Multidimensional Screening in a Monopolistic Insurance Market
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Abstract
We consider a population of individuals who differ in two dimensions, their risk type (expected loss) and their risk aversion, and solve for the profitmaximising menu of contracts that a monopolistic insurer puts out on the market. Our findings are threefold. First, it is never optimal to fully separate all the types. Second, if heterogeneity in risk aversion is sufficiently high, then some highrisk individuals (the risktolerant ones) will obtain lower coverage than some lowrisk individuals (the riskaverse ones). Third, because women tend to be more risk averse than men (in that the risk aversion distribution for women firstorder stochastically dominates that for men), gender discrimination may lead to a Pareto improvement.
Keywords
insurance markets asymmetric information screening gender discrimination positive correlation testIntroduction
Individuals who seek insurance differ from each other in many respects. At least two of these differences are of central importance for insurance companies and for insurance market outcomes: the distribution of losses that insurance takers face, and their willingness to bear the risk of those losses.^{1} Empirically, heterogeneity in the second characteristic is not negligible. For example, Aarbu and Schroyen^{2} find that the degree of relative risk aversion among Norwegians averages around 3.7 with a standard deviation of 2.1.
Insurance market theory has primarily focused on the consequences of private information on the loss distribution, and to a lesser extent on the case in which information on risk aversion is private. The study of situations in which private information applies to both characteristics is much more scant.^{3} Moreover, analysis of the twodimensional private information problem has been restricted to competitive markets; that is, a setting in which several insurers compete for clients. In this paper, we study the opposite setting by asking how a monopolist would design a contract menu intended to attract agents who hold not only private information on their loss distribution, but also on their risk preferences.
Our monopolistic setup encompasses (admittedly, in an extreme way) the presence of market power in insurance markets. Several empirical studies have recently documented the presence of such market failure or of one of its causes: significant search and switching costs for different lines of insurance. Honka^{4} estimates the average search cost in U.S. car insurance to lie between USD45 (online quote) and USD110 (offline quote), and the average switching cost at USD85. She argues that these costs may be an important cause for the high customer retention rates in that industry. In their study of a firm offering automobile insurance in Israel, Cohen and Einav^{5} argue that the firm has market power and that a monopoly insurance model better describes this situation than a competitive one. Also, health insurance markets show symptoms of low competitive pressure. In the Swiss health insurance market, premiums exhibit large variability even within the same canton despite coverage being strictly regulated. Indeed, Lamiraud^{6} reports that in the Geneva canton the difference between the least expensive insurance contract and the most expensive one amounted to 1,919 in 2011, a difference of 39 per cent. For the U.S. health insurance market, Dafny^{7} provides evidence of direct price discrimination of insurees in local markets, and Bates et al.^{8} find evidence that health insurers do exercise their market power to raise premiums.^{9} Although it is true that several firms may be present in a given geographical area, this does not guarantee a competitive outcome. Several reasons for this market failure have been put forward and shown to be consistent with empirical observations. These include promotion expenditures, firstmover advantages, exclusive control over final service provider networks, and—as already mentioned—switching costs. These switching costs, in turn, could be explained by choice overload (be it cognitive or psychological), status quo bias, bundling of basic and supplemental coverage, or lack of transferability of premium bonuses for low claims during a given period.^{10}
Adding risk aversion heterogeneity to the analysis of insurance markets calls for a multidimensional hidden information model. Such an analysis is technically not straightforward because the existence of private information in two or more dimensions implies that the ordering of agents according to their willingness to pay for extra coverage becomes endogenous. In other words, the ordering depends on the contract. To see this, consider two contracts: one with very partial coverage and one with almost full coverage. When offered the former contract, a highly riskaverse agent facing a low risk may be more willing to pay for additional coverage than a risktolerant agent facing a high risk, while the situation could be the other way around for the latter contract. Technically, the indifference curves of these “ intermediate” insurance takers cross twice, and this invalidates standard solution methods.^{11}
The literature on solutions to multidimensional screening problems is not that wide. One branch of this literature is methodological and deals with a principal–agent setting, as we do—see, for example, the “user’s guide” by Armstrong and Rochet.^{12} It turns out, however, that our insurance problem does not lend itself to being solved by the techniques proposed therein, the main reason being that our problem has two hidden characteristics, but only one instrument—the degree of coverage.^{13} A second branch of literature deals with multidimensional screening in insurance markets, but restricts itself to competitive markets. In this literature, it is usually assumed that each insurance company offers a single contract.^{14}^{,}^{15} In a monopolistic setting such as ours, such a restriction would render the analysis somewhat unrealistic, as market power allows the insurer to screen via menus without the fear of undercutting by rivals. By assuming that the monopolist offers a menu of contracts, the relative proportions of the nonintermediate types play a role that is as crucial as the nonsingle crossing of intermediate types’ indifference curves. Hence, the problem of the failure of the singlecrossing condition—brought about by the intermediate types—is compounded in the monopolistic setting by the necessity of dealing with nonintermediate types in the design of the optimal menu of contracts.
Our main objective is to characterise this optimal menu. We establish three results: (i) it is always optimal to pool some of the types (i.e. full separation of types is never optimal); (ii) unlike in the onedimensional case, exclusion of some highrisk individuals from insurance may be optimal; and (iii) some lowrisk individuals may end up with more coverage than some highrisk individuals.
Next, we address two issues that recently have received much attention. The first one is methodological. In testing for the presence of asymmetric information in insurance markets, the question is whether the absence of significant positive correlation between risk and coverage (i.e. the absence of adverse selection) should be taken as indicative of the absence of asymmetric information. Chiappori et al.^{16} (CJSS henceforth) derive the testable prediction that in a sufficiently competitive insurance market with asymmetric information, the observable risk should be related to coverage in a positive monotone way. Notice that this is stronger than requiring a positive correlation between coverage and risk. We show when this result goes through in our monopolistic setting and when it does not. In the latter case, we also show when risk and coverage can be statistically, positively correlated and when they cannot. In this sense, our results corroborate the role of the sufficient competition assumption for the result in CJSS. Our analysis also adds the combination of market power and preference heterogeneity to the list of possible explanations for the lack of evidence supporting the existence of asymmetric information.^{17} Other explanations in the (growing) list are: (i) endogenous heterogeneity in risks because of moral hazard (see, e.g. Cutler et al.^{18}); (ii) endogenous wealth heterogeneity (Netzer and Scheuer^{19}); and (iii) the insurer having privileged information on risks (Villeneuve^{20}).
The second issue concerns the possible welfare consequences of a ban on the use of gender discrimination in insurance. Such a ban took effect in December 2012 in the European Union, extending to the insurance industry the principle of equal treatment of women and men in the access to and the supply of goods and services.^{21} This will affect the insurance sector, because of the common practice of differentiating premiums according to gender when underwriting life, health and car accident risks. Regarding life insurance, it has been argued that if one controls for lifestyle, environmental factors and social class, “the difference in average life expectancy between men and women lies between zero and two years” and therefore that “the practice of insurers to use sex as a determining factor in the evaluation of risk is based on ease of use rather than on real value as a guide to life expectancy” (p. 6).^{22} We show that even if—as the Commission claims—gender does not provide any information on the underlying risk, if it does provide (imperfect) information on an individual’s risk aversion (as empirical research suggests), then allowing the monopolist to condition the terms of the insurance contract on gender may be Pareto improving. We provide sufficient conditions for such an improvement to arise.
From a technical point of view, we have taken a new approach to the analysis of screening insurance takers that simplifies the problem and is appealing from a modelling point of view. Rather than following the standard setup in which the individual faces the possibility of a single monetary loss, we assume that the loss is normally distributed and that agents differ in their expected losses, which can be high or low. If the insurance indemnity is linear in the loss, as is the case under a reimbursement scheme with a constant coinsurance rate, final income will also be normally distributed. Endowing agents with a utility function that displays constant absolute risk aversion, which also can be high or low, means that their preferences over uncertain income prospects can be represented as meanvariance preferences. An important consequence of this approach is that preferences over insurance contracts become quasilinear in the insurance premium and therefore in the information rent. Readers familiar with contract theory will acknowledge the usefulness of linearity in the information rent in specifying the incentive compatibility constraints. An additional advantage of meanvariance preferences is that they allow for an explicit characterisation of the optimal menu of contracts.
The limitations of our approach follow immediately from these assumptions. We do not consider insurance contracts with either a deductible or a cap because such features would destroy the normality of net income. Second, the normality assumption implies a positive likelihood of negative losses, although this problem may be rendered of secondary importance by considering sufficiently high means and/or low variances for the losses. Perhaps the most important objection is that we have no skewness in the loss distribution, and in particular no strictly positive probability mass for a zero loss. Nevertheless, these are minor limitations when compared with the considerable advantages the approach offers for characterising the solution to a twodimensional screening problem.
The remainder of the paper is organised as follows. In the next section, we model the preferences of insurance takers and specify reimbursement contracts. In the subsequent section, we set up the problem faced by a monopolistic insurer. In the section “Onedimensional screening”, we characterise the optimal menu of contracts when insurees only differ in risk levels or risk aversion, as well as considering the case of perfect positive correlation. In the section “Twodimensional screening”, we assume that insurees differ in both respects simultaneously and discuss the five menus that may be optimal. We determine which menu is dominating for which part of the parameter space. In the section “The positive correlation test”, we interpret the testable prediction of CJSS in the light of our results. In the subsequent section, we trace out the consequences of allowing the monopolistic insurer to gender discriminate. The final section concludes.
Except when otherwise stated, we have relegated all proofs to a technical companion paper that is on the Geneva Risk and Insurance Review’s website.
Insurance takers and reimbursement contracts
Insurance takers
We assume that individuals are endowed with initial wealth e and a negative exponential von Neumann–Morgenstern utility function defined on final wealth y: u(y)=−exp(−ry), where r>0 is the (constant) degree of absolute risk aversion. Initial wealth is subject to a random loss z that follows a normal distribution with mean μ and variance σ^{2}.
which ex ante is also normally distributed. We will express a contract C as a pair of a coinsurance rate c and a premium P: C=(c, P).
From now on, we write Open image in new window and assume that this product can be either high or low, and likewise for the expected loss: μ∈{μ_{ L }, μ_{ H }} and ν∈{ν_{ L }, ν_{ H }}, where μ_{ L }<μ_{ H } and ν_{ L }<ν_{ H }. The model can thus be interpreted in two ways: either individuals are equally risk averse but their losses have different variances, or the loss variance is identical but individuals have different degrees of risk aversion. Throughout, we adhere to the second interpretation and will refer to ν as risk aversion.
A person with characteristics (μ_{ i }, ν_{ j }) is said to be of type ij. The share of ij individuals in the population is given by α_{ ij } (i, j=H, L, ∑_{i, j}α_{ ij }=1). We denote by α_{k⋅} the fraction of individuals with expected loss μ_{ k } (α_{k⋅} =α_{ kL } + α_{ kH }); similarly, α_{⋅k} is the fraction of individuals with risk aversion ν_{ k } (α_{⋅k}=α_{ Lk }+α_{ Hk }).
Incentive compatible contracts
Hence, the rent decreases with the coinsurance rate both via the expected loss and via risk aversion (if c>0).
which increases linearly in c.
Thus, by pretending to be type kl, type ij can obtain type kl’s rent plus δ.
Thus, the rent for ij goes down to the extent that ij is mimicking a type with a lower risk or with lower risk aversion. Intuitively, when raising the copayment of a lowrisk (or risktolerant) individual, the decrease in the premium needed to compensate him is not too large, because of the small likelihood of needing that copayment (or because of the low valuation of the increase in the variance of final wealth). However, a person with a higher risk level or greater risk aversion who is tempted by this contract will dislike this change. Thus, increasing a coinsurance rate for some types will lower the rents of all those mimicking (and the mimickers of these mimickers) who have a higher risk, and will increase the rent of all those mimicking (and the mimickers of these mimickers) who have lower risk aversion.
with Open image in new window and where it is assumed that c⩾0 (on which more below).
The ratio D measures, in a unitfree fashion, the difference in risk between two types.^{24} The ratio x measures the degree of similarity along the riskaversion dimension. Using this notation, the locus of tangency points is therefore located at Dx/(1−x), so that for sufficiently small x, the tangency of the intermediate types’ indifference occurs at a coinsurance rate below unity. This makes it possible that both crossings become relevant for the analysis.
The insurance company
Therefore, the isoprofit associated with type ij has slope −μ_{ i } in the contract space (c, P).
With full information, the monopolist will provide ij with full insurance (c_{ ij }=0) at a premium that sets her rent equal to zero. Hence, using (1), P_{ ij }=μ_{ i }+(1/2)ν_{ j }. This yields a per capita payoff equal to π=(1/2)ν_{ j }. The tangency line in Figure 1 thus corresponds to the highest feasible isoprofit line, and the profit that the insurer makes can be read off from the dashed vertical axis on the righthand side. Under full information, the insurer can extract the entire risk premium ν/2. In what follows, we will characterise the optimal coinsurance rates and the optimal rents. The corresponding premiums can then be found with the help of (1).
Given (6), the insurer’s total profit is equal to ∑_{i, j}α_{ ij }π^{ ij }(c_{ ij }, P_{ ij }). From (1) and (6)—both evaluated at (c_{ ij }, P_{ ij })—and recalling that we can write R_{ ij } for R_{ ij }(c_{ ij }, P_{ ij }) (i.e. type ij’s rent when truthful), we can express the insurer’s total profit as ∑_{i, j}α_{ ij }[(1/2)[1−c_{ ij }^{2}]v_{ j }−R_{ ij }]. This objective function is to be maximised with respect to (c_{ ij }, R_{ ij }) (ij=H, L), subject to the usual voluntary participation and incentive compatibility constraints.
As in most of the literature, to these constraints we add two additional sets of constraints that are needed to avoid false claims (see, e.g. Picard^{25}). If a coinsurance rate is negative, the insurer refunds more than 100 per cent of the losses, and the insuree will obviously have a strong incentive to overstate the size of the loss. On the other hand, if a coinsurance rate exceeds unity, the agent will have to be paid to accept such a contract (i.e. a negative premium). Once the agent has accepted the insurance, he would have to pay the insurer as well as bearing the loss once it occurs. It is clear that he would have strong incentives to understate the size of the loss (or even hide the loss altogether). Hence, we constrain coinsurance rates to lie in the interval [0, 1].
The first set of constraints ensures voluntary participation, while the second ensures that all types selfselect. The third set comprises the (reduced form) ex ante and ex post moral hazard constraints.
Theorem 1

At the optimum, (i) c_{ HH }=0 and (ii) R_{ LL }=0.
Before characterising the rest of the solution to the twodimensional screening problem, it is useful to first consider the onedimensional case.
Onedimensional screening
Theorem 2

When all agents have the same risk aversion, the optimal menu has c_{H⋅}=0 and c_{L⋅}=min{Dα_{H⋅}/(1−α_{H⋅}), 1}.
The full insurance contract giving L zero rent would be selected by H as well. At a zero coinsurance rate, the slope of H’s indifference curve is steeper than that of L. If the insurer increases c_{L⋅} above zero, this will create a secondorder reduction in profit from L, but a firstorder gain in profit from H, because the latter can be charged a strictly higher premium (for full insurance). Hence, it pays to start distorting L’s contract. The optimal coinsurance rate balances the gain in profit from H (α_{H⋅}Δμ) with the loss in profits from L ((1−α_{H⋅})ν). Notice that it may pay to exclude type L whenever α_{H⋅}⩾1/(1+D); that is, whenever the proportion of low loss agents is sufficiently small—as expected.
Theorem 3

When all agents face the same expected loss, the optimal menu has c_{⋅H}=0, and Open image in new window
This result is less standard. With only differences in risk aversion, the optimal solution is always at the corner. Either the low type is excluded or he receives full insurance. The reason for this “bangbang” solution is that, unlike in the different risk scenario, at a zero coinsurance rate, both H’s and L’s indifference curves are tangential to one another. Hence, distorting L’s contract by raising the coinsurance rate now results in a secondorder gain in profit from H, and it is the secondorder condition that determines whether c_{⋅L}=0 is a local maximum or minimum.
Theorem 4

When the two characteristics are perfectly positively correlated, the optimal menu has c_{ HH }=0 and Open image in new window
We now turn to the twodimensional screening problem.
Twodimensional screening
From now on, we let individuals not only differ in their risk levels, but also in their risk aversion. The insurance company then faces the following bivariate probability distribution of types:
In what follows, we let ρ represent the numerator of the correlation expression: viz., Open image in new window and refer from now on to ρ as the degree of correlation; it plays a central role in the analysis.
It turns out that D and x are sufficient to describe the problem—we can discard the original parameters μ_{ i } and ν_{ j } (i, j=H, L).^{26}
Lemma 1

Suppose Order 1 applies with c_{ HH }<c_{ LH }. Then it is optimal to pool HL with HH if x>(α_{ HH }/α_{ H⋅ }); otherwise HL should be pooled with LH.
This result follows from Theorem 3. With Order 1, the only type who may want to mimic HL is HH. Thus, the choice of c_{ HL } is only governed by weighing the profits from these two types. Because they have the same risk levels, we can apply Theorem 3 to this subgroup. Given that the fraction of highly riskaverse individuals in this group is α_{ HH }/α_{H⋅}, the result follows. Thus no menu having Order 1 will have full separation of types.
Lemma 2

Suppose that Order 2 applies. Suppose also that (i) HH is indifferent between her own contract and that for LH, but strictly dislikes that for HL; (ii) LH is indifferent between her own contract and that for HL, but strictly dislikes that for LL; and (iii) HL is indifferent between her own contract and that for LL. Then, profit can be increased by pooling HL with either LL or LH.
In this way, it will never turn optimal to exclude type LL and therefore any other type under an Order 1 menu.^{27} We will later comment on how our results change when risk heterogeneity is larger than Open image in new window
The five equilibrium menus^{a}
Menu (order)  c_{ LL }−c_{ LH }  Optimal coinsurance rates (if different from zero)  Range for x 

A (1)  =0  
Bp (1)  =ℓ  
Bf (1)  =ℓ  
C (1)  >ℓ  
E (2)  <ℓ  Open image in new window ^{ b }  
This policy corresponds to one under which individuals differ only in their risk dimensions (Theorem 2). If x⩾1−α_{ LL }>α_{H⋅}, which we shall argue below is the optimal range to make use of A, the pooling of the lowrisk types happens at a “low” coinsurance rate, viz., c_{L·}^{ A }<Dx/(1−x)(=Δμ/Δv).
In menu E, separation of LH from LL is once more weak: c_{ LL }−c_{ LH }<ℓ. Under Order 1, separation of LL from LH is carried out to increase the profits from HH, HL and LH at the cost of a lower profit from LL. Under Order 2, HL is pooled with LL so as to extract more rent from the highly riskaverse types, HH and LH. Across these two types, rent extraction is optimised in the standard way (cf. Theorem 2).
Proposition 1

Suppose that Open image in new window Then, the optimal menu structure as a mapping from Open image in new window into the menu set is as illustrated in Figure 7. For each menu, the optimal coinsurance rates are as given in Table 1.
The formal proof is in the technical companion paper that is on the Review’s website; a sketch of the “proof strategy” is given in the appendix. Here, we restrict ourselves to a heuristic explanation of Proposition 1. Suppose that x equals 1 (no heterogeneity in risk aversion). Then it is optimal to design the menu as if there were only two groups—low and highrisk people—which is exactly what menu A does: the highrisk types obtain full insurance while the lowrisk types face a coinsurance rate as prescribed by Theorem 2. When x falls below 1, ν_{ H } starts to exceed ν_{ L }. This makes it optimal to start screening the LL from the LH types: by providing LL with less coverage (at a lower premium), LH (and therefore also the highrisk types HH and HL) can be charged a higher premium. However, because LL was initially pooled with LH at the lefthand crossing, a marginal increase in c_{ LL } is impossible for incentive compatibility reasons (see Figure 3). What is possible is to move LL from the lefthand crossing to the coinsurance rate corresponding to the righthand crossing, and adjusting her premium to keep her rent at zero. This nonmarginal “reform” involves a big loss in profit from LL and will only be compensated by a larger profit from the “upper” types when x<1−α_{ LL }; then, menu Bp takes over.
As explained earlier, a common feature of all menus of Order 1 is that HL insurees are provided with at least as much coverage as LH insurees. At the same time, the incentive compatibility constraints between these two types require that the order can only be reversed by providing far less coverage to HL than to LH. This is costly, because HL agents, despite their low risk aversion, have a large willingness to pay for being relieved from the high risk they face. It is when risk aversion heterogeneity gets very large that the benefit of reversing the order outweighs this cost: although LH agents face a low risk, their excessive risk aversion (and that of HH) endows them with a large risk premium which the insurer wants to extract. This is when it pays off to reverse the order and substitute menu E for menu C. In the technical companion paper that is on the Review’s website, we show that there exists a function, x_{ CE }(α_{H⋅},α_{ HH }, ρ, D), nonincreasing in D, such that π^{ E }>π^{ C }⇔x<x_{ CE }(α_{H⋅}, α_{ HH }, ρ, D).
Remark 1
 The transition between the menus of Order 1 is continuous in the sense that at least one of the coinsurance rates of the lower types is continuous in x. However, the optimal coinsurance rates show a discontinuity in x when Order 1 is replaced by Order 2. This is illustrated in Figure 9 depicting the optimal values of c_{ LL } and c_{ LH } as a function of x. Thus x_{ CE }(α_{H⋅}, α_{ HH }, ρ, D) is found by direct comparison of the maximal profit under menu C with the maximal profit under menu E.^{28}
Remark 2

Because there is continuity when switching from B to C, whereas there is discontinuity when switching from C to E, one may wonder whether π^{ E } may exceed π^{ C } for any x that makes C dominate B. In other words, does it make more sense to compare π^{ E } with π^{ B }? This is illustrated in Figure 8. The profit function π^{ E } intersects with π^{ C } at Open image in new window while the function Open image in new window dominates π^{ C }, indicating that once x falls short of Open image in new window menu B should be replaced by menu E. In the appendix, we substantiate the following claim:
Claim 1

The subset of distributions (α_{H⋅}, α_{ HH }, ρ) in Open image in new window for which menu C is entirely dominated by menus B and E is almost negligible. In particular, if ρ⩽−0.089, there does not exists a feasible pair (α_{H⋅}, α_{ HH }) where this is the case.
Remark 3

So far, we have restricted the heterogeneity in risk (D) below Open image in new window . Recall that D measures the incentive for μ_{ H }type individuals to mimic μ_{ L }type individuals, normalised by (twice) the risk premium of the latter. A high coinsurance rate on μ_{ L }types discourages μ_{ H }types from applying for the contracts intended for the latter, and thus allows insurers to charge the former group more for full insurance. For D exceeding Open image in new window it will turn optimal to start excluding LL types. In the appendix, we have redrawn Figure 7 for D∈[0, (1−α_{H⋅})/α_{H⋅}].^{29} The only new difference w.r.t. Figure 7 is that a new menu labelled M appears as a slice in between menus A and B. This menu excludes LL type as well, but has a degree of separation lower than that of B and higher than that of A (0<c_{ LL }^{ M }−c_{ LH }^{ M }<ℓ).
This concludes the discussion of the optimal menu choice. In the next two sections, we will discuss the implications for testing for the presence of asymmetric information and the normative implications for banning gender discrimination.
The positive correlation test
CJSS^{16} showed that a common prediction of any model of a competitive insurance market with asymmetric information is a strictly positive relationship between the degree of coverage and the expected loss across contracts. This is quite a strong result, and we refer to it as positive monotonicity (PM). This property implies a positive correlation between coverage and risk, but the converse is of course not true.
In the empirical literature on testing for asymmetric information in insurance markets, researchers typically rely on estimating the correlation coefficient between coverage and the expected loss, and then use a onesided test to determine whether this coefficient is statistically significantly positive (see, e.g. Cohen and Siegelman^{30}; Finkelstein and McGarry^{31}). The empirical evidence on positive correlation is somewhat weak; there is even evidence of negative correlation in some markets.^{32} This is quite surprising because the result of CJSS is general; conditional on the competition assumption, it holds for any combination of moral hazard and adverse selection in underlying risk.^{33}
As mentioned in the introduction, several theoretical explanations for this lack of evidence have been proposed. One is the socalled “cherry picking argument” (Chiappori and Salanié^{34}) or “propitious selection” (Hemenway^{35}), which combines adverse selection in risk preference (but not in the underlying risk) with moral hazard. The argument is that, if individual A is more risk averse than B, then A is willing to purchase more coverage than B. But, being more risk averse, A may ex post take more precautions than B. This may then result in a negative correlation between observed risk and coverage.^{36}
Because the optimal menu in a monopoly market with twodimensional screening may display Order 2, PM will cease to hold; for a subset of types (LH and HL), coverage is negatively related to risk size. This of course does not imply that the correlation between risk and coverage is negative, because PM does hold for other subsets of types (between HH on the one hand, and LH and LL on the other). This also suggests that a sufficiently negative correlation between risk and risk aversion (i.e. a sufficiently negative ρ) ensures a negative correlation between coverage and risk size. This is shown below.
Also CJSS point out that the PM property may be violated in a monopoly. They do this by starting with a model in which only preference heterogeneity exists (cf. the section “Onedimensional insurance”), and then introduce an infinitesimal amount of exogenous risk heterogeneity that is perfectly negatively correlated with risk aversion heterogeneity (i.e. the more riskaverse agents have a slightly smaller accident probability). Below, we show that the PM property does not hold whenever menu E applies, even if the underlying risk and risk preference are independently distributed.
Translated into our setting, the CJSS proposition may be stated as follows:
Consider two contracts C_{ a } and C_{ b } that are offered on the market. Suppose that: (i) C_{ a } gives more coverage then C_{ b }, that is, c_{ a }<c_{ b }; and (ii) the per capita profit generated by contract a does not exceed that of contract b, π(C_{ a })⩽π(C_{ b }). Then, (iii) the expected loss to those consumers signing up for contract a should exceed the expected loss of those consumers signing up for contract b, that is, μ(C_{ a })⩾μ(C_{ b }).
Because C_{ LH }^{ E }<C_{⋅L}^{ E }, it follows that π(C_{ LH })>π(C_{⋅L}), irrespective of which optimal values the coinsurance rates take under menu E.
Proposition 2

(i) For a sufficient degree of heterogeneity in risk aversion, such that menu E prevails, some lowrisk individuals (LH) purchase more coverage than do some highrisk individuals (HL). (ii) In the case of menu EI, cov(1−c(C_{ ij }), μ(C_{ ij }))<0 if and only if ρ<−α_{ HH }(x−α_{ HH }/α_{H⋅}) (<0).
In other words, the advantageous selection among LH and HL, described by part (i), may exactly offset the standard adverse selection, such that any correlation between risk and coverage vanishes. Finkelstein and McGarry^{31} show that the longterm care insurance market may suffer from asymmetric information, despite the absence of evidence for a positive correlation between risk and coverage. Our model helps in interpreting this evidence.
Gender discrimination
Crocker and Snow^{37} have shown that imperfect categorical discrimination in insurance—such as gender discrimination—always expands the efficiency frontier. Hoy^{38} showed how categorisation based on a signal may lead to a Pareto improvement in a competitive insurance market if the signal conveys information about the level of risk. In this section, we ask when such efficiency gains arise in a monopolistic market structure. We show that a Pareto improvement is possible if the signal, such as gender, is informative about risk aversion.
Lemma 3

If the likelihood function p(⋅) satisfies Condition S and if μ and ν are independently distributed, then μ and g are also independently distributed: p(μg)=p(μ).
Thus, these assumptions support the conclusion reached by the European Commission—that gender is insignificant in explaining risk type.
This condition is satisfied when the proportion of lowrisk individuals, α_{L⋅}, is not too small in relation to x.
where the first equality sign follows from Condition S and the second follows from independence. Thus, while α_{L⋅} and α_{H⋅} do not change when gender is observed, α_{ LL } does.
Proposition 3

Suppose that Condition S holds, and that μ and ν are independently distributed. Suppose that (12) holds. For given values of x, α_{L⋅}, and D, allowing gender discrimination will lead to a Pareto improvement in the insurance market if and only if conditions (13) and (14) hold.
Intuitively, because men are on average less risk averse, the “male” market consists of more LL types than does the overall market. This makes the distortion of the LL contract that was optimal for the entire market too costly for the “male” market: offering LL men a lower coinsurance rate (in return for a higher premium) increases profits from this market segment sufficiently to compensate for the lower rents extracted from the “higher” male types. Hence, all men benefit, and so does the monopolist.
It is worth mentioning that a similar result can be derived in the case where riskaverse and risktolerant agents all face the same expected loss.^{44} Then the optimal menu has c_{⋅H}=0, while c_{⋅L}=1 if and only if x⩽α_{⋅H} (or α_{⋅L}⩽1−x) (cf. Theorem 3). In other words, all types receive full insurance—which implies that riskaverse agents receive a rent—except if risktolerant individuals are sufficiently few in number. In this latter case, risktolerant agents are excluded and the riskaverse ones receive no rent. Hence, if one starts from a situation with few risktolerant agents (and where no agent obtains any rents), gender discrimination may create a “male” market with sufficiently many risktolerant agents so that these no longer become excluded. This in turn raises the rents of riskaverse men in that market. The “female” market is not affected since gender discrimination leads to an even lower percentage of risktolerant women, reinforcing the optimality of their exclusion. This result can be illustrated using a figure analogous to Figure 11, now with α_{⋅L} on the horizontal axis and c_{⋅L} on the vertical axis. The curve depicting c_{⋅L} is flat at 1 to the left of point 1−x and jumps down to zero to the right of 1−x. Starting from an overall proportion α_{⋅L} to the right of 1−x, gender discrimination may result in α_{⋅Lw}<1−x<α_{⋅Lm}, and thus in a Pareto improvement.
Conclusion
In this paper, we studied the outcome in a monopolistic insurance market when the insurer is only aware of the statistical distribution of the expected loss and the level of risk aversion of its customers. We formulated a meanvariance model that results in quasilinear preferences over contracts; we identified the five contract menus that emerge in equilibrium; and for each menu, we derived the optimal coinsurance rates. Next, we identified for each menu the subset of parameter values for which that menu is optimal. We did this under nonpositive correlation between the two characteristics.

it is never optimal to fully separate all the types. In other words, there will always be some pooling of types in equilibrium;

the greater is the heterogeneity in the expected loss, the more it pays to screen the lowrisk from the highrisk types, by imposing a high coinsurance rate on the former;

the greater is the heterogeneity in terms of risk aversion, the more it pays to screen the low risk averse from the highly risk averse by imposing a high coinsurance rate on the former; and

the property of positive monotonicity between coverage and expected loss need no longer hold—neither does the property of positive correlation.
We also identified an open set of parameter values such that when the female distribution of risk aversion firstorder stochastically dominates the male distribution, allowing gender discrimination results in a Pareto improvement in this market. Hence, our analysis shows that one should be careful when abolishing gender categorisation; even when gender itself does not (statistically) affect the expected level of losses or claims, it may affect the outcome in an imperfectly competitive insurance market so that nobody gains and some participants become worse off.
Footnotes
 1.
A third factor, that will not be discussed here, is the moral stance of insurees, determining the amount of false claims that insurers have to deal with each year.
 2.
 3.
Rothschild and Stiglitz (1976) analyse a perfectly competitive insurance market with private information on the distribution of losses. Stiglitz (1977, Sections 3 and 4), and Landsberger and Meilijson (1996) analyse a monopolist insurer. Stiglitz (1977, Section 5) and Landsberger and Meilijson (1994) analyse the outcomes under monopoly when private information is held on risk attitude. The latter paper shows that the first best (full extraction of consumer surplus by the insurer) can be achieved arbitrarily close through a very nonlinear contract. High riskaverse agents receive their certainty equivalent wealth; low riskaverse agents get, with probability, almost 1 of slightly more than their certainty equivalent, but receive with a tiny probability a very negative wealth. For the more riskaverse agent, this second contract is unattractive. Hence, full insurance with almost complete extraction of the risk premiums is achieved.
 4.
 5.
 6.
 7.
 8.
 9.
Dafny et al. (2012) find that mergers leading to an increase in market concentration were associated to a 7 per cent increase in premiums.
 10.
 11.
Jullien et al. (2007) analyse whether the singlecrossing property holds in the general monopolistic screening model with moral hazard and in which agents differ in their risk preferences. See also De Donder and Hindriks (2009).
 12.
 13.
Armstrong and Rochet (1999) consider an agent with quasilinear and separable preferences over two action levels and a transfer. The principal has similar preferences but she does not know whether the agent has a high or low valuation for either of the two activities. A contract specifies a transfer and two activity levels. Thus they have a “two instruments, two common values” problem (see also Dana, 1993). Our problem has only one instrument, that is, insurance coverage. The agent’s willingness to pay for coverage depends on both her risk level and risk aversion. On the other hand, the insurer’s willingness to offer coverage depends on the level of risk, but not on the agent’s risk aversion. Risk aversion only indirectly determines contract profitability through the rents that must be left for incentive compatibility reasons. Thus our problem is of the “one instrument, one common value” type. This also makes our problem different from that of Armstrong (1999) (one instrument and two common value characteristics).
 14.
In onedimensional screening models, competition in menus has been considered by, for example, Miyazaki (1977) and Spence (1978) using the Wilson (1977) equilibrium concept.
 15.
This de facto means that the main results are driven by the lack of order between what we refer to as “intermediate types”, that is, by those whose indifference curves cross twice. This explains why some authors only consider these intermediate types (e.g. Wambach, 2000). Although Smart (2000) and Villeneuve (2003) consider the full set of types, they maintain the assumption that each company offers a single contract per risk class. In some models with multidimensional private information, it is possible to reduce the dimensionality by a socalled type aggregator—see, for example, De Donder and Hindricks (2007) for a political economy social insurance context. In a context closer to ours, Crocker and Snow (2011) assume that an insurance taker may face different perils (fire, theft, etc.). Since the probability for each kind of loss is private information, the insurer must engage in multidimensional screening. Because risk classification based on observables is assumed sufficiently fine, the problem can be reduced to single dimensional private information.
 16.
 17.
CJSS propose a local argument for a negative correlation between risk and coverage to arise in the case of monopoly. Our analysis provides instead a full characterisation.
 18.
 19.
 20.
 21.
Council Directive 2004/113/EC of 13 December 2004 implementing the principle of equal treatment between men and women in the access to and supply of goods and services (Official Journal of the European Union 2004 L 373, p. 37). Originally, this directive provided for a derogation that allowed member states to permit genderspecific differences in insurance premiums and benefits in so far as gender is a determining risk factor that can be substantiated by relevant and accurate actuarial and statistical data. In March 2011, however, the European Court of Justice declared this derogating provision in the Directive to be invalid on the grounds that the use of risk factors based on sex in connection with insurance premiums and benefits is incompatible with the principle of equal treatment for men and women under European Union Law (European Court of Justice, 2011).
 22.
 23.
We have Open image in new window . Hence, the size of the lens is Open image in new window , a dimensionless number.
 24.
Because the coefficient of absolute risk aversion (r) measures twice the risk premium per unit of variance, we can conclude that the risk premium of a lowriskaverse type (RP_{ L }, say) equals (1/2)ν_{ L }. Therefore, D=Δμ/(2RP_{ L })=(Δμ/μ_{ L })/(2RP_{ L }/μ_{ L }).
 25.
 26.
This is because (i) we can normalise ν_{ L } to unity, and (ii) in the monopolist problem only Δμ matters—see (7).
 27.
Under Order 2, exclusion of risktolerant types will turn optimal, even for small values for D.
 28.
It is the lower root of a quadratic equation in x.
 29.
(1−α_{H⋅})/(α_{H⋅}) is the largest value for D for which menu A will still offer coverage to the lowrisk types (cf. Theorem 2).
 30.
 31.
 32.
The later phenomenon is termed “ advantageous or propitious selection”. In regard to life insurance, see, for example, Cawley and Philipson (1999) and McCarthy and Mitchell (2003), and in regard to longterm care, see, for example, Finkelstein and McGarry (2006). Olivella and VeraHernández (2013) show that, in duplicate private health insurance systems (in which the public and the competitive private insurance sectors offer the same portfolio of services), if heterogeneity is in risks only then propitious selection into private insurance should be observed if and only if information on risks is symmetric.
 33.
Moral hazard can encompass two distinct phenomena. One relates to bettercovered individuals having less of an incentive to undertake precautionary behaviour, which makes them observationally more risky. The other arises because one does not necessarily observe actual risk but the usage of, say, health services. Because coverage implies a lower cost of accessing these services, individuals may use more of these services because they have more coverage, not because they are more risky. Notice that both types of moral hazard reinforce the positive correlation. Of course, one of the econometric issues is that, even after observing some positive correlation, it is hard to disentangle the adverse selection and either of the two moral hazard effects.
 34.
 35.
 36.
See, for example, Jullien et al. (2007); De Donder and Hindriks (2009); and Finkelstein and McGarry (2006).
 37.
 38.
 39.
We assume that the support of the distribution of types does not vary with the signal. Alternatively, the support could be made dependent on the signal. This, in effect, amounts to assuming that the support consists of more than four (μ, ν) pairs, some of which have zero probability mass, depending on the observation of the signal.
 40.
Two equivalent formulations of Condition S are: p(gμ, ν)=p(gν) and p(μ, νg)=p(μν)⋅p(νg).
 41.
By Lemma 3, the marginal probability distribution of the risk size (α_{L⋅}, α_{H⋅}) is fixed.
 42.
 43.
Because the optimal menu for men is of the type Bp, the rents are given as follows: R_{ HH }=R_{ HL }+1/2Δν, R_{ HL }=R_{ LL }+(1−c_{ LL })Δμ, R_{ LH }=R_{ LL }+(1/2)(1−c_{ LL }^{ 2 })Δν, and R_{ LL }=0. Since c_{ LL } falls under discrimination, R_{ ij } (ij≠LL) rises.
 44.
We thank the Editor, Achim Wambach, for pointing this out to us.
 45.
Since x_{ CE }(α_{H⋅}, α_{ HH }, ρ, D) is decreasing in D, it is sufficient to consider x_{ CE }(α_{H⋅}, α_{ HH }, ρ, D)_{small D}.
 46.
That is, if (α_{ H· }, α_{ HH })∈ Open image in new window (ρ) for some ρ⩽0, then (α_{ H· }, α_{ HH }, ρ)∈ Open image in new window .
Notes
Acknowledgements
We are grateful to two anonymuous referees and the editor (Achim Wambach) for their detailed comments and suggestions. The paper has benefited from presentations at various seminars and conferences: Boston University, CORE (LouvainlaNeuve), ECARES (Brussels), HECER (Helsinki), NHH (Bergen), Toulouse School of Economics, the 6th European Health Economics Workshop, the 6th Public Economic Theory Meeting, the 5th IHEA World Congress, the 33rd EGRIE meeting, the 2009 HEBHERO Health Economics Workshop. Particular thanks to Catarina Goulão, Jean Hindriks, Eirik Kristiansen, Eric Nævdal and Gaute Torsvik for discussions and comments. Olivella acknowledges support from the Government of Catalonia project 2005SGR00836 and the Barcelona GSE Research Network, as well as from the Ministerio de Educación y Ciencia, project ECO2012–31962 and CONSOLIDERINGENIO 2010 (CSD2006–0016). Schroyen acknowledges the hospitality of CODE (Universitat Autónoma de Barcelona) where this project was started, and CORE (Université catholique de Louvain) where this version was completed, and financial support from Health Economics Bergen through an SNF grant.
Supplementary material
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