Two-Persons Eﬃcient Risk-Sharing and Equilibria for Concave Law-Invariant Utilities

Eﬃcient risk-sharing rules and equilibria between two agents with utilities in a class that contains the Rank Dependent Expected Utility (RDU) are fully characterized. Speciﬁc attention is given to the RDU. Call-spreads and contracts with mixed regimes are shown to be eﬃcient. Closed-form solutions are obtained for several examples.


Introduction
A fundamental issue of economics of uncertainty has been the allocation of contingent goods among consumers. In the literature, agents have mainly been assumed to be expected utility maximizers (denoted EU from now on) with homogeneous probability beliefs. Risk-sharing and pricing rules have extensively been discussed in that framework and numerous applications given to asset pricing and insurance (see Huang and Litzenberger [14] and Gollier's [12] textbooks and the references therein). However, there are a number of insurance indemnity schedules that are commonly offered in insurance markets, which don't seem to be efficient for the differentiable von-Neumann Morgenstern model. For example, the contract offered by the FDIC to reinsure American banks againt their losses is a call-spread and gives no reimbursement for low values of the loss, has an upper limit and is linear in between. Swiss health insurance companies provide no reimbursement for small values of the loss and, for high values of the loss, the reimbursement equals the loss minus some fixed amount. In contrast with a standard deductible contract, the reimbursement is a nonlinear function for intermediate values of the loss.
We argued in [2] that the rank dependent expected utility model with continuous or discontinuous distortions (denoted RDU from now on) could account for the coexistence of upper limits and deductibles. In this paper, we go one step further: we fully characterize risk-sharing and pricing rules between two agents for a class of utility functions that contains the RDU. This class accounts for call spreads, contracts with mixed regimes that include full insurance and no insurance for some intervals of the risk. In [3], we showed that the same class of utility functions could be used to model strangled demand functions. In this paper, we show that efficient risk-sharing rules are also frequently strangled.
The class considered is the set of concave utilities of the form that are second-order stochastic dominance preserving (henceforth SSD). In (1), X is a random variable on a non atomic space, (Ω, B, P ), with distribution function F X and F −1 X is a version of the inverse of F X or quantile of X. The term g(F −1 X (0)) = g(essinf X) accounts for a specific weight given to the minimal value of X. We refer to Chew and Epstein [6], Green and Jullien [13] and Chew and Wakker [8] for axiomatic foundations. Following Chew and Epstein [6], we will call utilities of type (1) Rank-Linear Utilities (henceforth RLU). We show that for a utility V of the form (1), SSD preservation is roughly equivalent to concavity.
The class of utilities of type (1) contains the set of Choquet integrals of U (X), a concave function of X with respect to a convex distortion f : [0, 1] → [0, 1] continuous or discontinuous at 1. When f is discontinuous, defines a utility of the type (1) with L(t, x) = f (1 − t)U (x) and g(x) = (1−f (1 − ))U (x) and f (1 − )) = lim x↑1 f (x). When the distortion is continuous, the first term is zero and g = 0.
In this paper, we consider efficient risk-sharing rules between two agents having SSD preserving utilities of type (1) and aggregate wealth X 0 with a continuous distribution function F X 0 . Any non comonotone sharing rule being dominated by a comonotone sharing rule (see Landsberger and Meilijson [15]), attention may be restricted to comonotone sharing rules, hence to pairs of non decreasing functions of aggregate risk. In [4], it is shown that the problem of efficient risk-sharing rules may be brought down a calculus of variations problem with two monotonicity constraints. This problem, new in the economics literature, may be solved by using a generalization of the ironing procedure of Mussa and Rosen [16]. More precisely, there is a partition of the set of values of aggregate risk into: -sub-intervals on which the consumption of agent 1 is constant hence, on which agent 2 bears the full risk, -sub-intervals on which the consumption of agent 2 is constant hence, on which agent 1 bears the full risk, -sub-intervals on which both consumptions are increasing functions of the risk and determined by first-order conditions.
As a by-product of our method, we are able to compute prices supporting efficient allocations. These prices are increasing functions of aggregate endowment. They are proportional to the marginal utility of the agent whose consumption is increasing in X 0 .
Special attention is paid in the paper to the RDU case. We recall that the Choquet expectation with respect to a convex distortion f corresponds to the multiple priors model where the priors are the densities that dominate f in the sense of second-order stochastic dominance (see [2]). Therefore, we shall say that two agents with respective (smooth) distortions f 1 and f 2 have similar beliefs (or that there is weak heterogeneity of beliefs) whenever the function (ln(f 2 )) − (ln(f 1 )) = f 2 f 2 − f 1 f 1 is small. Defining agent i's uncertainty aversion index as f i /f i , agents thus have similar beliefs whenever they have similar uncertainty aversion indices.
As an application of our method, we prove that when there is weak heterogeneity of beliefs, agents behave as if they were expected utility maximizers with heteregeneous beliefs having densities f i (1 − F X 0 (X 0 )), i = 1, 2 with respect to P . In particular, when f 1 = f 2 (no heterogeneity), agents behave as EU maximizers with probability f (1 − F X 0 (X 0 )). If, on the contrary, there is very high heterogeneity of agents' beliefs, then the more uncertainty averse agent is fully insured by the other. For example if one RDU agent has a high uncertainty aversion index while the other is EU, then the RDU agent is fully insured by the EU agent.
The results just stated assume global properties of agents' distortions. Focusing on local properties of agents' distortions, we show that, when one agent distorts much more small probabilities than the other, she is fully insured for high values of aggregate endowment.
In the case of discontinuous distortions, we prove that one of the agent must be fully insured for low values of aggregate endowment. In the case where one agent is averse to the worse state and the other is not, then the former is fully insured for low values of aggregate endowment. Aversion to the worse state or high distortion of large probabilities induces a minimal consumption while high distortion of low probabilities induces satiation. For a general pair of distortions, efficient risk-sharing exhibit several regimes as described in the ironing procedure above.
The paper is organized as follows. In section 2, we introduce utilities of type (1) and state some of their properties. We characterize SSD preserving utilities of type (1) and discuss their differentiability properties. In section 3, the risk-sharing and equilibria problems are brought down to a calculus of variations problem with two monotonicity constraints. Section 4 provides optimality conditions and first applications. Section 5 is devoted to risksharing between two RDU. In section 6, we study two examples of risksharing rules and equilibria: an RLU example and an example of logarithmic utilities and power distortions.
2 On rank linear utility functionals

Definitions
Given as primitive is a probability space (Ω, B, P ). Let X be a random variable and let F X (t) = P (X ≤ t), t ∈ R be its distribution function. The generalized inverse of F X is defined by: We recall that X dominates Y in the sense of second order stochastic dominance (SSD), denoted X 2 Y , if E(U (X)) ≥ E(U (Y )), for every utility function u : R → R concave nondecreasing and X strictly dominates Y in the sense of SSD (notation X 2 Y ) if E(U (X)) > E(U (Y )) for every strictly concave nondecreasing utility function U . We also recall that X 2 Y if and only if and that X 2 Y if and only if the inequality is strict in (3), for every decreasing function g.
The fact that two random variables X on (Ω, B, P ) and Y on (Ω , B , P ) have the same probability law will be denoted X d ∼ Y . For a map V : Since X d ∼ Y is equivalent to X 2 Y and Y 2 X, SSD preserving functions are law invariant. As X + Y 2 X for any Y ≥ 0, Y = 0, SSD preserving functions (strictly SSD preserving functions) are monotone (strictly monotone). The converse is not true. For a counterexample, see Dana [9]. However, we have the following result proven in Dana [9]: Proposition 1 Let (Ω, B, P ) be non-atomic and let V : L ∞ (Ω) → R∪{−∞} be concave, σ(L ∞ (Ω), L 1 (Ω)) upper semi-continuous. Then V is SSD preserving if and only if V is law invariant and monotone.
In the remainder of the paper, we shall assume that (Ω, B, P ) is nonatomic, that is, there exists a random variable U on (Ω, B, P ) uniformly distributed on [0, 1].

Stochastic dominance
In the paper, we consider utilities of the form: with Necessary and sufficient conditions for a utility V L defined by (5) to be SSD preserving are next provided (the proof is given in the appendix) . For the sake of simplicity, L is assumed to be smooth. Proposition 2 Let (Ω, B, P ) be non atomic and V L be of type (5). Let L ∈ C 2 ([0, 1] × R). The following are equivalent: If, in addition, L(t, .) is strictly concave for every t ∈ [0, 1] or ∂ tx L < 0, then V L is strictly SSD preserving.
Necessary and sufficient conditions for X → g(F −1 X (0)) to be SSD preserving read as follows.
Proof. Since SSD preserving functions are monotone, the monotonicity of g is a necessary condition. To prove the sufficiency of this condition, let us recall that X 2 Y can be characterized by the inequalities

Choquet integral and RDU
Important examples of utilities of type (4) are given by the Choquet integral with respect to a (possibly discontinuous) convex distortion (Yaari utility) and by Rank-Dependent Utility (RDU). A convex distortion is a convex increasing map f : Since f is nondecreasing, convex and finite, it is differentiable a.e. and f ∈ L 1 + [0, 1]. When f is continuous, one has . Since E f is translation invariant and X + X ∞ ≥ 0, we may assume that X ≥ 0. We then have: Hence a Choquet integral with respect to a discontinuous convex distortion is a utility of type (4) with L(t, It may also be written as an ε-contamination of the Choquet integral with respect to the continuous convex distortionf , Ef (X): Given a utility index U , the RDU is defined by E f (U (X)), hence E f (U (X)) = )dt if f is continuous and if f is discontinuous. An RDU is therefore a utility of type (4) with L(t, x) = f (1 − t)U (x) and g(x) = (1 − f (1 − ))U (x). From proposition 2, the RDU is SSD preserving if and only if U is concave nondecreasing (see also the seminal paper of Chew et al [7]).

Differentiability properties
In this subsection, we identify the superdifferential of a convave utility of type (5) and the set where it is Gâteaux differentiable. We assume that L is of class C 2 (see [5] for milder assumptions and proofs) and satisfies: The superdifferential of V L at X ∈ L ∞ (Ω) denoted ∂V L (X) is defined by: Let us recall that V L is Gâteaux-differentiable at X if the map: defines a continuous linear form on L ∞ (Ω), denoted V L (X).
The following theorem summarizes the results from [5] that will be used in the paper: Theorem 1 Let X ∈ L ∞ (Ω), then the following holds: 2. ∂V L (X) := co{∂ x L(Z, X), Z uniformly distributed, comonotone with X} where co denotes closed convex hull operation for the L 1 (Ω) topology, 3. any element of ∂V L (X) is anticomonotone with X,

5.
V L is Gâteaux-differentiable at X if and only if F X is continuous, in which case, one has: Let us consider the RDU case. In that case L(t, x) = f (1 − t)U (x). We thus obtain (see also [2]) that when V L is Gâteaux-differentiable, one has: and in general, Let us now consider the case of terms involving F −1 X (0) = essinfX in the utility function. Given g a concave function of class C 1 on R such that g > 0 on R, let us define V 0 (X) := essinfX, and V (X) := g(V 0 (X)), ∀X ∈ L ∞ .
It is straightforward to check that V 0 and V are concave and continuous (for the norm topology), hence superdifferentiable. Let us denote by S the simplex of (L ∞ ) : The following result gives differentiability properties of V 0 and V (see the appendix for a proof) Proposition 3 Let X ∈ L ∞ , then ∂V 0 (X) = {µ ∈ S : µ, X = essinfX}, moreover ∂V 0 (X) = ∂V 0 (g(X)) and Note that ∂V 0 (X) ∩ L 1 may be empty. When ∂V 0 (X) ∩ L 1 = ∅, one has: Since the superdifferential of the sum of continuous concave functions is the sum of their superdifferentials, we deduce from proposition 3 and theorem 1 the superdifferential of utilities of form (1).

A review of known results
We consider a two agents exchange economy under uncertainty. Let W i ∈ L ∞ + , i = 1, 2 be agent's i initial endowment and X 0 := W 1 + W 2 be the aggregate endowment. We assume that F X 0 is continuous. Agent i is characterized by a utility, V i : L ∞ + → R assumed to be strictly SSD preserving, σ(L ∞ , L 1 ) upper semi-continuous and concave. A feasible allocation is a pair for every i (with a strict inequality for some i). A Pareto efficient allocation is a feasible allocation which is not is strictly dominated. A We recall that whenever utilities are superdifferentiable, (X * 1 , X * 2 , Ψ * ) (with (X * 1 , X * 2 ) feasible) is an interior equilibrium with transfer payments iff there exists λ ∈ (0, 1) and α > 0 such that In particular this implies that (X * 1 , X * 2 ) solves the problem hence is Pareto efficient. We finally recall that, from Walras' law, a triple feasible is an equilibrium if it is an equilibrium with transfer payments such that Since comonotone feasible allocations play a crucial role in risk-sharing theory when utilities are SSD preserving, we next recall a few basic results on comonotonicity.
A feasible allocation (X 1 , X 2 ) is comonotone (see Denneberg [10]) if and only if there exists a pair of non decreasing functions (h 1 , h 2 ) on R such that h 1 + h 2 =Id, X 1 = h 1 (X 0 ), and X 2 = h 2 (X 0 ) a.e.. Hence, in this case, one has: Therefore if (X 1 , X 2 ) is comonotone, as h 1 + h 2 =Id, we obtain that Introducing the set: we deduce from (8) and (9), that the feasible allocation (X 1 , X 2 ) is comonotone if and only if there exists x ∈ A with such that X 1 = x(F X 0 (X 0 )) and In the next proposition, we gather the previous facts and a number of useful well-known results on Pareto optimal allocations.
for some x ∈ A.
2. Any non comonotone feasible pair is strictly dominated by a comonotone feasible pair.
Domination by comonotone pairs was originally proven by Landsberger and Meilijson [15]. Strict dominance has been proven by Carlier and Dana [1,2]. Assertion 3 is well-known.

Reduction to quantiles problems
From proposition 4, attention may be restricted to random variables of the form Defining A by (10), let us consider the quantile problem: The next proposition shows that Pareto optimal allocations may be described in terms of the solutions to (P λ ).
) and x λ is a solution of (P λ ).
Proof. Assume that X * λ = x(F X 0 (X 0 )) solves (P λ ). Let y ∈ A and Y := y(F X 0 (X 0 )). We have Conversely, assume that x λ ∈ A is a solution of (P λ ) and let X * λ is a solution of (P λ ).

RLU equilibria
In the preceding subsections, utilities were only assumed to be strictly SSD preserving, σ(L ∞ , L 1 ) upper semi-continuous and concave. In the remainder of the section, in order to do explicit computations, utility functions are assumed to be of the form These assumptions ensure that V i is strictly SSD preserving and superdifferentiable.
The next results show that, as standard in Negishi's method, interior equilibria with transfer payments may be parametrized by the utility weight λ and that they may easily be obtained from the solutions of (P λ ). Proposition 6 Assume that g 1 = g 2 = 0, then (X * , X 0 − X * , Ψ * ) is an interior equilibrium with transfer payments iff there exists λ ∈ (0, 1) such that 1. X * = x λ (F X 0 (X 0 )) and x λ is a solution of (P λ ), Proposition 6 enables us to compute prices supporting efficient allocations: they are are increasing functions of aggregate endowment and more precisely, they are proportional to the marginal utility of the agent whose consumption is increasing in X 0 . The proof of proposition 6 may be found in the appendix. The previous characterization may be extended to the case g 2 = 0, g 1 = 0 (see the appendix for a proof): Proposition 7 Assume that g 2 = 0 and g 1 > 0, then (X * , X 0 − X * , Ψ * ) is an interior equilibrium with transfer payments iff it satisfies the conditions of proposition 6. Moreover, in that case, Using (6) and proposition 6, interior equilibria may now easily be characterized: ) and x λ is a solution of (P λ ) for some λ ∈ (0, 1), It follows from the previous characterization that existence of equilibria amounts to find a λ ∈ (0, 1) such that condition 3. in the previous statement is satisfied. Existence of such a weight is guaranteed by the intermediate value theorem.
Remark. In the RDU case with continuous distortions As an application of proposition 8, the equilibrium price q(Z) of a risky asset Z is where λ is a solution of the equation of assertion 3. Normalizing the expected pricing density to one, we obtain the risk premium of Z: 4 Efficiency conditions 4

.1 Optimality conditions
It follows from the preceding subsections that for computing risk sharing rules and equilibria, we are brought down to solving (P λ ). This section is devoted to optimality conditions when utility functions are of the form (5), equivalently:ṽ From now on, we assume the following: • the following Inada-type conditions hold: either lim x→0 + g i (x) = +∞, or: Let us define Problem (P λ ) is then a calculus of variations problem with two monotonicity constraints: Existence of a solution x λ to (13) follows from Ascoli's theorem and x 0 being Lipschitz. Uniqueness of the solution, x λ , follows from the strict concavity assumptions. Since elements of A are Lipschitz and Lipschitz functions are a.e. differentiable, x λ is a.e. differentiable. Finally, the Inada conditions (11) or (12) ensure that the solution is interior: Condition (12) is a weakening of the Inada condition ∂ x L i (t, 0 + ) = ∞; it is motivated by cases such as L i (t, x) := ln(t + x), i = 1, 2 (see subsection 6.1), indeed, this utility satisfies (12) whereas (12) is equivalent to the usual Inada condition U i (0 + ) = +∞.
Letx λ be the maximizer of J when the monotonicity constraints are relaxed, i.e.x(t) solves: Whenx(t) is interior, it is characterized by Our next aim is to provide a method to explicitely solve (P λ ).
The proof of proposition 9 may be found in [4]. If Λ λ = Λ λ (1) on an interval I, thenΛ λ (t) = ∂ x L λ (t, x λ (t)) = 0 on I, therefore x λ =x λ on I. Hence, from (i) and (ii), x λ is constant on the connected components of Λ λ > Λ λ (1), x 0 − x λ is constant on the connected components of Λ λ < Λ λ (1), x λ =x λ on the connected components of Λ λ = Λ λ (1). The previous proposition may therefore be viewed as a generalization of Mussa-Rosen's ironing procedure. It implies that [0, 1] can be partitioned into • sub-intervals on which the quantile of the first agent's consumption quantile x λ is constant. Agent 2 bears the risk.
• sub-intervals on which the quantile of the second agent's consumption x 0 − x λ is constant. Agent 1 bears the risk.
• sub-intervals on which x λ and x 0 − x λ are increasing and x λ coincides withx λ .
The transversality condition (iii) is useful when for instance agent 2 is averse to the worse event (g 2 > 0) while agent 1 is not (g 1 = 0). In this case Since Λ λ (0) = 0 and Λ λ is continuous, we deduce from (ii) that x 0 − x λ is constant for low values of t. In other words, agent 1 insures agent 2 for low values of aggregate risk.

Direct applications
We now apply proposition 9 to the case wherex λ or x 0 −x λ is monotone.
Proposition 10 Assume g 1 = g 2 = 0. Then Hence, ifx λ ∈ A, it is the optimal solution to the constrained problem. Ifx λ (respectively x 0 −x λ ) is decreasing, the first (respectively the second) agent gets full insurance.
Agents' aversions to the worse events are taken into account in the next statement.

Ifx
To understand the meaning of the first two assertions, assume that agent 1 is averse to the worse event (g 1 > 0) while agent 2 is not (g 2 = 0) and that x λ ∈ A. Then either agent 1 gets full insurance or she gets full insurance for low values of aggregate risk. She thus avoids consumptions that are below some endogeneously determined value. Ifx λ is decreasing, then either the first agent is fully insured or she is fully insured for high values of aggregate risk.

Risk-sharing between two RDU
In this section, we focus on the RDU case. Agents have utility indices U 1 , U 2 and distortions f 1 , f 2 . In the regimes where both consumptions are strictly monotone, risk-sharing rules are those of EU maximizers with utility index U i and heterogeneous beliefs with densities f i (1 − F X 0 (X 0 )).

The case of the same distortion
As a benchmark, let us first discuss risk-sharing rules and equilibria between two (epsilon contaminated) RDU agents with same distortion f and utility index U i , i = 1, 2: The uncontaminated case is obtained for ε i = 0, i = 1, 2. From proposition 5, the optimal sharing rule associated to λ is (x λ (F X 0 (X 0 )), X 0 − x λ (F X 0 (X 0 ))) where x λ solves: From (16),x λ is interior and fulfills hence (17) is independent of f and also characterizes risk-sharing rules when agents are EU with same probability and utility index (1 − ε i )U i , i = 1, 2.
From standard risk-sharing arguments,x λ ∈ A. Using (17), we obtain If ε 1 = ε 2 = 0, from proposition 10 assertion 1, x λ =x λ . Thus risk-sharing rules between RDU agents are independent of f . Equilibrium weights are the solutions of the following equation: hence equilibria depend on f . Furthermore, (19) is the equation characterizing equilibrium weights of EU with same probability having density f (1 − F X 0 (X 0 )) with respect to P and utility index U i , i = 1, 2.
Let us summarize the previous results in a proposition.

Proposition 12
Consider two (epsilon contaminated) RDU agents with same distortion f . If ε 1 = ε 2 = 0, risk-sharing rules and equilibria are those of EU maximizers with utility index U i , i = 1, 2 and same probability with density f (1 − F X 0 (X 0 )) with respect to P . If ε 1 > ε 2 , then either the first agent is fully insured or she is fully insured up to some value of the risk and shares risk above that value as if agents were EU maximizers with utility index (1 − ε i )U i , i = 1, 2.

The case of two different continuous distortions
We now introduce some heterogeneity in beliefs and consider risk-sharing rules and equilibria between two RDU agents with utility index U i , i = 1, 2 and continuous distortions f i , i = 1, 2. We assume that U i i = 1, 2 fulfills (16) for i = 1, 2. The RDU risk-sharing problem with utility weight λ > 0 corresponds to the case From proposition 5, the optimal sharing rule is (x λ (F X 0 (X 0 )), X 0 −x λ (F X 0 (X 0 ))) where x λ solves: The functionx λ that maximizes pointwise the integrand in (22) is defined by It follows from the optimality conditions that whenever x λ and x 0 − x λ are strictly monotone on some interval, then x λ =x λ on that interval. In other words, wheneverx / ∈ A, one of the monotonicity constraints is binding somewhere: there is a range of values of the risk on which one agent is fully insured by the other. Thus, we next discuss whetherx λ belongs to A.
Differentiating (23) (assuming that all the data are smooth) yields that x λ (t) has the same sign as the quantity: Similarly (x 0 −x λ ) (t) has the sign of: Defining: We therefore havex λ ∈ A iff Condition (24) depends (in a complicated way) on the distribution of aggregate endowment, on the tolerances to risk of both agents, on the differences of their index of distortion and finally on utility weights. When agents have same beliefs (f 1 = f 2 ), then (24) is satisfied for any λ as in the previous paragraph. If inf x 0 > 0 and if beliefs are similar (i.e. f 1 − f 2 is sufficiently small in the C 2 norm so that f " 2

) then by continuous dependence of
x on (f 1 , f 2 ), we deduce that (24) is still satisfied for similar distortions. On the contrary, if agent one distorts probabilities much more than agent 2 and furthermore agent 2 has high tolerance to risk, then the condition is likely to be satisfied. From proposition 10 assertion 2, agent 1 is then fully insured. A symmetric condition may of course be given for agent 2.

Proposition 13
1. If (24) is fulfilled (in particular if there is weak heterogeneity of beliefs), then x λ =x λ . Agents thus behave as if they were expected utility maximizers with probabilities with densities f i (1 − F X 0 (X 0 )) and index U i , i = 1, 2.

If (25) is fulfilled (in particular if there is high heterogeneity of beliefs
with high distortion from agent 1), then agent 1 is fully insured.

Example
Let us consider the case of CARA utilities and exponential distortions with α 2 > α 1 > 0. We recall that in the expected utility model with CARA utilities, risk-sharing rules are piecewise linear, the interior pieces of the first agent having constant slope ρ 2 ρ 1 +ρ 2 . Assuming thatx λ is interior (which is not always the case since CARA utilities do not satisfy the Inada conditions), we obtain: Since α 2 > α 1 ,x λ is nondecreasing, while x 0 −x λ is nondecreasing iff This is in particularly true if α 2 is close to α 1 (case of weak heterogeneity of beliefs) or if ρ 1 is large. In that case, The first term corresponds to the standard risk-sharing rule between two EU agents with CARA utilities. The second term is due to the uncertainty effect. If α 2 > α 1 (i.e. agent 2 is more uncertainty averse than agent 1), then, in addition to the standard linear risk-sharing rule, X * λ includes a term which is comonotone with X 0 . In other words, agent 1 is more exposed to the variations of the aggregate risk X 0 than in the EU case and partially insures agent 2, by taking the additional risk (α 2 − α 1 )F X 0 (X 0 )/(ρ 1 + ρ 2 ). Furthermore, (26) can be interpreted as a three-fund result: at equilibrium, at most three assets are traded: the risk-free asset and assets with payoff X 0 and F X 0 (X 0 ). A two-fund result is obtained if X 0 is uniformly distributed.
Let ρ and α be respectively the aggregate risk aversion and the aggregate distortion indices defined by 1 The pricing density supporting an interior efficient allocation (or an interior equilibrium) being proportional to the marginal utility of any agent with strictly increasing sharing rule, it is proportional to In other words, at equilibrium, there is an RDU representative agent with constant risk tolerance ρ and power distortion with exponent α. It can be shown in this case that the risk premium is increasing in ρ and α.
If x 0 (t) < (α 2 − α 1 )/ρ 1 , for all t ∈ [0, 1], then the consumption of agent 2 is constant. This is true in the case where α 2 − α 1 is large (case of heterogeneous agents) or if the tolerance to risk of the first agent is high.
As the previous example and the next section show, (24) is not in general fulfilled on [0, 1]. When (24) is violated on some interval I, there are ranges of values of aggregate endowment containing x 0 (I) for which the consumption of one of the two agents is constant. This is the case when one of the agent distorts much more than the other small or large probabilities.
Proof. Let us assume that f 1 (0) = 0 and f 2 (0) = 0. From proposition 9, one has: therefore Λ λ decreases for t close to 1 and Λ λ (t) > 0. Hence x λ is constant for t close to 1. Thus, the consumption of the first agent is constant for high values of aggregate endowment. If f 1 (1) = ∞ and f 2 (1) < ∞, one obtains in a similar way that Λ λ (t) > 0 for small t : the consumption of the first agent is constant for low values of aggregate endowment.

The case of discontinuous distortions
We now consider risk-sharing between two epsilon contaminated RDU agents with discontinuous distortions f i , contaminations ε i and utility index U i fulfilling (16) for i = 1, 2. From proposition 5, the optimal sharing rule associated to λ solves: Letx λ be defined by As a consequence of proposition 11, one obtains:

Proposition 15
1. Ifx λ fulfills (24) (in particular if there is weak heterogeneity of beliefs) and if ε 2 1, then either the first agent is fully insured or she is fully insured up to some value of the risk. Above that value, agents behave as if they were expected utility maximizers with probabilities with densities f i (1−F X 0 (X 0 )) and index (1−ε i )U i , i = 1, 2.
2. Ifx λ fulfills (25) (in particular if there is high heterogeneity of beliefs, agent 1 is either fully insured or fully insured for high values of aggregate risk. As the next section shows, condition (24) or (25) may not be fulfilled. We therefore make below less stringent assumptions.

Proposition 16
1. If f 1 (0) = 0 and f 2 (0) = 0, then for any utility weights, the first agent is fully insured for high values of aggregate endowment.
2. If ε 2 = 0 and ε 1 > 0, then for any utility weights, the first agent is insured for low values of aggregate endowment.
Proof. The proof of the first assertion is as in proposition 14. To prove the second, let ε 2 = 0 and ε 1 > 0, then Therefore Λ λ (1) < 0 = Λ λ (0). By the optimality conditions, x λ is constant for low values of t and the first agent is fully insured for low values of aggregate endowment.
Propositions 14 and 16 provide foundations for insurance contracts with deductibles and upper limits.

Risk-sharing and equilibria: examples
The aim of this final section is to obtain closed-form solutions for some examples. In subsection 6.1, we solve the risk-sharing and equilibrium problems in the case of two agents with the same logarithmic RLU. In subsection 6.2, we consider the class of RDU's with logarithmic utility and power distortions. We study risk-sharing and equilibria both in the uncontaminated and contaminated cases.

6.1
Risk-sharing and equilibria between symmetric agents with RLU utilities We first provide an example of risk-sharing between two symmetric agents with utility function V L (X) = 1 0 ln(F −1 X (t) + t)dt. We further assume that aggregate endowment is uniformly distributed on [1,2] (hence x 0 (t) = t + 1 and F X 0 (X 0 ) = X 0 − 1). The optimal sharing rule is (x λ (X 0 − 1), X 0 − x λ (X 0 − 1)) where x λ solves: The next result is proven in the appendix.
If 1 3 ≤ λ ≤ 2 3 , then both agents hold a share of aggregate risk and of the riskless asset. For other values of λ, one of the agent is fully insured. We thus obtain a two-funds result.
In assertion 2 (resp. 3), a is determined as a function of λ by solving the equation We end this subsection by computing the equilibria when the first agent has a share 0 < k < 1 of X 0 . As a benchmark, we recall that, in the EU case with logarithmic utilities, the equilibrium weight is λ * = k. Hence for any k, there is a unique no-transaction equilibrium.
Since x λ (respectively x 0 −x λ ) is continuous and strictly increasing for λ > 1 3 (respectively λ ≤ 1 3 ), from proposition 8, the price supporting an efficient allocation is proportional to the marginal utility of the first (respectively second) agent. The properties of equilibrium are summarized in the next proposition (see appendix for proofs): Proposition 18 Equilibrium is unique. If k ≥ 3+2 ln 2 3+4 ln 2 , then agent 1 insures agent 2. If 2 ln 2 3+4 ln 2 ≤ k ≤ 3+2 ln 2 3+4 ln 2 , then both agents hold a share of aggregate risk and of the riskless asset. If k ≤ 2 ln 2 3+4 ln 2 , then agent 1 insures agent 2.

An example of risk-sharing rules and equilibria between two RDU agents
We now consider risk-sharing rules and equilibria between two RDU agents with utility index U (x) = ln(x) and distortions f 1 (t) = t α , α ≥ 1 f 2 (t) = (1 − ε)t β , t < 1, β > α. The first agent is thus less uncertainty averse than the second who furthermore displays aversion to the worse state. The risksharing problem thus depends on four parameters (α, β, ε, λ). Risk-sharing between an expected utility maximizer agent and a RDU agent is obtained for α = 1, ε = 0. We shall first describe risk-sharing rules for ε = 0 and fixed (α, β). We then study the effect of introducing ε. We again assume that aggregate endowment X 0 is uniformly distributed on [1,2]. As a benchmark, we recall that, if the first agent is a risk neutral EU and the second a strictly SSD preserving Yaari utility or a RDU, then efficient contracts are pairs of the form ((X 0 − d) + , min(X 0 , d)) (the first agent gets a call option on X 0 ). In the RDU model we consider, the more risk-averse agent is fully insured by the other for high values of X 0 .
Hence if the weight of the less uncertainty averse agent is large enough, she plays the role of an insurer and provides the other agent with full insurance.
If λ is small enough, below some threshold value for X 0 , agents behave as if they were expected utility maximizers with heterogeneous beliefs given by densities α(1 − F X 0 (X 0 )) α−1 and β(1 − F X 0 (X 0 )) β−1 and logarithmic utilities. Above that threshold, the first agent provides full insurance to the second. Hence, in this example, high values of the risk are always fully insured by the less uncertainty averse agent.

Equilibria in the uncontaminated case
We now compute the equilibria of the economy of the previous subsection assuming that the first agent has a share 0 < k < 1 of X 0 . As a benchmark, we recall that, in the EU case with logarithmic utilities, the equilibrium weight is λ * = k. Hence for any k, there is a unique no-transaction equilibrium. Since from proposition 19, x λ is continuous and strictly increasing for all λ, the price is proportional to the marginal utility of the first agent. From proposition 8, an equilibrium weight is characterized by The properties of equilibrium are summarized in the next proposition (see appendix for proofs): Proposition 20 1. Equilibrium is unique and the equilibrium utility weight λ * verifies λ * = k.
2. If k ≥ λ 1 with λ 1 as in proposition 19, then agent 1 insures the other agent. Her equilibrium consumption is a + X 0 − 1 for some 0 < a < 1 independent of β.
In the RDU case, contrary to the EU case where the equilibrium utility weight λ * verifies λ * = k and there is no trade at equilibrium, there is trade at equilibrium.

The case of epsilon-contamination
We now assume that the second agent has an epsilon-contaminated RDU utility Letx λ satisfy ∂ x L λ (t,x λ (t)) = 0 for all t. We havẽ From (iii) of proposition 9, Since Λ λ (1) > Λ λ (0) = 0, from (ii) of proposition 9, x λ (t) = t + x 0 in a neighborhood of 0. Then according to the values of the parameters (α, β, ε, λ) only two cases are possible: either x λ (t) = t + x 0 for all t in which case Λ λ (1) > Λ λ (t) for all t or there exists (t 0 , t 1 ) with 0 < t 0 < t 1 < 1 such that for t ≥ t 1 . In this later case, Λ λ increases on [0, t 0 ], Λ λ (t) = Λ λ (1) on [t 0 , t 1 ] and Λ λ (1) > Λ λ (t) on ]t 1 , 1[. Symmetrically either the second agent is fully insured for all values of the risk or her consumption is constant for low values of the risk and for high values of the risk. In other words, the second agent insures herself a minimal amount and reaches satiation for high values of the risk. When the first agent has a share 0 < k < 1 of X 0 , as in the uncontaminated case, we obtain: Proposition 21 Equilibrium is unique and the equilibrium utility weight λ * verifies λ * = k.

Appendix
Proof of proposition 2 Let us show that 1 implies 2. Let us first prove that ∂ x L ≥ 0 on [0, 1] × R. Let x 1 < x 2 < x 3 and p i > 0, i = 1, .., 3 be such that p 1 + p 2 + p 3 = 1. Let Y be a random taking values x i with probability p i . Let ε > 0 and δ > 0 be small and X be a random variable such that P (X = x 1 ) = p 1 , P (X = x 2 ) = p 2 − δ, P (X = x 2 + ε) = δ, and P (X = x 3 ) = p 3 . It may easily be verified that X 2 Y . Hence we obtain Dividing by δ and letting δ → 0 + , we have that for every ε small enough whuch proves the claim of monotonicity. Let and one easily verifies that for every Dividing by a − b and letting b → a + , one obtains that for every 0 < a < 1 and x 1 < x 3 , Hence L(t, .) is concave for every t ∈ [0, 1]. It remains to show that ∂ tx L ≤ 0 on [0, 1] × R. Let x 1 < x 2 < x 3 and ε > 0 be such that x 1 + ε < x 2 < x 3 − ε. Let X and Y be two random variables such that P (Y = x 1 ) = p 1 , P (Y = x 2 ) = p 2 , P (Y = x 3 ) = p 3 and P (X = x 1 ) = p 1 − u, P (X = x 1 + ε) = u, P (X = x 2 ) = p 2 , P (X = x 3 − ε) = u, P (X = x 3 ) = p 3 − u with u ≤ min{p 1 , p 3 }. It may easily be verified that E(X) = E(Y ) and that X 2 Y . Hence we obtain Dividing by u and letting u → 0 + , we have that for every ε small enough Dividing by ε and letting ε → 0 + , we obtain that ∂ x L(p 1 , x 1 ) ≥ ∂ x L(p 1 + p 2 , x 3 ), for all (p 1 , p 2 ) and x 1 < x 3 As x 3 → x 1 , one gets that ∂ x L(., x) is nonincreasing for every x as was to be proven.
Let us show that 2 implies 3. Since ∂ tx L ≤ 0, a submodular version of Hardy-Littlewood's inequality (see [1]) yields: Furthermore since (Ω, B, P ) is non-atomic, for every random variable X, there exists U X uniformly distributed such that X = F −1 X (U X ) P -a.e. Hence 1 0 L(t, F −1 X (t))dt = E(L(U X , X)) and Since for every U , E(L(U, .)) is concave and σ(L ∞ , L 1 ) upper-semi-continuous, it follows from (35) that V L is concave and σ(L ∞ , L 1 ) upper-semi-continuous which proves assertion 3. Finally, the fact that 3 implies 1 follows from proposition 1. If, in addition to the assumptions of the previous proposition, we either assume that L(t, .) is strictly concave for every t ∈ [0, 1], or that ∂ tx L < 0 then V L is strictly S.S.D. preserving. Indeed, assume first that L(t, .) is strictly concave and let X and Y be in L ∞ (Ω, B, P ) with X 2 Y . By strict concavity and since Let g(t) := ∂ x L(t, F −1 X (t)). By assumption, g is nonnegative. Since F −1 X can be approximated by smooth nondecreasing functions in the a.e. convergence, we may assume that F −1 X := x is a smooth nondecreasing function. Hence and g is nonincreasing. From (3), we deduce that V L (Y ) < V L (X), hence V L is strictly S.S.D. preserving. Finally, if ∂ tx L < 0, then g defined above is decreasing, hence if X 2 Y , then
We next check whether the equilibrium consumption can be of the type x λ (t) = a + t. Then we must have: where the second equality follows from (39). Hence λ * = k. Assertions 2 and 3 then follow from proposition 19.