Abstract
In this chapter we present some of the more significant results in the literature on adverse selection in insurance markets. Sections 10.1 and10.2 introduce the subject and Sect. 10.3 discusses the monopoly model developed by Stiglitz (Rev Econ Stud 44:407–430, 1977) for the case of single-period contracts extended by many authors to the multi-period case. The introduction of multi-period contracts raises many issues that are discussed in detail: time horizon, discounting, commitment of the parties, contract renegotiation, and accidents underreporting. Section10.4 covers the literature on competitive contracts. The analysis is more complicated because insurance companies must take into account competitive pressures when they set incentive contracts. As pointed out by Rothschild and Stiglitz (Q J Econ 90:629–650, 1976), there is not necessarily a Cournot–Nash equilibrium in the presence of adverse selection. However, market equilibrium can be sustained when principals anticipate competitive reactions to their behavior or when they adopt strategies that differ from the pure Nash strategy. Multi-period contracting is discussed. We show that different predictions on the evolution of insurer profits over time can be obtained from different assumptions concerning the sharing of information between insurers about individual’s choice of contracts and accident experience. The roles of commitment and renegotiation between the parties to the contract are important. Section10.5 introduces models that consider moral hazard and adverse selection simultaneously and Sect. 10.6 covers adverse selection when people can choose their risk status. Section10.7 discusses many extensions to the basic models such as risk categorization, multidimensional adverse selection, symmetric imperfect information, reversed or double-sided adverse selection, principals more informed than agents, uberrima fides, and participating contracts.
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Notes
- 1.
- 2.
For a detailed analysis of participation constraints, see Jullien (2000).
- 3.
As in the perfect discrimination case, the monopolist charges a price of insurance to each consumer equal to marginal cost. All potential consumer surplus is collected as monopoly profits, so there is no dead weight loss. This result would not be obtained with a proportional loading.
- 4.
Since individuals of different types have the same degree of risk aversion, at each point in the figure, the absolute value of the slope of the high-risk indifference curve is lower than that of the low-risk individual. For example, at point \({C}^{0},{U}^{{\prime}}(W)(1 - p_{H})/{U}^{{\prime}}(W - D)p_{H} < {U}^{{\prime}}(W)(1 - p_{L})/{U}^{{\prime}}(W - D)p_{L}\). At equilibrium points C H ∗ and C L ∗, the respective slopes (in absolute values) are \(\left (1 - p_{H}\right )/p_{H}\) and \(\left (1 - p_{L}\right )/p_{L}\). This is true because under full insurance, the insured of type i has \(W - p_{i}D - z_{i}^{{\ast}}\) in each state.
- 5.
- 6.
- 7.
Technically the preference structure of the model implies that indifference curves of individuals with different risks cross only once. This single-crossing property has been used often in the sorting literature (Cooper 1984).
- 8.
There is always a separating equilibrium in the monopoly case. However, the good-risk individuals may not have any insurance coverage at the equilibrium. Property 4 in Stiglitz (1977) establishes that C L ∗∗ = { 0, 0} when q H ∕q L exceeds a critical ratio of high- to low-risk individuals where q i is the proportion of individuals i in the economy. The magnitude of the critical ratio is function of the difference in accident probabilities and of the size of the damage. Here, to have \(C_{L}^{{\ast}{\ast}}\neq \{0, 0\}\), we assume that q H ∕q L is below the critical ratio.
- 9.
- 10.
Townsend (1982) discussed multi-period borrowing–lending schemes. However, his mechanism implies a constant transfer in the last period that is incompatible with insurance in the presence of private information.
- 11.
This type of “no-claims discount” strategy was first proposed by Radner (1981) and Rubinstein and Yaari (1983) for the problem of moral hazard (see also Malueg 1986 where the “good faith” strategy is employed). However, because the two problems of information differ significantly, the models are not identical. First the information here does not concern the action of the agent (moral hazard) but the type of risk which he represents (adverse selection). Second, because the action of the insured does not affect the random events, the sequence of damage levels is not controlled by the insured. The damage function depends only on the risk type. Third, in the adverse selection model, the insured cannot change his declaration and therefore cannot depart from his initial risk announcement although he can always cancel his contract. Therefore, the stronger conditions used by Radner (1981) (robust epsilon equilibrium) and Rubinstein and Yaari (1983) (“long proof”) are not needed to obtain the desired results in the presence of adverse selection only. The Law of the Iterated logarithm is sufficient.
- 12.
In fact their formal analysis is with a continuum of risk types.
- 13.
In general, introducing discounting in repeated games reduces the incentives of telling the truth and introduces inefficiency because players do not care for the future as they care for the current period. In other words, with discounting, players become less patient and cooperation becomes more difficult to obtain. See Sabourian (1989) and Abreu et al. (1990) for detailed discussions of the discount factor issues in repeated contracts.
- 14.
Radner’s (1981) contribution does not allow for revisions after the initial trigger. However, revisions were always present in infinite horizon models (Rubinstein and Yaari 1983; Dionne 1983; Radner 1985; Dionne and Lasserre 1985). A trigger strategy without revision consists in offering a premium corresponding to a risk declaration as long as the average loss is less than the reasonable average loss corresponding to the declaration. If that condition is not met, a penalty premium is offered for the remaining number of periods. With revisions, the initial policy can be reinstate.
- 15.
See also Gal and Landsberger (1988) on small sample properties of experience rating insurance contracts in the presence of adverse selection. In their model, all insureds buy the same contracts, and experience is considered in the premium structure only. They show that the monopoly’s expected profits are higher if based on contracts that take advantage of longer experience. Fluet (1999) shows how a result similar to Dionne and Lasserre (1985) can be obtained in a one-period contract with fleet of vehicles.
- 16.
- 17.
However, separating equilibria are possible with discounting because future considerations are less relevant. In a model with commitment and renegotiation, Dionne and:̧def:̧def Doherty (1994) obtain a similar result; when the discount factor is very low a separating equilibrium is always optimal in a two-period framework. Intuitively, low discount factors reduce the efficiency of using intertemporal transfers or rents to increase the optimal insurance coverage of the low-risk individuals by pooling in the first period. See Laffont and Tirole (1993) for a general discussion on the effect of discounting on optimal solutions in procurement when there is no uncertainty. See Dionne and Fluet (2000) for a demonstration that full pooling can be an optimal solution when the discount factor is sufficiently high and when there is no commitment. This result is due to the fact that, under no commitment, the possibilities of rent transfers between the periods are limited.
- 18.
- 19.
A Riley or reactive equilibrium leads to the Rothschild–Stiglitz separating equilibrium regardless of the number of individuals in each class of risk.
- 20.
Multiple equilibria are the rule in two-stage signaling models. However, when such equilibria are studied, the problem is to find at least one that is stable and dominates in terms of welfare. For a more detailed analysis of signaling models see the survey by Kreps (1989). On the notion of sequential equilibrium and on the importance of consistency in beliefs see Kreps and Wilson (1982).
- 21.
Jaynes (1978) and Hellwig (1988) analyze the consequences of relaxing this assumption. Specifically, they specify the conditions under which an equilibrium exists when the sharing of information about customers is treated endogenously as part of the game among firms. They also contend that it is possible to overcome Rothschild–Stiglitz’s existence problem of an equilibrium if insureds cannot buy more than one contract. Finally, Hellwig (1988) maintains that the resulting equilibrium is more akin to the Wilson anticipatory equilibrium than to the competitive Nash equilibrium.
- 22.
This is a consequence of the exclusivity assumption. Because we consider static contracts, observing accidents or claims does not matter. This conclusion will not necessarily be true in dynamic models.
- 23.
Partial coverage is generally interpreted as a monetary deductible. However, in many insurance markets, the insurance coverage is excluded during a probationary period that can be interpreted as a sorting device. Fluet (1992) analyzed the selection of an optimal time deductible in the presence of adverse selection.
- 24.
On the relationship between the coverage obtained by a low-risk individual under monopoly compared to that under the pure Nash competitive equilibrium, see Dahlby (1987). It is shown, for example, that under constant absolute risk aversion, the coverage obtained by a low-risk individual under monopoly is greater than, equal to, or less than that obtained under competition because the monopolist’s expected profit on a policy purchased by low-risk individuals is greater than, equal to, or less than its expected profit on the policy purchased by high-risk individuals.
- 25.
- 26.
See also Fagart (1996a) for another specification of the game. She extends the work of Rothschild and Stiglitz. Her article presents a game where two principals compete for an agent, when the agent has private information. By considering a certain type of uncertainty, competition in markets with asymmetric information does not always imply a loss of efficiency.
- 27.
See Grossman (1979) for an analysis of the Wilson-type equilibrium with reactions of insureds rather than reactions of sellers.
- 28.
For a proof that the equilibrium can never imply subsidization from high-risk individuals to low-risk individuals, see Crocker and Snow (1985).
- 29.
- 30.
On the relationship between risk aversion and the critical proportion of high risks so that the Rothschild–Stiglitz equilibrium is second-best efficient, see Crocker and Snow (2008). Their analysis shows that, when the utility function U becomes more risk averse, the critical value of high risks increases if U exhibits nonincreasing absolute risk aversion.
- 31.
- 32.
In other words, they implicitly assume that the conditions to obtain a Nash separating equilibrium in a single-period contract are sufficient for an equilibrium to exist in their two-period model.
- 33.
The Rothschild and Stiglitz contracts are not necessarily the best policy rival firms can offer. Assuming that outside options are fixed is restrictive. This issue is discussed in the next section.
- 34.
The authors limited their focus to separating solutions.
- 35.
But not on their contract choice.
- 36.
The corresponding expected utility of the low-risk individual who did not have an accident in the first period is strictly greater at equilibrium than that corresponding to the entrant one-period contract.
- 37.
Cross-subsidizations between risk types remain inconsistent with equilibrium, such that problems for equilibrium existence also exist in a multi-period context.
- 38.
They let insurers offer contracts specifying a per-unit premium for a given amount of coverage.
- 39.
On limited commitment and randomized strategies, see also Dionne and Fluet (2000).
- 40.
This concept implies that the set of strategies satisfies sequential rationality given the system of beliefs and that the system of beliefs is obtained from both strategies and observed actions using Bayes’ rule whenever possible.
- 41.
Put differently, q ia (x i ) and q in (x i ) are the probabilities at the beginning of the second period that, among the insureds having chosen the pooling contract in the first period, an insured belongs to the i-risk class if he has suffered a loss or no loss in the first period, respectively.
- 42.
Cromb (1990) considered the effects of different precommitment assumptions between the parties to the contract on the value of accident history. Under fully binding contracts, the terms of the contract depend only on the number of accidents over a certain time horizon, while under other assumptions (partially binding and no binding) the timing of accidents becomes important.
- 43.
When an individual quits a company A and begins a new relationship with a company B, he is considered by the latter as a new customer on the insurance market.
- 44.
- 45.
- 46.
However, it is possible to establish that a competitive insurance market always has an equilibrium, due to the compatibility of cross-subsidization with equilibrium, as opposed to the result in static models. The economic intuition is the following: an additional instrument can serve to make rival offers less attractive. It consists in informed insurers’ offering of unprofitable contracts in the second period. This instrument is possibly used in a case of commitment with renegotiation but cannot be enforced in no-commitment situations.
- 47.
See Cohen and Siegelman (2010) for a survey and a discussion about empirical work on the coverage–risk correlation that the pure asymmetric information model predicts. See also Chiappori and Salanié (2014) for a more recent review of empirical models on asymmetric information and Dionne et al. (2013) for a model that separates moral hazard from adverse selection and learning.
- 48.
Without loss of generality, Fombaron and Milcent (2007) obtain the same conclusion in a model of health insurance in which they introduce a gap between the reservation utilities. This formulation implies that the low risks may be more inclined to buy insurance than the high risks when loss probabilities are symmetric information. This finding suggests that preference heterogeneity may be sufficient in explaining the opposite selection of insurance coverage in various markets.
- 49.
Cutler et al. (2008) present empirical evidence in life insurance and in long-term care insurance in the USA that is consistent with this negative correlation (those who have more insurance are lower risk because they produce more prevention). However, in annuity markets, for example, higher-risk people seem to have more insurance, as the standard theory would predict. See also Finkelstein and McGarry (2006) and Fang et al. (2008).
- 50.
For an overview of regulations and policy statements, see Hoel and Iversen (2002) and Viswanathan et al. (2007). The latter describes four major regulatory schemes for genetic information in many states, from no regulation to the most strict regulatory structure: in the Laissez–Faire approach, insurers have full freedom to request new tests, disclosure of existing tests, and use tests results in underwriting and rating; under the Disclosure Duty approach, individuals have to disclose to insurers the result of existing tests but cannot be required to undergo additional tests, while under the Consent Law approach, consumers are not required to divulge genetic tests results, but if they do, insurers may use this information. Finally, in the Strict Prohibition approach (there is a tendency in most countries to adopt this regulation of information in health insurance policies), insurers cannot request genetic tests and cannot use any genetic information in underwriting and rating.
- 51.
Because insurance companies do not share information about the amount of insurance purchased by their customers in the context of life insurance, price–quantity contracts are not feasible. As a consequence, insurers can only quote a uniform (average) premium for all life insurance contracts. However, contrary to standard insurance setting, consumers can choose the size of loss and this loss is positively dependent on the probability of death. Hence, increasing symmetric information about risk type leads to changes in the demand for life insurance and in the average price quoted by insurers, contrary to standard setting.
- 52.
This case may stretch plausibility slightly because it is difficult to imagine an insurer being able to verify that someone claiming to be uninformed is not really an informed high risk. However, we will present the case for completeness.
- 53.
See also Fagart and Fombaron (2006) for a discussion of the value of information under alternative assumptions about what information is available to insurers in a model with preventive measures.
- 54.
As in Doherty and Thistle, an individual decides whether or not he wishes to obtain the information from testing.
- 55.
More precisely, voluntary health insurance is considered as a supplement to compulsory insurance.
- 56.
- 57.
In contrast, Cohen (2005) does not reject residual asymmetric information with data from Israel.
- 58.
- 59.
Similar conclusions would be obtained if individuals differed in the size of losses (in addition to the difference in risk).
- 60.
In contrast, Smart restricts the entry by a fixed barrier.
- 61.
In this model, one case in which profitable self-selection contracting arises, with patient types only partially covered exerting high effort (while impatient types exert low effort and receive a lower coverage).
- 62.
See Cohen (2012) for a model of asymmetric learning among insurers on insured risk.
- 63.
Even if their approach differs from Smith and Stutzer because the problem is one of cooperative game theory.
- 64.
The conditions under which this separating equilibrium exists are analogous to those under which a separating equilibrium exists in the standard Rothschild–Stiglitz model.
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Dionne, G., Fombaron, N., Doherty, N. (2013). Adverse Selection in Insurance Contracting. In: Dionne, G. (eds) Handbook of Insurance. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0155-1_10
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