WORKING PAPER N°
Mots clés :
Codes JEL :
Efficient risk-sharing under adverse
selection and subjective risk perception
Arnold Chassagnon,
Bertrand Villeneuve
2002-19
Efficient risk-sharing under adverse selection and
subjective risk perception
∗
Arnold Chassagnon
†GREMAQ Universit´
e Toulouse I
DELTA Ecole Normale Sup´
erieure
48 boulevard Jourdan
75014 Paris
FRANCE
arnold@cict.fr
Bertrand Villeneuve
‡IDEI Universit´
e Toulouse I
Place Anatole France
31042 Toulouse cedex
FRANCE
bertrand.villeneuve@cict.fr
December 2nd 2002
Abstract
The present paper thoroughly explores second-best efficient allocations in an adverse selection insurance economy. We start from a natural extension of the classical model, assuming less than perfect risk perceptions. We propose first and second welfare theorems, by means of which we describe efficiency-enhancing policies. Notions of weak and strong adverse selection are promising for interpreting real world insurance arrangements.
Keywords : Principal-agent, adverse selection, welfare theorems, second-best optimum.
∗Thanks to David Bardey, Fran¸cois Salani´e for useful comments and Gerald Traynor. Earlier
versions of this paper were presented in Strasbourg, Lisbon, Rio de Janeiro and Venice.
†University of Toulouse (GREMAQ) and DELTA (Ecole Normale Sup´erieure, Paris). ‡University of Toulouse (CEA, IDEI, LEERNA).
1
Introduction
Inexistent and inefficient equilibria threaten the effective operation of competitive in-surance markets in the presence of adverse selection. After this finding of Rothschild and Stiglitz (1976), inexistence was initially considered to be particularly puzzling. Several authors proposed alternative equilibrium concepts relying on more sophis-ticated, but ad hoc, interactions between insurers. In a technically difficult paper, Dasgupta and Maskin (1986) prove that Nash equilibria in mixed strategies exist but their structure is still unclear. However, another branch of the literature soon con-sidered that the inability of market mechanisms to reach an efficient allocation was the primary problem. Indeed, the argument goes, the Rothschild-Stiglitz equilibrium concept fails because, under certain circumstances (too many low risk policyholders), the allocations the market can propose are necessarily inefficient. Why ? Because the market is unable to perform transfers between types, in other words to make low risk policyholders subsidize high risk ones. See Crocker and Snow (2000) for a review of this literature.
Completing an idea of Dahlby (1981), Crocker and Snow (1985a,b) and Henriet and Rochet (1987) showed that public regulation (compulsory uniform basic cover-age, taxes or price regulation) was able to implement a welfare improving transfer without superior information about policyholders on the part of public authorities. The market, released from the burden of transferring between types, is stabilized and the equilibrium allocation reached is efficient.
We propose a development of these results in a framework in which we distinguish between the policyholder’s perception of his idiosyncratic risk and the risk perfor-mance in his category, as measured by the insurer. A simple interpretation, in line with non-expected utility models, considers that policyholders may be optimistic or pessimistic in their choices, and that high risk and low risk may also differ from each other in this respect. For the sake of simplicity, we only consider two types.
What are the consequences of this generalization of the classical model for our understanding of efficiency and public policy? What can be said in a context in which efficient contracts do not necessarily give full insurance and in which different types
have different views about what is an efficient contract ?1 In this paper, we analyze
in detail the properties of incentive compatible optimal allocations in connection with wealth transfers. This is essential for any attempts to evaluate market mechanism and public policy. In particular, we are in a position to prove when and why the first welfare theorem for economies with asymmetric information does not work (equilibria may not be constrained optima); we also characterize redistributive policies that decentralize a chosen constrained optimum (the second welfare theorem).
To compare with the other insurance papers that combine adverse selection with nonexpected utility, note that our project differs from Young and Browne (2001), who are interested in the Rothschild and Stiglitz equilibria, and from Jeleva and Villeneuve (2002), who study the monopolist’s problem. We address normative questions in the most general terms rather than search for particular solutions to the optimal allocation problem.
In the conclusion, we discuss the debt we owe to our predecessors (Crocker and Snow 1985a,b and Henriet and Rochet 1987). One of our contributions is that we char-acterize in detail the whole set of second-best allocations, meaning that we are able to compare them in terms of coverage quality, binding constraints and cross subsidies; the latter is crucial for implementing a desired outcome with minimal informational requirements.
Another contribution is that we base all our proofs on revealed preference ar-guments. Avoiding analysis of the Lagrangean means we can propose constructive arguments that rely essentially on the single crossing property rather than on a par-ticular specification.
Finally, we propose a classification of adverse selection economies that relies on a comparison between risk disparity and risk perception disparity. When types perceive themselves as more different than they objectively are, we are in a situation of weak adverse selection in which a continuum of first-best allocations can be implemented despite asymmetric information. When types have an underdeveloped perception of themselves as being different, we are in a situation of strong adverse selection. In this
1Indeed, in Rothschild and Stiglitz (1976), all efficient contracts provide full insurance. Two
different efficient contracts can only differ by the level of uniform consumption they offer, therefore the one offering more wealth is preferred to the other by any type of policyholder.
case, no first-best allocation is implementable and a continuum of second-best efficient allocations offers pooling contracts, which, generically, offer too little coverage for one type and too much for the other.
The paper is organized as follows. Section 2 presents the insurance model with subjective belief and asymmetric information. To base the welfare analysis of trans-fers, the limitations of redistribution and the frontier of the feasible allocations are fully characterized in section 3. We show in particular that there is a trade-off be-tween insurance coverage quality and expected consumption. In section 4, we study the impact of risk perception on the feasible menus of contracts. We then propose a classification of economies based on the degree of adverse selection (weak or strong). The conclusion draws consequences of our analysis on market failures and remedies.
2
Model
2.1
Preferences, technology and contracts
Let us assume that there is a continuum of two types of consumers H and L in pro-portions λH and λL respectively (λH+ λL= 1), and one commodity in the economy.
Each consumer faces an individual risk, with two individual states s = 1 (no loss), and s = 2 (loss). The objective probability pi of being in state 2 (loss) for a type-i
individual is statistically known to insurers (pi = pH, pL with pH > pL). Individual
risks are assumed to be independently distributed, and both types have the same ini-tial contingent endowment ω = (ω1, ω2) ∈ R∗2+. We suppose that consumers evaluate
“expected” utility with the same VNM utility function u defined over R∗+. However,
they use different subjective probabilities (qH and qL, respectively). Subjective
prob-abilities could be seen as a monotonic transformation of the true probprob-abilities as in Kahnemann and Tversky (1979) or Quiggin (1982); to be more general, we do not assume a priori that qH and qL are ranked.2 To sum up, “expected” utility ex ante
is:
ui(ω) = (1 − qi) u(ω1) + qi u(ω2), i = H, L (1)
2Though the model is compatible with other non-expected utility models, we restrict ourselves
to smooth indifference curves (no “kinks” on the 45◦ line). However, provided preferences exhibit single-crossing, our analysis is still valid.
Consumers can purchase insurance provided by a risk neutral insurer. Insurance contracts consist in an exchange by the policyholder of risk ω for a conditional con-sumption plan x = (x1, x2).
Following Prescott and Townsend (1984), contracts allow state specific distribu-tions over outcomes, meaning that x1 and x2 must be seen as lotteries, or probability
distributions over R+. There are costs and benefits in using these abstract contracts.
One benefit is that we are more general from a normative point of view, but most significantly, many proofs are neatly simplified. Indeed, decision variables (of the social planner or the insurer) are now N -tuples of probability distributions over the consumption set (e.g. a pair of contracts is a quadruple of distributions). This implies that choice sets are linear, as well as the feasibility constraints, with respect to deci-sion variables. Linear programming results, like uniqueness or continuity with respect to exogenous parameters, can be invoked (see also Landsberger and Meilijson 1999).3 The cost is that they are not systematically used in the insurance literature and they do not seem to be observed empirically. The following shows that the latter argument does not contradict our approach since optimal contracts are always “degenerate”, i.e. consumption will depend on the state s, and on no other artificial random factor.
Given a contract x, the insurer’s payoff πi(x) depends on the consumer’s type:
πi(x) = (1 − pi) (ω1− Ex1) + pi (ω2− Ex2), ∀i = H, L (2)
and the consumer’s utility is
ui(x) = (1 − qi) Eu(x1) + qi Eu(x2), ∀i = H, L (3)
The expectation operator E only recalls that lotteries are allowed.
2.2
Optimal coverage
We define the coverage rate of a deterministic contract x by c(x) = u 0(x 1) u0(x 2) (4)
3An interpretation is that non-convexities do not concern adverse selection problems, a point on
With the usual terminology, full insurance means a coverage rate of 1, underinsurance a coverage rate of less than 1 and overinsurance a coverage rate of more than 1.
In any Pareto optimal allocation (xH, xL), no lotteries are used and each type’s
marginal rate of substitution is equal to that of the insurer, i.e. 1 − qi qi u0(xi1) u0(x i2) = 1 − pi pi , ∀i = H, L (5) A contract xi satisfying the above condition is said to be i-efficient. An i-efficient
contract covers type i at a unique rate c∗i such that c∗i = qi 1 − qi · 1 − pi pi (6) Efficiency does not mean full insurance when objective probability and beliefs differ. An optimistic consumer (qi < pi) has an optimal coverage strictly lower than the full
coverage rate (c∗i < 1), and the rate of coverage is higher than 1 for a pessimistic consumer (c∗i > 1).
The curve of contracts ensuring a certain coverage rate is an income expansion path (IEP) for all types. In the model, each commodity being normal, IEPs are strictly increasing in the plane x1−x2 and they never cross. IEPs are “perpendicular”
to profit curves in the sense that when we move from one contract of given coverage to another, profit varies in the same direction, whatever the policyholder’s type.
For a given profit πi made on type i, we denote by xi the i-efficient associated
contract. The following statements are equivalent: 1. Expected wealth Exi increases.
2. πi(xi) decreases.
3. For j 6= i, ui(xi) and uj(xi) increase.
2.3
Feasible allocations
When types are not observed by the insurer, it is generally impossible to implement Pareto optimal allocations just by assigning an i-efficient contract to type i and check-ing that profits are non-negative. Indeed, a pair of efficient contracts x• = (xH, xL)
is likely to violate one (or more) incentive compatibility constraints (Rothschild and Stiglitz 1976).
We apply the revelation principle to reason directly on menus, the idea being that any feasible allocation can be implemented via a direct mechanism in which con-sumers are offered the menu, and each individual, following his preferences, chooses the preferred option. We denote by F the set of feasible menus, i.e. menus that are incentive compatible and that satisfy the resource constraint:
x• ∈ F ⇔ λH πH(xH) + λL πL(xL) ≥ 0 uH(xH) ≥ uH(xL) uL(xL) ≥ uL(xH) (7)
F , comprising quadruples of probability distributions (each of the two contracts com-prises two lotteries), is linear and convex.
For any allocation x• = (xH, xL), there are two numbers πH and πL such that
πH(xH) = πH and πL(xL) = πL; wealth transfers from one type to the other can be
summarized by π• = (πH, πL).
For a given profit profile π• = (πH, πL), we define
Fπ• = {(yH, yL)|πH(yH) ≥ πH ; πL(yL) ≥ πL; uH(yH) ≥ uH(yL) ; uL(yL) ≥ uL(yH)}
(8) as the set of menus for which profits π• are feasible. By the same argument as before,
this set is linear and convex.
For a given x•, u• = (uH, uL) represents the respective utilities of type-H and
type-L consumers. The set of feasible payoffs u•, denoted by u(F ), is convex, since
the image of the linear convex set F by the expected utility linear operator is convex.
3
Welfare analysis of transfers
3.1
Redistribution constrained optima
The second fundamental theorem of welfare states that any redistribution is compat-ible with efficiency, provided that the Walrasian market mechanism determines the allocation. The following definition will serve to show how second-best economies depart from first-best economies.
Definition 1 (RCO) x• is a redistribution constrained optimum (RCO) relative to
The redistribution constrained Pareto criterion is the usual Pareto relationship re-strained to Fπ•. Under symmetric information, an RCO is always a Pareto optimum.
Under asymmetric information, this concept of efficiency is weaker than second-best optimality, since one optimizes the allocation, given the transfer from one type to the other, but we ignore for the moment whether the profits we consider are compatible with second-best efficiency.
The proposition shows similarities (1, 2 and 3) and differences (4) between first-and second-best economies. In particular we show that extreme transfers are never feasible in a second-best world. Results on second-best optimal allocations are set forth in the next subsection.
Π will denote the set of feasible profit profiles under adverse selection that redis-tribute all the wealth to policyholders.
Proposition 1 Under adverse selection, for all π•,
1. The budget constraint for each type is binding at the RCO, which means that profit profiles that are not in Π are never socially desirable. Π is a segment in R2.
2. At the RCO, each type receives maximum utility in Fπ•. The unique menu that
implements the RCO will be denoted by ˆx• = (ˆxL, ˆxH). Moreover, the application
Π → F , π• 7→ ˆx• is continuous.
3. Any RCO is made up of degenerate lotteries.
4. πi < (1 − pi)ω1+ piω2 (i = H, L) exists such that for all π• which is feasible and
redistributes all wealth to policyholders, πi < πi. In other words, redistribution
can never be extreme.
It is never socially desirable that the insurer remains with positive profit. The rest of the paper will essentially work with profit profiles in Π. These profiles are interpreted as transfers from one type to the other. Point 4 implies that extreme transfers of wealth are not feasible under adverse selection.
The Rothschild and Stiglitz (RS) allocation, that is the unique candidate equi-librium in the standard model, is in fact the RCO associated with the profit profile
(0, 0) (see point 2 of the proposition). For the very reason that an implementable profit profile may not be optimal, the RS allocation may not be a second-best optimum.
3.2
Efficient redistribution of expected wealth
Certain feasible transfers may not be compatible with second-best efficiency. In the following, an optimal transfer is a transfer for which the associated RCO is a second-best Pareto optimum.
Clearly, second-best allocations are on the frontier of u(F ). Nevertheless, the location of the other RCOs remains to be determined, though we know they are on the frontier of u(FΠ), the convex set of zero-profit feasible allocations.4 The theorem
shows the relationships between RCOs, the feasibility frontier and transfers in Π. Theorem 1 Denote by U the set of payoffs implemented by RCOs distributing all expected wealth to consumers.
1. U is a connected part of the frontier of u(F ).
2. The application Π → U , π• 7→ u• is one-to-one and continuous.
UNE (the North-East part of U ) is the set of second-best allocations.
INSERT FIGURE 1 ABOUT HERE
A corollary of this theorem is that the set of optimal transfers is a segment included in Π. Another corollary is that, when optima are sought via weighted sums of the types’ utilities, the greater the weight assigned to a type, the greater will be the subsidies this type receives. Inefficient RCOs correspond to negative weights given to one of the types.
3.3
Trade-off between expected wealth and coverage quality
The following theorem shows that the less favored type (in terms of expected wealth) has to be compensated with better quality of coverage. This trade-off is necessary for incentives to work properly. Moreover, for a given redistribution, first-best contracts
4Convexity comes from the fact that zero profit is just an additional linear constraint imposed
(in particular who envies whom) provide valuable information on the constraints that are binding at the RCO.
Theorem 2 Consider transfers in Π
1. Type i must be assigned an i-efficient contract at the RCO when such a contract could be given to all types without loss.
2. If type i gets an i-efficient contract at the RCO for a given transfer, then it also gets an i-efficient contract at the RCO for a less favorable transfer.
3. If type i gets an i-efficient contract at the RCO for a given transfer, type-i can increase its utility only through more favorable transfers.
4. Type i will not receive an i-efficient contract when first-best contracts xi and xj
violate type j’s incentives.
A corollary of this theorem is that there are two thresholds in redistribution levels, each separating, for a given type i, RCOs assigning i-efficient contracts from RCOs assigning i-inefficient contracts. Points 2 and 3 provide implicit bounds on the values taken by these thresholds. In the Rothschild–Stiglitz model (qH = pH and qL= pL),
the two thresholds are identical and there are only two regimes: (1) H receives low transfers but is assigned an efficient contract, and L is not fully insured, (2) H receives large transfers but is not fully insured, whereas L is fully insured.
In our more general setting, we retrieve this idea for relatively low and relatively high transfers. In RCOs for which transfers are sufficiently favorable to type i, type j gets an efficient contract (with a binding incentive constraint) and type i’s contract is inefficient. For intermediate transfers, depending on the ranking between thresholds, either both types or neither get efficient contracts at the RCO, but these cases are still unclear at this stage of the analysis. We come back to this in the following section.
4
The effects of risk perception
4.1
Feasibility and efficiency with polarized beliefs
Intuitively, differences in tastes facilitate the implementation of different contracts since envy-free conditions are easier to satisfy. Interpreted in terms of risk perception,
this idea suggests that, other things equal, increasing the disparity between beliefs alleviates incentive constraints.
In this section, we consider changes of the parameters which increase differences between consumers beliefs, without affecting the objective parameters pi and pj.
Definition 2 Consider beliefs Q = (qi, qj). Beliefs Qe = (qie, qje) are a polarization
of Q if, when qi > qj then qie ≥ qi and qj ≥ qje (with at least one strict inequality), and
conversely, when qi < qj then qie ≤ qi and qj ≤ qje (with at least one strict inequality).
Proposition 2 Let beliefs Qe be a polarization of beliefs Q.
1. The set of feasible menus associated with beliefs Qe is greater than the one associated with Q.
2. If the contract that type i gets at the RCO relative to beliefs Q and profits π• is
i-efficient, then the contract this type gets at the RCO relative to beliefs Qe and profits π• is also i-efficient.
One result of this proposition is that policyholders who consider themselves risky benefit from increased optimism of the other policyholders. Conversely, low risk policyholders prefer, as far as their direct interest is concerned, that high risk policy-holders consider themselves to be risky beyond their true risk.
In the particular but significant case where qH = qL(risk perception is independent
of the true risk), there is a unique RCO which is second-best optimal. This is a consequence of Theorem 1 and of the fact that types necessarily receive the same utility. This implies that there is a unique optimal redistribution between types that can be implemented. Any policy that improves risk perception (qH approaches pH and
qL approaches pL) extends the set of policies that the decision-maker may consider.
4.2
Weak and strong adverse selection
Risk perception is not only important for determining possible redistribution, but also plays a crucial role in defining the qualitative properties of the contracts implemented via RCOs.
The economy is said to exhibit weak adverse selection when the intersection of first-best efficient allocations and second-best allocations is non-empty (“informa-tional constraints may not matter”), and to exhibit strong adverse selection when the
intersection is empty (“informational constraints always matter”). Under weak ad-verse selection, intermediate RCOs comprise efficient contracts (no adad-verse effects of adverse selection, at least locally), while under strong adverse selection, intermediate RCOs comprise inefficient contracts for both types.
Incentive constraints imply that the type that gives more value to marginal cov-erage (i.e. with the higher q) gets more covcov-erage. Whether or not this consequence of the single crossing property is met by first-best allocations discriminates weak from strong adverse selection.
Proposition 3 The economy exhibits weak (strong) adverse selection if and only if subjective accident probability and optimal coverage are positively (negatively) corre-lated (qH − qL) · (c∗H − c
∗
L) ≥ (<)0.
Fix pH and pL. We show the different regimes where the model can be found in the
unit box where qH and qL vary from 0 to 1 . Given the condition of the proposition,
the curves qH = qL and c∗H = c ∗
L are the frontier between the two regimes. Notice
that c∗H = c∗L is a section of an ellipse passing through (1,1) and (0,0).5
INSERT FIGURE 2 ABOUT HERE
Figure 2 exhibits 3 different regions. Quite intuitively, strong adverse selection corresponds to a situation in which the contradiction between first-best requirements and feasibility constraints is stronger (for example if type-H policyholders believe that their accident probability is larger than type-L policyholders’, but they need less coverage at the first best). This is represented in the intermediary area.
The Rothschild and Stiglitz model is represented by the unique point qH = pH and
qL = pL. In any neighborhood of that economy, strong and weak adverse selection
are possible, which shows its non-generic nature in that respect.
Strong adverse selection is an instance of the phenomenon that Guesnerie and Laffont (1984) name nonresponsiveness. If, for example, the insurer imposes transfers via uniform compulsory insurance, and if the resulting position offers excess coverage for H and insufficient coverage for L, then the types will remain pooled. None of
5In factorized (non-polynomial) form, the equation is indeed
p H 1 − pH 1 − qH qH − p L 1 − pL 1 − qL qL = 0 (9)
the types is assigned an efficient contract. Our analysis shows that with the non-transferable utility functions that we have here, nonresponsiveness leads to pooling allocation only if the transfer between types is not extreme. Indeed, if the initial risk is high and if transfers are low, then each type will receive a supplementary insurance along its actuarially fair budget constraint.
5
Conclusion
As has already been recognized in the literature, the origin of inefficiency in the Rothschild and Stiglitz model hinges on the market’s inability to perform transfers between types. To overcome this obstacle, extra constraints on the insurers’ programs must be imposed. The simplest policy is as follows: the Government chooses the optimum they want to implement; they impose a basic uniform coverage performing the exact transfers required (by definition compatible with efficiency) and leave the market to reach the related RCO, now an RS allocation that is necessarily stable. Public insurance is not the only possible method, and taxes achieve the same goal (indeed, the two approaches are equivalent, see Crocker and Snow 1985a).
One contribution of this paper is a characterization of the structure of the set of efficiency compatible transfers. Crocker and Snow (1985a), who must be credited for sketching this program and proposing first developments, show that some transfers are always possible, but they remained unclear as to the degrees of freedom left for public choice.
Starting with a weaker notion than second-best optimality, we provide a new and useful understanding of the frontier of the feasible menus in a world where risk percep-tion may differ from objective probabilities. We have proven that possible transfers constitute an interval that excludes extreme transfers. Negative public insurance is even theoretically conceivable and is compatible with efficiency provided the RS allocation related to the initial risk is efficient.
The results on who receives and who does not receive an efficient contract come from the fact that first-best efficient contracts do not necessarily provide full insurance and that efficient contracts in general depend on the type of the policyholder. This imposes a thorough reconsideration of incentive constraints as we know them from the
Rothschild-Stiglitz model. Though the classical lesson remains, by and large, valid when transfers between types are extreme, the structure of the incentive constraints is upset when public policy seeks inequality reduction. Indeed, weak adverse selection illustrates that a certain degree of freedom in public choice between first-best alloca-tions is compatible asymmetric information, provided differences in risk percepalloca-tions are sufficiently large. Conversely, strong adverse selection shows that when percep-tions are close, optimal allocapercep-tions pool types. This means that certain policyholders are over-covered (according to their taste) and the others are under-covered.
A
Appendix
Technical note. In all the proofs, we adopt the weak topology for lotteries. To simplify, we never use the restriction “almost surely”. Two lotteries are considered as equal if their consequences can differ only for events of null probability (for practical purposes and economic interpretations, they are equal). Given that the optimal lotteries we find are always degenerate, and that the natural state of space is discrete (accident/no accident), the simplification is legitimate.
A.1
Proof of Proposition 1
A.1.1 Proof of point 1
We reason by contradiction. Suppose that RCO (ˆxi, ˆxj) is such that the insurer
makes a larger profit than πi when type i gets ˆxi = (˜x1, ˜x2). Then, as u is
con-cave, the insurer would also make a larger profit than πi when type i gets the
degenerate lottery (u−1(Eu(x1)), u−1(Eu(x2))). There is an open ball B around
(u−1(Eu(x1)), u−1(Eu(x2))) in which πi(·) > πi. Now we define the (degenerate)
contract xε,η = (x1, x2) by the following equations:
u(x1) = Eu(˜x1) + ε (10)
Profit functions being continuous, ε and η exist such that xε,η is in B and verifies
(1 − qH) ε + qH η > 0 (12)
(1 − qL) ε + qL η < 0 (13)
It follows that (xε,η, ˆxj) satisfies incentive constraints, belongs to Fπ•, and ui(xε,η) >
ui(ˆxi), a contradiction. Finally, note that Π is bounded, convex and constrained by
the linear zero profit condition. This implies that Π is a segment. A.1.2 Proof of point 2
Fix π• and suppose that Fπ• is non-empty. Define Cπ• as the set of contracts within
a menu of Fπ• (x• ∈ Fπ• ⇒ xL ∈ Cπ• ; xH ∈ Cπ•). Define x
M
i ∈ arg maxx∈Cπ• ui(x)
for i = H, L. These contracts are in Cπ•, since, by continuity of u, Cπ• is closed.
By definition, xM
H is in a feasible menu, therefore there is a contract XH ∈ Cπ•
such that uH(XH) ≥ uH(xMH) and πH(XH) ≥ πH (note that possibly, XH = xMH).
Similarly, xML is in a feasible menu, therefore there is a contract XL ∈ Cπ• such that
uL(XL) ≥ uL(xML) and πL(XL) ≥ πL(possibly, XL = xML). Moreover, xMH and xML are
such that uH(xMH) ≥ uH(XL) and uL(xML) ≥ uL(XH). All the preceding conditions
imply that menu (XH, XL) ∈ Fπ• dominates (weakly) any other menu of Fπ•. This
implies that there is at least a maximum element of Fπ• which is, necessarily, an
RCO.
For given transfers π•, RCOs are unique in terms of utilities implemented.
There-fore they are all maximum elements of Fπ•, meaning that they are all solution of the
following optimal program (the objective could be any other function increasing in uH and uL): max xH,xL uH(xH) + uL(xL) (14) s.t. uH(xH) ≥ uH(xL) uL(xL) ≥ uL(xH) πH(xH) ≥ πH πL(xL) ≥ πL
is never parallel to a constraint,6 therefore the solution is necessarily at a corner,
which implies that it is unique, and that π• → (ˆxH(π•), ˆxL(π•)) is continuous.
A.1.3 Proof of point 3
We reason by contradiction. Let ˆx• be the RCO associated with π•, where ˆxH =
(˜xH1, ˜xH2) and ˆxL = (˜xL1, ˜xL2), and where one of the four lotteries at least is
nonde-generate. Replacing each coordinate by its certainty equivalent,7 we get the “degen-erate” contracts XH = (XH1, XH2) and XL = (XL1, XL2) respectively, that give the
same utilities to type H and L, and that are strictly cheaper due to risk aversion. Uniqueness of the RCO (see point 2) proves that (XH, YL) is an RCO, a contradiction.
A.1.4 Proof of point 4
Consider a sequence of transfers (πn
•)n≥0 such that πHn converges to πH, and the
cor-responding RCOs (ˆxn•)n≥0. By continuity of function ˆx•(π•) (see point 2), sequence
(ˆxn
•)n≥0 converges to a pair of contracts,8 and this pair is necessarily a menu
be-longing to Fπ•. This proves that the set of feasible transfers is closed (the usual
convexity/linearity argument proves that this is an interval).
Equation πH(xH) = (1 − pH)ω1 + pHω2 implies that xH = 0. Therefore any pair
(0, xL) satisfying incentive constraints must satisfy xL = 0. However, menu (0, 0)
does not verify the zero profit constraint, meaning that the maximum value of πH,
which is attainable (previous paragraph), is strictly lower than (1 − pH)ω1 + pHω2.
The proof is similar for the other bound.
6For the profit conditions, this is evident: the expectation operator is not collinear to the expected
utility operator. For the incentive constraints, note that the two independent operators uH and uL
are combined differently to generate the objective and the constraints. This proves independence.
7For example, we replace ˜x
H1by XH1where u(XH1) = Eu(˜xH1), and so on for other lotteries. 8The set of feasible contracts is compact since the conditional wealth the insurer can give to a
A.2
Proof of Theorem 1
A.2.1 RCO payoffs are on the frontier of u(F )
Consider an RCO ˆx• = (ˆxH, ˆxL) distributing all expected wealth whose payoffs are
ˆ
u• = (ˆuH, ˆuL). We prove that ˆu• cannot be in the interior of u(F ). We reason by
con-tradiction: assume that ˆu• is in the interior of u(F ). Consider an open neighborhood
ˆ
u included in u(F ). Choose two points (ˆuH, ˆuL+ ε) and (ˆuH + η, ˆuL) in this
neigh-borhood such that ε > 0 and η > 0. We denote by y• = (yH, yL) (resp. z• = (zH, zL))
a menu implementing (ˆuH, ˆuL+ ε) (resp. (ˆuH + η, ˆuL)).
One can readily see that (ˆxH, yL) and (zH, ˆxL) satisfy incentive constraints. The
Pareto optimality of (ˆxH, ˆxL) in Fπ• implies that these pairs of contracts cannot
belong to Fπ•, and we must conclude that:
πL(yL) < πL(ˆxL) (15)
πH(zH) < πH(ˆxH) (16)
which in turn implies that (because global profit of (ˆxH, ˆxL) is zero):
πH(yH) > πH(ˆxH) (17)
πL(zL) > πL(ˆxL) (18)
Now consider the menu of lotteries (lHα, lαL) which pays y• with probability α and z•
with probability 1−α. (More precisely, this means that if state 1 occurs, a policyholder of type i will confront lottery yi1with probability α and zi1with probability 1−α, etc.)
Menu (lHα, lLα) belongs to F and Pareto dominates (ˆxH, ˆxL); moreover, by continuity
of profit functions, α0 exists such that
πH(lHα0) = πH(ˆxH) (19)
which implies that
πL(lLα0) ≥ πL(ˆxL) (20)
in contradiction with the fact that (ˆxH, ˆxL) is a maximum element in Fπ•.
A.2.2 U is connected in Fπ•
It has been proven in proposition 1 (point 2) that application π• 7→ ˆx• is continuous.
image by a continuous function is connected.
A.2.3 Application Π → U , π• 7→ u• is one-to-one and continuous
Each redistribution level corresponds to a unique RCO (proposition 1, point 2), and to a unique element of U . In addition, each point of U corresponds to a unique RCO. Suppose, on the contrary, that two RCOs (ˆxH, ˆxL) and (ˆyH, ˆyL) implement the same
payoffs (ˆuH, ˆuL). Without loss of generality, suppose that πH(ˆxH) > πH(ˆyH). This
condition implies that (ˆxH, ˆyL) implements the same utility as the CPOs, is feasible
and makes strictly positive profits. This is impossible.
A.3
Proof of Theorem 2
A.3.1 Proof of point 1
Suppose that for a given transfer, the j-efficient contract xj could be given to all
types without loss. Deviation of agent-i on the j-efficient contract does not cause negative profit and consequently, (xj, xj) is a feasible menu (it satisfies global profit
constraint and trivially, revelation constraints). Theorem 1 implies that (xj, xj) is
Pareto dominated by (ˆxi, ˆxj). Particularly, uj(xj) ≤ uj(ˆxj). Then, as xj is j-efficient,
we conclude ˆxj = xj. Type j gets an efficient contract at the RCO.
A.3.2 Proof of point 2
Point 2 is proved for i = H. Symmetric arguments for L are not developed. We will use the following notations. Consider two transfer systems π• = (−(λL/λH)π, π)
and π•+ = (−(λL/λH)π+, π+) such that H is increasingly subsidized (π < π+). We
define ˆx• and ˆx+• as the corresponding RCOs, and x• and x+• as the respective pairs
of efficient contracts. To shorten statements, we say that a transfer system is singular for H if H obtains an H-efficient contract at the RCO and if, at the same time, there is at least one other transfer system less favorable to H, for which type H does not get an H-efficient contract. In fact, we prove that singular transfers do not exist. If π+
• is singular, then, for any transfer π• less favorable to H such that
ˆ
condition ˆxH 6= xH implies that (xH, ˆxL) cannot be a menu. However, this pair would
make a non-negative profit if H took xH and L took ˆxL. Moreover type-H’s incentive
constraint is verified: indeed, by definition uH(ˆxL) ≤ uH(ˆxH) and uH(ˆxH) ≤ uH(xH).
Therefore the problem must come from type L’s incentive constraint, i.e.
uL(ˆxL) < uL(xH) (21)
Consider the pair (xH, ˆx+L). It generates positive profits,9 and it satisfies
type-L’s incentive constraint (indeed, uL(ˆx+L) ≥ uL(x+H), and from inequality uL(x+H) >
uL(xH), we deduce that uL(ˆx+L) > uL(xH)). However, it cannot be a menu since
it would be an element of F(−(λL/λH) π,π) giving more utility to type H than ˆxH.
Therefore type-H’s incentive constraint is necessarily violated, i.e. uH(xH) < uH(ˆx+L).
It follows that
uH(ˆxH) < uH(ˆx+L) (22)
If π+
• is singular, then, there is an open neighborhood of π+• such that
any transfer π• in that neighborhood less favorable to type H gives at the
RCO an H-efficient contract to type H. Suppose by contradiction that there is a sequence of transfers, (πn
•)n∈N∗, less favorable to type H than π+•, converging to π+•,
and such that the contract assigned to H is not H-efficient. We denote by (ˆxnH, ˆxnL) the respective RCOs (by continuity, the sequence converges to (x+H, ˆx+L)). We can apply (21) and (22) to each pair of transfers (π+
•, π•n) (for any n ∈ N). At the limit,
we obtain by continuity of application ˆx•(π•)
uL(ˆx+H) ≤ uL(ˆx+L) ≤ uL(x+H) (23)
uH(ˆx+L) ≤ uH(ˆx+H) ≤ uH(ˆx+L) (24)
(lower bounds come from incentive constraints). But ˆx+H = x+H (π+
• is singular), hence:
uL(ˆx+H) = uL(ˆx+L) (25)
uH(ˆx+H) = uH(ˆx+L) (26)
which implies, because of the single crossing property: ˆ
x+L = ˆx+H = x+H (27)
9If H took x
One implication of (27) is that contract x+H could be given to all types without loss. That should also be the case for any H-efficient contract corresponding to transfers less favorable to H,10 in particular for xn
H, n ∈ N. Point 1 of theorem 2 implies that
type H gets an H-efficient contract for the transfer system πn
•, n ∈ N, which is a
contradiction.
There is no singular transfer Suppose that π+
• is singular and consider the
nonempty set of transfers, T , less favorable to type H, such that type H does not get an H-efficient contract at the RCO. Define transfer π•∞ = (−(λL/λH) π∞, π∞)
as the upper limit of T (the most favorable to H): π∞ ≤ π+. Either type H gets
an H-efficient contract at the RCO for transfer π∞• , or not. If type H gets an H-efficient contract at the RCO for transfer π•∞, that transfer would be singular with elements of T in any of its neighborhoods: this is impossible (preceding paragraph). So, type H must not get an H-efficient contract at the RCO for transfer π•∞ and π∞ < π+. Consequently, all transfer system between π∞
• and π•+ must give an
H-efficient contract to type H at the RCO. However, it is immediately apparent that the set of transfers such that type H obtains an H-efficient contract is closed. So π•∞ must give an H-efficient contract to type H at the RCO, which is clearly absurd. A.3.3 Proof of point 3
Suppose that type i gets an i-efficient contract at the RCO for a given transfer. Then Theorem 2 (point 2) has proved that the RCO associated with transfers less favorable to type i are also i-efficient. However, the maximum utility that type i can get decreases with its expected wealth. Consequently, any RCO associated with transfers less favorable to type i gives less utility to type i: only transfers more favorable to type i can increase this type’s utility.
A.3.4 Proof of point 4
Suppose that for a given transfer, type j prefers the i-efficient contract xi to the
j-efficient contract xj: uj(xi) > uj(xj). However for that transfer system, the RCO is
such that uj(xj) ≥ uj(ˆxj). This implies uj(xi) > uj(ˆxj), which is impossible. Type i
cannot obtain an i efficient contract at the RCO.
A.4
Proof of Proposition 2
In what follows, for any contract z, ue
i(z), uej(z), ui(z), uj(z) and uk(z) represent the
policyholder’s utility for respective beliefs qie, qej, qi, qj and qk.
In this proof we use the fact that the preferences of different types are ranked following the single crossing property: if two contracts are similarly ranked for two different beliefs, they will also be ranked in the same way for any “intermediate” belief. We summarize that point using the following notation:
SCP(qi, qk, qj) : ui(Xi) ≥ ui(Xj) uj(Xi) ≥ uj(Xj) (qi− qk)(qj− qk) < 0 =⇒ uk(Xi) ≥ uk(Xj) (28) In particular SCP(qe
i, qi, qej) and SCP(qie, qj, qje) are true.
A.4.1 Proof of point 1
Suppose that (Xi, Xj) is a menu relative to parameters (Q, π•). Incentive constraints
are satisfied:
ui(Xi) ≥ ui(Xj) (29)
uj(Xi) ≤ uj(Xj) (30)
Then, one must conclude from SCP(qe
i, qi, qje) and SCP(qie, qj, qje) that:
uei(Xi) ≥ uei(Xj) (31)
uej(Xi) ≤ uej(Xj) (32)
Equations (31) and (32) mean that incentive constraints relative to contracts Xi
and Xjand to parameter (Qe, π•) are verified. Feasibility constraints are not modified.
A.4.2 Proof of point 2
We denote by Xi the i-efficient contract relative to parameter (Q, π•) and by Yi the
i-efficient contract relative to parameter (Qe, π•). We denote by (Xi, ˆXj) the RCO
relative to parameter (Q, π•) and by ( ˆYi, ˆYj) the one relative to parameter (Qe, π•).
We treat the case qi = qei separately from the case qi 6= qei.
Case qi = qie. (Xi, ˆXj) is a menu relative to parameter (Q, π•). Then, Theorem 2
(point 1) implies that this pair of contracts is a menu relative to parameters (Qe, π•).
Proposition 1 (point 2) implies that this menu is Pareto dominated by the RCO ( ˆYi, ˆYj). Formally
uei( ˆYi) ≥ uei(Xi) (33)
However, as the preference parameters qi and qei are identical, Xi is i-efficient relative
to parameter (Qe, π
•). Then, it transpires that equation (33) implies: ˆYi = Xi = Yi
; type i gets an i-efficient contract for the RCO relative to parameter (Qe, π •).
Case qi 6= qie. The difference between qi and qei implies that agents of respective
subjective risk perception qi and qie prefer their own i-efficient contracts:
uei(Xi) < uei(Yi) (34)
ui(Xi) > ui(Yi)
Thus it follows from SCP(qe
i, qi, qje) that
uej(Xi) > uej(Yi) (35)
We know from point 1 of this proposition that menu (Xi, ˆXj) is feasible relative to
parameters (Qe, π•), hence:
uei(Xi) ≥ uei( ˆXj) (36)
uej(Xi) ≤ uej( ˆXj) (37)
No further data are needed to verify that (Yi, ˆXj) is a menu relative to
parame-ters (Qe, π
•). Feasibility constraints are immediate. It is also proved that incentive
compatibility constraints are satisfied:
uei(Yi) ≥ uei( ˆXj) (38)
Indeed, (38) is deduced from (34) and (36), while (39) is deduced from (35) and (37). ¿From Proposition 1 (point 2), we know that the RCO ( ˆYi, ˆYj) relative to
parame-ters (Qe, π
•) Pareto dominates any menu that is feasible for parameters (Qe, π•), and
particularly, the menu (Yi, ˆXj). This means:
uei( ˆYi) ≥ uei(Yi) (40)
which implies that ˆYi = Yi ; type i gets an i-efficient contract for the RCO relative
to parameter (Qe, π •).
A.5
Proof of Proposition 3
Under weak adverse selection, there is at least one transfer system such that both types get an efficient contract at the RCO, and then c(ˆxH) = c∗H and c(ˆxL) = c∗L.
This implies that (qH − qL) · (c∗H − c∗L) ≥ 0. For the reciprocal, there are two cases to
be considered.
qH > qL and c∗H > c ∗
L. Denote by xI the contract at the intersection of the two
curves of equations c(x) = c∗H and λH πH(x) + λL πL(x) = 0. Consider profit profile
(πH(xI), πL(xI)) and the RCO for this profile. Clearly xI = xH therefore ˆxH = xH is
H-efficient.
We apply the same argument for yI, the contract at the intersection of the two
curves of equations c(x) = c∗Land λH πH(x) + λLπL(x) = 0. The corresponding profit
profile assigns an L-efficient contract at the CPO to type-L policyholders. However, since c∗H > c∗L, the transfer system implicitly defined by yI is more favorable to type
L than (πH(xI), πL(xI)). So from Theorem 2 (point 2), we deduce that type-L
poli-cyholders also get an L-efficient contract at the RCO associated to (πH(xI), πL(xI)).
Consequently (xH, xL) is the RCO associated with this transfer. We are in a situation
of weak adverse selection.
qH < qL and c∗H < c∗L. We apply the intermediate value theorem to define implicitly
a transfer system so that for that particular transfer, the associated H- and L-efficient contracts are such that uH(xH) = uH(xL). Given that c(xH) < c(xL), it follows that
implies that uL(xH) < uL(xL), which proves that (xH, xL) is feasible. We are in a
References
[1] Bisin A. and Gottardi P. (2000): “Efficient Competitive Equilibria with Ad-verse Selection”, Universit`a di Venezia, mimeo.
[2] Briys E. and Louberge H. (1985): “On the Theory of Rational Insurance Purchasing: A Note,” Journal of Finance, 40, 577–581.
[3] Crocker, Keith J. and Arthur Snow (1985a): “The Efficiency of Competi-tive Equilibria in Insurance Markets with Asymmetric Information,” Journal of Public Economics, 26, 207–219.
[4] Crocker, Keith J. and Arthur Snow (1985b): “A Simple Tax Structure for Competitive Equilibria and Redistribution in Insurance Markets with Asymmet-ric Information,” Southern Economic Journal, 51, 1142–50.
[5] Crocker, Keith J. and Arthur Snow (2000): “The Theory of Risk Classifica-tion,” Chapter 8 in Handbook of Insurance, edited by Georges Dionne, Kluwer Academic Press.
[6] Dahlby, B. G. (1981): “Adverse Selection and Pareto Improvements through Compulsory Insurance,” Public Choice, 37, 547–558.
[7] Dasgupta, Partha and Eric Maskin (1986) : “The Existence of Equilibrium in Discontinuous Economic Games, II: Applications,” Review of Economic Studies, 53, 27–41.
[8] Dionne, G. and Fombaron N. (1996): “Non-Convexities and the Efficiency of Equilibria in Insurance Markets with Asymmetric Information”, Economics Letters, 52, 31–40.
[9] Doherty N.A. and Eeckhoudt L. (1995): “Optimal Insurance without Ex-pected Utility: The Dual Theory and the Linearity of Insurance Contracts,” Journal of Risk and Uncertainty, 10, 157-179.
[10] Guesnerie R. and Laffont J.J. (1984): “A Complete Solution to a Class of Principal-Agent Problems with an Application to the Control of the Self-Managed Firm,” Journal of Public Economics, 25, 329–369.
[11] Henriet, D. and J.-C. Rochet (1987): “Price Regulation in Insurance Markets with Asymmetric Information,” Economics Letters, 24, 327–329.
[12] Jeleva, M. and Villeneuve B. (2002): “Insurance Contracts with Imprecise Probabilities and Adverse Selection”, mimeo.
[13] Kahneman D. and Tversky A. (1979): “Prospect Theory: An Analysis of Decision under Risk,” Econometrica, 47, 263–292.
[14] Landsberger M. and Meilijson I. (1999): “A General Model of Insurance under Adverse Selection,” Economic Theory, 14, 331–352.
[15] Prescott, E.C. and Townsend R.M. (1984): “Pareto Optima and Compet-itive Equilibria with Adverse Selection and Moral Hazard”, Econometrica 52, 21–45.
[16] Quiggin J. (1982): “A Theory of Anticipated Utility”, Journal of Economic Behavior and Organization, 3, 324-343.
[17] Rothschild M. and Stiglitz J.E. (1976): “Equilibrium in Competitive Insur-ance Markets: An Essay on the Economics of Imperfect Information.” Quarterly Journal of Economics, 90, 629–649.
[18] Schmeidler D. (1989): “Subjective Probability and Expected Utility without Additivity”, Econometrica, 57, 571–587
[19] Yaari M. (1987): “The Dual Theory of Choice under Risk”, Econometrica, 55, 95–105.
[20] Young, Virginia R. and Mark J. Browne (2001): “Equilibrium in Competitive Insurance Markets Under Adverse Selection and Yaari’s Dual Theory of Risk,” Geneva Papers on Risk and Insurance Theory 25, 141–157.
uH
u(F) uL
Figure 1 : Feasible utility set and RCOs
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