is generally
inefficient. That means in our context: the individuals
invest «too little» in prevention relative to the socially
efficient level.
Let us now introduce compulsory insurance.A monopolistic insurer may act like a social planner and in this way increase the social prevention level by setting appropriate prevention incentives via insurance premiums.
COMPULSORY INSURANCE
Assume a (risk-neutral) monopolistic insurer with no transaction costs. Insurance is compulsory, hence there is full market participation and coverage is provided to all individuals. Let us also assume that insurance premiums are subject to public regulation in the sense that the expected profits of the monopolist are limited (for example by government) to some constant. Otherwise, of course, the insurer would demand infinitely high premiums given insuance is compulsory.
By concluding insurance contracts, individuals substitute their expected uncertain loss-expenses for the payment of a certain insurance premium.An insurance policy consists of a premium paid by an individual regardless of state and an indemnification payment in case of loss.
Without loss of generality, let us assume that losses are completely reimbursed after an accident occurs, i.e. there is no risk sharing between the insurer and its customers. Thus the indemnification payment in case of loss equals the size of the loss. For the time being, we suppose that the monopolist offers an actuarially fair price, so the premium equals the individual expected loss of a policyholder (i.e. the expected loss with and without an investment in loss prevention, respectively).We therefore begin our analysis, as a point of reference, by assuming zero expected profits of the monopolist.
The individuals can be divided into two groups: «low cost» individuals who tend to invest in loss prevention and «high cost» individuals who don´t.While the insurer cannot, in general, observe individual prevention cost, it may, however, assign fair premiums to the individuals when it has information about the social prevention level and about to which group a policyholder belongs. Suppose that the insurer can neither observe individual prevention cost nor an individual investment in prevention. It has, however, an idea about the distribution of prevention costs in the economy.We make the reasonable assumption that the individuals voluntarily provide evidence of their preventive measures to the insurer in order to get the lower (fair) premium based on their lower risk. Assuming further that risk revelation imposes no additional cost to a policyholder, an individual with ¡ìlow cost¡í and for whom prevention is worthwhile has indeed an incentive to truthfully reveal himself as being a ¡good risk!.
Clearly, given insurance is available at fair premiums, the social prevention level is not optimal. Again, externalities are not internalized in the equilibrium. As a consequence, we obtain a similar result as shown in figure 2. The insurer may now act like a social planner by demanding premiums that do not only depend on the actuarial value of the policy, i.e. the expected loss, but depend also on the ¡ìprevention behavior¡í of the individuals. In this way, the social prevention level would rise and individual expected loss of each individual would decrease. For this reason, risk categorization can increase allocative efficiency.
Suppose the insurer engages in premium discrimination in order to raise the social prevention level. In particular, it may design different contracts for different risk types, relying on the policyholders¢¥ selfcategorization: it may offer a premium rebate ¡©relative to the fair premium¡© for low risk individuals (i.e. those investing in prevention), and/or it may impose a premium loading for high risk individuals (i.e. those not investing in prevention), and let individuals voluntarily decide whether or not to invest in loss prevention.The sequence of the considered game between the insurer and its customers may then be seen as follows. At a first stage, the insurer offers appropriate contracts including a premium loading and/or rebate on fair premiums. At a second stage, the individuals choose a contract and decide simultaneously on the basis of their individual cost whether or not to invest in prevention.
In particular, the monopolist may shift the social prevention level in the economy up to the optimal level by engaging in premium discrimination. For individuals who do not invest in prevention, the insurer may offer a premium E(x) + ð, where ð > 0 denotes a premium penalty (loading) and E(x) denotes expected loss of the insurance policy. For individuals who do invest in prevention, the insurer may offer a premium E(x) - ð, where ð > 0 denotes a premium rebate.
The insurer can achieve the socially optimal prevention level c** in the economy by offering a combination of rebates and penalties on fair premiums, and at the same time make non-negative expected profits.
Hence, the insurer engages in premium discrimination so that the individuals individually place themselves into one of two categories: those who invest in prevention and those who don'Lt.
It can be shown in the same way that the socially optimal prevention level is achievable by imposing a premium loading on fair premiums for high risk individuals (those who do not invest in prevention) and by insuring low risk individuals (those who do) at their fair premium.Theoretically, the optimum may also be induced by offering a premium rebate ƒÂ for low risk individuals and by insuring high risk individuals at their fair premium. However, a rebate is not a plausible solution to the externality problem since it implies negative expected profits for the insurer.
The social optimum can also be induced in a setting where low risk individuals pay a zero premium and high risk individuals pay a high loading. The idea behind this setting is that the insurer may set prices according to causation.Then the áoriginatorsâ (high risk individuals) are made fully responsible for the externalities caused by them. In this special case, the damaged parties (i.e. low risk individuals who would not suffer any loss if not for those high risk individuals) would pay zero premiums.
The previous arguments demonstrate that a monopolistic insurer can shift the social prevention level in the economy up to the optimum and thereby ensure an efficient risk allocation.This outcome can be shown under very general conditions, since constraining the monopoly profit to be equal to some constant rather than zero has no qualitative impact on the results. Mathematical proof is given in Hofmann (2003).
We can further derive a theoretical explanation of empirical findings. As mentioned above, the social prevention level in markets with (compulsory) insurance monopolies tends to be substantially higher (and therefore the claims levels tend to be lower) than in competitive insurance markets. An inefficient outcome may result because in competitive insurance markets with interdependent risks, no allocative efficiency seems to be attainable, whether insurance is compulsory or not. The rationale for the latter is as follows. Suppose a market with many risk neutral insurers being in competition to attract customers, and suppose insurers act so as to maximize expected profits. The only policies that will survive in the market are those that yield zero expected profits to insurers and, given this constraint, the highest possible expected utility to individuals. Arbitrarily risk-averse individuals will buy full insurance coverage if the price of insurance is actuarially fair. As a consequence, full insurance at fair premiums results and the competitive equilibrium corresponds to c*.
However, we have shown that in an environment involving externalities of loss prevention, social welfare is not optimal at c*. Due to the possibility of adverse selection in this market, an insurer acting in a competitive market environment cannot induce high risk individuals to pay a premium loading, i.e. other insurers may undercut the demanded price by ignoring externalities.
The only premium for which there is no incentive to undercut a rival firm is the fair premium. As a result, externalities cannot be internalized in a competitive insurance market.
A further intuitive argument is that in a competitive insurance market, when an insurer undertakes actions that reduce the riskiness of its customers (i.e. engaging in premium discrimination), it will run the risk of competing insurers free-riding on its achievements in reducing the risks.Therefore, an insurer acting in a competitive insurance market will have little or no incentive (compared to the monopolist) to engage in premium discrimination, given that competitors may enjoy the benefits at no cost. Hence, there seems to be a strong argument for a monopolistic insurer in an insurance market with interdependent risks.
CONTINUOUS PREVENTION
Loss prevention has been considered as a discrete choice, i.e. an individual can either invest in loss prevention or not. An extension to a world where loss prevention is a continuous variable would change the results. If loss prevention is continuous, individuals who invest in prevention –those who did not impose any externality on others before– will then impose some externality on others, too. This results since the probability of a direct risk cannot (always) be reduced to zero. In the discrete case, some individuals –those with prevention costs higher than the equilibrium but lower than the social optimum– did not behave efficiently, while the others did. In a world with continuous loss prevention, however, all individuals would undertake less than optimal prevention since they would not take into account the social benefit of their individual prevention effort. Therefore, the equilibrium prevention level would be inefficient, as well.When the insurer has full information, it might set premiums according to individual efforts, and might in this way induce the social optimum.
LIABILITY LAW
An alternative instrument to internalize externalities of loss prevention is liability law. Liability requirements tend to improve the individuals´ incentives to reduce risks. An optimal liability rule sets incentives for efficient loss prevention (called care). Common liability rules in economic theory of liability law are «strict liability» and the «negligence rule». Under strict liability, injurers must pay for all losses that they cause, regardless of their level of taking care. Under the negligence rule, an injurer is held liable for losses he or she causes only if he or she was negligent, i.e. only if his or her level of care is less than a level specified by courts, the «level of due care». Generally, when the probability of loss depends on preventive efforts of both injurer and victim, the negligence rule is optimal when the level of due care is set equal to the socially efficient level. In contrast, in this case, strict liability is not optimal since there is no incentive for the victim to behave efficiently.
In our setting above, however, only the injurer was able to reduce the probability of loss (i.e. we analyzed the case of unilateral losses).Therefore, the socially efficient level of care may be induced via strict liability, as well, since a potential injurer would in any case internalize the entire liability risk. As a result, strict liability and the negligence rule seem to be equivalent.
Nevertheless, liability law may not work any more in order to internalize externalities of loss prevention when the injurer cannot be identified. In a setting with interdependent risks, loss prevention generally spreads and the actual injurer may not always be found. As a consequence, liability law is indeed an alternative instrument in order to internalize externalities of loss prevention. But under more general conditions involving information problems, there might be situations where it seems more reasonable to internalize prevention externalities ex ante (before a loss actually takes place) via insurance premiums, and not ex post (after a loss has occurred) via liability law where the injurer has to be identified. A monopoly insurer may overcome such information problems.
INSURANCE REGULATION
In a competitive insurance market, insurance regulation may solve the externality problem, as well.A regulator may offer a tax cut on insurance contracts for low risks and impose higher tax rates for high risks. Regulation would then provide the same incentives in the insurance market as an insurance monopoly engaging in premium discrimination. However, an insurer will generally be better informed about prevention costs, risk aversion and loss probabilities of its policyholders than a regulator.Therefore, it might be socially desirable to implement a monopoly insurer to solve this task by regulating only the insurer´s expected profit.
CREATING EFFICIENCY
Finally, the efficiency gain of price discrimination by the insurer should be weighted against disadvantages that are generally associated with monopolies. Since a monopolist is not forced by competitive threat to operate at lowest cost, it may not be as productively efficient as an insurer in a competitive insurance market. Furthermore, it may not have the same innovation incentive and therefore may not want to reduce costs in the same way. Given that innovation seems to play an important role in some lines of insurance only, these shortcomings seem not so important in insurance markets when compared to other markets.
The efficiency will always be improved in the sense of Kaldor-Hicks efficiency, but not necessarily in the sense of Pareto efficiency.Thus, a more efficient outcome might in fact leave some individuals worse off.Therefore, losses in equity due to premium discrimination need to be weighted against the social costs of the market failure that it seeks to overcome.
Generally, the potential efficiency gain of the insurance monopoly relative to a competitive insurance market will depend on the magnitude of externalities in the market we consider.Thus, the more important externalities in the market are, the more an insurance monopoly will make sense.
CONCLUSIONS
In an economy with interdependent risks, the number of individuals who invest in loss prevention is too low relative to the socially efficient number. This is because the individuals ignore marginal external costs or benefits conferred on others.
When risks are interdependent, the social prevention level –both in case of no insurance and with (compulsory) insurance at actuarially fair premiums– is not socially efficient. The indirect risk in the economy is «too high» from a social planner´s viewpoint.
Given interdependent risks in an insurance market, the social prevention level can be improved by a monopolistic insurer engaging in premium discrimination. Premiums are then optimally chosen to reflect social costs or benefits. For instance, an individual who invests in loss prevention may enjoy a premium rebate not only for the reduced individual expected loss, but also for the reduction in expected loss to others.The insurer designs different contracts for different risk types, relying on the individuals´ incentives.
REFERENCES
EHRLICH, I./BECKER, G. S. (1972). Market Insurance, Self-Insurance, and Self-Protection, Journal of Political Economy, 80, 623-648.
HOFMANN,A. (2005). Internalizing Externalities of Loss Prevention through Insurance Monopoly: An Analysis of Interdependent Consumer Risks,Working Paper, University of Hamburg.
KUNREUTHER, H. C./HEAL, G. (2003). Interdependent Security, Journal of Risk and Uncertainty (Special Issue on the Risks of Terrorism), 26, 231-249.
UNGERN-STERNBERG,T. v. (2003). State Intervention on the Market for Natural Damage Insurance in Europe, CESifo Working Paper 1067.
UNGERN-STERNBERG,T. v. (2004). Efficient Monopolies - The Limits of Competition in the European Property Insurance Market. USA: New York.