In this section, I provide some preliminary properties of the set of incentive compatible allocations that will prove useful in the analysis. Note that a special case of the model is the Rothschild and Stiglitz [55] model, where T = 2 and Ω = 1. Type 1 is the high-risk type and type 2 is the low-risk type. To better illustrate the argument, I will sometimes employ this spe- cial case. Figure2.2 provides a graphical illustration of the Rothschild and Stiglitz [55] model. In the same figure, we can also see the well-known Roth- schild and Stiglitz [55] allocation, often called the Rothschild and Stiglitz
2.3. Some Preliminary Properties of Incentive Compatible Allocations 25 [55] equilibrium.
p
b
O
d
1ψ
1RSWp=θ
11b
p=θ
12b
ψ
2RSWFigure 2.2: The RSW allocation for the special case of T = 2 andΩ = 1.
Following Maskin and Tirole [43], the generalisation of the Rothschild and Stiglitz [55] allocation will be called from now on the RSW allocation; an acronym for Rothschild-Stiglitz-Wilson.10
Definition: The RSW allocation is denoted by ψ·RSW = {ψRSWt }Tt=1
and can be derived by solving the following recursive program: Program R(1): max
ψ1 U
1(ψ1) subject to
π1(ψ1)≥0 and for everyt= 2, ..., T:
Program R(t): max
ψt U
t(ψt) subject to
Ut−1(ψtRSW−1 )≥Ut−1(ψt) πt(ψt)≥0
For each t = 1, ..., T, the constraint set of Program R(t) is a closed subset of a compact set and therefore is compact. Therefore a solution
always exists. It is easy to show that, with strictly increasing utility in- dexes, for each t = 1, ..., T all constraints are satisfied with equality. In fact, for t = 1, ψRSW1 = (PΩ
ω=1θtωdω, d1, ..., dΩ); the lowest in the rank
type’s RSW contract coincides with his “perfect-information” contract. For each typet= 2, ..., T,ψtRSW provides less than full insurance and moreover, Ut(ψRSWt ) > Ut(ψtRSW−1 ). Note that because of the sorting assumption, we can neglect global incentive constraints and solve each program using the in- centive constraint for the upward-adjacent type. Because of strict concavity of the utility index, Ut(ψRSWt )> Ut
The RSW allocation plays a significant role in this paper as well as in any competitive market with adverse selection. This is because it is the only incentive compatible allocation that maximises the utility of all types and it is also ex post individually rational. In the spirit of Myerson [46], the RSW allocation is a “safe” or incentive compatible “type-by-type” allocation (or mechanism). A safe mechanism is one which would be incentive compatible even if the sellers knew the type of the buyer.
The following lemma is a preliminary result which will be extensively used in all the lemmas that follow:
Lemma 2.3.1. For every ψ·,ψ˜· ∈ ΨIC, such that ψ˜ strictly Pareto dom- inates ψ·, there exist 0 < < 1 and ψ¯· ∈ ΨIC that also strictly Pareto dominates ψ· andΠ( ¯ψ·)> Π(ψ·) + (1−)Π( ˜ψ·).
Proof: Take ψ·,ψ˜· ∈ ΨIC, such that ˜ψ strictly Pareto dominates ψ·.
Consider the following random allocation: Every typetis offered a contract that after the realisation of the state of nature ω, there is a lottery which with probability pays−pt+btω and with probability 1−,−p˜t+ ˜btω. The expected utility of type t from this random contract can be written as: Ut=PΩ
ω=1θωt[u(W−dω−pt+btω) + (1−)u(W −dω−p˜t+ ˜btω)]. For every 0< <1 and every ω we can find −p¯t+ ¯btω (the certainty equivalent) such that
u(W −dω−pt+btω) + (1−)u(W −dω−p˜t+ ˜btω) =u(W −dω−p¯t+ ¯btω) Because of the strict concavity of the utility function and by Jensen’s inequality, W −dω−p¯t+ ¯btω< (W −dω−pt+btω) + (1−)(W −dω−p˜t+ ˜btω) or ¯ pt−¯btω > (pt−btω) + (1−)(˜pt−˜btω) Therefore, πt( ¯ψt)≡p¯t− Ω X ω=1 θtω¯btω> (pt− Ω X ω=1 θtωbtω)+(1−)(˜pt− Ω X ω=1 θωt˜btω)≡πt(ψt)+(1−)πt( ˜ψt)
2.3. Some Preliminary Properties of Incentive Compatible Allocations 27 Summing up overt: T X t=1 λt0πt( ¯ψt)> T X t=1 λt0πt(ψt) + (1−) T X t=1 λt0πt( ˜ψt) or Π( ¯ψ·)> Π(ψ·) + (1−)Π( ˜ψ·)
Sinceψ·,ψ˜·∈ΨIC, for any, the random allocation (⊗ψ·,1−⊗ψ˜·) is also incentive compatible or (⊗ψ·,1−⊗ψ˜·)∈ΨIC. This necessarily means that ¯ψ· ∈ ΨIC. Moreover, for any 0 < < 1, Ut(ψt) < Ut( ¯ψt) < Ut( ˜ψt), therefore ¯ψ· strictly Pareto dominates ψ·. Q.E.D.
Many of the proofs will be based on the following important property of IC allocations:
Lemma 2.3.2. For everyψ·∈ΨICRandδ >0withΠ(ψ·)>0, there exists
¯
ψ·∈ΨICR that strictly Pareto dominates ψ· and Π( ¯ψ·)>Π(ψ·)−δ.
Proof: Take allocation ψ· ∈ ΨICR with Π(ψ·) > 0. Consider the
following complete-risk-pooling allocation ˜ψ·, where ˜ψt = (˜p, d1, ..., dΩ) for
each t = 1, ..., T and ˜p < PT
t=1λt0
PΩ
ω=1θtωdω. From Lemma 2.3.1, there exists 0< <1 and ¯ψ·that strictly Pareto dominates ψ·such that Π( ¯ψ·)> Π(ψ·) + (1−)Π( ˜ψ·). Forδ= (1−)[Π(ψ·)−Π( ˜ψ·)],and ˜pappropriately chosen, we obtain the result. Q.E.D.
If we recall the definition of incentive efficiency, it is not hard to see that given risk-neutrality on behalf of the sellers, and Lemma2.3.2, every (weak) incentive efficient allocation must be zero-profit.
Corollary 2.3.3. Every ψ·∈ΨW IE is such that Π(ψ·) = 0.
Another important property of incentive compatible allocations is the following:
Lemma 2.3.4. For everyψ·∈ΨICR, withΠ(ψ·) = 0 andψ·∈/ ΨSIE, there exists ψ¯·∈ΨICR that strictly Pareto dominates ψ· withΠ( ¯ψ·)>0.
Proof: Case 1. Assume first that ψ· ∈/ ΨW IE, with Π(ψ·) = 0. By
definition, there exists ˜ψ·∈ΨW IE that strictly Pareto dominatesψ·. From Corollary 2.3.3, Π( ˜ψ·) = 0. From Lemma 2.3.1, there exists ¯ψ·∈ΨIC that strictly Pareto dominates ψ·with Π( ¯ψ·)> Π(ψ·) + (1−)Π( ˜ψ·) = 0.
Case 2. Assume that ψ· ∈ ΨW IE but ψ· ∈/ ΨSIE. There exists ˜ψ· ∈
ΨSIE that weakly Pareto dominatesψ·. Let the set of types whose utility remains the same in both allocations beT1and those whose utility is strictly
higher under ˜ψ·beT2. By following the same logic as in the proof of Lemma
Ut( ¯ψt) = Ut(ψt), for all t∈T
1. Moreover, Π( ¯ψ·) >0. From Lemma 2.3.2,
for any δ >0, there exists ˆψ· ∈ΨIC that strictly Pareto dominates ¯ψ· and Π( ˜ψ·)>Π( ¯ψ·)−δ≥0, forδ small enough. Q.E.D.
Corollary 2.3.5. The sets of strict and weak incentive efficient allocations coincide, or: ΨSIE =ΨW IE.