2.1.2.1 Adverse Selection Equilibrium

Figure 2 provides a graphical representation of the adverse selection insurance equilibrium for the “textbook case” we have just outlined. The relative price (or cost) of contract is on the vertical axis. Quantity (i.e., share of individuals in the market with contract ) is on the horizontal axis; the maximum possible quantity is denoted by . The demand curve denotes the relative demand for contract . Likewise, the average cost () curve and marginal cost () curve denote the average and marginal incremental costs to the insurer from coverage with contract relative to contract .

Because agents can only choose whether to purchase the contract or not, the market demand curve simply reflects the cumulative distribution of individuals’ willingness to pay for the contract. The difference between willingness to pay and is the risk premium, and is positive for risk averse individuals.

Because of the “textbook” assumption that individuals are homogeneous in all features of their utility function—i.e., includes only characteristics relating to one’s expected claims —willingness to pay for insurance is increasing in risk type. This is the key feature of adverse selection: individuals who have the highest willingness to pay for insurance are those who, on average, have the highest expected costs. This is represented in Figure 2 by drawing a downward sloping curve. That is, marginal cost is increasing in price and decreasing in quantity. As the price falls, the marginal individuals who select contract have lower expected cost than infra-marginal individuals, leading to lower average costs.

The link between the demand and cost curve is arguably the most important distinction of insurance markets (or selection markets more generally) from traditional product markets. The shape of the cost curve is driven by the demand-side consumer selection. In most other contexts, the demand curve and the cost curve are independent objects; demand is determined by preferences and costs by the production technology. The distinguishing feature of selection markets is that the demand and cost curves are tightly linked since the individual’s risk type not only affects demand but also directly determines cost.

As noted, the efficient allocation is to insure all individuals whose willingness to pay is at least as great as their expected cost of insuring them. In the textbook case, the risk premium is always positive, since by assumption all individuals are risk averse and there are no other market frictions. As a result, the demand curve is always above the curve and, as shown in Figure 2, it is therefore efficient for all individuals to be insured . The welfare loss from not insuring a given individual is simply the risk premium of that individual, or the vertical difference between the demand and curves.

The essence of the private information problem is that firms cannot charge individuals based on their (privately known) marginal cost, but are instead restricted to charging a uniform price, which in equilibrium implies average cost pricing. Since average costs are always higher than marginal costs, adverse selection creates underinsurance, a familiar result first pointed out by Akerlof (1970). This underinsurance is illustrated in Figure 2. The equilibrium share of individuals who buy contract is (where the curve intersects the demand curve), while the efficient number is ; in general, the efficient allocation is determined where the curve intersects the demand curve, which in the textbook case is never (unless there are people with risk probability of zero or who are risk neutral). The fundamental inefficiency created by adverse selection arises because the efficient allocation is determined by the relationship between marginal cost and demand, but the equilibrium allocation is determined by the relationship between average cost and demand.

The welfare loss due to adverse selection arises from the lost consumer surplus (the risk premium) of those individuals who remain inefficiently uninsured in the competitive equilibrium. In Figure 2, these are the individuals whose willingness to pay is less than the average cost of the insured population, Integrating over all these individuals’ risk premia, the welfare loss from adverse selection is given by the area of the “dead-weight loss” trapezoid CDEF.

The amount of underinsurance generated by adverse selection, and its associated welfare loss, can vary greatly in this environment. As illustrated graphically in Einav and Finkelstein (2011), the efficient allocation can be achieved despite a downward sloping marginal cost curve if average costs always lie below demand. In contrast, if average costs always lie above demand, the private market will unravel completely, with no insurance in equilibrium.

2.1.2.2 Public Policy in the Textbook Case

One can use the graphical framework in Figure 2 to evaluate the welfare consequences of common public policy interventions in insurance markets that alter the insurance allocation. The comparative advantage of the public sector over the private sector is that it can directly manipulate either the equilibrium quantity of insurance (through mandates) or the equilibrium price of insurance (through either tax/subsidy policy or regulation of insurance company pricing). We briefly discuss each in turn.

Mandates. The canonical solution to the inefficiency created by adverse selection is to mandate that everyone purchase insurance, a solution emphasized as early as Akerlof (1970). In the textbook setting, mandates produce the efficient outcome in which everyone has insurance. However, the magnitude of the welfare benefit produced by an insurance purchase requirement varies depending on the specifics of the market since, as noted, the amount of underinsurance produced by adverse selection in equilibrium can itself vary greatly.

Tax subsidies. Another commonly discussed policy remedy for adverse selection is to subsidize insurance coverage. Indeed, adverse selection in private health insurance markets is often cited as an economic rationale for the tax subsidy to employer provided health insurance, which is the single largest federal tax expenditure. We can again use Figure 2 to illustrate. Consider, for example, a subsidy toward the price of coverage. This would shift demand out, leading to a higher equilibrium quantity and less underinsurance. The gross welfare loss would still be associated with the area between the original (pre-subsidy) demand curve and the curve, and would therefore unambiguously decline with any positive subsidy. A large enough subsidy (greater than the line segment in Figure 2) would lead to the efficient outcome, with everybody insured.

Of course, the net welfare gain from public insurance subsidies will be lower than the gross welfare gain due to the marginal cost of the public funds that must be raised to finance the subsidy; this may be quite large since the subsidy must be paid on all the infra-marginal consumers as well as the marginal ones. Given a non-zero deadweight cost of public funds, the welfare maximizing subsidy would not attempt to achieve the efficient allocation. It is possible that the welfare maximizing subsidy could be zero. That is, starting from the competitive allocation (point C), a marginal dollar of subsidy may not be welfare enhancing. Although given the equilibrium distortion the welfare gain will be first order, the welfare cost of raising funds to cover the subsidy is first order as well. Hence, the benefits of subsidies are again an empirical question.

Restrictions on characteristic-based pricing. A final common form of public policy intervention is regulation that imposes restrictions on the characteristics of consumers over which firms can price discriminate. Some regulations require “community rates” that are uniform across all individuals, while others prohibit insurance companies from making prices contingent on certain observable risk factors, such as race or gender. For concreteness, consider the case of a regulation that prohibits pricing on the basis of gender. Recall that Figure 2 can be interpreted as applying to a group of individuals who must be given the same price by the insurance company. When pricing based on gender is prohibited, males and females are pooled into the same market, with a variant of Figure 2 describing that market. When pricing on gender is allowed, there are now two distinct insurance markets—described by two distinct versions of Figure 2—one for women and one for men, each of which can be analyzed separately. A central issue for welfare analysis is whether, when insurance companies are allowed to price on gender, consumers still have residual private information about their expected costs. If they do not, then the insurance market within each gender-specific segment of the market will exhibit a constant (flat) curve, and the equilibrium in each market will be efficient. In this case, policies that restrict pricing on gender unambiguously reduce welfare because they create adverse selection where none existed before. However, in the more likely case that individuals have some residual private information about their risk that is not captured by their gender, each gender-specific market segment would look qualitatively the same as Figure 2 (with downward sloping and curves). In such cases, the welfare implications of restricting pricing on gender could go in either direction. Depending on the shape and position of the gender-specific demand and cost curves relative to the gender-pooled ones, the sum of the areas of the deadweight loss trapezoids in the gender-specific markets could be larger or smaller than the area of the single deadweight loss trapezoid in the gender-pooled market.³ See Einav and Finkelstein (2011) for a numerical illustrative example.

Comment: Pareto improvements. It is important to note that while various policies may be able to increase efficiency or even produce the efficient outcome—such as mandates—they are not, in this environment, Pareto improving. Consider for concreteness the case of mandates. The insurance provider (be it the government or the private market) must break even in equilibrium, and therefore the cost of providing the insurance must be recouped. The total cost is equal to the market size times the average cost of insurance provision to individuals, which is given by point G. Suppose the government uses average cost pricing, effectively issuing a lump sum tax on individuals equal to the average cost of insuring all individuals (given by the vertical distance at point G). While this policy achieves the efficient allocation, those whose willingness to pay is less than the price level at point G are made strictly worse off. Other financing mechanisms may generate welfare gains for a larger set of individuals, but assuming that the government does not observe the private information about individuals’ costs, the government—like the private sector—cannot price insurance to individuals based on their (privately known) marginal cost.

The inability for mandates to produce a Pareto improvement are a direct consequence of the Akerlovian modeling framework which has fixed the contract space. Some models that endogenize the contract offers generate Pareto improving mandates (e.g., Wilson, 1977) or Pareto improving tax-transfer schemes (Rothschild & Stiglitz, 1976). Crocker and Snow (1985)discuss the assumptions under which the decentralized equilibrium is constrained Pareto efficient in models with endogenous contracts.⁴

2.1.3.1 Production Costs (Loads)

Consider first the supply-side assumption we made above that the only costs of providing insurance to an individual are the direct insurer claims that are paid out. Many insurance markets show evidence of non-trivial loading factors, including long-term care insurance (Brown & Finkelstein, 2007), annuity markets (Friedman & Warshawsky, 1990; Mitchell, Poterba, Warshawsky, & Brown, 1999; Finkelstein & Poterba, 2002), health insurance (Newhouse, 2002), and automobile insurance (Chiappori, Jullien, Salanié, & Salanié, 2006). While these papers lack the data to distinguish between loading factors arising from administrative costs to the insurance company and those arising from market power (insurance company profits), it seems a reasonable assumption that it is not costless to “produce” insurance and run an insurance company.

We therefore relax the textbook assumption to allow for a loading factor on insurance, for example in the form of administrative costs associated with selling and servicing insurance. In the presence of such loads, it is not necessarily efficient to allocate insurance coverage to all individuals. Even if all individuals are risk averse, the additional cost of providing an individual with insurance may be greater than the risk premium for certain individuals, making it socially efficient to leave such individuals uninsured. This case is illustrated in Figure 3, which is similar to Figure 2, except that the cost curves are shifted upward reflecting the additional cost of insurance provision.

In Figure 3, the curve crosses the demand curve at point E, which depicts the socially efficient insurance allocation. It is efficient to insure everyone to the left of point E (since demand exceeds marginal cost), but socially inefficient to insure anyone to the right of point E (since demand is less than marginal cost).

2.1.3.2 Implications for Policy Analysis

The introduction of loads does not affect the basic analysis of adverse selection but it does have important implications for standard public policy remedies. The competitive equilibrium is still determined by the zero profit condition, or the intersection of the demand curve and the curve (point C in Figure 3), and in the presence of adverse selection (downward sloping curve) this leads to underinsurance relative to the social optimum , and to a familiar deadweight loss triangle CDE.

However, with insurance loads, the qualitative result in the textbook environment of an unambiguous welfare gain from mandatory coverage no longer obtains. As Figure 3 shows, while a mandate that everyone be insured recoups the welfare loss associated with underinsurance (triangle CDE), it also leads to overinsurance by covering individuals whom it is socially inefficient to insure (that is, whose expected costs are above their willingness to pay). This latter effect leads to a welfare loss given by the area EGH in Figure 3. Therefore whether a mandate improves welfare over the competitive allocation depends on the relative sizes of triangles CDE and EGH. These areas in turn depend on the specific market’s demand and cost curves, making the welfare gain of a mandate an empirical question. It may also depend on factors outside of our model—such as the administrative costs of (publicly provided) mandatory insurance relative to private sector competition. Naturally, if government-mandated or provided insurance has lower loads—e.g., because of less spending on marketing—then the welfare gains of a mandate could be larger.

2.1.3.3 Preference Heterogeneity and Advantageous Selection

Our “textbook environment”—like the original seminal papers of Akerlof (1970) and Rothschild and Stiglitz (1976)—assumed that individuals varied only in their risk type. In practice, however, consumers of course may also vary in their preferences. Thus the vector of consumer characteristics that affects both willingness to pay and expected costs may include consumer preferences as well as risk factors.

Recent empirical work has documented not only the existence of substantial preference heterogeneity over various types of insurance, but the substantively important role of this preference heterogeneity in determining demand. Standard expected utility theory suggests that risk aversion will be important for insurance demand. And indeed, recent empirical evidence suggests that heterogeneity in risk aversion may be as or more important than heterogeneity in risk type in explaining patterns of insurance demand in automobile insurance (Cohen & Einav, 2007) and in long-term care insurance (Finkelstein & McGarry, 2006). In other markets, there is evidence of a role for other types of preferences. For example, in the Medigap market, heterogeneity in cognitive ability appears to be an important determinant of insurance demand (Fang, Keane, & Silverman, 2008); in choosing annuity contracts, preferences for having wealth after death play an important role (Einav, Finkelstein, & Schrimpf, 2010c); in annual health insurance markets, heterogeneity in switching costs can also play an important role in contract demand (Handel, 2011).

Such heterogeneity in preferences can have very important implications for analysis of selection markets. In particular, if preferences are sufficiently important determinants of demand for insurance and sufficiently negatively correlated with risk type, the market can exhibit what has come to be called “advantageous selection.”

2.1.3.4 Equilibrium and Public Policy with Advantageous Selection

In our graphical framework, advantageous selection can be characterized by an upward sloping marginal cost curve, as shown in Figure 4. This is in contrast to adverse selection, which is defined by a downward sloping marginal cost curve.⁵ When selection is advantageous, as price is lowered and more individuals opt into the market, the marginal individual opting in has higher expected cost than infra-marginal individuals. Note that preference heterogeneity is essential for generating these upward sloping cost curves. Without it, willingness to pay must be higher for higher expected cost individuals. Marginal costs must be upward sloping because the individuals with the highest willingness to pay are highest cost.⁶

Since the curve is upward sloping, the curve lies everywhere below it. If there were no insurance loads (as in the textbook situation), advantageous selection would not lead to any inefficiency; the and curves would always lie below the demand curve, and in equilibrium all individuals in the market would be covered, which would be efficient.

With insurance loads, however, advantageous selection generates the mirror image of the adverse selection case; it also leads to inefficiency, but this time due to overinsurance rather than underinsurance. This can be seen in Figure 4. The efficient allocation calls for providing insurance to all individuals whose expected cost is lower than their willingness to pay—that is, all those who are to the left of point E (where the curve intersects the demand curve) in Figure 4. Competitive equilibrium, as before, is determined by the intersection of the curve and the demand curve (point C in Figure 4). But since the curve now lies below the curve, equilibrium implies that too many individuals are provided insurance, leading to overinsurance: there are individuals who are inefficiently provided insurance in equilibrium. These individuals value the insurance at less than their expected costs, but competitive forces make firms reduce the price in order to attract these individuals, simultaneously attracting more profitable infra-marginal individuals. Intuitively, insurance providers have an additional incentive to reduce price, as the infra-marginal customers whom they acquire as a result are relatively good risks. As we discuss below, such advantageous selection is quite important empirically. Cutler, Finkelstein, and McGarry (2008) summarize some of the findings regarding the presence of adverse compared to advantageous selection in different insurance markets.

We can characterize the welfare loss from overinsurance due to advantageous selection as above. The resultant welfare loss is given by the shaded area CDE, and represents the excess of over willingness to pay for individuals whose willingness to pay exceeds the average costs of the insured population. Once again, the source of market inefficiency is that consumers vary in their marginal cost, but firms are restricted to uniform pricing.

From a public policy perspective, advantageous selection calls for the opposite solutions relative to the tools used to combat adverse selection. For example, given that advantageous selection produces “too much” insurance relative to the efficient outcome, public policies that tax existing insurance policies (and therefore raise toward ) or outlaw insurance coverage (mandate no coverage) could be welfare improving. Although there are certainly taxes levied on insurance policies, to our knowledge advantageous selection has not yet been invoked as a rationale in public policy discourse, perhaps reflecting the relative newness of both the theoretical work and empirical evidence. To our knowledge, advantageous selection was first discussed by Hemenway (1990), who termed it “propitious” selection. de Meza and Webb (2001) provide a theoretical treatment of advantageous selection and its implications for insurance coverage and public policy.

Advantageous selection provides a nice example of the interplay between theory and empirical work in the selection literature. Motivated by the seminal theoretical papers on adverse selection, empirical researchers set about developing ways to test whether or not adverse selection exists. Some of this empirical work in turn turned up examples of advantageous selection, which the original theory had precluded. This in turn suggested the need for important extensions to the theory.

2.2.1.1 Positive Correlation Test for Asymmetric Information

The graphical depiction of adverse selection in Figures 2 and 3 suggests one natural way to test for selection: compare the expected cost of those with insurance to the expected cost of those without. More generally, one can compare the costs of those with more insurance to those with less insurance. If adverse selection is present, the expected costs of those who select more insurance should be larger than the expected costs of those who select less insurance.

Figure 5 illustrates the basic intuition behind the test. Here we start with the adverse selection situation already depicted in Figure 3, denoting the curve shown in previous figures by to reflect the fact that it averages over those individuals with the higher coverage contract, . We have also added one more line: the curve. The curve represents the average expected cost of those individuals who have the lower coverage contract . That is, the curve is derived by averaging over the expected costs of those with coverage (integrating from to a given quantity ) while the curve is produced by averaging over the expected costs of those with coverage (integrating from the given quantity to ).

A downward sloping curve—i.e., the existence of adverse selection—implies that is always above . Thus, at any given insurance price, and in particular at the equilibrium price, adverse selection implies that the average cost of individuals with more insurance is higher than the average cost of those with less insurance. The difference in these averages is given by line segment CF in Figure 5 (the thick arrowed line in the figure). This basic insight underlies the widely used “positive correlation” test for asymmetric information. The positive correlation test amounts to testing if point C (average costs of those who in equilibrium are insured) is significantly above point F (average costs of those who in equilibrium are not insured).

Chiappori and Salanie (2000) formalized this intuition and emphasized that the basic approach requires some refinement because it does not clearly differentiate between individual characteristics that are observable and those that are not.⁷ In particular, one must stratify on the consumer characteristics that determine the contract menu offered to each individual. Implementing the positive correlation test requires that we examine whether, among a set of individuals who are offered the same coverage options at identical prices, those who buy more insurance have higher expected costs than those who do not. In the absence of such conditioning, it is impossible to know whether a correlation arises due to demand (different individuals self select into different contracts) or supply (different individuals are offered the contracts at different prices by the insurance company). Only the former is evidence of selection. As a result, some of the most convincing tests are those carried out using insurance company data, where the researcher knows the full set of characteristics that the insurance company uses for pricing. Absent data on individually customized prices, which are sometimes difficult to obtain, one may instead try to flexibly control for all individual characteristics that affect pricing.

Chiappori and Salanie’s work has led to a large literature studying how average costs vary across different coverage options in several insurance markets, including health, life, automobile, and homeowners insurance. The widespread application of the test in part reflects its relatively minimal data requirements. The test requires that one observe the average expected costs of individuals (who are observationally identical to the firm) with different amounts of insurance coverage.

A central limitation in interpreting the results of the positive correlation test is that it is a joint test of the presence of either adverse selection or moral hazard. Even in the absence of selection (i.e., a flat marginal cost curve), moral hazard (loosely, an impact of the insurance contract on expected claims) can produce the same “positive correlation” property of those with more insurance having higher claims than those without. Intuitively, individuals with more generous insurance coverage may choose to utilize more services simply because their marginal out-of-pocket cost is lower. These two very different forms of asymmetric information have very different public policy implications. In particular, in contrast to the selection case where government intervention could potentially raise welfare, the social planner generally has no comparative advantage over the private sector in ameliorating moral hazard (i.e., in encouraging individuals to choose socially optimal behavior). Thus distinguishing between adverse selection and moral hazard is crucial.⁸

2.2.1.2 Cost Curve Test of Selection

Faced with the challenge of how to interpret the results of the correlation test, researchers have taken a variety of approaches. One is to test for selection in insurance markets where moral hazard is arguably less of a concern, such as annuity markets. More generally, researchers have used experimental or quasi-experimental variation in prices that consumers face to try to separate selection from moral hazard (see e.g., Abbring, Chiappori, & Pinquet, 2003a; Abbring, Heckman, Chiappori, & Pinquet, 2003b; Adams, Einav, & Levin, 2009; Cutler & Reber, 1998; Einav et al. (2010a); Karlan & Zinman, 2009).

The intuition for how pricing variation that is exogenous to demand (and hence by definition to costs since demand depends on expected costs) allows one to separate selection from moral hazard is easily seen in our simple graphical framework. Consider an experiment that randomly varies the relative price at which the contract is offered (relative to the contract) to large pools of otherwise identical individuals. For each relative price, we observe the fraction of individuals who bought contract and the average realized costs of the individuals who bought contract .⁹ We thus can trace out the demand curve as well as the average cost curve in Figure 3. From these two curves, the marginal cost curve is easily derived. Total costs are the product of average costs and demand (quantity), and marginal costs are the derivative of total cost with respect to quantity. The features of the marginal cost curve then provide direct evidence on selection. Specifically, rejecting the null hypothesis of a constant marginal cost curve is equivalent to rejecting the null of no selection. Moreover, the sign of the slope of the estimated marginal cost curve informs us of the nature of any selection. A downward sloping marginal cost curve (i.e., a cost curve declining in quantity and increasing in price) indicates adverse selection, while an upward sloping curve indicates advantageous selection. Einav et al. (2010a) develop and discuss in more detail this “cost curve” test of selection.

Crucially, this “cost curve” test of selection is unaffected by moral hazard. Conceptually, variation in prices for a fixed contract allows us to distinguish selection from moral hazard. To see this, recall that the curve is estimated using the sample of individuals who choose to buy contract at a given price. As prices change, the sample changes, but everyone always has the same coverage. Because the coverage of individuals in the sample is fixed, the estimate of the slope of the cost curve is not affected by moral hazard, which only affects costs when coverage changes.¹⁰

2.2.2.1 Annuities

In return for an up-front lump sum premium, annuities provide an individual with a survival-contingent income stream. They therefore offer a way for a retiree facing stochastic mortality to increase welfare by spreading an accumulated stock of resources over a retirement period of uncertain length (Davidoff, Brown, & Diamond, 2005; Yaari, 1965). Yet private annuity markets remain quite small. As a result, annuities have attracted a great deal of interest in discussions involving the design and reform of public pensions. Many of these public pension systems, including the current US Social Security System, provide benefits in the form of mandatory, publicly provided annuities. A major economic rationale for this form of benefit provision is the potential for adverse selection to undermine the functioning of private annuity markets, making it important to determine whether selection actually exists in these markets.

Several studies have implemented variants of the positive correlation test for selection in annuity markets. In the context of annuities, higher risk (i.e., higher expected claim) individuals are the ones who are longer lived than expected; adverse selection therefore is expected to generate a positive correlation between annuitization and survival. Results from a number of studies all point to evidence of a positive correlation in annuity markets on both the extensive margin—individuals who purchase annuities tend to be longer lived than those who do not—and on the intensive (i.e., contract feature) margin—individuals who purchase annuity contracts with shorter gaurantee periods tend to be longer lived than those who purchase less. These findings obtain conditional on the characteristics of individuals used to price annuities, namely age and gender. The positive correlation has been documented in several countries including the United States (Mitchell et al., 1999), the United Kingdom (Finkelstein & Poterba, 2002, 2004, 2006), and in Japan (McCarthy & Mitchell, 2003).

In the case of annuities, it may be reasonable to assume that annuitization does not induce large behavioral effects. Indeed, work in this literature tends to assume that moral hazard—an impact of income in the form of an annuity on the length of life—is likely to be quantitatively negligible even though theoretically possible (see Philipson & Becker, 1998). As a result, evidence of a positive correlation between annuitization and survival can be interpreted as clear evidence of adverse selection in this market (Finkelstein & Poterba, 2004). Finkelstein and Poterba’s (2004) empirical findings also illustrate how selection may occur not only along the dimension of the amount of payment in the event the insured risk occurs, but also in the form of selection on different insurance instruments, such as the length of a guarantee period during which payments are not survival-contingent; see Sheshinski (2008) for a theoretical discussion of this point.