First-Best Equilibrium in Insurance Markets with Transaction Costs and Heterogeneity
Author(s): Jerry W. Liu and Mark J. Browne
Source: The Journal of Risk and Insurance, Vol. 74, No. 4 (Dec., 2007), pp. 739-760
Published by: American Risk and Insurance Association
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? The Journal of Risk and Insurance, 2007, Vol. 74, No. 4,739-760
First-Best Equilibrium in Insurance Markets
With Transaction
Costs and Heterogeneity
Jerry W. Liu
Mark J. Browne
Abstract
We
of
adverse
the market.
off
We
to either
heterogeneity,
nor
low-risk
we
(1976) (RS)
of transaction costs and buyer heterogeneity
or wealth.
aversion
risk
and Stiglitz
In RS,
information.
low-risk
asymmetric
to an
created
buyers
externality
by high-risk
un
in insurance
behavior
buyers'
changes
under
owing
critical
find
der the joint assumptions
respect
of the classic Rothschild
selection
are worse
customers
in
extensions
investigate
model
find a separating equilibrium
are
individuals
due
penalized
in which
to information
with
costs
transaction
Combining
and
neither high-risk
asymmetry.
Introduction
on markets
is an important
information
issue in the eco
impact of asymmetric
nomics
literature. In 2001, Joseph Stiglitz, George Akerlof,
and Michael
Spence were
the Nobel Prize in Economics
for their analyses of markets under asymmet
awarded
ric information.
In their seminal paper, Rothschild
and Stiglitz
(1976) (RS) provide a
for analyzing
the problem of adverse selection in insurance markets. Their
framework
causes markets
to deteriorate. When
central finding is that information
asymmetry
a separating
"The
individuals
exists,
(low
etc.)
[exert] a
equilibrium
high-risk
ability,
so that the low-risk
dissipative
externality on the low-risk (high ability) individuals/'1
are worse off due to the presence of
customers
customers.
RS
Furthermore,
high-risk
find that competitive
insurance markets may have no equilibrium.
The
In spite of the pessimistic
tion well under asymmetric
of RS, we observe
conclusions
that many markets
func
information:
labor markets,
credit markets,
and security
JerryW. Liu is at the Krannert
is at the School
via
e-mail:
Suzanne
Papai,
of Business,
Bellezza,
Georges
Robert
Picard,
seminar participants
for helpful
comments.
improving
the article.
1
Rothschild
Purdue University.
of Wisconsin-Madison.
[email protected]
Pierre
School of Management,
University
and [email protected].
Dionne,
Edward
Puelz,
Cheng-zhong
Frees,
We
Thomas
Qin,
Larry
authors
The
thank
at the 2003 Risk Theory Society meeting, University
The
All
and Stiglitz
of
suggestions
errors
are strictly
two
anonymous
referees
J. Browne
can be
Thomas
Donald
Gresik,
Samuelson,
Mark
contacted
Cosimano,
Szilvia
Hausch,
Virginia
Young,
and
ofWisconsin-Madison
were
ours.
(1976, p. 468).
739
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especially
helpful
in
The Journal of Risk and
740
Insurance
are obvious
the results of empirical
studies are not
examples. Moreover,
consistent
the
with
RS
For
in
insurance
RS,
always
predictions.
example,
high-risk
more
customers
customers.
insurance
than
low-risk
However,
always buy
Chiappori
that customers with higher accident
and Salanie
(2000) find no statistical evidence
contracts with more comprehensive
In this article,
coverage.
purchase
probabilities
we modify
several of RS's assumptions
and offer new insights into why the RS results
are not always consistent with empirical
findings.
markets
The most important contribution
of this article is that we show conditions under which
remain efficient. Whereas
RS assume that there
markets with asymmetric
information
are no transaction
are identical
costs in the insurance markets
and that customers
we
are
costs
for
loss
evaluate
when
there
transaction
except
probability,
equilibrium
customers
and when
in addition
differ with respect to risk aversion or endowment,
In the RS framework,
to loss probability.
all customers
In our
prefer full coverage.
are
all
when
there
costs,
buyers prefer partial coverage. Our
setting,
proportional
or wealth
to
with
risk
aversion
indicate that
of
respect
assumptions
heterogeneity
a
in
We
the
insurance
decision.
relevant factors other than loss probability
part
play
are different
new
if
in
and
there
find that if customers
these
dimensions,
enough
are transaction
there are no negative
exists in which
costs, a separating
equilibrium
externalities.
to two areas of the literature that have evolved
from RS. First,
This article contributes
it extends previous work on the effect of transaction costs in insurance markets. Many
the premium
contains a fixed-percent
pre
papers
investigate
optimal choice when
uses
to show the optimality
of a
mium
calculus
of
variations
(1971)
loading. Arrow
a
insurance
that
Mossin
theorem
whereas
deductible,
(1968) proposes
stating
partial
and Vanasse
In another important study, Dionne, Gourieroux,
is optimal.
(1998) make
an attempt to show that the RS result remains unchanged
when different
risk types
costs into the RS
cost structure. We integrate transaction
have the same proportional
two cost structures: constant costs and pro
framework by systematically
evaluating
from proportional
individuals
suffer more
costs.
We
that
argue
high-risk
portional
and the RS externality
result persists.
costs than low-risk individuals,
Second, our study extends the literature on heterogeneity
among insurance customers.
are
assumes
in
endowment
risk aversion,
that
RS
model
The
homogeneous
people
we
not
offer an
and
be
These
and
loss
realistic,
wealth,
may
assumptions
severity.
new
that
alternative
setting
yields
insights.2
and Webb
evaluated
risk aversion has been previously
by De Meza
Heterogeneous
more
to
assume
insurance
risk-averse
tend
that
coverage,
(2001). They
buy
people
In their setting, it is
reckless people put less value on insurance protection.
whereas
in which
to reach a partial pooling
some, but not all, low-risk
equilibrium,
possible
are worse
risk
aversion has a different
off. In our model,
individuals
heterogeneous
2Our article is
the RS model by
closely related to Doherty and Jung (1993). They modify
is that first
in
to
conclusion
and
their
addition
differences
probability
considering severity
best
we
solutions
focus
on
are
feasible.
the externality
Although
conclusion
and Jung concentrate
Doherty
of the RS model
by revising
on
a
multiple
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single
assumption,
assumptions.
First-Best Equilibrium in Insurance Markets
effect: we
mation
and the infor
may reach a new separating equilibrium
irrelevant. This result, however,
relies on the simultaneous
costs and heterogeneous
risk aversion.
find that the market
becomes
presence
741
problem
of proportional
two scenarios: Scenario A,
In our framework with proportional
costs, we analyze
are
more
customers
in which high-risk
risk averse (less wealthy)
than low-risk cus
are
more
in
customers
which
low-risk
risk averse (wealthier)
tomers, and Scenario B,
In
customers.
than high-risk
in risk aversion
Scenario A, when
the difference
(wealth)
between high- and low-risk customers
is great enough,
the optimal contracts of the
In the
creating a separating
high- and low-risk customers will diverge,
equilibrium.
a
new separating
on
neither
risk
the
group imposes
externality
negative
equilibrium,
we call this an NE (no externality)
In
other group. Henceforth,
Scenario
B,
equilibrium.
an NE
in risk aversion
exists when
the difference
(wealth) is large and the
equilibrium
in loss probabilities
if consumers
is small. Moreover,
differ inwealth,
difference
rather
than in risk aversion,
the same no-externality
result occurs (high wealth
is equivalent
to low-risk
aversion).
on the assumption
It appears that the RS negative
conclusion
that
externality
depends
are
on
customers
and
low-risk
identical
all
than
dimensions
other
loss
highprobabil
are so similar that the risk groups prefer the same
ity; customers
policy. They do not
have other reasons for preferring
that are different. Our results suggest that
policies
costs and consumer heterogeneity
the interaction between
transaction
may explain,
at least in part, why we observe markets
and
information
thriving under asymmetric
why
empirical
results do not always
confirm
the RS predictions.
as follows: first is an overview
of this article is organized
of the RS
an
the
costs
in
next
followed
of
effect
of
transaction
the
section.
model,
analysis
by
are presented
The case of heterogeneous
risk aversion and the NE equilibrium
next,
are
and then the effect of heterogeneous
is investigated.
wealth
Finally, conclusions
The
remainder
presented.
The Model
Because our analysis
is based on the RS model, we begin with a brief review of their
model
and its assumptions.
There are two types of insurance customers
in the market,
>
low-risk and high-risk. They have different accident probabilities,
pH
pL. Customers
know their accident probabilities,
but the insurance company does not. RS also make
the following
four simplifying
assumptions:
1. There
2. Both
3. Every
are no transaction
types of customers
costs and the insurer makes
have
client has an identical
4. If there is an accident,
both
the same
endowment
risk groups
zero expected
profit.
level of risk aversion.
W.
have
the identical
In this article, we modify
the first three assumptions,
dissimilar
than they are in the original model. We
from the RS model.
loss amount
d.
thus making
the two groups more
the
adopt
assumptions
following
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The Journal of Risk and
742
Insurance
a contract a = (ct\, 012),where a\ is the
The insurance company writes
premium
amount and a2 the net indemnity
amount
is deducted).
If an accident
(premium
The price of in
of ot\ +a2.
occurs, the individual will receive a total indemnity
surance is q(a) = a\/a2.
nor the indemnity
Since neither
the insurance premium
should be negative, we call an insurance contract (pt\, aj) admissible if and only if
a\ > 0 and a2 > 0.3
(accident or no accident)
the agent's wealth
is the
in the two states of nature
The individual
agent's wealth
a
is represented
vector
(YI\, W2). If there is no accident,
by
endowment
minus
the premium paid to the insurer,
W1
If an accident
occurs,
=
W-a1.
(1)
a loss d and receives
he suffers
W2 = W-d+a2.
a reimbursement
a\
-f a2,
(2)
share
Insurers behave as if they are risk neutral. RS argue that insurance company
holders hold well-diversified
portfolios.
customers
is defined as a set of contracts such that, when
choose
The equilibrium
no
contract
in
set
contracts tomaximize
the
makes
expected utility: (1)
equilibrium
no contract outside
set
the equilibrium
negative
expected profits, and (2) there is
that, if offered, will make a nonnegative
profit.4
All insurance customers have absolute risk aversion,
U"(x)
<3)
'"-m-
x. In
or an increasing function of wealth
where r (x)may be a constant, a decreasing
constant
to
is
be
assumed
the
role
of
risk
this article, when
aversion,
r(x)
studying
that r(x) decreases with wealth
in order to control for wealth
effects. We postulate
wealth.
the effect of heterogeneous
while discussing
=
contract
t individual
The expected utility of a risk-type
(t H, L), who purchases
a (ai, a2) is
V*(ai,a2)
From
(4), the marginal
=
(!
-
-
rate of substitution
MRS'
3
This
restriction
when
transaction
4
result
The main
such
most
allocation
of
is necessary
costs are
since
is robust
without
to alternative
for type
it the
optimal
equilibrium,
the concept
equilibrium
-d+
a2). (4)
t is
dW2 Q-WM)
extremely
high.
no
of this article,
externality
is an
whatever
equilibrium
the article
e*i)+ pfU(W
Pl)U(W
(5)
premium
relies
(among
could
become
negative
on
allocations;
any
envy-free
In this respect,
standard
ones).
concepts.
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First-Best Equilibrium in Insurance Markets
> 1~? ) the low-risk
^?^~
curve.
than the high-risk
indifference
It is clear to see that
steeper
Transaction
curve
indifference
743
is everywhere
Costs
there are no transaction costs and insurance firms make zero expected
for each insurance policy, the expected claim, p(a\ + oti), is equal
profits. Consequently,
as
to the premium
a\. The insurer's expected profit can be expressed
In the RS model,
n = a\ -
= 0.
p(?i + a2)
(6)
In reality, insurance contracts cost more than the expected value of their claims. Allard,
a distribution
the RS model
cost,
Cresta, and Rochet
(1997) modify
by introducing
an
to
cost
and
contract.
which
of
insurance
the
corresponds
designing
marketing
is that the total cost is assumed
to be a constant, and
The key feature of their model
cost decreases when more customers buy the contract. This
therefore the per-customer
a counterforce
to adverse
and it leads to their
of scale provides
selection,
economy
most
exist
conclusion:
the set-up cost is
when
important
"Pooling equilibria always
large
enough."
In this article, the cost structure has both
or reimbursement:
tional to their premium
a constant
term and terms
cost = a + boi\ + cp(pt\ + a?).
that are propor
(7)
The cost function can be added to the right side of the budget Equation
(6). Below, we
discuss
the constant and proportional
of the general cost function, and
components
impacts on insurance buyers' behavior.
analyze their respective
Constant
Cost
Insurance
maintenance,
have many
fixed costs, such as rent, technical hardware
companies
a constant
and utilities.
Let us first look at the effect of having
and
cost
alone,
cost = a.
With
a fixed cost, the individual's
Max
: E(U) =
(8)
optimization
(1
-
p)U(W-
problem
is (hereafter
Problem
I)5
c*i)+ pU(W-d+a2)
5 In
Problem I, the first condition is the budget constraint and the second condition states
that people need to see an improvement in their welfare before they pay for any amount of
insurance.
The
detailed
The
final
solution
condition
to Problem
the
guarantees
I is available
insurance
from
premium
the authors
will
on
not
request.
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become
negative.
744
The Journal of Risk and
Insurance
to :
Subject
?
a\(l
p)
?
a2p
= a
(1 p)U(W ax) + pU(W d + a2) > (1 p)U(W) + pU(W d)
on >
0,
0.
a2>
is optimal when
costs are constant.
Previous
studies have shown that full coverage
As long as the fixed cost is not too high, people will pay a premium
of pd + a and
?
? a
an
occurs.
will have wealth W
of
whether
Without
accident
any
pd
regardless
?
a fixed cost a, the
W
and
level
is
when
is
wealth
there
costs, customers'
optimal
pd,
to W ? pd ? a. We can view the constant
in
amount
is reduced
load as a reduction
all insurance buyers start with reduced assets and make the same choice
endowment;
as in a perfect market without
costs.
transaction
case
if the fixed cost is too high. An interesting
People will not buy any insurance
are
out
cost
market
but
the
is
that
low
risks
of
the
when
fixed
such
the
forced
emerges
the high risks get their first best contract and
the high risks still stay. If this happens,
the low-risk contract in the second best is the same as in the first best. The high risks
reaches a special
do not bring any negative
effects to the low risks and the market
separating
equilibrium.
Cost
Proportional
Insurance
also
companies
variable
have
costs,
such
as
commissions,
premium
taxes,
so we can assume that costs also
and litigation expenses,
claim adjustment
expenses,
costs is premium
An
these
of
include a variable
taxes, which
component.
example
on
state
4
2
to
the
of
from
vary
Expenses
percent
regulations.
premium,
depending
are closely correlated with the total claim amount
associated with claim settlements
Consequently,
our cost function
two proportional
includes
Cost = ba\ + pc(ot\ + a2),
where
b and
c are constants.
The
seeks
components:
(9)
to maximize
insurance
buyer
p)U(W-
on) + pU(W-d+a2)
his expected
utility (Problem II):
Max:
E(U)
Subject
=
(1
to: a\(l
?
p)
ot\ >
?
0,
a2p
a2
= bot\ +
po.(a\ + a2)
>
0.
is optimal
if
that partial insurance
Previous
studies, such as Smith (1968), conclude
In
the
context
Problem
of
in
the
there are proportional
II,
partial
premium.
loadings
state than in the
in the non-loss
have more wealth
that individuals
insurance means
in the interest of brevity. Figure 1 illustrates
The proof is omitted
loss state, Wl>W^.
in the no-accident
insurance
becomes
optimal. The customer's wealth
why partial
in the accident state is on the y-axis. The 45-degree
state is on the x-axis and his wealth
is the same in both states of nature.
the buyer's wealth
full coverage:
line represents
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First-Best Equilibrium in Insurance Markets
745
Figure 1
Optimal
Choice
W2
Accident
/
Under
Costs
Proportional
No Cost /
A .
.
With Cost
B
XV
^
s
I
/
/
yy
\^
*
N%
\
"'X..\
B
/
|/^45_
No Accident
or market
odds
endowment
is E. The lines AE and BE are budget,
customer's
transaction
dollar. Proportional
lines. Their slope is the dollar coverage per premium
costs reduce the coverage per premium dollar, causing the budget
lines to flatten out.
a shallow slope; they get less coverage per premium
dollar
customers
face
High-risk
there is no cost, the customer buys full coverage,
than do low-risk customers. When
a*. When
there is a proportional
cost, the customer's
utility curve is tangent to the
more shallow budget line at a point P*, below the full coverage
line; the customer buys
worse
customer
is
Without
off if an accident
full
the
coverage,
only partial coverage.
The
occurs.
Externality Under Proportional Costs
Let us first recall the situation in the RS model
both risk types
of asymmetric
information,
transaction costs. In the absence
without
take full insurance aH and aL, as shown
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746
The Journal of Risk and
Insurance
Figure 2
Separating Equilibrium in theOriginal RS Framework
Accident
/
/
/
a*
rv^F-?
A
/\
v*
No Accident
in Figure 2, as their first-best choice. With asymmetric
information,
high-risk buyers
more
contract
in both states of
it
the
low-risk
because
aL
consumption
provides
prefer
nature. Since insurers cannot sort out bad customers
from good, they offer only a par
tial coverage contract to the low-risk customers. This contract, yL, is at the intersection
curve and the low-risk budget
line. Thus, the presence of
of the high-risk
indifference
a
on
low-risk
individuals.
individuals
inflicts
externality
negative
high-risk
costs, the optimal contracts for both risk groups are below the 45
proportional
costs can reverse the
in
the incorporation
of transactions
line
degree
Figure 2.Whether
and Vanasse
becomes
conclusion
(1998)
interesting. Dionne, Gourieroux,
externality
on
costs. They ana
transaction
with
first
adverse
selection
the
(DGV) conduct
study
cost function
under the proportional
lyze optimal deductibles
Under
Cost =
CJ)((X\+ ot2), (10)
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First-Best Equilibrium in Insurance Markets
747
Figure 3
Externality
w*
Accident
in the RS Model With
!j
Proportional
Transaction
Costs
/
L (withCosts)
H (withCosts)
/
\
TJH/
A/
\
\
P\
U*
1
a
/
\v\\ff'
/ a"* ^|
/
^^
O *.4.^
?
^45_
No Accident
and
their conclusion
is that the RS externality
use
an
we
alternative
insight,
approach to show
cost structure bct\ + pc(ot\ + o^).
result persists. To provide
additional
the cost effects under the more general
1: Under the cost function bot\ + cp{ct\ + ct2), high-risk
Proposition
low-risk first-best contract to their first-best contract.
individuals prefer the
in Figure 3. Point E is the in
Proof: Our mainly
is illustrated
graphical procedure
surance buyer's
are the low
and
LE
and
HE
endowment
lines
point,
respectively
curve
and high-risk
costs.
lines
with
The
indifference
budget
proportional
high-risk
to
at
and
it
is
and
the
line
A,
C,
F,
passes
tangent
through points
high-risk
budget
the point aH*, the high-risk
first-best contract. Our goal is to show that the low-risk
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The Journal of Risk and
748
contract
first-best
aL*
prefer
aL* must
Insurance
C and F.6 Therefore,
fall between
customers
the high-risk
to aH*.
As a first step, we show that aL* will fall to the southeast of point C. It is obvious
that,
curve is steeper than LE, the low-risk budget line.
at point C, the high-risk
indifference
curve is steeper than the
From equation
(5) we know that the low-risk indifference
curve and the low-risk
one.
the
The
low-risk
indifference
tangent point of
high-risk
must
to
fall
of
C.
line
southeast
the
ctL*
budget
step is to prove that aL* will
in Problem
II:
Our goal in the second
the first-order condition
U'(W-ai(p))
= l-b-p(l+c)
(l+c)(l-p)
U'(W-d+a2(p))
Taking
the derivative
of both
be above
sides with
respect
to p, we
point
F.We
start from
'( '
have
+
d*i = g[U'(W2) IT'(WQ] (1 p)(ai a2)U?(Vh)
'
K }(m
+
dpg2W{]N2) p{l-p)U'\]N1)
+
U" < 0 (decreasing marginal
c). Since W > 0 (increasingutility),
1~b~^1
< 0, which means
> W2
it is trivial to show that
(partial coverage),
utility), and W\
-^
than the high
that the low-risk first-best contract offers a larger net reimbursement
line with
the
risk contract. In Figure 3, point D is the point on the low-risk budget
same net reimbursement
the low-risk first-best contract aL* is
a2 as aH*. Obviously
whereg
=
to the northwest
of F, the optimal contract aL* is above
above D. Since D is positioned
curve and the low-risk budget
of
intersection
the
indifference
the
high-risk
point F,
line.
line.
fall between
C and F on the low-risk budget
that aL* must
proved
of
is
if
instead
consumers'
a?L*
the
Therefore,
they purchase
utility
higher
high-risk
We
have
aH*.
costs are much
like taxes imposed on insurance buyers. All else equal,
individuals
ot\ (a2),
(receive less net reimbursement),
pay higher premiums
high-risk
the cost structure bot\ + cp{ot\ + a2), high-risk
individuals.
Under
than do low-risk
is reduced to
suffer more from the proportional
individuals
cost, and their coverage
contract
a greater extent than low-risk individuals'
the
low-risk
Therefore,
coverage.
it is trivial to show the RS
to the high-risk
individuals. Moreover,
is more attractive
cost structures,
such as
transaction
result holds under other conventional
externality
and
+
ba\
ot2)7
cp(a\
Transaction
6 It is also
possible
low-risk
that point C, the intersection
line,
budget
falls
above
A,
point
the
of the high-risk
intersection
of
indifference
the high-risk
curve and the
indifference
curve
and the 45-degree line. In that case, the graph is different and it is obvious that aL*would be to
the southeast of point C, the intersection of the high-risk indifference curve and the 45-degree
line.
7
We would
schemes.
like
For
to point
example,
out
if cost
that
=
the RS
ca2,
the
can be reversed
result
externality
more
be affected
low risks will
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under
than
certain
the high
cost
risks.
First-Best Equilibrium in Insurance Markets
749
Risk Aversion
Insurance buyers might have different levels of risk aversion. This possibility
has been
in an information
evaluated
researchers. RS
asymmetry
setting by several previous
are less risk averse. RS discuss
that low-risk individuals
point out that it is possible
are
more
averse
risk
that high-risk
than low-risk individuals
the possibility
people
cannot
exist.
De
Meza
and Webb
that a pooling
and conclude
(2001) find
equilibrium
individuals
tend to buy more coverage
that low-risk, highly
risk-averse
than high
In their setting, a pooling
individuals.
is possible.
risk, less risk-averse
equilibrium
risk aversion to the RS model.
Smart (2000) and Villeneuve
(2003) add heterogeneous
In Smart's model,
there are four customer groups: high risks with high risk aversion
with
low risk aversion
(hh), high risks
(hi), low risks with high risk aversion
(lh),
is that the indifference
and low risks with low risk aversion
His
(11).
major conclusion
curves of the hi and lh types may cross twice and different
risk types may be pooled
assumes
in one contract. Villeneuve
that one risk class is more
risk averse than the
are
that positive
other risk class in the market. He reaches the conclusion
profits
for the low-risk contract and that random insurance contracts may also
sustainable
exist. Our article differs from Smart and Villeneuve
in two critical ways: First, we
in
to
transaction
addition
risk aversion,
into the RS
costs,
incorporate
heterogeneous
we
contest
model.
Second and most
the
conclusion
of the RS
important,
externality
model.
In this section, we
1.
There
extend
are proportional
the basic RS model
transaction
with
a new
costs:
Cost = boti + pc(ai + a2).
2.
set of assumptions:
(13)
on risk aversion. The
customers'
utility functions depend
respective
are UH(x) and UL(x),
of
the
and
low-risk
customers
functions
utility
high-risk
are rH(x) and rL(x). We examine the
and their respective
risk aversion coefficients
two possible
are more risk averse
scenarios: In Scenario A, high-risk
individuals
>
than low-risk
in
B the relation is
whereas
Scenario
individuals,
r(x)H
r(x)L,
Insurance
reversed.
3.
It is usually
on wealth
assumed
that absolute
risk aversion depends
level. We
neutralize
the wealth
effect by assuming
that customers have constant absolute
risk aversion
is not critical; our result could be eas
(CARA). This assumption
cases.
to
to Pratt (1964), constant risk aversion
other
ily generalized
According
a
generates
negative
exponential
utility function,
U*(x)
We find numerical
examples
inwhich
=
-e~rt\
t = H,L.
(14)
the high risks are no longer interested in the low-risk
contract.
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The Journal of Risk and
750
Insurance
Are These Scenario
Realistic?
Assumptions
are
Both Scenario A and Scenario B
in different circumstances.
Customers
reasonable
with a history of serious, extended
illnesses in their families often have a relatively high
likelihood of contracting
these illnesses. It is sensible to think that customers with such
also be more risk averse than others in making
their health
family histories would
This gives us a realistic situation of Scenario A. In alternative
insurance decisions.
we can find real-world
situations
for Scenario B. For example,
reckless
circumstances,
to
minimum
and
tend
be
less
risk
drivers
averse,
may
might
buy only
(high-risk)
since both recklessness
and low coverage are consistent with disregard
for
coverage,
averse
more
risk
and
cautious
also
be
drivers
risk. Likewise,
(low-risk)
might
buy
are consistent with risk aversion.
more coverage,
since both behaviors
Indifference Curves
RS, our procedure
Following
is represented
by the marginal
Under CARA, MRS is [(1
and low-risk MRS is
ismainly
graphical. The slope of the indifference
rate of substitution,
pt)/pt]e"''^-w^.
MRSH=
MRSL (Izrl
Therefore,
the ratio of high-risk
MRS
(i6)
llz?\e-frM)xw^
V PH I
curve
PL J
this ratio is greater than one; this will tell us whether
customers have a greater MRS. Clearly, the first factor
on the right side of the equation,
is positive
and less than one. Is the
l~S /?/?-,
~r
at the exponent
of this
second factor, e~(r
)x(Wi-w2)^ ajso jess faan one? Locking
in the loss state than in
should not have more wealth
factor, we know that customers
is positive. Thus, whether
the no-loss state, W2 < Wi, so the last factor in the exponent
on
or
is
than one depends
in
than
less
the exponent
the risk aversion
factor
greater
two
risk
aversion
of
the
risk
levels
of
the relative
groups.
to determine whether
or low-risk-aversion
We wish
the high-
-
Scenario A
High-risk
individuals
are more risk averse than low-risk individuals.
is less than one and MRSH < MRSL. The
In this case, rH > rL. So e^rH-rL^w'-w^
curve.
curve is always flatter than the low-risk indifference
indifference
high-risk
Scenario B
In
this
-
case,
High-risk
rH
<
individuals
rL and
e~(r
are less risk averse than low-risk individuals.
-rL)x(Wi-w2)
js greater
than
one.
It is unclear
whether
the product of the two factors in (16) is greater than or less than one, and therefore,
~r )x(wi-w2)
curve is flatter. Ifwe hold pH and pL fixed, then e~(r
which
indifference
rH and rL, and with the difference between Wi
increases with the difference between
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First-Best Equilibrium in Insurance Markets
751
are sufficiently
the
large, ^-(rH-rL)x(wi-w2) dominates
> 1. In this case, the
curve
is
than
steeper
high-risk-indifference
T^|tr
curve. On the other hand, when
the differences
between
the low-risk-indifference
rH
curve is steeper.
and rL and between
]N\ and W2 are small, the low-risk indifference
and W2. When
these differences
first factor and
If the indifference
curves
indifference
risk type are flatter
curves
indifference
of one risk type is always flatter, as in Scenario A, the two
curves of each
if the indifference
will cross only once. However,
in some circumstances
and steeper in others, as in Scenario B, the
may cross twice.
curve
Risk Aversion and No Transaction Costs
Equilibrium Under Heterogeneous
we find that, in most
these assumptions,
there cannot be a
circumstances,
note
at
with
which
is
consistent
RS.
RS
that
any possible pooling
pooling equilibrium,
curves have different
point the two indifference
slopes. Any new contract that lies
curves will attract the low-risk
the two indifference
between
type away from the
In our Scenario
the
unstable.
point, making
potential
equilibrium
pooling
pooling
curve
is
the
of
low-risk
indifference
than
the
the
A,
steeper
slope
slope. In
high-risk
curve is flatter than the high-risk
that the low-risk indifference
Scenario B, it is possible
curve. In this case, we can find a destabilizing
contract that lies to the west
indifference
curve and below the
of the point of intersection,
above the low-risk indifference
high
risk indifference
curve, as shown in Figure 2 of the original RS paper. This contract
a
will attract the low-risk individuals
away from the potential pooling point, making
a
to
In
sustain.
low
stands
for
equilibrium
Figure 2, rjL
pooling
impossible
possible
curve is flatter than that of the
contract if the low-risk indifference
risk equilibrium
Market
Under
high-risks',
which
scenario
is discussed
later in the article.
can emerge, however, when
curves
A special pooling equilibrium
the two indifference
are tangent to each other. This case can only be found in Scenario B and it
requires
curve lies below the low-risk indifference
curve. More
that the high-risk
indifference
that the tangent point falls on the aggregate
over, it is also necessary
line,
budget
zero profit when
which makes
both high- and low-risk
individuals
buy one con
curves will attract only the
tract. Any contract that lies between
the two indifference
a
it
to
and
will
lead
individuals,
negative
high-risk
profit. As noted by Wambach
rare
this
of
is
since
it calls for special parameter
(2000),
type
pooling
equilibrium
values.
it comes to the separating
the introduction
of heterogeneous
risk
equilibrium,
aversion alone does not change the RS argument. The main reason for this is that both
risk types take full insurance as their first-best choice. Thus, as in the cases discussed
in the "Transaction Costs"
of high-risk
individuals
inflicts a
section, the presence
on
low-risk
individuals.8
negative
externality
When
8Villeneuve
curve might be flatter
(2003) suggests that, in Scenario B, the low-risk-indifference
than that of the high-risk group at yL, as in Figure 2. In this case, the low-risk group will prefer
another
point
nL, the
tangent
point
of
to the insurer since it is located below
the
two
indifference
the low-risk budget
curves.
The
contract
line.
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?yLis profitable
752
The Journal of Risk and
Insurance
Risk Aversion and Proportional Costs
Equilibrium With Heterogeneous
new
the
under
costs and heterogeneous
Overall,
separating equilibrium
proportional
risk aversion differs from the RS equilibrium
in two ways. First, both risk groups pre
fer to have less than full coverage; their optimal contracts are below the 45-degree
line.
costs. Second, we know from Schlesinger
This is due to the presence of proportional
costs, whichever
(2000) that with proportional
group has low risk aversion prefers
on the
contract
less insurance;
their optional
is closer to the endowment
budget
Separating
line.
The consumer's
optimizing
contract ar(ai,
the optimal
is as follows. For a type t individual
decision
as the solution
to the following
a2) is defined
(t
= H,
L)
problem
(Problem III):
Max:
E(U)
Subject
=
(1
to: ?i(l
-
pf)lW
-
pl)
a2pf
ct\ > 0,
and
?i) + p'tfiW
=
- d
+ a2)
ba\ + pc(pt\ + a2)
a2 > 0.
The expected utility of a type s (s = H, L) individual
from a type t 's optimal contract
as
a
Vs (a**). For example,
is expressed
individual's
high-risk
expected utility from
the low-risk optimal contract is
VH[aL*]= (1 pH)U[W al(pL)] + pHU[W -d+
a2*(pL)].
1: A separating equilibrium inwhich neither group imposes a negative external
Definition
to
the
other
group, which we call an NE (no externality) equilibrium, satisfies thefollowing
ity
conditions:
VH[aH*]
>
VH[aL*]
and
(17)
VL[aL*]> VL[aH*l (18)
1 says that an NE equilibrium
exists if each risk type prefers the contract de
market
because
for
their
The
insurance
reaches a separating
type.
signed
equilibrium
a
risk
and
neither
risk
of self-selection
group imposes
type,
negative
externality
by
on the others.
Definition
Lemma
1: VL[aL*] > VL[aH*] always holds in the presence of a linear cost function.
Proof:
See the Appendix.
1 indicates that if there are proportional
costs, low-risk individuals will always
contract
for
their
risk
the
(18) is always
group, and thus condition
designed
prefer
it
to
true. Therefore,
to show that a separating
is
exists,
equilibrium
only necessary
customers prefer their contract, condition
show that high-risk
(17).
Lemma
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First-Best Equilibrium in Insurance Markets
The Existence
753
of an NE Equilibrium.
3: With a proportional cost function, Cost = ba\ + pc{a\ + ot2), an NE equi
Proposition
librium exists in Scenario A under condition 1 below. In Scenario B, both conditions 1 and 2
below are needed for an NE equilibrium to exist:
1. The difference
2. The difference
Proof:
averse
in risk aversion between the groups
in loss probabilities
between
We first consider Scenario
than low-risk individuals.
A,
is substantial.
the groups
in which
is small.
high-risk
individuals
are more
risk
group's optimal coverage contract, aH*. See Figure 4.
Identify the high-risk
curve that passes through aH* must intersect the low-risk
The high-risk
indifference
some
at
line
point; call it C.
budget
In Fig
As the low-risk customer's
risk aversion declines, he will buy less coverage.
ure 4, the low-risk optimal contract, aLn, moves
toward
the
endowment
southeast,
point, E.
We can always find a risk-aversion
level for the low-risk-aversion
customer
that
to make aLn fall between C and E. When
is low enough
this happens,
high-risk
contract
individuals
the
for
than the contract
rather
them, arH*,
prefer
designed
for the low-risk group, aul.
designed
an NE
the difference
exists for Scenario A when
between
Therefore,
equilibrium
the risk aversion of the two risk groups is large enough.
The argument
for an NE
equilibrium
in Scenario
B is similar.
curve that passes
The high-risk
indifference
through aH* intersects the 45-degree
line at a single point; call itA.
in the difference
To show the effect of a reduction
between
the loss probabilities,
rotate the low-risk budget line counterclockwise.
When
the low-risk loss probability
is close enough to the high-risk
loss probability,
the low-risk budget
line intersects
the 45-degree
line at a point below A and it intersects
the high-risk
indifference
curve at another point, which we call B.
As the low-risk group's
risk aversion
in
low-risk customers buy more
increases,
surance coverage and the optimal point, aL*2, moves
toward the 45-degree
line. At
a certain level of risk aversion, aL*2 falls above B on the low-risk
line. When
budget
this occurs, we have an NE equilibrium.
In summary, we find that when high-risk
customers are more risk averse than low-risk
an
NE
exists
if
the
in risk aversion between
difference
the risk
customers,
equilibrium
are less risk averse,
is
On
the
other
if
customers
hand,
groups
great enough.
high-risk
in loss probabilities
then an NE equilibrium
exists if the difference
is small enough
and if the difference
in risk aversion
is great enough.
can emerge in Scenario B under
As we discussed
earlier, a pooling equilibrium
special
a
values.
establishes
the fact that a pooling
However,
parameter
simple argument
cannot exist when
there is an NE equilibrium;
the low-risk individuals
equilibrium
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754
The Journal of Risk and
Insurance
Figure 4
NE Equilibrium With
Accident
Proportional
Costs
and Heterogeneous
Risk Aversion
/
\/\A5_*
No Accident
the contract ah*2, as in Figure 4, to any other contract since it is the first
prefers
contract is
best contract under the low-risk budget constraint. Any potential pooling
to
and
will
the
contract
choose to
the low-risk individuals
aL*2
unstable;
pooling
prefer
individuals
leave. The contract aL*2 has no appeal to the high-risk
since, to a high-risk
exists.
customer, aH* is better than aL*2 when an NE equilibrium
in which
traits
insurers use observable
In the insurance markets,
risk categorization,
customers
is often used to
into risk groups,
that are related to risk for separating
are close, as discussed
in the NE
adverse selection. When
the loss probabilities
mitigate
in
is
Scenario
of
limited
risk
classification
B,
practical value; not
probably
equilibrium
are
to
but
too
variables are
the
risk
the
levels
close
risk
distinguish
categorization
only
to other variables,
not particularly
sensitive
such as risk aversion. Risk categorization
other than risk.9
ismost important when consumers do not differ much on dimensions
9
We
thank
an anonymous
referee
for suggesting
this point.
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First-Best Equilibrium in Insurance Markets
755
a normal
exist within
range of risk-aversion
parameters?
equilibrium
aversion
relative
risk
indicate that the level
of
the
studies
of
magnitude
Empirical
In a study of investment
from person to person.
of risk aversion varies substantially
find
that
estimates
of relative risk aversion
and Singleton
behavior, Hansen
(1993)
an
in
et
to
al.
from
0.359
58.25.
find,
(1997)
range
experimental
Barsky
study, that
to
of relative risk aversion
from
0.7
15.8.
The
estimates
range
example
following
this range of risk-aversion
that an NE equilibrium may exist within
demonstrates
Will
an NE
parameters.
Examples of NE Equilibria
in the insurance
Let every customer
?W
relative risk aversion is R(W) =
is set to one, absolute
Since wealth
an
that when
aversion. We assume
0.5, and that both
U(x)
=
W = 1. By definition,
and absolute risk aversion is r(W) =
jp^
?jp^
risk aversion has the same range as relative risk
accident occurs, the individual
suffers a loss d =
market
the low- and high-risk
have
customers
initial wealth
have negative
exponential
utility
-e~rx.
are more
customers
risk averse than low-risk
First, consider Scenario A: high-risk
= 0.1 and
customers.
individuals
have a loss probability
Let low-risk
high-risk
pL
=
a probability
risk aversion of the high-risk,
0.15. Let the absolute
individuals
pH
individuals
be rH = 2, and let rL = 0.5 for the low-risk, low-risk
high-risk-aversion
aversion
individuals.
=
If insurance companies provide actuarially
fair policies
(cost
0), both low- and high
a
full
in
to
risk individuals
and
coverage,
transparent market,
prefer
they are willing
a
are
of
and
customers
0.05
When
loss
0.5
of
occurs,
0.075,
pay premiums
respectively.
with asymmetric
the high-risk
individuals
information,
fully reimbursed. However,
in
will
the
low-risk
if
0.0077
gain
prefer
policy. They
expected utility
they buy the low
are forced to offer a
risk policy. Thus, insurance companies
partial coverage policy to
are worse off due to the presence of
the low-risk customers. The low-risk customers
customers.
high-risk
consider
the case in which
the premium
includes a proportional
transaction
Now,
cost bai + pc(a\ + a2). Let both b and c be 10 percent. No one will buy full insurance
because the premium
is not actuarially
fair. The high-risk
customers pay a premium a
J1
= 0.07,
= 0.5 occurs,
case.
a
to
in
the
zero-cost
If
0.075
d
loss
compared
they will receive
a reimbursement
customers will scale back their premium payment
of 0.38. Low-risk
case of a loss, they will recover only 0.049, about 10
to
in
from
0.05
and
0.0059,
a\
ismuch
lower than the
percent of the loss amount. Because the low-risk best coverage
best
individuals
receive
coverage, high-risk
greater expected utility10 from
high-risk
their own policy VH[aH*] = -0.162
to
low-risk policy cut, VH[aL*] =
the
compared
?0.167.
Low-risk
customers
also prefer their own policy V^a^*] = ?0.624
to the
=
an
to
leads
NE
self-selection
Thus,
-0.631).
high-risk policy (VL[aH*]
equilibrium.
10
The
level
utility
it
is
utility,
always
expected
expected
U(x) is equivalent
is negative
permissible
because
to add
= -e~rx.
U(x)
a fixed
positive
If the
reader
number.
to the utility function U(x) = AU(x) + B, with A > 0.
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The
prefers
utility
positive
function
756
The Journal of Risk and
Insurance
are more risk averse, there
low-risk customers
Similarly, under Scenario B, in which
an
an absolute risk-aversion
NE
Let
customers
also
be
have
may
equilibrium.
high-risk
coefficient
rH = 0.5, and let low-risk customers have a coefficient
that
rL = 2. Assume
= 0.105, and that low-risk customers
a
customers
have
loss
high-risk
probability
pH
= 0.1. In this case,
have a loss probability
that is close to the low-risk probability,
pL
are
a
on
customers
to
0.0059
whereas
low-risk
high-risk
willing
spend
premium,
customers will pay a premium
customers
of 0.047. In case of a loss, d = 0.5, high-risk
will only receive a payment
of 0.046, whereas
low-risk customers will recover 0.39.
?
= 0.0047, and
own
customers
low-risk
their
VL[o?H*]
Again,
prefer
policy, V^a11*]
?
=
customers prefer a high-risk policy, VH[arH*]
0.0002, and an NE
V,H[aL*]
high-risk
exists.
equilibrium
Wealth
RS assume
that all agents have the same endowment,
but it is possible
that one risk
than the other. For example, when use of age is prohibited
in health
class iswealthier
are in
insurance underwriting,
individuals
both
lower
risk
and less
young
general
on
affluent than middle-aged
individuals. Alternatively,
individuals maybe,
disabled
and less healthy than the general population.
In states such
average, both less wealthy
as New
use
and
status
New
where
is
of
York,
Vermont,
Jersey,
disability
prohibited,
this gives a perfect example of negative
association
between
risk and wealth
level.
are added to the RS model,
costs and heterogeneous
When both proportional
wealth
an NE
occur.
Insurance
absolute
risk aversion
should vary
buyers'
equilibrium may
In
and we assume decreasing
with wealth
absolute risk aversion
this situa
(DARA).11
influences agents' behavior
risk
under
aversion;
DARA,
tion, wealth
greater
through
to lower risk aversion, and vice versa. The analyses under hetero
is equivalent
wealth
are identical to those under
risk aversion, and are not
geneous wealth
heterogeneous
in
the
interest
of
repeated
brevity.
is probably observable,
could be
Itmight be argued that since wealth
risk classification
are
reasons
two
to control for the divergence
in endowment.
there
utilized
However,
we
a self
case.
not
in
relevant
find
that
be
this
that risk classification
First,
may
is
irrelevant
when
this
NE
and
risk
often
classification
exists,
equilibrium
selecting
occurs. Second, classification
in
is becoming
the
United
States.
Various
problematic
state laws and regulations
insurers from using certain characteristics,
such
prohibit
or mental
as a person's
to
discriminate
sex, age, and physical
among
impairment,
individuals
applying
for insurance.
Conclusion
In this article, we investigate
several extensions
of the RS model
of adverse selection
are without
in insurance markets.
RS assume that markets
transaction
costs and that
are
at the least,
in
that
conclude
loss
except
agents
probability.
They
homogenous
consumers
one risk category, and at the worst,
in
the
harm
externalities
negative
market may not be able to reach an equilibrium. We analyze the same problem under
11
DARA
increasing
we
what
is a
commonly
accepted
absolute
risk aversion
see under
DARA,
on the difference between
with
assumption
in the
literature.
An
is
assumption
a mirror
of
image
relies
Since our result
alternative
behavior
would
Individual's
(IARA).
reversed.
the roles of the two risk types
be
the two types, the NE equilibrium may also exist under IARA.
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First-Best Equilibrium in Insurance Markets
757
that transaction costs are present and that the risk types differ on
the joint assumption
in addition
to risk level. We summarize
another dimension
(risk aversion or wealth)
our study as follows:
1.
to the assumptions
costs do ex
Transaction Costs. Contrary
of RS, transaction
costs are present,
ist. When proportional
transaction
individuals
choose partial
2.
costs leads to fundamental
Adverse Selection. Inclusion of transaction
changes in
more
the dynamic
of informational
of the RS model. Analysis
problems
closely
costs are considered.
transaction
resembles actual market dynamics when
3.
on
First, insurance buyers may differ in risk aversion depending
Heterogeneity.
their risk types. Practical examples are found inmedical
insurance and automobile
one risk type may also be more wealthy
insurance. Moreover,
than another risk
have heterogeneous
loss severity. This case has
type. Finally, risk types might
and Jung (1993), and their conclusion
is consistent with
been studied by Doherty
4.
NE Equilibrium. The interaction of transaction
costs and consumer heterogeneity
consumers
to
lead
self-selection
risk
may
type, and in this separating
by
equilib
rium there may be no negative
externalities.
Risk Categorization. Our study suggests
that risk classification
is likely to be most
on relevant dimensions
customers
do not differ significantly
useful when
other
an
in
than risk. When
do
differ
other
then
NE
occur,
ways,
they
equilibrium may
risk assessment
irrelevant.
making
coverage.
ours.
5.
It has been long noted that RS negative
and market
results
externality
equilibrium
are not consistent with many
observable
and
results.
phenomena
empirical
Many
such as labor and financial markets,
in the presence
function well
of infor
markets,
mation
in contrast to the pessimistic
RS prediction.
and Salani?
asymmetry,
Chiappori
show no evidence
(2000) find that customers who have higher accident probabilities
contracts with more comprehensive
some
of choosing
coverage. Our results provide
into
the
forces
within
markets
that may partially
insight
explain these phenomena.
The role of interactions between market
characteristics
and consumer heterogeneity
on multiple
a fruitful avenue for future
dimensions
in
may provide
empirical work
markets
characterized
information
by
asymmetry.
Appendix
Proof
of Lemma
aL*[ai(pL),a2(pL)]
The
low-risk
are comparing
andaH*[ai(pH),a2(pH)].
1: We
individual's
optimal
Max:
=
the value
solution,
aL*,
of function
VL[a]
is the solution
to the following
problem:
E(U)
Subject
(1
-
pL)U(W
c*i)+ pLU(W
to: a?i(l ? pL) ? a2pL =
ba\
a\
> 0,
ct\
-hex
-
d + a2)
pL(a\ + u\)
> 0.
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All use subject to JSTOR Terms and Conditions
at two points,
The Journal of Risk and
758
Insurance
Figure Al
Low-risk Individuals Do Not
Prefer the High-Risk Optimal
Contract
Accident
/
\
/\\
Vs.
V
\ Ul
/^\
ctH
/ r?jy
\\
/
'
X45 if
\
i
a
I
B
9
4
D
Wx
No Accident
We
can rewrite
as
the first constraint
l-fr_pL(1+c)
=
a2 -T7T-^-
2
the value of E(U) increases with a2/ we
restatement
of the first constraint:
following
Since
Max:
E(U)
=
}
Subject
x
pL(l + c)
(1
-
pL)U(W
l
<-.
to: ak2 "
ot\
al
l
can reformulate
ax) + pLU(W
+ c)
l-b-p\l
,/ \pL(l + c)
> 0,
a\
Problem
- d
+ a2)
L
x ot\1
> 0.
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I using
the
First-Best Equilibrium in Insurance Markets
Let SL be the set of feasible
AEGF.
The comparable
problem
SH =
This
f
to this problem.
solutions
customer
for the high-risk
x a1/ttl
set is the area BEGI in the figure. By definition,
this set is the area
is
1
< l-b-vH(l+c)
(c*i,cx2) a2
-hm+c)--
It is clear to see that Sf c
In Figure Al,
759
> 0,a2 > 0
.
aH* e SH.
SL.
VL[aL*] is the point of maximum
utility for the function VL[a] over the set SL, and is
to all contracts in SL, including point A. Using
therefore preferred
the notation X > Y
to indicate that the low-risk customer prefers contract X to contract Y, this result is:
aL* > any point in SL, including
aL* >-A.
specifically:
Let us divide
SH into two subsets:
Therefore,
SH = S2HU Sf.
It is easy
to see that low-risk
Sf, which
customers
is CEGF, and
S^1,which
prefer aL* to any contract
is BCFL
in set
>
S^: aL*
S^.
to contract A, any contract within
in both states of
less wealth
Compared
S|* yields
nature. Therefore,
low-risk customers prefer A to any contract in the set
A >
S^:
S^.
Therefore,
aL* >-A > any point
in SH, including
aH*, and
VL[aL*]
>
yL[aH*].
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