Splitting Risks in Insurance Markets With Adverse Selection

Splitting Risks in Insurance Markets With Adverse Selection

Pierre PICARD

August 24, 2017

Cahier n° 2017-01

ECOLE POLYTECHNIQUE

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE

DEPARTEMENT D'ECONOMIE

Route de Saclay 91128 PALAISEAU CEDEX

(33) 1 69333410

http://www.portail.polytechnique.edu/economie/fr weronika-natalia.leduc@polytechnique.edu

Splitting Risks in Insurance Markets with Adverse Selection

Pierre Picard August 24st, 2017

Abstract

We characterize the design of insurance schemes when policyhold- ers face several insurable risks in a context of adverse selection. Split- ting risks emerges as a feature of second-best Pareto-optimal alloca- tions. This may take the form of risk-speci…c contracts, or of contracts where risks are bundled but subject to di¤erential coverage rules - such as risk-speci…c copayments -, combined with a deductible, an out-of- pocket maximum or a cap on coverage.

CREST, Ecole Polytechnique, 91128, Palaiseau Cedex, France. Email:

pierre.picard@polytechnique.edu

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1 Introduction

Most people are simultaneously a¤ected by several insurable risks, including property, casualty, liability and health risks. It is a fact that these various categories of risks may require speci…c underwriting and claim-handling skills that are not uniformly shared by all insurers. This is a su¢ cient reason why we may cover di¤erent risks through di¤erent insurers, and thus, for instance, why many people purchase automobile and medical insurance from di¤erent insurers.

Understanding why similar risks incurred by the same economic unit (e.g., a family) are not covered by the same insurance contract is less obvious. To take a concrete example, why is it the case that, most of the time, the automobile risks of both members of a couple who drive di¤erent cars are covered by separate contracts, with speci…c premium and deductibles, and similarly, why the health care risks of these two people may be covered by independent contracts?¹

These very simple questions do not have simple answers. When there are no transaction costs, risk averse individuals would optimally purchase full coverage with actuarially fair insurance premium, and, obviously, in such a case it does not matter whether the risks of both members of a couple are covered through speci…c contracts or through a single contract. This is no longer true when insurers charge loaded premiums and households react by purchasing partial coverage. A variant of the wellknown "theorem of the deductible" says that, under proportional loading, households should be covered by an "umbrella policy" with full marginal coverage of the total losses beyond an aggregate deductible.² The fact that they may choose to be

1These questions are legitimate insofar as both members of the couples have identical access rights to the same set of contracts, and if they jointly manage their budget. Matters should be seen di¤erently in a collective approach to family decision making, where a household is described as a group of individuals with their own preferences, and among whom there is a coperative or non-cooperative decision process; see Bourguignon and Chiappori (1992).

2Umbrella insurance usually refers to coverage provided as a complement to other policies, particularly automobile and homeowners’liability policies. Like others before us, we use this terminology because the principle of umbrella policies is to globally protect policyholders against uncovered risks, without reference to the speci…city of these risks.

Following Eeckhoudt et al. (1991), who considered the particular case of a binomial distribution, more generally, Gollier and Schlesinger (1995) established the optimality of an umbrella policy under linear transaction costs. As shown by Gollier (2013), this property

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covered by separate insurance policies is inconsistent with this conclusion.

A similar question arises with respect to the coverage of various risks in- curred by an individual. This is particularly the case in medical insurance.

In an umbrella policy approach, the insurance indemnity depends on total medical expenses measured on a per period basis. However, much more fre- quently, various medical services (e.g. visits to general practitioners, visits to specialists, inpatient care, dental care, etc) are bundled in a single contract, but the insurance indemnity depends on service speci…c copayments, coinsur- ance rates or policy limits, frequently combined with an aggregate deductible, an out-of-pocket maximum or a cap on the indemnity. Here also, this casual observation of medical insurance contracts is at odds with the optimality of an umbrella policy. Likewise, homeowner insurance usually provides bundled coverage for both property and liability risks, with di¤erent deductibles and policy limits for these risks.

The fact that risk-speci…c policies and di¤erential coverage of risks bun- dled in a contract are so frequent suggests that splitting risks is often an optimal answer to the insurance choice problem. Umbrella policies may be suboptimal for various reasons that are ignored in the standard model of insurance demand, and that would be worth further exploring. Most of them are probably related to asymmetries of information between insurer and in- sured, for instance when the two members of a couple do not react to …nancial incentives in the same way under moral hazard, or when medical expenses can be more or less easily monitored by insurers, depending on the type of health care service (e.g. visit to general practionners vs inpatient care).³

In this paper, we will analyze how the splitting of risks is a rational answer to the adverse selection problem. We will consider a setting where households face several risk exposures (only two, for the sake of simplicity), with hidden information about the type of each of them: they may be high risk or low risk. For instance, in the case of health insurance the two risk

as well as the optimality of a straight deductible contract hold when policyholders dislike any zero-mean lottery that would be added to their …nal wealth, which is more general than the case of risk aversion under the expected utility criterion.

3Cohen (2006) considers a related issue. She analyzes the determinants of the deductible in insurance contracts that cover a risk that may materialize more than once during the life of the policy. She shows that aggregate deductibles may produce higher veri…cation costs and moral hazard costs than per-loss deductibles. In a similar perspective, Li. et al.

(2007) analyze the case of automobile insurance in Taiwan, and provide evidence of the incentive e¤ects of increasing per-claim deductibles on the policyholders’behavior.

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exposures may correspond to the medical expenses of each member of a couple, or to di¤erent types of medical services. We will have high-risk households with two high-risk exposures, medium-risk households with a high risk and a low risk (with two subgroups according to the identity of the high- and low-risk exposures), and low-risk households with two low risks. An umbrella policy provides coverage for the aggregate losses that may result from the two risk exposures, while a risk-speci…c policy only covers the loss of a particular risk. In this setting, the coverage of risks may be split by purchasing insurance through two risk-speci…c policies, such as two automobile insurance policies for the two cars owned by the household, or separate health insurance contracts for each member of a couple. Bearing in mind the case of health insurance, we may also consider that policyholders are individuals who face two additive risks, each of them corresponding to a speci…c service, such as visits to general practitioners and dental care. In this case, splitting risks refers to the bundling of insurance coverage with service-speci…c coverage, possibly combined with a deductible, an out-of- pocket maximum or a cap on indemnity.

Crocker and Snow (2011) study a related but di¤erent problem. They analyze how the bundling of distinct perils (seen as factors that may be at the origin of a loss) in a single contract with di¤erential deductibles improves the e¢ ciency of insurance markets under adverse selection. They consider a setting with two types of policyholder (high risk and low risk) where a loss may result from various perils, and they show that bundling the coverage of these perils in a single contract facilitates the separation of high risks and low risks. If the probabilities that the loss results from di¤erent perils (conditionnally on a loss occurring) di¤er between high-risk and low-risk individuals, then bundling perils with di¤erential deductibles allows insurers to implement a multidimensional screening that yields a Pareto e¢ ciency gain and makes the existence of a Rothschild-Stiglitz equilibrium more likely.

Our perspective is di¤erent. We consider a setting in which each poli- cyholder faces two risks that may be high or low, and potentially one loss per risk. We thus have four types of policyholder and 0,1 or 2 losses. The issue we consider is whether the aggregate loss that may result from mul- tiple risk exposures should be covered through an umbrella policy, and if not, whether risk splitting takes the form of risk speci…c policies where risk exposures are unbundled, or of a single policy where risks are bundled but di¤erent indemnity rules apply for each risk.

We will not try here to analyze the market equilibrium that may arise 4

when there is adverse selection about multiple insurable risks, although this should be an ultimate research objective. Our objective will be more mod- estly limited to the characterization of e¢ cient allocations. E¢ ciency will be in a second-best sense, meaning that allocations should satisfy the in- centive compatibility conditions inherent in a hidden information setting.

Implicitly, we presume that either competition forces or market regulation lead to such a second-best Pareto optimal allocation. Our analysis will be carried out within the framework of a model that belongs to the class of multi- dimensional screening models surveyed by Armstrong and Rochet (1999) and Rochet and Stole (2003). As in these models, technical di¢ culties arise from the fact that no single crossing assumption exists in such models, and thus that - contrary to what occurs in the standard Rothschild-Stiglitz (1976) set- ting with a single risk exposure, extended to an arbitrary number of types by Spence (1978) - the state-contingent wealth allocated to a household may be jointly a¤ected by the incentive constraints of several other households with di¤erent types of risk exposure.

The main lesson learned from the characterization of e¢ cient allocations will be that risk splitting facilitates the separation of risk types. The intu- ition for this result is as follows. As in the Rothschild-Stiglitz (1976) model, unsurprisingly, at any second-best Pareto-optimal allocation high-risk house- holds are fully covered, which may correspond to an umbrella policy or to two risk-speci…c policies. They are deterred from choosing the risk coverage of the "upward adjacent" type of household - which is the medium risk in the present model - by providing partial coverage to this type. In our multiple risk setting, partial coverage may correspond to several insurance schemes, such as, for instance, a uniform proportional coinsurance for the two risk ex- posures. It turns out that full coverage for the high-risk exposure and partial coverage for the low-risk exposure is the most e¢ cient way to separate high- and medium-risk households, since the existence or the absence of such a low risk exposure is the only di¤erence between these two categories of house- hold. This requires that medium-risk households be covered by risk-speci…c insurance policies, with full coverage for their high risk and partial coverage or even no coverage for their low risk.

The situation is di¤erent for low-risk households, since their two risk ex- posures are homogeneous (both are low risks), and their insurance policy, to facilitate the separation from medium-risk households, should include de- creasing coverage at the margin for both risk exposures. Risk speci…c policies are not optimal for low-risk households, but splitting the coverage of risk ex-

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posures is again optimal because of the (possibly uneven) pressure of the two types of medium-risk policyholders. It takes a form that will sound very familiar in a health economics perspective: risk-speci…c contributions of the policyholder in the case of a loss (i.e., in the terminology of health insurance, risk-speci…c copayments) should be combined with an aggregate deductible, an out-of-pocket maximum or a cap on the insurance indemnity.

We will also consider the case when actuarial premiums are loaded up because of transaction costs. This extension is of particular interest because of the theorem of the deductible that states the optimality of an aggregate deductible when there is proportional loading and no asymmetry of infor- mation. Taking into account insurance loading will modify our conclusions about the optimality of full coverage for high-risk households and of risk- speci…c insurance contracts for medium-risk households, but splitting risks will remain an optimal answer to the adverse selection problem. In particular, high-risk households should be protected by a straight deductible contract, while medium- and low-risk households should be covered by a policy that includes risk-speci…c copayments combined with an aggregate deductible or an out-of-pocket maximum.

The rest of the paper is arranged as follows. Section 2 presents our model of an insurance market with multiple risks and hidden information about risk types. Section 3 shows why and how risk splitting may be an answer to adverse selection. Section 4 analyzes the e¤ect of transaction costs on insurance schemes. Section 5 concludes. The proofs are in an appendix.

2 Model

2.1 Notations

We consider an economy where households are a¤ected by two additive …nan- cial risk exposures (more brie‡y, risks), indexed by r2 f1;2g and identi…ed by observable characteristics.⁴ We can think of the health care expenses incurred by both members of a couple, hence the "household" terminology.

The model may also be interpreted as the case of a single person who faces

4Hence, households cannot present their risk exposure1as belonging to category2, and vice versa. The analysis would be di¤erent if both risk exposures could be substituted, as for instance in the case of property owners seeking …re insurance for a group of two buildings, without special features making it possible to distinguish between them.

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two risks, such as doctors’ visits and inpatient care in the case of health insurance, or accident and theft risk in the case of automobile risk. For the sake of notational simplicity, we assume that each risk r may lead to the same loss L, with probability p_{`</sub> or p<sub>h</sub>, with 0 < p<sub>`} < p_h < 1, and that the occurrence of these losses are independent events. Hence, r is a high risk when the corresponding probability of loss is p_h and it is a low risk when the probability is p<sub>`</sub>. Thus, there are four household types characterized by a double index ij, with i; j 2 fh; `g, where i and j are the types of risk expo- sures r = 1 and 2, respectively. Types hh and `` are high-risk and low-risk households because their two risks are either high or low. Types h` and `h are medium risk households, because one of their risk exposures is high and the other is low. We denote ij the proportions of type ij households with

hh+ _{h`</sub>+ <sub>`h}+ _`` = 1. Households have private information about their type and all of them have the same initial wealth w₀.

The state-contingent …nal wealth of a household is written asw= (w⁰⁰; w⁰¹; w¹⁰; w¹¹), where w^xy denotes its …nal wealth in state (x; y)2 f0;1g², where x = 0 when risk r = 1 does not lead to a loss and x= 1 when it does, with a similar interpretation of index y for risk r = 2. For instance, w¹⁰ is the household’s …nal wealth when only risk 1 is at the origin of a loss.

Households are decision units: they are risk averse and they maximize the expected utility of their aggregate …nal wealth w_f, with utility function u(w_f), such thatu⁰ >0; u⁰⁰<0. To obtain more speci…c results, we may also assume that households are prudent, which corresponds to the additional assumption u⁰⁰⁰ > 0, but we will mention it explicitly when this assumption is needed. The expected utility of a type ij household is denoted U_ij(w) for all i; j 2 fh; `g, with

U_hh(w) (1 p_h)²u(w⁰⁰) +p_h(1 p_h)[u(w⁰¹) +u(w¹⁰)] +p²_hu(w¹¹);

U_{h`</sub>(w) (1 p<sub>h</sub>)(1 p<sub>`})(w⁰⁰) +p_{`</sub>(1 p<sub>h</sub>)u(w<sup>01</sup>) +p<sub>h</sub>(1 p<sub>`})u(w¹⁰) +p_hp_`u(w¹¹);

U_{`h</sub>(w) (1 p<sub>h</sub>)(1 p<sub>`})(w⁰⁰) +p_h(1 p_{`</sub>)u(w<sup>01</sup>) +p<sub>`}(1 p_h)u(w¹⁰) +p_hp_`u(w¹¹);

U``(w) (1 p`)<sup>2</sup>u(w<sup>00</sup>) +p`(1 p`)[u(w<sup>01</sup>) +u(w<sup>10</sup>)] +p<sup>2</sup><sub>``</sub>u(w¹¹):

The state-contingent …nal wealth of a type ij household is denoted by w_ij = (w_ij⁰⁰; w_ij⁰¹; w¹⁰_ij; w_ij¹¹), with expected utility U_ij(w_ij). An allocationA= fw_hh; w_{h`</sub>; w<sub>`h}; w_``g speci…cies the state-contingent …nal wealth for each type of household.

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Insurance schemes⁵ are characterized by the non-negative indemnity T^xy paid by the insurer to the household when there are x and y losses, in risk exposuresr = 1and2, respectively, with(x; y)2 f0;1g²andT⁰⁰ = 0, and by the premium P paid in all states by the policyholder to the insurer. If type ij households are protected by an insurance scheme with coverage schedule T_ij (T_ij⁰¹; T_ij¹⁰; T_ij¹¹) and premiumP_ij, then we have

w_ij^xy =w (x+y)L P_ij +T_ij^xy for all x; y 2 f0;1g; (1) with T_ij⁰⁰= 0.

In what follows, for the sake of simplicity and realism, we will restrict at- tention to insurance schemes where the indemnity is comonotone with losses and does not increase more than losses, and thus we assume⁶

0 T_ij^x1 T_ij^x0 L if x2 f0;1g; 0 T_ij^1y T_ij^0y L if y2 f0;1g; or, equivalently

L w^x0_ij w^x1_ij 0 if x2 f0;1g; L w^0y_ij w_ij^1y 0 if y2 f0;1g; for all i; j 2 fh; `g.

2.2 Typical multirisk insurance schemes

Risk-speci…c policies and umbrella policies are particular forms of insurance schemes that restrict the set of feasible allocations. A risk-speci…c insurance policy provides coverage for a particular risk. A policy that covers risk 1 is de…ned by an indemnity I¹ paid to the household, should risk 1 lead to a loss, and by an insurance premium Q¹, with similar notations I²; Q² for a

5We use the terminology "insurance scheme" instead of "insurance policy" or "contract"

because, in what follows, an insurance scheme may correspond to the combination of two risk speci…c insurance policies.

6In addition to the fact that overinsurance may be prohibited by law, policyholders would be incentivized to deliberately create damage (e.g. by commiting arson in the case of …re insurance) if coverage were larger than losses.

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policy that covers risk 2. Hence, if type ij households are covered by two risk-speci…c policies (I_ij¹; Q¹_ij) and (I_ij²; Q²_ij), then we have

T_ij¹⁰ =I_ij¹; T_ij⁰¹ =I_ij²; T_ij¹¹ =I_ij¹ +I_ij²; P_ij =Q¹_ij +Q²_ij; which induces an allocation that satis…es

w_ij¹¹=w_ij⁰¹+w_ij¹⁰ w⁰⁰_ij for all i; j 2 fh; `g: (2) This characterization can be reversed: any allocation that satis…es (2) is induced by an insurance scheme where type ij households are covered by two type-dependent risk-speci…c policies.

Under an umbrella policy, the coverage is a function of the household’s total loss, and the policy may be written as (J¹; J²; P), where J¹ and J² denote the indemnity in the case of one or two losses, respectively, and, as before, P is the total premium paid for the coverage of the two risks. Thus, when type ij households are covered by an umbrella policy (J_ij¹; J_ij²; P_ij);we have

T_ij¹⁰=T_ij⁰¹=J_ij¹; T_ij¹¹=J_ij²: The corresponding allocation satis…es

w_ij⁰¹=w_ij¹⁰ for all i; j 2 fh; `g; (3) with, here also, an inverse relationship: any allocation that satis…es (3) is induced by an umbrella policy for type ij households.

If an umbrella policy (J_ij¹; J_ij²; P_ij) provides the same state contingent wealth w_ij = (w_ij⁰⁰; w⁰¹_ij; w¹⁰_ij; w_ij¹¹) as two risk-speci…c policies (I_ij¹; Q¹_ij) and (I_ij²; Q²_ij), then we have

w¹¹_ij w_ij⁰¹=w_ij¹⁰ w⁰⁰_ij =w⁰¹_ij w_ij⁰⁰; from (2) and (3), and thus

J_ij² = 2J_ij¹: 9

Hence, the indemnity is doubled when the loss is doubled, which means that linear coinsurance is an umbrella policy equivalent to two separate risk- speci…c policies, both of them providing linear coinsurance at the same rate.

However, risk-speci…c policies and umbrella policies are far from being the only forms of multirisk insurance schemes. Consider the case of risk-speci…c copays c¹_ij and c²_ij combined with a deductible D_ij, and assume c¹_ij +D_ij < L and c²_ij +D_ij < L.⁷ This gives

T_ij¹⁰ = L c¹_ij D_ij; T_ij⁰¹ = L c²_ij D_ij; T_ij¹¹ = 2L c¹_ij c²_ij D_ij; Using (1) yields

c¹_ij =w⁰¹_ij w_ij¹¹; c²_ij =w¹⁰_ij w_ij¹¹;

D_ij =w⁰⁰_ij +w_ij¹¹ (w_ij¹⁰+w_ij⁰¹):

with D_ij 0if

w_ij⁰⁰+w_ij¹¹ w¹⁰_ij +w⁰¹_ij; (4) and we have T_ij⁰¹; T_ij¹⁰; T_ij¹¹ > 0 if w⁰⁰_ij w_ij⁰¹ < L and w⁰⁰_ij w_ij¹⁰ < L.⁸ Con- versely, under condition (4), w_ij can be sustained by non-negative copays c¹_ij; c²_ij and deductible D_ij. This equivalence between condition (4) and the sustainability ofw_ij through copays and a deductible also holds whenT_ij⁰¹; T_ij¹⁰ and/or T_ij¹¹ = 0, with w_ij⁰⁰ w_ij⁰¹=L; w_ij⁰⁰ w¹⁰_ij =L and/or w_ij⁰⁰ w¹¹_ij = 2L, respectively.⁹ Thus, more generally, an allocation that satis…es condition (4)

7In health insurance, a copay (or copayment) is a set out-of-pocket amount paid by the insured for health care services. We use this terminology although our approach is not limited to health insurance. We assume, that the amounts paid in copays do not count towards meeting the deductible. In other words, the insurance indemnity is equal to the di¤erence between the part of out-of-pocket costs over the copays and the deductible if this di¤erence is positive.

8Using (4) then givesw⁰⁰_ij w¹¹_ij <2L.

9In such cases, c¹_ij; c²_ij and/or Dij are not uniquely de…ned as a function of wij. For instance, we have T_ij¹⁰ = 0; T_ij⁰¹ >0; T_ij¹¹ >0 when w⁰⁰_ij w¹⁰_ij =L; w⁰⁰_ij w_ij⁰¹ < L and w⁰⁰_ij w¹¹_ij <2L. Such an allocation may be sustained byc¹_ij; c²_ij2[0; L]andD_ij 0such thatc¹_ij+Dij > L; c²_ij+Dij< Landc¹_ij+c²_ij+Dij > L. This is possible under condition (4), by choosing c¹_ij =w⁰¹_ij w¹¹_ij; c²_ij +D_ij =w⁰⁰_ij w_ij⁰¹ and w⁰⁰_ij +w_ij¹¹ w_ij¹⁰ w⁰¹_ij Dij w_ij⁰⁰ w_ij⁰¹. In particular, c¹_ij =w⁰¹_ij w¹¹_ij; c²_ij = 0; Dij =w_ij⁰⁰ w_ij⁰¹ is a possible solution.

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may be sustained by an incentive scheme T_ij (T_ij⁰¹; T_ij¹⁰; T_ij¹¹) such that T_ij^xy = supf0; x(L c¹_ij) +y(L c²_ij) D_ijg;

for x; y 2 f0;1g.

Note that condition (4) may be rewritten as

w_ij⁰⁰ w_ij¹¹ (w⁰⁰_ij w_ij¹⁰) + (w⁰⁰_ij w_ij⁰¹);

which means that the loss w⁰⁰_ij w_ij¹¹ retained by the household (i.e., not covered by insurance) in the case of two losses a¤ecting simultaneouslyr = 1 and r = 2 is smaller than the sum of the retained losses w⁰⁰_ij w¹⁰_ij and w⁰⁰_ij w⁰¹_ij for the two events where a single loss a¤ects eitherr= 1 orr = 2.

In this sense, combining a deductible with copays provides a larger coverage for large risks than for small risks: the average indemnity per $ loss is larger when losses a¤ect both risk exposures simultaneously than when a single loss a¤ects either r = 1 orr = 2 with equal probabilities.¹⁰

For illustrative purposes, Figures 1 and 2 show how changing the values of the deductible and of the copays allows the insurer to either cover more or less extensively the occurrence of two simultaneous losses or to prioritize a particular risk. Figure 1 illustrates the case where the coverage of large losses is reduced without a¤ecting the coverage of small losses by increasing the deductible and decreasing the copays simultaneously. Figure 2 shows how the coverage of a single loss a¤ecting risk r= 1 can be reduced without changing the coverage of a single loss a¤ecting r = 2, and of a large loss a¤ecting r= 1 and r= 2 simultaneously.

Figure 1 Figure 2

Consider now the case where the insurance scheme of typeij households includes risk-speci…c copays c¹_ij; c²_ij and an out-of-pocket maximum M_ij in- stead of an aggregate deductible. Let us focus on the case wherec¹_ij+c²_ij M_ij

10In the case of two losses, the indemnity per dollar loss is(w_ij⁰⁰ w¹¹_ij)=2L. If there is one single loss a¤ecting r= 1 or r= 2 with equal probabilities, the expected indemnity per dollar loss is(w⁰⁰_ij w¹⁰_ij)=2L+ (w⁰⁰_ij w⁰¹_ij)=2L.

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and c¹_ij; c¹_ij < M_ij, which means that the out-of-pocket maximum is reached in the case of two losses. In that case, we have

T_ij¹⁰ = L c¹_ij; T_ij⁰¹ = L c²_ij; T_ij¹¹ = 2L M_ij; and (1) yields

c¹_ij =w⁰⁰_ij w_ij¹⁰; c²_ij =w⁰⁰_ij w_ij⁰¹; M_ij =w_ij⁰⁰ w_ij¹¹:

Usingc¹_ij+c²_ij M_ij shows thatw_ij can be sustained by non-negative copays c¹_ij; c²_ij and an out-of-pocket maximum M_ij if (4) holds, and thus, here also, we may conclude that mixing an out-of-pocket maximum with risk-speci…c copays allows the insurer to prioritize the coverage of large risks. Choosing c¹_ij =L; c²_ij =Land/orM_ij = 2Lleads to zero coverage whenT_ij⁰¹; T_ij¹⁰and/or T_ij¹¹= 0, respectively. Overall, such an insurance scheme is de…ned by

T_ij^xy = supfx(L c¹_ij) +y(L c²_ij);2L M_ijg; for x; y 2 f0;1g.

Finally, consider the case of risk-speci…c copaysc¹_ij andc²_ij combined with anupper limit on coverage U L_ij. The case where this upper limit is reached in the case of two losses corresponds to L c¹_ij U L_ij; L c²_ij U L_ij and U L_ij 2L c¹_ij c²_ij. In that case, we have

T_ij¹⁰ = L c¹_ij; T_ij⁰¹ = L c²_ij; T_ij¹¹ = U L_ij; and (1) gives

c¹_ij =w⁰⁰_ij w¹⁰_ij; c²_ij =w⁰⁰_ij w⁰¹_ij;

U L_ij = 2L+w_ij¹¹ w⁰⁰_ij; 12

with U L_ij 2L c¹_ij c²_ij if

w_ij⁰⁰+w_ij¹¹ w¹⁰_ij +w⁰¹_ij; (5) which is the condition under which w_ij can be sustained by non-negative copays c¹_ij; c²_ij and an upper limit on coverage U Lij, with an interpretation which is symmetric to (4): mixing risk-speci…c copays with an upper limit on coverage is a way of prioritizing the coverage of losses that may a¤ect one single risk exposure. Including the cases where no indemnity is paid yields

T_ij^xy = inffx(L c¹_ij) +y(L c²_ij); U L_ijg; for x; y 2 f0;1g.

2.3 Incentive compatibility and feasibility

The incentive compatibility of an allocationA=fw_hh; w_{h`</sub>; w<sub>`h}; w_``grequires that any typeij household weakly prefers w_ij tow_i0j⁰ if i6=i⁰ and/orj⁰ 6=j, i.e.,

U_ij(w_ij) U_ij(w_i0j⁰); (6) for all i; j; i⁰; j⁰ 2 fh; `g such that i6=i⁰ and/or j 6=j⁰.

Let C_ij(w) be the expected wealth of a type ij household, for state- dependent wealth w= (w⁰⁰; w⁰¹; w¹⁰; w¹¹), with

C_hh(w) (1 p_h)²w⁰⁰+p_h(1 p_h)(w⁰¹+w¹⁰) +p²_hw¹¹; C_{h`</sub>(w) (1 p<sub>h</sub>)(1 p<sub>`})w⁰⁰+p_`(1 p_h)w⁰¹

+ph(1 p`)w<sup>10</sup>+php`w¹¹;

C_{`h</sub>(w) (1 p<sub>h</sub>)(1 p<sub>`})w⁰⁰+p_h(1 p_{`</sub>)w<sup>01</sup> +p<sub>`}(1 p_h)w¹⁰+p_hp_`w¹¹

C_``(w) (1 p_{`</sub>)<sup>2</sup>w<sup>00</sup>+p<sub>`}(1 p_{`</sub>)(w<sup>01</sup>+w<sup>10</sup>) +p<sup>2</sup><sub>`}w¹¹;

By de…nition, the allocationA isbudget-balanced if the average wealth in the population is lower or equal to the average wealth in the absence of any redistribution between households. This is written as

X

i;j2fh;`g

ijCij(wij) w; (7)

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where

w=w₀ L[2 _hhp_h+ ( _{h`</sub>+ <sub>`h})(p_h+p_{`</sub>) + 2 <sub>``</sub>p<sub>`}]:

Finally, we say that the allocationAisfeasible if it is incentive compatible and budget-balanced.

Any feasible allocation can be induced by an insurance scheme where insurers o¤er menus of insurance schemes and make non-negative expected pro…t when households choose their best scheme. Indeed, assume that each typeij household chooses a scheme(T_ij,Pij), among the o¤ers available in the insurance market. In this case, the induced allocationA=fwhh; wh`; w`h; w``g is de…ned by Equation (1) for all i; j 2 fh; `g and this allocation is incentive compatible. Conversely, any incentive compatible allocationA is induced by a menu of insurance contracts with coverage Ti (T_ij⁰¹; T_ij¹⁰; T_ij¹¹) and pre- mium P_ij such that P_ij = w_ij⁰⁰ w₀ and T_ij^xy = w_ij^xy w⁰⁰_ij + (x +y)L if (x; y)6= (0;0).

Assume that insurers o¤er the menu of insurance schemesf(Tij; Pij); i; j = h; `gand that(T_ij; P_ij)is chosen by typeij households in this menu. Assume moreover that policyholders are evenly spread among insurers. Then insurers make non-negative expected pro…t if

hh P_hh p_h(1 p_h)(T_hh⁰¹+T_hh¹⁰) p²_hT_hh¹¹

+ _{h`</sub> P<sub>h`} p_{`</sub>(1 p<sub>h</sub>)T<sub>h`}⁰¹ p_h(1 p_{`</sub>)T<sub>h`}¹⁰ p_hp_{`</sub>T<sub>h`}¹¹ + _{`h</sub> P<sub>`h} p_h(1 p_{`</sub>)T<sub>`h}⁰¹ p_{`</sub>(1 p<sub>h</sub>)T<sub>`h}¹⁰ p_{`</sub>p<sub>h</sub>T<sub>`h}¹¹ + _{``</sub> P<sub>``} p_{`</sub>(1 p<sub>`})(T_{``</sub><sup>01</sup>+T<sub>``}¹⁰) p²_`T_``¹¹

0 (8)

A satis…es (7) if and only if (8) holds, which shows the correspondence be- tween the insurers’ break-even condition and the budget balance of alloca- tions induced by insurance schemes. The results obtained so far are summa- rized in the following proposition.

Proposition 1 Conditions (6) and (7) de…ne the set of feasible allocations that can be induced by insurance schemes o¤ered by insurers that break even.

Under condition (2) typeij households can be covered by risk-speci…c policies.

Under condition (3) they can be covered by an umbrella policy. Under condi- tion (4) - respect. condition (5) - they can be covered by an insurance policy that combines risk-speci…c copays with either a deductible or an out-of-pocket maximum - respect. an upper limit on indemnity -.

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2.4 Second-best Pareto-optimality

By de…nition, an allocation is second-best Pareto-optimal if it is feasible and if there does not exist another feasible allocation with expected utility at least as large for all types, and larger for at least one type. More explicitly, let us consider a feasible allocation A = fw_hh; w_{h`</sub>; w<sub>`h}; w_{``</sub>g, where w<sub>ij</sub> = (w<sup>00</sup><sub>ij</sub> ; w<sub>ij</sub><sup>01</sup> ; w<sub>ij</sub><sup>10</sup> ; w<sup>11</sup><sub>ij</sub> ) for i; j 2 fh; `g. A is second-best Pareto-optimal if, for all i; j, Uij(wij) reaches its maximum at A in the set of feasible allocations A=fw<sub>hh</sub>; w<sub>h`</sub>; w<sub>`h</sub>; w<sub>``}g such that U_i0j⁰(w_i0j⁰) U_i0j⁰(w_i0j0) for all i⁰; j⁰ 2 fh; `g; i⁰ 6=iand/or j⁰ 6=j.

Investigating the competitive interactions that could lead to a particu- lar second-best Pareto-optimal allocation is out of the scope of the present paper. In what follows, we will limit ourselves to analyzing the insurance contracts that sustain second-best Pareto-optimal allocations. Thus, we im- plicitly presume that, in the absence of trade restraint (such as barriers to entry or exogenous constraints on contracts), competitive forces should lead to a second-best Pareto optimal allocation, which is of course a debatable assumption. We may also have in mind a regulated insurance market where second-best Pareto-optimality results from public intervention, for instance through taxation of contracts.

3 Risk splitting as an answer to adverse se- lection

A second-best Pareto-optimal allocationA =fw_hh; w_{h`</sub>; w<sub>`h}; w_{``</sub>gmaximizes U<sub>``}(w_{``</sub>) in the set of feasible allocations A = fw<sub>hh</sub>; w<sub>h`</sub>; w<sub>`h</sub>; w<sub>``}g with ex- pected utility equal to u_ij U_ij(w_ij)fori; j 2 fh; `g; i6=` and/orj 6=`. We also denote u_{``</sub> U<sub>``}(w_``). Incentive compatibility implies that lower risk types reach weakly higher expected utility.¹¹ We will assume

u_hh<minfu_{h`</sub>; u<sub>`h}g maxfu_{h`</sub>; u<sub>`h}g u_``, (9)

11This is true because overinsurance is excluded. In the Rothschild-Stiglitz (2016) set- ting, with only one risk exposure per type, second-best Pareto-optimal allocations exist where high-risk individuals are overcovered and reach a higher expected utility than low- risk individuals, and such allocations could also exist in the present setting if overinsurance were allowed; See Crocker and Snow (1985).

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where the strict inequality u_hh < minfu_{h`</sub>; u<sub>`h}g is made to exclude alloca- tions where high-risk households are pooled with medium-risk households, i.e. where they have the same insurance coverage.¹²

Thus,A maximizesU_{``</sub>(w<sub>``})with respect tow_hh; w_{h`</sub>; w<sub>`h}andw_``, subject to

-Incentive compatibility constraints:

U_ij(w_ij) U_ij(w_i0j⁰); (IC_ij^i0j0) for all i; j; i⁰; j⁰ 2 fh; `g, with i6=i⁰ and/orj 6=j⁰,

-Individual rationality constraints:

U_ij(w_ij) = u_ij; (IRij) for all i; j 2 fh; `g with i6=` and/or j 6=`,

-Break-even constraint: X

i;j2fh;`g

ijC_ij(w_ij) w; (BC)

and sign and no-overinsurance constraints.

This problem belongs to the class of multi-dimensional screening models surveyed by Armstrong and Rochet (1999) and Rochet and Stole (2003).

Propositions 2 and 3 characterize its optimal solution, and the associated insurance schemes.

Proposition 2 Any second-best Pareto-optimal allocation is sustained by in- surance schemes such that : (i) high-risk households are fully covered, (ii) medium-risk households are covered either by two risk-speci…c policies with full coverage for the large-risk exposure and partial coverage for the low-risk exposure, or only by a risk-speci…c policy with full coverage for the large-risk exposure, without coverage for the low-risk exposure, (iii)low-risk households are covered by a policy with partial coverage at the margin.

12Such pooling allocations may be second-best Pareto-optimal. For instance, in the standard Rothschild-Stiglitz model, the pooling allocation with full coverage at average actuarial price is a second-best Pareto-optimum. Our results can be extended to the case u_hh = minfu_{h`</sub>; u<sub>`h}g. Types hh and h` (respect. hh and `h) would be pooled with the same coverage if u_hh=u_{h`</sub> (respect. ifu<sub>hh</sub>=u<sub>`h}). See the comments on Proposition 2.

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Figure 3

The proof of Proposition 2 is in two stages illustrated in Figure 3. At the …rst stage, we characterize the optimal solution of the relaxed prob- lem where only the "weakly upward" incentive compatibility constraints IC_hh^{h`</sup>; IC<sub>hh</sub><sup>`h}; IC_hh^{``</sup>; IC<sub>h`</sub><sup>`h</sup>; IC<sub>h`</sub><sup>``}; IC<sub>`h</sub><sup>h`</sup> and IC_`h^`` are taken into account. At the second stage, we show that the optimal solution of the relaxed problem sat- is…es the incentive compatibility constraints that have been ignored at stage 1 and thus that it is the solution to the complete problem.¹³ This leads us to the following results:

w_hh⁰⁰ = w¹⁰_hh=w_hh⁰¹ =w¹¹_hh; (10) w_{h`</sub><sup>00</sup> = w<sup>10</sup><sub>h`}> w⁰¹_{h`</sub> =w<sup>11</sup><sub>h`}; (11) w_{`h</sub><sup>00</sup> = w<sup>01</sup><sub>`h}> w¹⁰_{`h</sub> =w<sup>11</sup><sub>`h}; (12) w_{``</sub><sup>00</sup> > minfw<sup>01</sup><sub>``}; w_``¹⁰g; (13)

and/or

w_{``</sub><sup>11</sup> < maxfw<sub>``}⁰¹; w¹⁰_``g: (14)

Proposition 2 expresses these results in terms of insurance contracts. It states that households are covered by insurance contracts that depend on their risk type, at the same time regarding the size and contractual form of coverage. Equation (10) shows that, as in the Rothschild-Stiglitz (1976) model, high-risk households are fully covered, which means thatT_hh⁰¹ =T_hh¹⁰= L and T_hh¹¹ = 2L. Indeed, the proof shows that the downward incentives constraints IC_{h`</sub><sup>hh</sup>; IC<sub>`h}^hh and IC_{``</sub><sup>hh</sup> are not binding, and consequently there is no reason to distort the state-dependent wealth of high-risk households in order to prevent lower-risk households from choosingw<sub>hh</sub>, instead of w<sub>h`</sub>; w<sub>`h</sub> orw<sub>``}. This is compatible either with an umbrella policy(J_hh¹ ; J_hh² ; P_hh)such

13This approach is usual in adverse selection problems with the single-crossing condition and an arbitrary number of types. It is less obvious here, because there are four possible outcomes for each household, and not only two as in more standard models. See Armstrong and Rochet (1999) for an application of this approach to a principal-agent model with multi-dimensional information about the agent’s type.

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that J_hh¹ = L and J_hh² = 2L, or with two risk-speci…c policies, (I_hh¹ ; Q¹_hh) and (I_hh² ; Q²_hh), for risks r = 1 and 2, respectively, with I_hh¹ = I_hh² = L and Q¹_hh+Q²_hh=P_hh.

Equation (11) shows that medium-risk households are covered either by two risk-speci…c policies with full coverage for the high risk and partial cover- age for the low risk, or by a single risk-speci…c policy with full coverage for the high risk, without coverage for the low risk. Hence, for type h` households, we have (I<sub>h`</sub>¹ ; Q¹_{h`</sub>) and (I<sub>h`}² ; Q²_{h`</sub>) such that I<sub>h`}¹ = L and 0 I_{h`</sub><sup>2</sup> < L, with I<sub>h`}² > 0 when w⁰⁰_{h`</sub> w<sup>01</sup><sub>h`} < L and I_{h`</sub><sup>2</sup> = 0 (i.e. no coverage for the low risk) when w<sub>h`}⁰⁰ w<sub>h`</sub><sup>01</sup> = L. The characterization is symmetric for type `h house- holds. This is not an astonishing conclusion if we have in mind the standard Rothschild-Stiglitz model, where the contract chosen by low-risk individuals provides partial coverage, which makes it unattractive to high-risk individu- als. It is striking, and of course logical, that the same objective of reducing insurance compensation should lead insurers to o¤er risk-speci…c policies, where the low coverage (or even the absence of coverage) is for the low-risk exposure only. Reducing the compensation for the high-risk exposure would just be a waste of risk protection, without any e¤ect on the separation of typeh`(or`h) fromhh, since both are equally a¤ected by this high risk.^{14 ,15} Finally, from equations (13) and (14), low-risk households are covered by an insurance scheme that provides partial coverage at the margin. More explicitly, we have partial coverage of a single loss a¤ecting either r = 1 or r = 2if w_{``</sub><sup>00</sup>>minfw<sup>01</sup><sub>``}; w_{``</sub><sup>00</sup>g. Likewise, we have partial coverage of a second loss when a …rst loss a¤ects either r = 1 or r = 2, if w<sub>``}¹¹ < maxfw⁰¹_{``</sub>; w<sub>``}¹⁰g. According to Proposition 3, low-risk households have partial coverage in at least one of these two cases.

The fact that low-risk households should be partially covered at the mar- gin is not surprising, insofar as more risky households (i.e., types hh; h`and

`h) should be detered from choosing w_`` and, as in the Rothschild-Stiglitz

14Providing partial coverage of the losses caused by risk exposure r = 2 to type h`

individuals (i.e., choosing w_{h`</sub><sup>01</sup>< w<sup>00</sup><sub>h`}) is a way to deter typehhand `hindividuals from pretending that their type is h`. Maintaining full coverage forr= 1does not incentivize type hhindividuals to pretend that their type is h`, because r = 1is a high risk for hh and h`. IfIC_hh<sup>h`</sup> andIC<sub>`h</sub><sup>h`</sup> are both binding, then it would even be optimal to over-cover risk exposurer= 1of type h`individuals (i.e. to choosew¹⁰_{h`</sub>> w<sub>h`}⁰⁰). Since over-coverage has been excluded, typeh`individual have full coverage for riskr= 1.

15It can be checked that the strict inequalities in (11) and (12) become equalities if u_{h`</sub> =u<sub>hh</sub> andu<sub>h`} =u_hh, respectively. In this case, medium risks are pooled with high risk policyholders and they have full coverage.

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model, this goes through partial coverage. Going beyond this conclusion requires additional assumptions, as shown in Proposition 3.

Proposition 3 At a second-best Pareto-optimal allocation, low-risk house- holds are covered by an insurance scheme that combines risk-speci…c copays c¹_{``</sub>; c<sup>2</sup><sub>``} with a deductible D_{``</sub>, an out-of-pocket maximumM<sub>``} or an upper limit on coverage U L_``. If households are prudent (i.e. u⁰⁰⁰ >0), then combining risk-speci…c copays with a deductible or with an out–of-pocket maximum is optimal when the fraction of high-risk households hh is small enough.

The low-risk state-dependent wealth w<sub>``</sub> should reach a compromise be- tween providing risk coverage to low-risk households and satisfying the in- centive compatibility constraints of higher risk types, namelyIC<sub>hh</sub><sup>``</sup>; IC_{h`</sub><sup>``</sup> and IC<sub>`h}^{``</sup>. Intuitively,IC<sub>h`</sub><sup>``} will be all the easier, when the coverage of risk expo- sure r = 1 is lower, i.e. when c¹<sub>``</sub> is larger, because r = 1 is a high risk for type h` and a low risk for type ``. The conclusion is reversed for IC_{`h</sub><sup>``</sup>, with here a reason for increasing c<sup>2</sup><sub>``</sub>. Which copay should be the larger depends on the relative importance of typesh`and`h, including, among other things, their weights h`; `h and expected utility levels u<sub>h`}; u_`h. The non-adjacent incentive compatibility constraint IC_hh^`` does not justify treating risks r = 1 and r = 2 di¤erently. Under the joint e¤ect of adjacent and non-adjacent constraints, two cases are generically possible. If

w_{``</sub><sup>00</sup>+w<sup>11</sup><sub>``} > w¹⁰_{``</sub> +w<sup>01</sup><sub>``}; (15) then we know from condition (4) that risk-speci…c copays combined with either a deductible D_{``</sub> or an out-of-pocket maximum M<sub>``} are optimal. On the contrary, if

w_{``</sub><sup>00</sup>+w<sup>11</sup><sub>``} < w¹⁰_{``</sub> +w<sup>01</sup><sub>``}; (16) then risk-speci…c copays combined with an upper limit on idemnity U L_``

are optimal. Prudent policyholders are downside risk averse, and in that case, it seems intuitive that some priority should be given to the coverage of large losses. With the conclusions of section 2.2 in mind, we expect that (15) should hold, and thus that the optimal coverage of low-risk individuals should mix risk-speci…c copays with a deductible or an out-of-pocket maximum.

However, this intuition is undermined when theIC_hh^`` constraint is taken into account. Indeed, for a type ij household, the probability q_ij of having two

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losses, conditionally on the occurrence of at least one loss, is such that q_hh = p²_h

ph(2 ph); q_{h`</sub> = q<sub>`h} = p_hp_`

p_h+p_{`</sub> p<sub>h</sub>p<sub>`}; q_`` = p²_`

p_{`</sub>(2 p<sub>`}); with

q_`` < q_{h`</sub>=q<sub>`h}< q_hh.

Thus, providing a generous coverage in the case of two losses is particularly attractive to high-risk households, and incentive constraint IC_hh^`` justi…es reducing the coverage of low-risk households in the case of two losses. In other words, for low-risk households, prudence favors the coverage of large losses, but deterring high-risk households from pretending that they are low risk goes in the opposite direction.¹⁶ Proposition 3 shows that the balance is tipped in favor of the coverage of large losses through a deductible or an out- of-pocket maximum, mixed with risk-speci…c copays, when the proportion of high-risk households is small enough, so that they exert a relatively small e¤ect on the optimal insurance coverage of low-risk households.¹⁷

16It may be objected that the same mechanism is also at work for medium-risk house- holds, i.e. through constraintsIC_{h`</sub><sup>``</sup> andIC<sub>`h}^``. This is true, but with a weaker intensity, so that this mechanism does not cancel the e¤ect of downward risk aversion. See the proof of Proposition 3 for details.

17See the proof of Proposition 3 for a more explicit statement of the "for hh small enough" condition. Note that u_hh; u_{h`</sub>; u<sub>`h}; u_`` are considered as data, but in practice they result either from competitive market interactions and/or from public regulation.

Whatever the mechanism we may have in mind, it will not determine u_hh; u_{h`</sub>; u<sub>`h}; u_``

independently from _hh; _{h`</sub>; <sub>`h}; _``. Hence, the second part of Proposition 3 should not be overinterpreted: it allows us to understand the mechanism that leads to a speci…c form of coverage, rather than explaining what is expected to occur in markets where the fraction of small-risk households is small.

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