Ambiguity and Asymmetric Information

Rachel J. Huang Graduate Institute of Finance

National Taiwan University of Science and Technology

Arthur Snow Department of Economics

University of Georgia

Larry Y. Tzeng Department of Finance

National Taiwan University

Preliminary version

Abstract The theoretical literature has found that advantageous selection could be observed in equilibrium when there is multi-dimensional heterogeneity of customers or both hidden action and hidden information are employed. This paper proposes a third approach for advantageous selection. Based on the classic Rothschild and Stiglitz (1976) model, we consider ambiguity-averse individuals who face a general risk and a specific risk in a perfectly competitive insurance market. The individuals have private information regarding the specific risk but have unbiased ambiguous beliefs regarding the general risk. Individuals' preferences are characterized by Klibanoff, Marinacci and Mukerji's (2005) smooth model, and they are homogeneous except for their risk type. We find that both pooling and separating equilibria can exist. Furthermore, we find that when the single crossing property does not hold, equilibrium could be determined based on adverse selection or advantageous selection. Keywords: advantageous selection, adverse selection, ambiguity. JEL classification: D80, G22, C30

1

Ambiguity and Asymmetric Information

1. Introduction

In Rothschild and Stiglitz's (1976) classic adverse selection model, high risk

types of individuals self-select a contract with higher coverage, whereas the low risk

types choose a contract with lower coverage in equilibrium. Their model thus predicts

a positive correlation between insurance coverage and accident occurrence. The

empirical evidence regarding the positive relationship is, however, mixed. In acute

health insurance and annuity markets, researchers have found empirical evidence that

is consistent with the prediction of adverse selection.1 On the contrary, a significant

negative correlation is supported in the markets for life insurance (Cawley and

Philipson 1999, McCarthy and Mitchell 2003), long-term care (Finkelstein and

McGarry 2006), medigap insurance (Hurd and McGarry 1997, Fang et al. 2006),

reverse mortgages (Davidoff and Welke 2007) and commercial fire insurance (Wang,

Huang and Tzeng 2009).

This phenomenon of a negative correlation between insurance and claim

frequency is documented as "advantageous selection". To provide theoretical support

for advantageous selection, the researchers have adopted the following two different

approaches. The first one is to consider the multi-dimensional heterogeneity of

customers. Liu and Browne (2007) exogenously assume that individuals are

heterogeneous with respect to risk type and risk aversion and find advantageous

selection in equilibrium when the insurance is not fair.2 Netzer and Scheuer (2010)

endogenize heterogeneity in wealth levels and assume that both risk type and patience

are private information. They show that a negative correlation between insurance

1 For health insurance, see Cutler and Zeckhauser (2000). Mitchell et al. (1999), Finkelstein and Poterba (2004) and McCarthy and Mitchell (2003) respectively examine the annuity market in the U.S., the U.K. and Japan. 2 Smart (2000), Wambach (2000), Villeneuve (2003) also assume that risk type and risk aversion are private information. Those models predict a positive correlation as in Rothschild and Stiglitz (1976).

2

coverage and risk type can be obtained.

The second approach is to integrate both hidden action and hidden information.

Individuals are homogeneous except for their degree of risk aversion (de Meza and

Webb 2001), patience (Sonnenholzner and Wambach 2009), overconfidence (Huang,

Liu and Tzeng 2010), or regret (Huang, Muermann and Tzeng 2011). This

one-dimensional heterogeneity of customers will affect the optimal choice of the

hidden action regarding self-protection and further cause heterogeneity in the risk

type. They respectively show that advantageous selection can appear in equilibrium

since risk-neutral, impatient, overconfident and regretful customers might spend less

on insurance and prevention, thereby becoming higher risk types.

In this paper, we propose a third approach by adopting ambiguity. Ambiguity

aversion describes the situation in which individuals prefer a lottery with a certain as

opposed to an uncertain probability even though the mean of the uncertain probability

is the same as that of the certain probability. To employee ambiguity, we first assume

that the risk of an individual can be decomposed into two parts: a general risk and a

specific risk. For example, the car accident probability of an individual depends on

the individual's driving skills (specific risk) and the traffic condition (general risk).3

The mortality rate is jointed determined by the health status (specific risk) and the

lethality of diseases (general risk).

We further assume that the specific risk is private information to the individuals

but the general risk is ambiguous to all participants in a perfectly competitive

insurance market. All participants in the market share the same ambiguous beliefs

regarding the general risk.4 In addition, we assume that individuals are

3 Huang, Tzeng and Wang (2012) show that a driver's car accident rate is positively significantly affected by both the driver's kilometer driven and the average kilometer driven of the society. The former could be viewed as a specific risk, whereas the later could be viewed as a general risk. 4 Our setting is close to but different from that of Seog (2009). Seog (2009) assumes that the general risk is privately known by the insurers and the individuals have precise beliefs regarding the general

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ambiguity-averse and are characterized by the same ambiguity preferences. In other

words, individuals are heterogeneous only in the specific risk. Note that we do not

assume that individuals can make any effort to change the loss probability. Thus, our

model is neither one that follows a multi-dimensional approach nor a model with both

hidden action and hidden information.

To model ambiguity aversion, we adopt Klibanoff, Marinacci and Mukerji's

(2005) smooth model.5 In their model, the ambiguity function captures the utility of

an ambiguity-averse individual and is set as a concave transfer of the individual's

traditional expected utility. Thus, under ambiguous beliefs, utility could be presented

as the expected ambiguity function over the ambiguous beliefs. This two-stage

decomposition of the decision process helps us to apply the well-developed

techniques under the expected utility framework to the analysis of problems involving

ambiguity aversion.

We find that a pooling contract might be optimal although the equilibrium occurs

only for very special parameter values. Furthermore, as in Rothschild and Stiglitz

(1976), we find that the market might settle on separating equilibrium. However, the

equilibria no longer exhibit a monotone relationship between insurance coverage and

risk in our model. The equilibrium might thus settle on adverse selection or

advantageous selection. It depends on insurance loading and how the degree of

ambiguity aversion varies when the expected utility increases.

Our paper contributes to the literature in the following ways. First, we propose a

different approach to advantageous selection. Second, to the best of our knowledge,

our paper is the first one to examine the effect of ambiguity aversion on competitive

risk. In other words, ambiguity is absent in his model. 5 Since Ellsberg (1961), researchers have proposed different settings to characterize ambiguity aversion preferences. For example, see Gilboa and Schmeidler (1989), Schmeidler (1989), Ghirardato, Maccheroni, and Marinacci (2004), and Klibanoff, Marinacci and Mukerji (2005).

4

insurance markets where the individuals have private information. There are several

papers in the literature that incorporate ambiguity in adverse selection problems. For

example, see Jeleva and Villeneuve (2004), Chassagnon and Villeneuve (2005) and

Vergote (2010). However, they all focus on a monopolistic rather than a competitive

insurance market. Third, our paper is the first one to adopt Klibanoff, Marinacci and

Mukerji's (2005) smooth model and to apply it to insurance markets under

asymmetric information.

The remainder of this paper is organized as follows. Section 2 introduces the

demand for insurance in the presence of ambiguity under full information. Section 3

employees asymmetric information and examines the equilibrium. Section 4

concludes the paper.

2. Full information: Demand for insurance in the presence of ambiguity

To prepare the background knowledge, we first examine the demand for

insurance in the presence of ambiguity under full information. In a perfectly

competitive insurance market, individuals face a binomial property risk with either a

fixed loss L or no loss. As in Seog (2009), we assume that the risk probability of an

individual can be decomposed into two parts: a general risk and a specific risk. To

introduce ambiguity, let us assume that the general risk ( r%) is ambiguous, whereas the

specific risk (π ) is unambiguous. r% is ranged in 0 1( , )r r , where 0r and 1r are

constant with 0 10 1r r≤ < < , and follows a cumulative distribution function ( )F r .

The probability of a loss equals to r π+% , where 1 1r π+ < .

Let ( ),C p q≡ be an insurance contract with coverage q and premium rate

p . Assume that the insurance companies are ambiguity neutral and risk neutral. Thus,

the premium rate is

( )( )1 ,p rλ π= + +

5

where λ denotes the insurance loading and ( )1

0

r

rr rdF r= ∫ denotes the expected

value of r%.

Let w be the initial wealth. The final wealth of an individual at the accident

state ( Aw ) and the no loss state ( Nw ) are respectively

Aw w pq L q= − − + , and Nw w pq= − .

Assume that individuals are ambiguity-averse. Following Klibanoff, Marinacci,

and Mukerji (2005), the utility function is then modeled as follows:

( ) ( ) ( ) ( )( ) ( )1

0

1r

A NrAEU r u w r u w dF rπ π= Φ + + − −∫ , (1)

where u denotes the traditional utility function with 0uʹ′ > and 0uʹ′ʹ′ < . Φ is the

differentiable ambiguity function with 0ʹ′Φ > . Since individuals are ambiguity averse,

we have 0ʹ′ʹ′Φ < .

Under full information, the optimal demand for insurance satisfies the following

first-order condition (FOC):

( ) ( )( ) ( ) ( ) ( ) ( )1

0

1 1

0,

r

A Nr

dAEUdq

Eu r p u w r pu w dF rπ πʹ′ ʹ′ ʹ′= Φ + − − − −⎡ ⎤⎣ ⎦

=

∫ (2)

where ( ) ( ) ( ) ( )( ) 1A NEu r r u w r u wπ π= + + − − . The second-order condition (SOC)

holds, i.e.,

( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( )

1

0

1

0

2

2

2

2 2

( ) 1 1

( ) 1 1

0,

r

A Nr

r

A Nr

d AEUdq

Eu r r p u w r pu w dF r

Eu r r p u w r p u w dF r

π π

π π

ʹ′ʹ′ ʹ′ ʹ′= Φ + − − − −⎡ ⎤⎣ ⎦

⎡ ⎤ʹ′ ʹ′ʹ′ ʹ′ʹ′+ Φ + − + − −⎣ ⎦

<

∫

∫

since both Φ and u are concave functions.

When the insurance is fairly priced, full coverage is optimal because

6

( )( ) ( ) ( ) ( )

( )( ) ( ) ( )

1

0

1

0

( ) ( )

( ) ( )

0.

p r

r

r

dAEUdq

u w r q r p u w r q dF r

u w r q u w r q rdF r p

π

π π π

π π π

= +

ʹ′ ʹ′= Φ − + + − − +⎡ ⎤⎣ ⎦

⎡ ⎤ʹ′ ʹ′=Φ − + − + + −⎢ ⎥⎣ ⎦=

∫

∫

It is obvious that if there is any insurance loading, then partial coverage is optimal.

When the insurance loading is positive, the relationship between risk probability

and demand for insurance is positive in the traditional expected utility framework

while keeping the price constant. However, in the presence of ambiguity, an increase

in π does not necessarily increase the demand for insurance. By the implicit

function theorem, we know

2

2

2

d AEUdq dqd

d AEUddq

ππ= − .

Since SOC holds, the sign of dqdπ

depends on the sign of 2d AEUdqdπ

, where

( ) ( ) ( )

( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1

0

0

2

1

( )

1 1

( ) 1

r

A Nr

A N

A Nr

d AEUdqd

Eu r u w u w

r p u w r pu w dF r

Eu r p u w pu w dF r

π

π π

ʹ′ʹ′= Φ −⎡ ⎤⎣ ⎦

ʹ′ ʹ′× + − − − −⎡ ⎤⎣ ⎦

ʹ′ ʹ′ ʹ′+ Φ − +⎡ ⎤⎣ ⎦

∫

∫

( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1

0

1

0

( )

( ) 1 1

( ) 1 ,

r

N Ar

A N

r

A Nr

u w u w R Eu r

Eu r r p u w r pu w dF r

Eu r p u w pu w dF r

π π

Φ= −⎡ ⎤⎣ ⎦

ʹ′ ʹ′ ʹ′×Φ + − − − −⎡ ⎤⎣ ⎦

ʹ′ ʹ′ ʹ′+ Φ − +⎡ ⎤⎣ ⎦

∫

∫

(3)

where ( ) ( )( )

( )( )

( )Eu r

R Eu rEu rΦ

ʹ′ʹ′−Φ=

ʹ′Φ denotes the degree of ambiguity aversion. The

assumption that both Φ and u are positive sloping functions ensures the second

7

term of the above equation positive. However, the sign of the first term is ambiguous.

From the FOC and the fact that ( )( ) ( ) ( ) ( )1 1A Nr p u w r pu wπ πʹ′ ʹ′+ − − − − increases

in r , there must exist an r̂ such that

( )( ) ( ) ( ) ( )ˆ ˆ1 1 0A Nr p u w r pu wπ πʹ′ ʹ′+ − − − − = .

Thus, the first term in Equation (3) can be rewritten as

( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( )

ˆ

ˆ

( )

( ) 1 1

( )

( ) 1 1 .

N Ar r

A N

N Ar r

A N

u w u w R Eu r

Eu r r p u w r pu w dF r

u w u w R Eu r

Eu r r p u w r pu w dF r

π π

π π

Φ≤

Φ>

Γ = −⎡ ⎤⎣ ⎦

ʹ′ ʹ′ ʹ′×Φ + − − − −⎡ ⎤⎣ ⎦

+ −⎡ ⎤⎣ ⎦

ʹ′ ʹ′ ʹ′×Φ + − − − −⎡ ⎤⎣ ⎦

∫

∫

If ( )( )R Eu rΦ is constant or increasing in r , then

( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( )1

0

ˆ

ˆ

ˆ( )

( ) 1 1

ˆ( )

( ) 1 1

ˆ( )

( ) 1 1

0.

N Ar r

A N

N Ar r

A N

N A

r

A Nr

u w u w R Eu r

Eu r r p u w r pu w dF r

u w u w R Eu r

Eu r r p u w r pu w dF r

u w u w R Eu r

Eu r r p u w r pu w dF r

π π

π π

π π

Φ≤

Φ>

Φ

Γ ≥ −⎡ ⎤⎣ ⎦

ʹ′ ʹ′ ʹ′×Φ + − − − −⎡ ⎤⎣ ⎦

+ −⎡ ⎤⎣ ⎦

ʹ′ ʹ′ ʹ′×Φ + − − − −⎡ ⎤⎣ ⎦

= −⎡ ⎤⎣ ⎦

ʹ′ ʹ′ ʹ′× Φ + − − − −⎡ ⎤⎣ ⎦

=

∫

∫

∫

Therefore, we have 2

0d AEUdqdπ

≥ , or 0dqdπ

≥ . On the other hand, to have 0dqdπ

< ,

the necessary condition is that ( )( )R Eu rΦ decreases in r . Note that ( )Eu r is a

decreasing function of r . We could conclude that if the ambiguity preferences

exhibit constant or decreasing ambiguity aversion with respect to expected utility,

then an increase in π will increase the demand for insurance. An increase in π will

decrease the demand for insurance only if the ambiguity preferences exhibit

increasing ambiguity aversion.

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3. Asymmetric information: Model setting and equilibrium

This section analyzes the market equilibrium under hidden information. Let us

assume that individuals have private information regarding the specific risk. For

simplicity, assume that there are two types of individuals: one, denoted as h with a

high loss probability of the specific risk ( hπ ); and the other, l , with a low risk

probability of the specific risk ( lπ ), 0 1l hπ π< < < . The insurance companies only

know the fraction of both types in the population. Let θ denote the fraction of type

h individuals in the population.

As in Rothschild and Stiglitz (1976), we assume that individuals are

homogeneous except their risk types. Therefore, the ambiguity function of type i

individuals, , ,i h l= could be denoted as

( ) ( ) ( ) ( )( ) ( )

( ) ( )

1

0

1

0

1

( ) .

r

i i A i Nr

r

ir

AEU r u w r u w dF r

Eu r dF r

π π= Φ + + − −

= Φ

∫

∫ (4)

The reason why there is a subscript i on Eu is because individuals are

heterogeneous in specific risk. The risk preferences among types are the same.

We will use graphs to analyze the existence of equilibria. In all of the figures, the

x -axis represents Nw , whereas the y -axis denotes Aw . The endowment is labeled

by point E . The slope of the indifference curve of type i individuals (denoted as

iI in all figures) is represented by the marginal rate of substitution

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1

0

1

0

1 ( )0

( )i

r

N i irAi r

N AEU A i ir

u w r Eu r dF rdwMRSdw u w r Eu r dF r

π

π

ʹ′ ʹ′− − Φ= − = >

ʹ′ ʹ′+ Φ

∫

∫. (5)

Furthermore, the indifference curves are convex. To see that, let the utility of type i

at point 1 1( , )N Aw w be the same as that at point 2 2( , )N Aw w , i.e.,

9

( ) ( ) ( ) ( )( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( ) ( )( ) ( )

1

0

1

0

1

0

1

0

1 1 1

1

2

2 2

2

1

1

.

r

i i A i Nr

r

ir

r

ir

r

i A i Nr

i

AEU r u w r u w dF r

Eu dF r

Eu dF r

r u w r u w dF r

AEU

π π

π π

= Φ + + − −

= Φ

= Φ

= Φ + + − −

=

∫

∫

∫

∫

Let [ ]0,1α∈ . Thus, the utility level at point 1 2 1 2( (1 ) , (1 ) )N N A Aw w w wα α α α+ − + − is

equal to ( ) ( )( ) ( ) ( )( )( ) ( )0

1 1 2 1 21 1 1i A A i N Nrr u w w r u w w dF rπ α α π α αΦ + + − + − − + −∫ .

By Jensen’s inequality, we have

( ) ( )( ) ( ) ( )( )( ) ( )

( )( ) ( )

( ) ( ) ( ) ( ) ( )

0

0

0

1 1 2 1 2

1 1 2

1 11 2

0

1

1 1 1

( ) 1 ( )

( ) 1 ( )

.

i A A i N Nr

i ir

i ir

i

r u w w r u w w dF r

Eu r Eu r dF r

Eu r dF r Eu r dF r

AEU

π α α π α α

α α

α α

Φ + + − + − − + −

≥ Φ + −

≥ Φ + − Φ

=

∫

∫

∫ ∫

Now, let us compare the indifference curves of types h and l . The

indifference curves of type h are flatter than these of type l if and only if

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 1

0 0

1 1

0 0

1 ( ) ( )

( ) 1 ( )

1.

r r

h h l lr rhr r

l h h l lr r

r Eu r dF r r Eu r dF rMRSMRS r Eu r dF r r Eu r dF r

π π

π π

ʹ′ ʹ′− − Φ + Φ=

ʹ′ ʹ′+ Φ − − Φ

<

∫ ∫

∫ ∫

Under full coverage, A Nw w= , we have ( ) ( )h lEu r Eu r= and

1 1,1

h h l

l h l

MRS r rMRS r r

π ππ π

− − += <

+ − −

which means that, under full coverage, the indifference curves of the high risk type

are flatter than these of the low risk type. Furthermore, if ( )( )hEu rʹ′Φ is large

enough than ( )( )lEu rʹ′Φ , then we might have 1h

l

MRSMRS

≥ . In other words, the single

crossing property may not hold. Figure 1 illustrates the indifference curves of both

types when the single crossing property does not hold.

10

Lines hP and lP denote the zero profit lines of the insurer based upon the

objective loss probabilities h rπ + and l rπ + , respectively. The slope of iP is then

11i rπ

−+

.

Let P denote the average pricing line. Thus, the slope of P is

( )

11(1 )h l rθπ θ π

−+ − +

.

Our equilibrium concept follows that of Rothschild and Stiglitz (1976). When the

single crossing property holds, it is obvious that our equilibrium is consistent with

Rothschild and Stiglitz (1976), i.e., (1) there is no pooling equilibrium; (2) if there is

an equilibrium, it will be a separating equilibrium with adverse selection; (3) when the

proportion of low risk type is high, there is no equilibrium.

In the following, we focus on the cases where the single crossing property does

not hold. To proceed the analyses, let ( )* *,h h hC r qπ= + denote the first-best contract

for type h individuals, *hI denotes the corresponding indifference curve of type h .

lCʹ′ denotes the intersection between *hI and lP . lI ʹ′ is the indifference curve of

type l under lCʹ′ . Let ˆlC be the tangent contract of *hI and lI , and let l̂I be the

corresponding indifference curve of type l . Furthermore, let C denote the contract

which maximizes type l 's utility under pooling pricing line.

Proposition 1 Suppose that the single crossing property does not hold. When the

insurance is not fair, if contract C also maximizes type h 's utility under pooling

pricing line, then C is a pooling equilibrium contract.

Proof Figure 2 shows that the equilibrium contract is at the point of tangency of

the average pricing line P and the indifference curves of both types of individuals,

hI and lI . From Figure 2, we can find that there is no way for any insurance

company to provide a profitable contract which could attract low risk types.

11

Although Proposition 1 shows that there could be a pooling equilibrium, the

equilibrium occurs only for very special parameter values. From the above analyses,

we know that if the insurance is fair, then the high risk type individuals will demand

full coverage. The average price is lower than the fair price for the high risk type.

Thus, the high risk type will demand partial coverage under the average price only

when the insurance loading is sufficiently large.

Proposition 2 Suppose that the single crossing property does not hold. When

ˆlC is located at the right-hand side of *

hC , and l̂I is located at the upper-region of

P ,

(1) contracts *hC and lCʹ′ constitute a separating equilibrium if ˆlC is located at

the upper-region of lP , and lI ʹ′ is at the upper-region of P and is flatter than

lP at lCʹ′ ;

(2) contracts *hC and ˆlC constitute a separating equilibrium if ˆlC is located

between lP and P , and l̂I is located at the upper-region of P .

Proof Let us assume a fair premium to illustrate the equilibrium. When the

premium is fair, the first-best contract *hC for the type h individuals contains full

coverage. When ˆlC is located at the upper-region of lP , and lI ʹ′ is at the

upper-region of P and is flatter than lP at lCʹ′ as shown in Figure 3, we cannot

find any other profitable contract to attract type l individuals. In this equilibrium,

firms cannot make a positive profit.

Figure 4 demonstrates the conditions that ˆlC is located between lP and P ,

12

and l̂I is located at the upper-region of P . It is easy to find that there is no any

other profitable contract to attract type l individuals. Note that ˆlC is a profitable

contract.

In the separating equilibrium outlined in Proposition 2, risk type is positively

correlated with coverage level. In other words, adverse selection is observed.

Proposition 3 Suppose that the single crossing property does not hold. When

ˆlC is located at the left-hand side of *

hC , and l̂I is located at the upper-region of

P ,

(1) contracts *hC and lCʹ′ constitute a separating equilibrium if ˆlC is located at

the upper-region of lP , and lI ʹ′ is at the upper-region of P and is steeper than

lP at lCʹ′ ;

(2) contracts *hC and ˆlC constitute a separating equilibrium if ˆlC is located

between lP and P , and l̂I is located at the upper-region of P .

Proof Figures 5 and 6 respectively illustrate conditions (1) and (2). It is easy to

find that there does not exist other profitable contract to attract type l individuals

besides the equilibrium contracts.

Proposition 3 provides the condition for advantageous selection, i.e., risk type is

negatively correlated with coverage level. In other words, in the presence of

ambiguity, both adverse selection and advantageous selection could exist. Whether

13

coverage is positively or negatively correlated with loss probability depends on

insurance loading and the degree of ambiguity aversion. When the premium is fair, it

is not possible to obtain advantageous selection or pooling equilibrium. If there is an

equilibrium, the market will be settled at separating equilibrium with adverse

selection.

When the premium is unfair, the degree of ambiguity aversion is important for

different types of equilibrium. As shown in the previous section, if ( )( )R Eu rΦ for

all individuals is constant or increasing in r (i.e., the preferences of the individuals

exhibit constant or decreasing ambiguity aversion), then type l will demand less

coverage than type h at *hC . Therefore, ˆlC will be at the right-hand side of *

hC .

Advantageous selection is not possible to be observed in this case. It will be observed

only when ( )( )R Eu rΦ for all individuals is decreasing in r , i.e., the preferences

of the individuals exhibit increasing ambiguity aversion.

In the above analyses, we assume that each insurer can offer only a single

contract as in Rothschild and Stiglitz (1976). Thus, we could observe profitable

contract in a perfectly competitive insurance market when single crossing property

does not hold. When insurers are allowed to offer more than one contract, profit

making contracts cannot exist in equilibrium.

4. Conclusion

This paper provides a new approach to advantageous selection. Based on the

classic Rothschild and Stiglitz (1976) model, we consider ambiguity-averse

individuals who face a general risk and a specific risk in a perfectly competitive

insurance market. The individuals have private information regarding the specific risk

but have unbiased ambiguous beliefs regarding the general risk. Individuals'

preferences are characterized by Klibanoff, Marinacci and Mukerji's (2005) smooth

14

model, and they are homogeneous except for their risk type.

We find that both pooling and separating equilibria can exist. Furthermore, we

find that when the single crossing property does not hold, the equilibrium could be

determined based on adverse selection or advantageous selection. The equilibrium

depends on the individuals' degree of ambiguity aversion. If the individuals'

preferences exhibit decreasing ambiguity aversion, then we might observe adverse

selection in equilibrium. However, if they exhibit increasing ambiguity aversion, then

advantageous selection might characterize the equilibrium.

15

References

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17

Figure 1: The indifference curves of the h types ( hI ) and the l types ( lI ).

Nw

Aw

hI

lI

450

0

18

Figure 2: Pooling equilibrium.

Nw

Aw

E

hI

lI

450

0

C

hP

lP

P

19

Figure 3: Adverse selection with zero profit.

Nw

Aw

E

*hI

lI ʹ′

450

0

*hC

lCʹ′

hP

lP

P

20

Figure 4: Adverse selection with positive profit.

Nw

Aw

E

*hI

l̂I

450

0

*hC

ˆlC

hP

lP

P

21

Figure 5: Advantageous selection with zero profit.

Nw

Aw

E

*hI

lI ʹ′

450

0

*hC

lCʹ′hP

lP

P

22

Figure 6: Advantageous selection with positive profit.

Nw

Aw

E *hI

l̂I

450

0

hP

lP

P

*hC

ˆlC