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Equilibrium in the insurance market with adverse selection and fraud
S. Hun Seog Associate Professor
Graduate School of Finance, KAIST 207-43 Cheongryangri-Dong Dongdaemun-Gu,
Seoul, 130-012, Korea Email : [email protected]
Phone : +82-2-958-3527 Fax : +82-2-958-3604
Chang Mo Kang*
PhD Candidate Graduate School of Management, KAIST
207-43 Cheongryangri-Dong Dongdaemun-Gu, Seoul, 130-012, Korea
Email : [email protected] Phone : +82-2-958-3427 Fax : +82-2-958-3604
This Draft : February 2007
Keywords : Insurance; Fraud; Audit; Adverse Selection; Competitive Contracts; Commitment; Risk Categorization JEL classification : K42; D80; D81; G22
* Corresponding Author
Abstract
This paper investigates the insurance market where policyholders have private
information on their risk type and some of them, opportunistic policyholders, may file
fraudulent claims. We assume that insurance companies cannot each type of
policyholders, but they uncover fraudulent claims by costly audit technology. Under
Rothschild and Stiglitz (1976) conjecture, policyholders select different contracts not
only based on their honesty type, but based on their risk type, if the market equilibrium
exists. Since high risk type policyholders are audited more frequently than low risk
type policyholders, they leave the insurance market first, as the monitoring cost
increases. In this case, the insurance market shuts down, even though low risk type
policyholders can Pareto improve their expected utility by fair insurance.
Consequently, the imperfect information on risk type makes the market inefficiency
from insurance fraud worse. The market equilibrium also depends on whether
insurance companies can commit to their audit strategy. The commitment can improve
the social welfare by completely preventing fraudulent claims.
1
Ⅰ. Introduction
In most insurance markets, insurance companies have suffered serious losses
from the insurance fraud. Many researchers have studied this issue with an economic
approach. Most of them regard it as a kind of moral hazard problem. If insurance
companies cannot observe the actual loss occurred to their policyholders, the
policyholders are likely to report exaggerated loss to receive excess coverage.
However, it is an implausible assumption that every policyholder may report their loss
falsely. Even though all policyholders without criminal records on fraud are applied to
the same monitoring policy, only some of them try to file fraudulent claims. Taking
this aspect into consideration, we treated the insurance fraud as a kind of adverse
selection problem like Picard (1996). Likewise, we consider the asymmetric
information on the risk type of policyholders, and find the market equilibrium based on
the conjecture of Rothschild and Stiglitz (1976).
Let us review related literatures before discussing the results of this paper.
Rothschild and Stiglitz (1976), hereafter RS, assumed that policyholders have private
information on their risk type and that the insurer is myopic like players in the Cournot-
Nash game. Then, if the proportion of low risk type is too large, the competitive
market equilibrium does not exist. Otherwise, each type of policyholders selects a
distinct contract in the equilibrium. To prevent high risk type policyholders’ deviation,
low risk type policyholders should tolerate some welfare loss as compared with perfect
information case. However, subsequent refinements of RS conjecture relaxed the
myopic behavior assumption. Among them, Wilson (1977) assumed that the insurance
firm withdraws their unprofitable policy from the market. By this modification, he
reached the conclusion that the market equilibrium always exists, whether it is a
2
separating equilibrium or a pooling one.
Researches on the insurance fraud have developed into two main approaches:
the costly state verification and the costly state falsification. The costly state
verification approach assumes that the insurer can verify the filed claims if it expends
some resources (monitoring cost). Gollier (1987) shows that a contract that is upward
continuous and overcompensates small losses is optimal under the costly state
verification with deterministic auditing. On the contrary, Huberman, Mayers and
Smith (1983) and Picard (1999) show that if the policyholders can inflate their claims
intentionally, the optimal policy is a straight deductible. Dionne and Gagne (2001)
empirically find that the deductible policy prevent the insurance fraud.
Picard(1996) regarded the insurance fraud as an adverse selection problem.
He assumed that a certain type of policyholders, called opportunists, file fraudulent
claims for their interests. Thus, policyholders can be classified into two groups, honest
policyholders and opportunists, and find the market equilibrium following
Wilson(1977)’s conjecture. As a result, he showed that the insurance market shuts
down, if the monitoring cost is too high or the proportion of opportunists in the
population of policyholders is too high.
On the other hand, policyholders are considered more active in the costly state
falsification hypothesis. For example, they can falsify their claims through collusion
with agents or insurance brokers. Bond and Crocker (1997) and Picard (1999) assume
that policyholders are able to manipulate monitoring cost by expending some resources.
Crocker and Morgan (1997) assume that policyholders may expend resources to falsify
their actual losses when the verification of claims is impossible. These studies
commonly emphasized on the necessity of coinsurance to prevent these manipulations.
3
We consider insurance fraud costly observable and analyze it with asymmetric
information on risk type. Like Picard (1996), we assume that a certain type of
policyholders, called opportunistic policyholders, file fraudulent claims. Since
policyholders can be classified by their risk type and honesty type, there are four types
of policyholders in our three stage model. At the first stage, insurance company and
policyholders make a policy on insurance premium, coverage, and audit strategy based
on the information that each party has. At the second stage, policyholders realize their
states and determine whether they report loss or not. Honest policyholders file claims
only when the loss occurs to them actually, while opportunistic policyholders may file
fraudulent claims with a fraud strategy. And at the last stage, insurance companies
audit claims through the contracted audit strategy, and fulfill the contract. If
opportunistic policyholders are uncovered to file fraudulent claims by an insurance
company, they should pay a fine to the company.
The main results of this paper are summarized as follows. First, if claims are
completely unobservable, the market equilibrium does not exist, and the market shuts
down. This implies that insurance companies cannot provide insurance contract
without auditing claims.
Second, whether insurance companies can commit to their audit strategy or not,
insurance contracts separate policyholders based on their risk type, and pooled them
based on honesty type in the market equilibrium. That is, which contract policyholders
select in the market equilibrium only depends on their loss probability.
Third, as the monitoring cost increases, high risk type policyholders are willing
to leave the market prior to low risk type policyholders. Since high risk type
policyholders are more frequently audited than low risk type policyholders in the
4
competitive market equilibrium, they are less patient to the monitoring cost. If they
leave the market, the market shuts down. Considering that low risk type policyholders
can get utility more than reservation one from fair insurance, this implies that the
imperfect information on risk type makes the market inefficiency from insurance fraud
worse.
Fourth, if the market equilibrium exists, only low risk type policyholders suffer
the welfare loss in comparison to the market with perfect information on risk type. As
in Rothschild and Stiglitz (1976), low risk type policyholders should take the separating
cost caused by the possible deviation of high risk type.
Finally, insurance companies’ commitment to their audit strategy improves the
social welfare by preventing B policyholders’ fraud completely. The market shuts
down at lower monitoring cost under no commitment than under commitment.
However, it does not affect on the separation of policyholders in the equilibrium. In
other words, policyholders still select their contracts based on their risk type in the
equilibrium.
The remainder of the paper is composed as follows. We describe the model in
Section 2. In section 3, we investigate the benchmark case where insurance companies
cannot audit the fraudulent claims even with cost. Section 4 contains a case in which
insurance companies can costly audit claims and commit to their audit strategy. In
section 5 the model is expanded to no commitment case. In the last section, we
summarize and conclude this paper. Appendix provides proofs of propositions and
lemmas.
5
Ⅱ. Model description
Risk-averse policyholders have initial wealth , which reduces by when
the loss occurred. The risk type of policyholders can be classified into high risk type
(hereafter, H policyholders) and low risk type (hereafter, L policyholders), whose loss
probabilities are
W L
Hδ and Lδ )( LH δδ > , respectively. Likewise, policyholders can be
divided into honest policyholders (hereafter, G policyholders) and opportunistic
policyholders (hereafter, B policyholders) in terms of their honesty type. G
policyholders report their loss to insurance companies only when the loss actually
occurred to them. On the other hand, B policyholders may file claims when they do
not suffer the loss as well as when they do. B policyholders can file fraudulent claims
randomly with a probability ]1,0[∈q , called ‘fraud strategy’ in this paper. By
combining risk type and honesty type, we can classify all policyholders into 4 types: BH,
BL, GH, GL type. The characteristics of each type of policyholders are summarized in
Table 1.
Table 1. Types of policyholders
Honesty type Risk Type
Honest (G policyholders) Opportunistic (B policyholders)
High risk type (H policyholders)
• Notation : GH type • Loss probability : Hδ • No Fraud
• Notation : BH type • Loss probability : Hδ • Fraud strategy : ]1,0[∈q
Low risk type (L policyholders)
• Notation : GL type • Loss probability : Lδ • No Fraud
• Notation : BL type • Loss probability : Lδ • Fraud strategy : ]1,0[∈q
6
Let us denote the proportion of population of B policyholders and H
policyholders as )1,0(∈α and )1,0(∈β , respectively. We assume that distributions
of risk type and honesty type are mutually independent. Thus, the proportion of
population of each type can be obtained as the product of the proportion of
corresponding risk type and honesty type. For example, the proportion of BH type
among all policyholders is αβ .
Risk-neutral insurance companies cannot distinguish each policyholder’s type.
However, they know the distribution of each type in the market, and they can audit
claims with monitoring cost γ . The insurance market is assumed to be competitive.
Given the above information, insurance contracts are made and fulfilled according to
the following 3 stages.
Stage 1. Considering the information they have, companies and policyholders make a
contract on insurance premium , coverage , and audit probability or
strategy . We will denote the insurance contract as .
)(P )(C
)( p ),,( pCP
Stage 2. The loss occurs and each policyholder determines whether they file a claim or
not. B policyholders may file fraudulent claims with probability q to
maximize their expected utility.
Stage 3. Companies audit policyholders’ claims with the contracted probability, p . If
a B policyholder is detected in fraud, he should pay a fine M , to insurance
company. 1 We assume that M is determined by law in the range of
, and the lawsuit cost is assumed to be zero. ],0( PW −
1 Picard (1996) distinguish a fine paid by defrauder from an award paid to the insurance company. But we assume that the compensation is enacted into law, and they do not go to court to save their legal fee.
7
We introduce the administration cost of insurance companies, which is denoted
as . For simplicity, it is assumed to be quadratic function of the coverage level of
each policy, that is, , where .
c
2kCc = 0>k 2
Each policyholder is assumed to have Von Neumann-Morgenstern expected
utility function. Thus, policyholders’ expected utility without insurance contract is
LHiWUWUU iiGi ,),()1()( 21 =−+= δδ , where LWW −=1 , WW =2 , regardless of
their honesty type. Since policyholders are risk averse, their utility function is concave,
that is, 0",0' <> UU . If G policyholders buy an insurance contract , their
expected utility becomes
)~,~,~( pCP
LHiWUWUU iiGi ,),~()1()~( 21 =−+= δδ , where LCPWW −+−=~~~
1 , PWW ~~2 −= .
Since B policyholders, however, file fraudulent claims with probability q~ , insurance
contract, )~,~,~( pCP , will change their expected utility into
)(~~)1()(~)~1)(1()()~1)(1()( 4321 WUqpWUqpWUqWUU iiiiBi δδδδ −+−−+−−+= , i = H, L,
where LCPWW −+−=~~
1 , PWW ~2 −= , CPWW +−= ~
3 , and MPWW −−= ~4 .
We follow Rothschild-Stiglitz conjecture (hereafter, RS conjecture) to find the
competitive market equilibrium. Additionally, we introduce the following assumptions
to guarantee reasonable and unique market equilibrium.
First, the gamble with insurance is prohibited, implying that . 0>> PC
Second, is high enough to guarantee the convexity of zero profit curve in
plane.
k ),( PC
3
2 Administration cost does not play a significant role in our model. But it simplify the analysis by making the profit function convex in (C, P) plane. We will discuss it in detail later. 3 Picard (1996) assumed that the equilibrium is singleton. This assumption is stronger but more explicit than Picard’s. Since this assumption implies that the slope of iso-profit curve in (C,P) plane is
8
Ⅲ. Benchmark Case : Insurance Fraud is unobservable
As a benchmark, we consider the case in which insurance fraud is unobservable
even with cost. Here, insurance companies cannot discover the truth of reported
claims as well as policyholders’ types. If the market equilibrium does exist, the
insurance market will not shut down even when auditing is completely impossible.
The RS equilibrium should satisfy some conditions represented by the following
constraints: Incentive Compatibility Constraints (ICC), Individual Rationality
Constraints (IRC) or participation constraints, zero profit constraints and RS conjecture.
In this case, the audit strategy need not be determined in policy, and B policyholders
always report the loss regardless of their actual state. Therefore, when each
policyholder selects an insurance contract , G policyholders’ expected utility is ),( CP
)()1()( PWULCPWUU iiG −−+−+−= δδ , LHi ,= [1]
, while B policyholders’ expected utility is
)()1()( CPWULCPWUU iiB +−−+−+−= δδ , LHi ,= [2]
Since B policyholders can earn PC − from the insurance contract regardless of their
state, they select the insurance contract only when 0≥− PC . Proposition 1
characterizes the RS equilibrium in this market.
Proposition 1 When insurance companies cannot observe the claims, the insurance
market equilibrium does not exist.
Proof. See the appendix.
Figure 1 shows that G policyholders and B policyholders should be separated in decreasing in C, it makes the equilibrium unique with concave utility function.
9
RS equilibrium, when auditing filed claims is impossible. For simplicity, we exclude
the administration cost in Figure 1. Insurance companies can earn strictly positive
expected profit by introducing new contracts that attract only G policyholders for all
pooling contract based on honesty type. Suppose that GH policyholders and B
policyholders select . Then the point G in Figure 1 presents the contingent
wealth of GH policyholders and B presents that of B policyholders. If new entrant
provides a contract
),( CP
),( εε −− CP , where 0>ε , only GH policyholders choose it
since B policyholders are indifferent between and new one. Then the
contingent wealth of GH-policyholders will move to G’ and since new entrant’s zero
profit line is DE, it can get positive profit from this policy. So cannot be the
equilibrium under RS conjecture.
),( CP
),( CP
W-P+C-L
E
W
W-L
W1
W2
W-P W-P+C
G
G’ B
D
Figure 1. Pooling contract based on honesty type cannot be offered
Proposition 1 implies that insurance market may be collapsed if there is no way to audit
the filed claims. Because B policyholders always file a claim regardless of their actual
10
state, insurance companies cannot offer contracts that Pareto improve policyholders’
expected utility with satisfying their zero profit constraints.
Ⅳ. The market equilibrium when insurance fraud is costly observable :
Commitment Case
Now insurance companies are assumed to audit the reported claims with cost.
We will find the market equilibrium following 3 steps described in the below box.
Step 1. Investigate the audit game ),,,( μσCPΦ , where μ andσ are the beliefs of
insurance companies on the proportion of H policyholders in a contract and the
proportion of B policyholders among policyholders without loss in a contract
, and obtain optimal audit strategy and fraud strategy are
determined
),,( pCP )( p )(q
Step 2. Given audit game results, constitute and check the conditions for the market
equilibrium under RS conjecture
Step 3. Find the market equilibrium by solving the maximization problem
For the audit game, we can investigate two cases : commitment and no commitment.
In the commitment case, since insurance companies can commit to their audit strategy,
they can determine it to maximize their expected profit given B policyholders’ best
response fraud strategy on it. Under no commitment, however, the Perfect Bayesian
audit game equilibrium will be determined where fraud strategy is optimal to B
policyholders given audit strategy, where audit strategy is optimal to insurance
11
companies given their belief on the probability with which a claim is fraudulent, and
where the belief of insurance companies are derived from fraud strategy and the loss
probability using Bayes’ rule.4 We consider the commitment case in this section, and no
commitment case in the next section.
1. How to separate policyholders in the market equilibrium
Suppose that denotes an insurance contract, ),,( pCP μ does insurance
companies’ belief on the proportion of H policyholders in this contract, and σ does
insurance companies’ belief on the proportion of B policyholders among policyholders
to whom the loss does not occur actually. For instance, if insurance companies believe
that only BH policyholders select this contract, 1== μσ . In equilibrium, these
beliefs should be consistent with the actual proportion of each type of policyholders in
this contract. Since we consider the commitment case in this section, we need to
obtain the best response of B policyholders given audit strategy of insurance companies.
For a contract , B policyholders try to maximize their utility by selecting
optimal fraud strategy , which is defined as
),,( pCP
*q
)]()1()()1()([arg* PWUqCPWUpqMPWqpUMaxq q −−++−−+−−≡ [3]
From [3], we can derive Lemma 1 easily.5
Lemma 1. Let be the insurance premium, coverage and audit strategy of an
insurance contract, . Then, B policyholder’s optimal fraud strategy, , is
determined as follows ;
pCP ,,
),,( pCP *q
4 The definition of perfect Bayesian equilibrium in each case is identical to that of Picard (1996). 5 Lemma 1 is equivalent to the discussion of Picard (1996) on the best response of opportunists
12
1*=q , if pp ~<
]1,0[*∈q , if pp ~=
0*=q , if pp ~>
, where )()(
)()(~MPWUCPWU
PWUCPWUp−−−+−
−−+−=
Proof. See the appendix.
Lemma 1 implies that is determined irrespective of the risk type of B
policyholders. Even though the critical audit strategy
*q
p~ varies across contracts, the
basic decision rule of B policyholders is unique. If insurance companies do not audit
claims to a sufficient level, B policyholders file fraudulent claims.
Given the best response of B policyholders described in Lemma 1, insurance
companies’ expected profit from a contract is ),,( pCP
2*)1(*])1([)]1(*)1([ kCpqMpqpqCPE −−+−+−−−+−=Π σδσδδγσδδ ,
where LH δμμδδ )1( −+= .
Insurance companies will determine the audit strategy to maximize their expected profit
described above. Lemma 2 shows the optimal audit strategy and fraud strategy in the
commitment case.
Lemma 26 When insurance companies can commit to their audit strategy and provide
an insurance contract with beliefs σ and μ described above, the optimal audit and
fraud strategy of audit game *)*,( qp ),,,( μσCPΦ is determined at
)1.0(*)*,( =qp , if δδσγ
pC
~)1( −
>
6 Lemma 2 is identical to Proposition 1 of Picard (1996) except for some differences in notations and the addition of risk type.
13
)0,~(*)*,( pqp = , otherwise
, where LH δμμδδ )1( −+= .
Proof. See the appendix.
Corollary 1
If insurance companies can commit to their audit strategy and believe that a contract,
is selected only by G policyholders, i.e. ),,( GGG pCP 0=σ , then the corresponding
optimal audit and fraud strategy, )1,0(),( ** =GG qp
Lemma 2 implies that if monitoring cost is too high to prevent the fraud
completely, insurance companies would rather allow B policyholders to file fraudulent
claims, and save the monitoring cost. Otherwise, they will monitor the claims strictly
enough to make B policyholders give up the fraud, and save the cost from inappropriate
coverage. Seeing that the monitoring cost, γ , is strictly positive, Corollary 1 can be
derived directly from Lemma 2. But we need to check its implication to understand
the behavior of insurance companies in this model. Insurance companies determine
their audit strategy to maximize their expected profit under their belief on the proportion
of H policyholders in a contract, and on the proportion of B policyholders among
policyholders without loss. Thus, if they believe only G policyholders will select the
contract, they need not pay the monitoring cost to find fraudulent claim. Given this
audit strategy, B policyholders always file fraudulent claims if they select the contract.
In other words, the audit strategy do not influence on the cost that B policyholders
should pay to mimic G policyholders, or to select G policyholder’s contract.
14
Based on the optimal audit and fraud strategy described in Lemma 2, we can
derive the constraints that each contract should satisfy to be the market equilibrium
under RS conjecture. Before solving the maximization problem, we need to consider
how policyholders will be separated in equilibrium. With the derived constraints and
the assumption that the contract should Pareto improve policyholder’s expected utility
over reservation utility, we can obtain Proposition 2, which characterizes the way each
type of policyholders select their contracts in the market equilibrium.
Proposition 2 When insurance companies can commit to their audit strategy, contracts
should separate policyholders based on risk type, but pool them based on their honesty
type in the equilibrium under RS conjecture.
Proof. See the appendix.
Corollary 2
(1) Regardless of their risk type, G policyholders and B policyholders cannot be
separated in the equilibrium
(2) H policyholders and L policyholders cannot be pooled in the equilibrium
Proof. See the appendix.
In fact, the proof of proposition 2 consists of two parts, each of them implies
Corollary 2. Let us investigate the intuition of Proposition 2, or equivalently,
Corollary 2. Suppose that and denote insurance
contracts for G policyholders and B policyholders, respectively. As we have shown in
Proposition 1, the market equilibrium does not exist if insurance companies do not audit
),,( GGG pCP ),,( BBB pCP
15
claims from . Thus, insurance companies will audit claims from this
contract enough to prevent B policyholders from fraud completely, while they do not
audit claims from by Lemma 2 and Corollary 1. This implies that B
policyholders can always receive the coverage, if they select instead of
. To prevent this mimicking of B policyholders, should be
selected by only GH policyholders, while is selected by BL policyholders.
The remaining GL and BH policyholders should be pooled. As we have shown in the
proof of proposition 1, the contract for GL and BH policyholders should be audited to
the extent that BH policyholders do not file fraudulent claims in the market equilibrum.
Otherwise, insurance companies can provide a contract selected by only GL
policyholders with getting strictly positive profit. Thus, BH policyholders have the
same expected utility with GH policyholders, implying that they select as
long as . Thus,
),,( BBB pCP
),,( GGG pCP
),,( GGG pCP
),,( BBB pCP ),,( GGG pCP
),,( BBB pCP
),,( GGG pCP
GG PC > 0== GG CP , and consequently, any contract cannot be
provided for the ICC of each type of policyholders.
The first of Corollary 2 implies that G and B policyholders should be pooled, when
H and L policyholders are also pooled. If insurance companies do not audit claims the
contract selected by both H and L policyholders, G policyholders will deviate to a new
contract as Proposition 1 implies. If insurance companies audit claims from the
pooling contract of H and L policyholders and B policyholders in this contract do not
deceive their states, we can apply RS conjecture to show the second part of Corollary 2.
Proposition 2 shows that if insurance companies can commit to their audit strategy,
they offer two different contracts selected by H policyholders and L policyholders,
16
respectively. In other words, insurance companies cannot distinguish G policyholders
from B policyholders by contracts, even though they can observe the truth of reported
claims.
2. The market equilibrium under commitment
Considering the separating way described in proposition 2, we will find the
market equilibrium. Before checking the market equilibrium, we should investigate
the audit game equilibrium. From Lemma 1 and Lemma 2, we can easily derived
Lemma 3, which shows the equilibrium of the audit game under the separating way.
Lemma 3 When insurance companies commit to their audit strategy and offer a
separating contract for H and L policyholders with pooling G and B policyholders,
, , the equilibrium of the audit game is determined at ),,( iii pCP LHi ,= ),( **ii qp
)1,0(),( ** =ii qp , if ii
ii
pC
δδαγ ~
)1( −>
)0,~(),( **iii pqp = , otherwise
, where )()(
)()(~MPWUCPWU
PWUCPWUpiii
iiii −−−+−
−−+−=
and the expected profit
}~,)]1([max{ 22iiiiiiiiiii kCpCPkCCPE −−−−−+−=Π δγδδαδ [4]
Proof. See the appendix.
Proposition 1 implies that if insurance companies do not audit any reported claim, the
market equilibrium does not exist under RS conjecture. For simplicity, however, we
17
introduce two additional assumptions on the proportion of B policyholders.
Assumption 17
The proportion of B policyholders in the market, α , is so large that if insurance
companies do not audit the claims, zero profit contracts cannot improve the utility of all
policyholder over the reservation one.
Assumption 2
The proportion of H policyholders, β , is so large that pooling contract for H and L
policyholders cannot break the RS separating equilibrium.
0)]1([ 2 =−−+− KKKkK kCCP δαδ
),( KKK CPU
0~ 2 =−−− KKKKKK kCpCP δγδ
U
Figure 2. Insurance companies cannot provide contract without monitoring
7 This assumption is also introduced in Picard(1996) for simplicity.
18
Figure 2 depicts the insurance market equilibrium under the assumption 1.
Expected profit of insurance companies is given by [4]. In Figure 2, zero profit
constraints of an insurance company who offers a contract without auditing
and with auditing are depicted as OA and OB, respectively. And the reservation utility
of a policyholder is depicted as
),,( iii pCP
U . Since α is so high that OA and U cross only
at the origin, the insurance company cannot offer an insurance contract without auditing.
In other words, if we find an insurance contract such as K in Figure 2, which satisfies
zero profit constraint with auditing, we need not doubt whether the insurance company
can get higher expected profit from allowing fraud. Additionally, the market
equilibrium may not exist without assumption 2, as Rothschild and Stiglitz (1977) show
in their research.
Despite the role of assumption 1 in finding the market equilibrium discussed
above, it is very reasonable. If the proportion of B policyholders in the insurance
market is small enough to provide a contract without auditing, insurance companies
would not complain that they have gotten serious loss from insurance fraud, and need
not try to find a way to prevent it. Under the above two assumptions, the market
equilibrium is characterized like Proposition 3.
Proposition 3
When insurance companies can commit to their audit strategy under assumption 1 and 2,
the market equilibrium should satisfy the following conditions;
(1) )0,~(),( ** pqp = HHH
, if insurance market is not collapsed.8
8 This condition can be derived without assumption 1 in RS equilibrium concept, because new entrant can always take out G policyholders if incumbent provides a contract without auditing. But in proof, we will
19
(2) There exists Hγ < Lγ such that if )1,0(),( ** =ii qp iγγ > , LHi ,= .
When Hγγ < , all policyholders are provided with insurance contracts. H
policyholders do not suffer welfare loss comparing with perfect information case on
risk type, while L policyholders do.
Otherwise, no insurance is taken out, and the market will be collapsed.
Proof. See the appendix.
Each case in proposition 3-(2) is depicted in Figure 3, 4, and 5. By
assumption 1, Insurance companies cannot provide a insurance contract without
auditing claims from the contract. But since the contract that improves the utilities of
L policyholders over the reservation utilities can always do so for H policyholders, the
insurance market shuts down if the contract for H policyholders cannot Pareto improve
H policyholders’ utilities, as proposition 3-(1) implies. Because H policyholders
should audit claims more frequently than L policyholders, they are more impatient to
the monitoring cost. Thus, they leave the insurance market prior to L policyholders, as
the monitoring cost increases. And the ICC of H policyholders implies that if they
cannot get utilities more than reservation one, the insurance market shuts down.
Figure 4 shows the case in which the market shuts down, even though L policyholders
are willing to buy some fair insurance. However, if the market equilibrium exists, L
policyholders suffer the welfare loss comparing to the market with perfect information
on risk type, because of the possible deviation of H policyholders. Figure 3 depicts the
market equilibrium under RS conjecture
In the market equilibrium, B policyholders do not file fraudulent claims if
use assumption 1 to show this condition.
20
insurance companies commit to their audit strategy. Thus, the final wealth or expected
utility depends not on the honesty type, but on the risk type under commitment. That
is, even though B policyholders are not distinguished from G policyholders by self-
selection mechanism, their fraudulent behavior can be prevented completely.
0),(~ 2 =−−− HHH
HH
HH kCCPpCP γδδ
),( HHH CPU
),( LLL CPU
0),(~ 2 =−−− LLL
LL
LL kCCPpCP γδδ
Figure 3. Both policyholders are provided with insurance contracts
LU
0),(~ 2 =−−− HHH
HH
HH kCCPpCP γδδ
HU
0),(~ 2 =−−− LLL
LL
LL kCCPpCP γδδ
21
Figure 4. No insurance is taken out
0),(~ 2 =−−− HHH
HH
HH kCCPpCP γδδ
HU
0),(~ 2 =−−− LLL
LL
LL kCCPpCP γδδ
LU
Figure 5. No insurance is taken out
Ⅴ. The market equilibrium when insurance fraud is costly observable : No
Commitment Case
If insurance companies cannot commit to their audit strategy, it should be
determined where it minimizes the expected cost of insurance companies given the best
response of B policyholders to it. In other words, to be Perfect Bayesian Equilibrium
under no commitment, the audit strategy and fraud strategy should be optimal to
insurance companies and B policyholders, respectively, and insurance companies’ belief
on the truth of claims should be obtained from the loss probability, the belief on the
proportion of B policyholders among policyholders without loss in the contract, and
their fraud strategy. Let us denote an insurance contract under no
commitment, and
),,( pCP
μ and σ denote insurance companies’ belief on the proportion of
22
H policyholders and on the proportion of B policyholders among policyholders without
loss in this contract, respectively. Then, Lemma 1 still can explain the way B
policyholders determine their fraud strategy under no commitment. If we denote
insurance companies’ belief on a claim to be fraudulent as π , which is determined by
Bayes’ rule, insurance companies determine their audit strategy like [5] to minimize the
expected cost.
})1(})1([{minarg* CpMCpp p −+−−+= ππγ
)]}([{minarg MCpCp +−+= πγ ,
where δδσ
δσπ+−
−=
)1()1(
qq , and LH δμμδδ )1( −+= [5]
From [5] and Lemma 1, PBE of audit game for each contract will be determined like
Lemma 4.
Lemma 49
When insurance companies cannot commit to their audit strategy and provide an
insurance contract with beliefs σ and μ as described above, the optimal audit and
fraud strategy of audit game ),( ** qp ),,,( μσCPΦ is determined at
)1,0(),( ** =qp , if )()1(
)1( MC ++−
−>
δδσδσγ
)~,~(),( ** qpqp = , otherwise
, where )()(
)()(~MPWUCPWU
PWUCPWUp−−−+−
−−+−= and
))(1(~
γδσγδ
−+−=
MCq
And the corresponding expected profit of insurance companies,
9 Like Lemma 2, Lemma 4 is identical to Proposition 2 of Picard (1996) except for the addition of risk type.
23
},))1((max{ 22 kCCMC
CPkCCPE −−+
−−−−+−=Πγ
δγδσδδ , if MC +≤γ
, otherwise 2))1(( kCCPE −−+−=Π σδδ
Proof. See the appendix
Corollary 3
If insurance companies cannot commit to their audit strategy and believe that a contract,
is selected only by G policyholders, i.e. ),,( GGG pCP 0=σ , then the corresponding
optimal audit and fraud strategy, )1,0(),( ** =GG qp
Since 1))(1(
~0 <−+−
=<γδσ
γδMC
q , Lemma 6 implies that B policyholders
always file fraudulent claims with positive probability even when insurance companies
threat to audit their claims with positive probability. This difference between Lemma
2 and Lemma 4 results from the commitment of audit strategy. Because insurance
companies cannot commit to their audit strategy, B policyholders have no reason to
believe that insurance companies will keep their promised audit strategy when no one
files fraudulent claims. Corollary 3 characterizes the optimal strategies of insurance
companies and B policyholders for a contract that insurance companies believe to be
selected by only G policyholders.
Like the commitment case, we should determine how policyholders will be
separated by considering the constraints that a contract should satisfy to be the market
equilibrium based on Lemma 4 and Corollary 3. Proposition 4 shows policyholders
are separated in the same way with commitment case.
24
Proposition 4 When insurance companies cannot commit to their audit strategy,
contracts should separate policyholders based on risk type, but pool them based on
honesty type in the market equilibrium under RS conjecture.
Proposition 4 is reasonable because nothing has been changed from
commitment case, except that B policyholders file fraudulent claims with strictly
positive probability q~ even when insurance companies audit the claims. As we have
discussed in Proposition 2, if G policyholders and B policyholders select different
contract, no policyholder can Pareto improve their utility by purchasing the insurance
over the reservation utility. For the pooling contract based on risk type, we can also
apply to the same logic with Proposition 2, if insurance companies do not audit claims.
However, when insurance companies audit claims of the contract with the probability
p~ , we should consider that B policyholders file fraudulent claims with strictly positive
strategy. B policyholders’ expected utility from a contract )~,,( pCP is
)()1()( PWULCPWUEU iii −−+−+−= δδ
, which is independent of the fraud strategy . Thus, B policyholders in this contract
can be regarded as G policyholders in terms of their expected utility. Furthermore,
since the expected profit from
q
)~,,( pCP is 2kCCMC
CPE −−+
−−=Πγ
δγδ by
Lemma 4, it is continuous in P and . Consequently, we can find a new contract
that attracts only L policyholders in
C
)~,,( pCP by RS conjecture like commitment case.
From Lemma 1 and Lemma 4, we can obtain Lemma 5 which characterizes the
audit game equilibrium when policyholders are separated as described in Proposition 4.
25
Lemma 5
When insurance companies cannot commit to their audit strategy and offer a separating
contract for H and L policyholders with pooling G and B policyholders ,
, the equilibrium of the audit game is determined at
),,( iii pCP
LHi ,= ),( **ii qp
)1,0(),( ** =ii qp , if )()1(
)1( MCiii
i ++−
−>
δδαδαγ
)~,~(),( **iiii qpqp = , otherwise
, where )()(
)()(~MPWUCPWU
PWUCPWUpiii
iiii −−−+−
−−+−= and
))(1(~
γδαγδ
−+−=
MCq
ii
ii
And the corresponding expected profit of insurance companies,
2))1(( iiiii kCCPE −−+−=Π αδδ , if MCi +≤γ
},))1((max{ 22ii
i
iiiiiiiii kCC
MCCPkCCPE −
−+−−−−+−=Π
γγδδαδδ , otherwise
Proof. See the appendix
Based on Lemma 5 and Proposition 4, we can find the market equilibrium
under no commitment, which is characterized in Proposition 5. Here, we also
introduce assumption 1 and assumption 2.
Proposition 5
When the insurance company cannot commit to their audit strategy under assumption 1
and 2, the RS equilibrium contract should satisfy the following conditions
(1) )~,~(),( ** qpqp = HHHH , if insurance market is not collapsed
26
(2) There exist nL
nH γγ < such that if )1,0(),( ** =ii qp n
iγγ > , LHi ,= .
If nγγ < H , all policyholders are provided with insurance contracts.
Otherwise, no insurance is taken out, and the market will be collapsed
(3) Comparing to the commitment case, LHiini ,, =< γγ
Like the commitment case, the market shuts down if H policyholders cannot get
utility more than reservation one. And since B policyholders file fraudulent claims
with strictly positive probability in equilibrium, the final wealth of G policyholders and
B policyholders is not identical, while their expected utilities are identical. This
dishonest behavior of B policyholders in equilibrium also causes insurance companies
to provide less preferable contract for the same monitoring cost than under commitment.
In other words, considering the reservation utilities of policyholders, they leave the
insurance market at lower monitoring costs under no commitment than under
commitment, as Proposition 5-(3) implies.
Ⅵ. Conclusion and Discussion
In this paper, we simultaneously investigate two key issues in insurance market,
insurance fraud and asymmetric information on loss probability. More precisely, we
consider two dimensional adverse selection problem, based on Picard(1996) and
Rothschild and Stiglitz(1976). Using RS conjecture, we find the insurance market
equilibrium characterized as follows. First, if insurance companies cannot or do not
audit claims from all contracts, the market equilibrium does not exist, and the insurance
27
market shuts down. Second, whether insurance companies can commit to their audit
strategy or not, insurance contracts should separate policyholders only based on their
risk types in the market equilibrium. Third, since the contract for H policyholders
should be audited more frequently in the equilibrium, they are less patient to the
monitoring cost than L policyholders. If they leave the insurance market for high
monitoring cost, the market shuts down even when L policyholders can improve their
expected utility by fair insurance. Fourth, only L policyholders suffer the welfare loss
comparing to the market with perfect information on risk type, if the market equilibrium
exists. And finally, the commitment to the audit strategy improves the social welfare
by preventing the fraudulent behavior completely.
These results imply that separating contract cannot address the adverse
selection problem caused by B policyholders, even though insurance companies can
find out the fraudulent claims by monitoring technology. And furthermore, the
imperfect information on risk type makes the market inefficiency from insurance fraud
problem worse. Even though L policyholders can get utility more than reservation one,
the deviation of H policyholders prevents insurance companies from providing contracts
for L policyholders
Appendix
Proof of Proposition 1
First, we investigate the pooling contract of G policyholders and B
policyholders. Suppose that is a contract selected by both G and B
policyholders in the market equilibrium. Likewise, let be the utility of type
),( KK CP
*ijEU ij
28
policyholders in the market equilibrium, where BGi ,= and LHj ,= . When each
type of policyholders select , their expected utility is like the followings. ),( KK CP
G policyholders
)()1()( KiKKiGi PWULCPWUEU −−+−+−= δδ , where LHi ,=
B policyholders
)()1()( KKiKKiBi CPWULCPWUEU +−−+−+−= δδ , where LHi ,=
Then, for policyholders who do not select in the equilibrium
by ICC. Now, let us assume that new contract
),( KK CP
),(*KKijji CPEUEU ≥ ),( εε −− KK CP
is provided, where 0>ε . Since ),(),( εε −−≥ KKBjKKBj CPEUCPEU , B
policyholders have no reason to select this new contract. On the contrary, since
),(),( KKGjKKGj CPEUCPEU >−− εε for all , G policyholders select j
),( εε −− KK CP rather than . ),( KK CP
If the expected utility of G policyholders who do not select in the
equilibrium, , there exists
),( KK CP
),(*KKGjjG CPEUEU > 0>ε such that
by the continuity of utility function.
Then, only G policyholders who select change their decision to
),(),(*KKGjKKGjGj CPEUCPEUEU >−−> εε
),( KK CP
),( εε −− KK CP . When δ is the average loss probability of G policyholders who
select , and ),( KK CP σ is the proportion of B policyholders, the expected profit from
before introducing ),( KK CP ),( εε −− KK CP is
, and the expected profit from 2})1({),( KKKKK kCCPCPE −+−−=Π σσδ
),( εε −− CP KK KKKKK is . Since 2)()()(),( εεδεεε −−−−−=−−Π CkCPCPE
29
),(),( KKKK CPECPE Π>−−Π εε , ),( εε −−Π KK CPE is strictly positive by zero
profit constraint of . Thus, any contract including cannot be the
market equilibrium under RS conjecture.
),( KK CP ),( KK CP
If for G policyholders who do not select ,
they also change their insurance contract to
),(*KKGjjG CPEUEU = ),( KK CP
),( εε −− KK CP . If only GH
policyholders select with some B policyholders, ),( KK CP
),(),( KKKK CPECPE Π>−−Π εε implying that cannot be the market
equilibrium under RS conjecture. However, in the case that GL policyholders select
, we cannot compare
),( KK CP
),( KK CP ),( εε −−Π KK CPE and ),( KK CPEΠ . Now, we set
1ε and 2ε such that HCP
LCP MRSMRS ,
2
1, <<
εε , where are the
marginal rate of substitution between insurance premium and coverage. Then,
HCP
LCP MRSMRS ,, ,
),( 21 εε −− KK CP is selected only by GL policyholders. Since
),(),( 21 KKKK CPECPE Π>−−Π εε , cannot be the market equilibrium in
this case.
),( KK CP
Since we have shown that the pooling contract of G and B policyholders cannot
be the market equilibrium under RS conjecture, we consider the separating contract of G
and B policyholders for the existence of market equilibrium. Suppose that
, are contracts for G policyholders and B policyholders, respectively.
From , insurance companies’ expected profit is
),( GG CP ),( BB CP
),( BB CP
),(,0 BBBBB CPCP ∀=−=Π
for the participation constraint of insurance company. Thus, B policyholders can get
merely reservation utility in the equilibrium. But ICC of B policyholders implies that
. With this condition, IRC of G policyholders can be satisfied )(,0 , GGGG CPPC ∀≤−
30
only when . Thus, separating contracts cannot be provided, and
consequently, the market shuts down.
0== GG CP
QED.
Proof of Lemma 1
The proof is trivial since the objective function of [3] is linear to x.
QED.
Proof of Lemma 2
For the equilibrium of audit game with 1=q , the audit strategy should be
determined in pp ~≤ . Since ΠE is linear to p , it is maximized at or 0=p p~ .
The expected profit in each case is
, if 21 ])1([ kCCPE −−+−=Π σδδ 0=p
22 ~)1(~)]1([)]~1()1([ kCpMppCPE −−+−+−−−+−=Π σδδσδγσδδ , if pp ~= .
If at the equilibrium of audit game, the corresponding audit strategy should
be
)1,0(∈q
p~ , and the expected profit is
23 ~)1(~])1([)]~1()1([ kCpqMpqpqCPE −−+−+−−−+−=Π σδσδδγσδδ
Finally, , only when the audit strategy 0=q pp ~≥ . Since the expected profit is
decreasing in p , it is maximized at pp ~= . Then we have
24 ~ kCpCPE −−−=Π γδδ
The concavity of utility function and the definition of p~ imply that
PWCPWpMPWp −>+−−+−− ))(~1()(~
31
or equivalently,
0)~1(~ <−− CppM [A1]
By [A1], and . Thus, the equilibrium of audit game will be
determined at if , or equivalently
24 Π>Π EE 34 Π>Π EE
)1,0(),( ** =qp 41 Π>Π EEii
ii
pC
δδαγ ~
)1( −> .
Otherwise, it is obtained at )0,~(),( ** pqp = .
QED.
Proof of Corollary 1
Suppose that is a contract selected by only G policyholders. From ,
the companies’ expected profit is
),( GG CP ),( GG CP
2GGGGGGG kCpCPE −−−=Π γδδ
, since they believe only G policyholders will select this contract. GEΠ is maximized at
, and the best response of B policyholders is by Lemma 1 0* =Gp 1* =Gq
QED.
Proof of Proposition 2.
(1) Separating contract for G and B policyholders
First, let us investigate contracts that separate G and B policyholders. We note
that the costly observable case is equivalent to unobservable one, if the equilibrium of
audit game ),,,( μσCPΦ is )1,0(*)*,( =qp . When and
denote contracts for G policyholders and B policyholders, respectively, the market
equilibrium under RS conjecture does not exist if .
),,( GGG pCP ),,( BBB pCP
)1,0(),(),( **** == BBGG qpqp
32
In the case where and )1,0(),( ** =GG qp )0,~(),( **BBB pqp = the ICC of B
policyholders who select is )~,,( BBB pCP
)()1()( GGiGGi CPWULCPWU +−−+−+− δδ
)()1()( BiBBi PWULCPWU −−+−+−≤ δδ , LHi ,= [A2]
And the ICC of G policyholders who select is )0,,( GG CP
)()1()( BjBBj PWULCPWU −−+−+− δδ
)()1()( GjGGj PWULCPWU −−+−+−≤ δδ , LHj ,= [A3]
By combining [A2] and [A3], we can obtain some conditions that the market
equilibrium should satisfy. If i = j, that is, risk type of B policyholders who select
and that of G policyholders who select are the same, then
. Then by the IRC of G and B policyholders and [A2],
),,( BBB pCP ),,( GGG pCP
0=GC 0=== BBG PCP . It
implies that no insurance is taken out. If i = H and j = L, then by [A2] and [A3],
, which is excluded by assumptions. Hence, the only possible way of
separating G and B policyholders is to provide selected only by GH
policyholders and selected only by BL ones. And it implies that the
remaining GL and BH policyholders should be pooled. If the pooling contract for GL
and BH policyholders, , is not audited in the equilibrium, insurance
companies can provide a new contract selected by only GL policyholders with getting
strictly positive profit as we have shown in Proposition 1, and
0<GC
),,( GGG pCP
),,( BBB pCP
),,( KKK pCP
0== KK CP . Then, all
other contracts cannot be provided for the ICC of policyholders. Now suppose that the
pooling contract is audited to the extent that B policyholders do not file fraudulent
33
claims. Then ICC of BH policyholders is
)()1()( GGHGGH CPWULCPWU +−−+−+− δδ
)()1()( KHKKH PWULCPWU −−+−+−≤ δδ
Thus, . Then, because of the ICC of GH policyholders and GL
policyholders,
0== GG CP
0== KK CP and 0== BB CP . Hence, G policyholders and B
policyholders cannot be separated in the market equilibrium under RS conjecture
(2) Pooling contract for H and L policyholders
Next, let us investigate the pooling contract of H and L policyholders, and
denote it as . Then, the audit game equilibria for this contract are ),,( KKK pCP
)0,~(),( ** pqp = KKK KK and . )1,0(),( ** =qp
Case 1. )0,~(),( ** pqp = KKK
Under this audit strategy, all policyholders do not file fraudulent claims. Thus,
we can find a contract that attracts only L policyholders from by the
same logic used in Rothschild and Stiglitz (1976). Suppose that and
denotes H and L policyholders’ MRS between the wealth in no loss state and
loss state when they select , respectively.
)~,,( KKK pCP
),( KK CPh
),( KK CPl
)~,,( KKK pCP
Since we assume a competitive market and the convexity of zero profit function,
the insurance company should get zero profit from the pooling contract. And KL δδ <
implies that
0~ 2 >−−− KKLLKK kCpCP γδδ
By continuity of the expected profit function, there exist 0>ε such that
0)('~'' 2' >−−− KKLLKK CkpCP γδδ
, where )),(,()','( εε KKKKKK CPhCPCP −−=
34
, )'()''(
)'()''('~MPWUCPWU
PWUCPWUpKKK
KKKK −−−+−
−−+−=
Since for all , ),(),( KKKK CPhCPl > ),( KK CP
BGipCPEUpCPEU KKKiLKKKiL ,),~,,()'~,','( =>
Hence attracts only L policyholders with generating strictly positive
profit, and cannot be a RS equilibrium.
)'~,','( KKK pCP
)~,,( KKK pCP
Case 2. )1,0(),( ** =qp KK
Since the insurance companies do not audit the claims in this contract, it is the
same with unobservable case. As we discussed in (1) of this proof, this pooling
contract should also include G and B policyholders. By Proposition 1, insurance
companies can provide a new contract that attracts G policyholders in , and
thus it cannot be the RS equilibrium. If
)0,,( KK CP
0== KK CP , we cannot find a new contract
that attracts only G policyholders in this contract, but H policyholders in this contract
will choose other contracts . Hence, in the equilibrium, the pooling contract for H and
L policyholders cannot be provided
By combining the results of (1) and (2), proposition 2 can be directly derived.
QED.
Proof of Corollary 2
It is already shown in the proof of Proposition 2.
Proof of Lemma 3
Seeing that the proportion of B policyholders in both H and L policyholders is
α , Lemma 3 can be directly derived by Lemma 1 and Lemma 2.
35
Proof of Proposition 3
Before solving the maximization problem, we need to check the optimal audit
strategy and fraud strategy in the market equilibrium. By assumption 1, insurance
companies cannot provide insurance contracts if they do not audit claims at all. And
since the expected profit function is convex and the expected utility function is concave,
and since the zero profit constraints and reservation utility functions cross at the origin,
we can determine the existence of the market equilibrium by comparing the slope of
zero profit constraints and reservation utility functions at the origin under the strictly
positive audit strategy and B policyholders’ response on it. If the slope of zero profit
constraints is greater than that of reservation utility functions, the insurance contract for
the risk type cannot be provided, and insurance companies will not audit them.
Otherwise, insurance companies audit them with strictly positive strategy. In sum,
)0,~(),( **HHH pqp = , if
)(')1()(')(')
)()()('1(
WULWULWU
MWUWUWU
HH
HH δδ
δγδ−+−−
<−−
+
)0,~(),( **LLL pqp = , if
)(')1()(')(')
)()()('1(
WULWULWU
MWUWUWU
LL
LL δδ
δγδ−+−−
<−−
+
, or equivalently
)0,~(),( **HHH pqp = , if H
HH
H
WULWUWUWULWUMWUWU γ
δδδγ =
−+−−−−−−
<)](')1()(')[('
)](')(')][()()[1( [A4]
)0,~(),( **LLL pqp = , if L
LL
L
WULWUWUWULWUMWUWU γ
δδδγ =
−+−−−−−−
<)](')1()(')[('
)](')(')][()()[1( [A5]
Here, LH γγ < , implying that even when )1,0(),( ** =qp HH )0,~(),( ** pqp = LLL
Case 1. )0,~(),( **iii pqp = for all LHi ,=
Then the Lagrange function is
36
])()1()(
])()1()([
)]()1()()()1()([
)]()1()()()1()([
]),(~[
]),(~[)()1()(
21[6
215
21214
21213
22
2121
HH
HH
H
LL
LL
L
HL
HL
LL
LL
LH
LH
HH
HH
LLLLLLL
HHHHHHHL
LL
L
UWUWU
UWUWU
WUWUWUWU
WUWUWUWU
kCCPpCP
kCCPpCPWUWUL
−−++
−−++
−−−−++
−−−−++
−−−+
−−−+−+=
δδλ
δδλ
δδδδλ
δδδδλ
γδδλ
γδδλδδ
, where LHiPWWLCPWW ii
iii ,,, 21 =−=−+−=
1) 05 =λ , 06 =λ
[A4], [A5] and the above discussion imply that 05 =λ , 06 =λ
2) 0,0 21 >> λλ
For the convexity of expected profit functions and the concavity of utility functions, the
solution of the above Lagrange function should be binding to zero profit constraints.
FOC
0)](')1()('[)](')1()('[]~
1[ 2142131 =−++−−−+∂∂
−=∂∂ H
LH
LH
HH
HH
HH
WUWUWUWUPp
PL δδλδδλγδλ
0)]('[)]('[]2~
[ 14131 =−++−∂∂
−−=∂∂ H
LH
HHH
HHH
WUWUkCCp
CL δλδλγδδλ [A6]
)](')1()('[]~
1[)(')1()(' 213221L
HL
HL
LL
LL
LL
WUWUPpWUWU
PL δδλγδλδδ −++
∂∂
−+−−−=∂∂
0)](')1()('[ =−−−+ LL WUWU δδλ 214 LL
0)]('[)]('[]2~
[)(' 141321 =+−+−∂∂
−−+=∂∂ L
LL
HLL
LLL
LL
WUWUkCCpWU
CL δλδλγδδλδ
3) 0,0 43 => λλ
Since 01 >λ , 03 >λ in [A6]. 06 =λ implies that 04 =λ since the expected utility
functions of H and L policyholders cross only at the origin in plane. ),( CP
37
By 2) and 3), is determined at ),( ** CP HH
)(')1()(')('
~1
2~
21
1H
HH
H
HH
HH
HH
HH
WUWUWU
Pp
kCCp
δδδ
γδ
γδδ
−+=
∂∂
−
+∂∂
+ [A7]
, which is the tangent of zero profit constraints of H policyholders and their expected
utility indifference curve. is determined at ),( ** CP LL
)(')1()(')('
~1
2~
21
1L
LL
L
LL
LL
LL
LL
WUWUWU
Pp
kCCp
δδδ
γδ
γδδ
−+<
∂∂
−
+∂∂
+. [A8]
Thus, only L policyholders suffer the welfare loss comparing to the perfect information
market in the above contracts. By assumption 2, we exclude the existence of pooling
contract which is preferred to these separating contracts by both H and L policyholders.
Thus, the contracts satisfy the above FOC with 1), 2) and 3) is the market equilibrium
under RS conjecture.
Case 2. and )1,0(),( ** =qp HH )0,~(),( ** pqp = LLL
In this case, LH γγγ << , 0== HH CP . Then the Lagrange function is
])()1()([)]()1()([
]),(~[)()1()(
213212
2121
LL
LL
LL
HL
HH
LLLLLLLL
LL
L
UWUWUWUWUU
kCCPpCPWUWUL
−−++−−−+
−−−+−+=
δδλδδλ
γδδλδδ
03 =λ , since LH γγγ << . And 01 >λ for the convexity of expected profit function
and the concavity of expected utility function.
0)](')1()('[]~
1[)(')1()(' 212121 =−++∂∂
−+−−−=∂∂ L
HL
HL
LL
LL
LL
WUWUPpWUWU
PL δδλγδλδδ
0)]('[]2~
[)(' 1211 =−+−∂∂
−−+=∂∂ L
HLL
LLL
LL
WUkCCpWU
CL δλγδδλδ
If 02 >λ , . 0== LL CP
38
If 0=2λ , is determined at ),( ** CP LL
)(')1()(')('
~1
2~
21
1L
LL
L
LL
LL
LL
LL
WUWUWU
Pp
kCCp
δδδ
γδ
γδδ
−+=
∂∂
−
+∂∂
+.
Since )()1()( LL WUWUU δδ −+< 21 HHH LL 2 at , this contradicts to ),( ** CP 0=λ .
Thus, the market equilibrium does not exist.
Case 3. for all )1,0(),( ** =ii qp LHi ,=
In this case, γγ <H , and for all 0== ii CP )1,0(),( ** =ii qp LHi ,= . It implies that
the insurance market shuts down.
QED
Proof of Lemma 4
Let δδσ
δσπ+−
−=
)1()1()(
qqq be an insurance company’s belief on claims from
policyholders of insurance contract to be fraudulent. Then ),( CP )(qπ is a
continuous decreasing function of . q
Case 1. )()1(
)1( MC ++−
−>
δδσδσγ
Since )1()( ππ <q for all 10 <≤ q , and )()1(
)1()1( MC ++−
−=
δδσδσπ ,
γπ <+ ))(( MCq for all 10 ≤≤ q . [5] implies that 0* =p is the optimal audit
strategy in this case. By Lemma 1, B policyholders determine their fraud strategy at
, if . Therefore, if 1* =q 0* =p )()1(
)1( MC ++−
−>
δδσδσγ , is the )1,0(*)*,( =qp
39
equilibrium of the audit game ),,,( μσCPΦ . The expected profit of insurance
companies under this auditing strategy is 2))1(( kCCPE −−+−=Π σδδ
Case 2. )()1(
)1( MC ++−
−≤
δδσδσγ
Suppose that ))(1()*)(( MCMCq +<<+ πγπ , for a given 1*0 <≤ q . Then, [5]
implies that . But for 0* =p 0* =p , 1* =q which contradicts to )1(*)( πγπ <<q .
Now suppose that )*)(( MCq +<πγ , for a given 1*0 <≤ q . Then [5] and Lemma 1
implies that and . But since 1* =p 0*=q 0)0( =π , ))(0( MC +>πγ , which
contradicts to )*)(( MCq +<πγ .
At last, suppose that )*)(( MCq +=πγ , for a given 1*0 ≤≤ q .
Then))(1(
),(~*γδμ
γδ−+−
≡=MC
CPqq which lies on if )1,0(
)()1(
)1( MC ++−
−<
δδσδσγ , and equals to if 1 )(
)1()1( MC ++−
−=
δδσδσγ . And every
can satisfy the condition of [5]. Considering these fraud strategy of B
policyholders, Lemma 1 implies that the audit strategy should be determined at
]1,0[∈p
pp ~= if
)()1(
)1( MC ++−
−<
δδσδσγ , and )],(~,0[ CPpp∈ if
δδσδσγ+−
−=
)1()1( . To simplify the
analysis, we set ),(~ CPpp = as the audit strategy of auditing equilibrium in this case.
The expected profit of insurance companies under this auditing strategy is
22 )~)1(( kCCqPkCCMC
CPE −−+−=−−+
−−=Π σδδγ
δγδ
QED.
Proof of Corollary 3
It can be directly derived from Lemma 4 by setting 0=σ .
40
Proof of Proposition 4
(1) Separating contract for G and B policyholders
Let us denote contracts for G policyholders and B policyholders as and
, respectively. And the corresponding fraud strategy is denoted as
and . We have already shown that the market equilibrium does not exist if
in Proposition 2. If and
),,( GGG pCP
),,( BBB pCP Gq
Bq
)1,0(),(),( **** == BBGG qpqp )1,0(),( ** =GG qp
)~,~(),( ** qpqp = BBBB , the ICC of B policyholders who select )~,,( pCP BBB is
)()1()( GGiGGi CPWULCPWU +−−+−+− δδ
)(~)~1)(1()()~1)(1()( BBiBiBBi CPWUqpPWUqLCPWU +−−−+−−−+−+−≤ δδδ
)(~~)1( MPWUqp Bi −−−+ δ
)()1()( BiBBi PWULCPWU −−+−+−= δδ , LHi ,=
And the ICC of G policyholders who select is )0,,( GG CP
)()1()( BjBBj PWULCPWU −−+−+− δδ
)()1()( GjGGj PWULCPWU −−+−+−≤ δδ , LHj ,=
By the same reason discussed in Proposition 2, GL and BH policyholders should be
pooled to provide separating contract for G and B policyholders. But because of the
ICC of GH and GL policyholders, this pooling contract cannot be provided, and
consequently, G policyholders and B policyholders cannot be provided in the market
equilibrium.
41
(2) Pooling contract for H and L policyholders
Let us denote the pooling contract as , and the corresponding fraud
strategy as .
),,( KKK pCP
Kq
Case 1. )1,0(),( ** =qp KK
This case is exactly the same with that under commitment, discussed in the proof of
Proposition 2.
Case 2. )~,~(),( ** qpqp = KKKK
By Lemma 4, the expected profit from is ),,( KKK pCP
2KK
K
KKKK kCC
MCCPE −
−+−−=Π
γγδδ , which is zero if is the market
equilibrium by the assumption of the competitive market and the convexity of expected
profit function. Then, since the expected profit function is continuous on
),,( KKK pCP
P and C ,
we can apply the logic used in the proof of Proposition 2 to find a new contract selected
by only L policyholders in with generating strictly positive profit. ),,( KKK pCP
Thus, the pooling contract of H and L policyholders cannot be the market
equilibrium under RS conjecture. And by combining the results of (1) and (2),
Proposition 4 is proved.
Proof of Lemma 5
Seeing that the proportion of B policyholders in both H and L policyholders is α ,
Lemma 5 can be directly derived by Lemma 1 and Lemma 4.
Proof of Proposition 5
Like commitment case, we first compare the slope of zero profit constraints under
42
strictly positive audit strategy and that of reservation expected utility at the origin.
)~,~(),( **HHHH qpqp = , if n
HH
LWUWULWUM γδγ =
−−−−
<)('
)](')('[)1(
)~,~(),( **LLLL qpqp = , if n
LL
LWUWULWUM γδγ =
−−−−
<)('
)](')('[)1(
Here, nn γγ < LH HH, implying that even when )1,0(),( ** =qp )~,~(),( ** qpqp = LLLL
Case 1. )~,~(),( **iiii qpqp = for all LHi ,=
In this case, the Lagrange function is
])()1()([
])()1()([
)]()1()()()1()([
)]()1()()()1()([
][
][)()1()(
216
215
21214
21213
22
2121
HH
HH
H
LL
LL
L
HL
HL
LL
LL
LH
LH
HH
HH
LLL
LLLL
HHH
HHHH
LL
LL
UWUWU
UWUWU
WUWUWUWU
WUWUWUWU
kCCMC
CP
kCCMC
CPWUWUL
−−++
−−++
−−−−++
−−−−++
−−+
−−+
−−+
−−+−+=
δδλ
δδλ
δδδδλ
δδδδλ
γγδδλ
γγδδλδδ
1) 05 =λ , 06 =λ
2) 0,0 21 >> λλ
For the convexity of expected profit functions and the concavity of utility functions, the
solution of the above Lagrange function should be binding to zero profit constraints like
the commitment case.
FOC
0)](')1()('[)](')1()('[ 2142131 =−++−−−+=∂∂ H
LH
LH
HH
HH
WUWUWUWUPL δδλδδλλ
0)]('[)]('[]2)(
[ 141321 =−++−−+
−−−=
∂∂ H
LH
HHH
HHH
WUWUkCMC
MCL δλδλ
γγγδδλ
[A9]
43
)](')1()('[)(')1()(' 213221L
HL
HL
LL
LL
WUWUWUWUPL δδλλδδ −+++−−−=
∂∂
0)](')1()('[ =−−−+ LL WUWU δδλ 214 LL
0)]('[)]('[]2)(
[)(' 1413221 =+−+−−+
−−−+=
∂∂ L
LL
HLL
LLL
LL
WUWUkCMC
MWUCL δλδλ
γγγδδλδ
3) 0,0 43 => λλ
Since 01 >λ , 03 >λ for [A9]. And by definition of separating contract, 06 =λ and
04 >λ is impossible. Thus, 0,0 43 => λλ .
For the same reason with commitment case, only L policyholders suffer the welfare loss
comparing to perfect information market in the contracts that satisfy the FOC with 1), 2),
and 3). And by assumption 2, these contracts are the market equilibrium under RS
conjecture.
Case 2. and )1,0(),( ** =qp HH )~,~(),( ** qpqp = LLLL
In this case, nn γγγ << LH , 0==CP HH . Then the Lagrange function is
])()1()([)]()1()([
][)()1()(
213212
2121
LL
LL
LL
HL
HH
LLL
LLLL
LL
LL
UWUWUWUWUU
kCCMC
CPWUWUL
−−++−−−+
−−+
−−+−+=
δδλδδλ
γγδδλδδ
03 =λ , since nL
nH γγγ << . And 01 >λ for the convexity of expected profit function
and the concavity of expected utility function.
0)](')1()('[]~
1[)(')1()(' 212121 =−++∂∂
−+−−−=∂∂ L
HL
HL
LL
LL
LL
WUWUPpWUWU
PL δδλγδλδδ
44
0)]('[]2~
[)(' 1211 =−+−∂∂
−−+=∂∂ L
HLL
LLL
LL
WUkCCpWU
CL δλγδδλδ
If 02 >λ , . 0== LL CP
If 0=2λ , is determined at ),( ** CP LL
)(')1()(')('2
)( 21
12 L
LL
L
LL
LL
LL WUWUWUkC
MCM
δδδ
γγγδδ
−+=+
−+−
+ .
Since )()1()( LL WUWUU δδ −+< 21 HHH LL 2 at , this contradicts to ),( ** CP 0=λ .
Thus, the market equilibrium does not exist.
Case 3. for all )1,0(),( ** =ii qp LHi ,=
In this case, γγ <nH , and for all 0== ii CP )1,0(),( ** =ii qp LHi ,= . It implies that
the insurance market shuts down.
Finally, too compare iγ and niγ , LHi ,= , we investigate the slope of zero profit
constraints at the origin under commitment and no commitment
By the concavity of the utility function, ))((')()( γ−>−− MWUMWUWU . Thus,
))()(
)('1()1(MWUWU
WUM ii −−
+>−
+ γδγ
γδ , LHi ,= , that is, the slope of zero profit
constraints under no commitment is steeper than under commitment. Since both sides
of this inequality are increasing in γ , ini γγ < , LHi ,=
QED
45
46
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