Research Paper
Also available as TILEC DP 2011-037 | http://ssrn.com/abstract=1914473
07
Jan Boone | Christoph Schottmüller
2011
Health insurance without single
crossing: why healthy people have
high coverage
Health Insurane without Single Crossing:
why healthy people have high overage
∗
†
Jan Boone and Christoph Shottmüller
July 7, 2011
Abstrat
Standard insurane models predit that people with high (health) risks have high insurane
overage. It is empirially doumented that people with high inome have lower health risks and
are better insured. We show that inome dierenes between risk types lead to a violation of
single rossing in the standard insurane model. If insurers have some market power, this an
explain the empirially observed outome. This observation has also poliy impliations: While
risk adjustment is traditionally viewed as an intervention whih inreases eieny and raises the
utility of low health agents, we show that with a violation of single rossing a trade o between
eieny and solidarity emerges.
Keywords: health insurane, single rossing, risk adjustment
JEL lassiation: D82, I11
∗
We gratefully aknowledge nanial support from the Duth National Siene Foundation (VICI 453.07.003)
†
Boone:
Department of Eonomis, University of Tilburg, P.O. Box 90153, 5000 LE, Tilburg, The Netherlands;
Tile, CentER and CEPR, email: j.booneuvt.nl. Shottmüller: Department of Eonomis, Tile and CentER, Tilburg
University, email: hristophshottmuellergooglemail.om.
1
1. Introdution
A well doumented problem in health insurane markets with voluntary insurane (like the US) is
1 Standard insurane models (inspired
that people either have no insurane at all or are underinsured.
by the seminal work of Rothshild and Stiglitz (1976) (RS) and Stiglitz (1977)) predit that healthy
people have less than perfet insurane or in the extreme no insurane at all. However, both popular
aounts like Cohn (2007) and aademi work like Shoen, Collins, Kriss, and Doty (2008) show that
2 We
people with low health status are overrepresented in the group of uninsured and underinsured.
develop a model to explain why sik people end up with little or no insurane. We do this by adding
two empirial observations (disussed below) to the RS model: (i) riher people tend to be healthier
and (ii) health is a normal good. Tehnially speaking, introduing the latter two eets an lead to
a violation of single rossing in the model.
Another indiation that the standard RS framework (with single rossing) does not apture reality
well is the following.
The empirial literature that is based on RS does not unambiguously show
that asymmetri information plays a role in health insurane markets. One would expet people to
be better informed about their health risks than their insurers think for example of preonditions,
medial history of parents and other family members or life style.
However, some papers, like for
example Cardon and Hendel (2001) or Dowd, Feldman, Cassou, and Finh (1991), do not nd evidene
of asymmetri information while others do, e.g.
Bajari, Hong, and Khwaja (2006) or Munkin and
Trivedi (2010). The test for asymmetri information employed in these papers is the so alled positive
orrelation test, i.e. testing whether riskier types buy more overage.
We show that the RS model with a violation of single rossing is apable of explaining why healthy
people have better insurane (in equilibrium) than people with a low health status. In partiular the
positive orrelation property no longer holds if single rossing is violated.
Consequently, testing for
this positive orrelation an no longer be viewed as a test for asymmetri information. As mentioned,
we use two well doumented stylized fats to motivate this violation of single rossing in the market
for health insurane.
First, riher people fae lower health risks, see for example Frijters, Haisken-DeNew, and Shields
(2005), Gravelle and Sutton (2009) or Munkin and Trivedi (2010).
1
Potential explanations for this
In empirial studies, underinsurane is dened using indiators of nanial risk.
To illustrate, one denition of
underinsurane used by Shoen, Collins, Kriss, and Doty (2008) is out-of-poket medial expenses for are amounted
to 10 perent of inome or more. In our theoretial model, underinsurane refers to less than soially optimal/eient
insurane.
2
In the words of Shoen, Collins, Kriss, and Doty (2008, pp. w303): underinsurane rates were higher among adults
with health problems than among healthier adults.
2
orrelation between inome and health inlude the following. High inome people are better eduated
and hene know the importane of healthy food, exerise et. Healthy food options tend to be more
expensive and therefore better aordable to high inome people.
Or (with ausality running in the
other diretion) healthy people are more produtive and therefore earn higher inomes.
Aording
to standard insurane models, this would imply that rih people buy less generous health insurane
(ompared to poor people). However, riher people have more generous health insurane, for example,
in the form of lower dedutibles or higher overage.
Evidene for this an be found in Munkin and
Trivedi (2010), Finkelstein and MGarry (2006), Kuttner (1999) or DeNavas-Walt, Protor, and Smith
(2008) where the extreme form of underinsurane is emphasized: Low inome itizens are more likely
to have no health insurane at all.
The seond stylized fat is that health is a normal good. The eet of inome on treatment hoie
is well doumented in the medial literature.
The main emphasis in this literature is that patients
with low inome annot aord treatment even if they have a presription by their dotor.
Piette,
Heisler, and Wagner (2004b, p. 384) for instane report that from a sample of hronially ill diabetes
patients A total of 19% of respondents reported utting bak on mediation use in the prior year due
to ost [. . . ℄. Moreover, 28% reported forgoing food or other essentials to pay mediation osts. By
extrapolating from their sample to the US population Piette, Heisler, and Wagner (2004a, p. 1786)
onlude that 2.9 million of the 14.1 million Amerian adults with asthma (20%) may be utting bak
on their asthma mediation beause of ost pressures. They also doument for a number of hronial
onditions that people from low inome groups are muh more likely to report foregoing presribed
treatment due to osts.
3 Further examples an, for instane, be found in Goldman, Joye, and Zheng
(2007).
For the eet of insurane overage on treatment hoie, Shoen, Collins, Kriss, and Doty (2008,
pp. w305) report that [b℄ased on a omposite aess indiator that inluded going without at least one
of four needed medial are servies, more than half of the underinsured and two-thirds of the uninsured
reported ost-related aess problems. A similar piture emerges in the international omparison by
Shoen, Osborn, Squires, Doty, Pierson, and Applebaum (2010).
4 Hene, low inome people with high
opayments will tend to forego treatment or hoose heaper treatment options.
Single rossing means that people with higher health risks have a higher willingness to pay for
3
For most hroni diseases people with inome less than $ 20000 are roughly 2 (5) times more likely to forego presribed
treatment due to osts than people with an inome between $ 20000 and $ 40000 (more than $ 60000); see table 3 in
Piette, Heisler, and Wagner (2004a) for details.
4
Li and Trivedi (2010) also show that marginal utility of health insurane overage is inuened by both inome and
risk fators.
3
marginally inreasing overage, e.g.
reduing opayments.
If this property holds for all possible
overage levels, a given indierene urve of a high risk type an ross a given indierene urve of a
low risk type at most one. To see why the stylized fats above an lead to a violation of single rossing,
onsider the following. At full overage (indemnity insurane that pays for all medial osts) high risk
(low health) types will tend to spend more on treatments than low risk types. Hene a small redution
in overage, leads to a bigger loss in utility for high risk types. Now onsider health insurane with
low overage where the insured faes substantial opayments. Beause health is a normal good, it is
possible that the rih-healthy type spends more on treatment than the low inome, low health type.
In that ase, a small hange in overage has a bigger eet on the utility of the healthy type than of
the low health agent. The healthy type will therefore have a higher willingness to pay for a marginal
inrease in overage than the low health type. This violates single rossing.
We show that in insurane models without single rossing higher health risks are not neessarily
assoiated with more overage while this predition is inevitable with single rossing. Not only leads
this to preditions that are loser to empirial observations (as doumented above), it also has lear
poliy impliations. We illustrate this with risk adjustment.
Risk adjustment is used by the sponsor (government or employer) of a health insurane sheme to
redue expeted ost dierenes between types. Based on observable harateristis (like age, gender
5 the health insurer is subsidized (taxed/subsidized less) for ustomers with high (low) expeted
et.)
osts. This is used in a number of ountries like Australia, Canada, Germany, Ireland (until 2008),
The Netherlands, South Afria and the USA (see Ellis (2007) and Armstrong, Paolui, MLeod, and
Van de Ven (2010)) in both mandatory state insurane and voluntary private insurane. The two main
goals of risk adjustment are eieny and fairness (or solidarity); see Van de Ven and Ellis (2000) for
an overview of the literature on risk adjustment and the way it is used in pratie.
Indeed, in the
standard RS model, these two goals go in hand in hand. Starting from zero risk adjustment, inreasing
the subsidy to the high risk onsumer inreases both eieny and the utility of people with low health
status. Hene the sponsor of the health insurane sheme does not need to hoose between these two
objetives. As we show below, this is no longer the ase with a violation of single rossing.
In partiular, when single rossing is violated, risk adjustment will either inrease eieny (by
reduing o-payments for high risk types) or inrease the utility of high risk types; but not both.
Hene when designing the risk adjustment sheme, the sponsor needs to be expliit whether the goal is
eieny or solidarity. In other words, with a violation of single rossing there is a trade o between
eieny and equity.
5
Clearly, observable harateristis are not a perfet preditor of risk type. Glazer and MGuire (2000) show how the
imperfetion of the signals should be taken into aount when designing the risk adjustment sheme.
4
The literature on violations of single rossing is relatively sare. There are three papers analyzing
perfetly ompetitive insurane markets with
2×2
types: People dier in two dimensions and both
dimensions an either take a high or a low value. In Smart (2000) the two dimensions are risk and risk
aversion. Netzer and Sheuer (2010) model an additional labor supply deision and the two dimensions
are produtivity and risk. Wambah (2000) models both wealth and risk. All papers have a pooling
result, i.e. if single rossing does not hold two of the four types an be pooled. Our paper ontributes
by deviating from the perfet ompetition assumption.
We show that under imperfet ompetition
types annot only be pooled but high risk types might get even less overage in equilibrium than low
risk types.
Jullien, Salanie, and Salanie (2007) take a dierent approah to answer the question why high risk
types might have lower overage in general insurane markets. They use a model where types dier
in risk aversion and single rossing is satised. Hene, types with higher risk aversion will have more
overage in equilibrium. At the same time more risk averse agents might engage more in preventive
behavior. If types are still separated in equilibrium and risk aversion dierenes remain the driving
fore, high risk aversion types will exhibit less risk (due to prevention) and higher overage. Similar
lines of reasoning an be found in Hemenway (1990) and De Meza and Webb (2001).
Sine risk in the health setor is exogenously dierent for dierent persons (e.g. due to genetis),
we follow RS and take a dierent starting point than Jullien, Salanie, and Salanie (2007). We assume
risk dierenes instead of risk aversion dierenes. The result that high risk people have low overage
is in our paper not the result of low risk aversion. The driving fore is the violation of single rossing
aused by empirially doumented inome dierenes between high risks and low risks.
Finally, our paper is related to the industrial eonomis literature on non-linear priing in oligopoly
(see, for instane, Armstrong and Vikers (2001), Stole (1995) and Stole (2007)). This literature fouses
on the welfare eets of non-linear priing in imperfetly ompetitive markets. Our ontribution to
this literature is to do omparative stati analysis with respet to risk adjustment. In other words, our
paper has positive (testable) impliations for non-linear priing in oligopoly. Moreover, whereas the
insurane papers mentioned above fous on either perfet ompetition or monopoly, we atually analyze
all three ases: perfet ompetition, monopoly and oligopoly. As one would expet, the assumptions
on the ompetitive situation matter for the results.
In the following setion, the model is introdued and illustrated with an example. Then equilibrium
in monopoly and oligopoly is derived. Thereafter we introdue risk adjustment and show the trade o
between eieny and solidarity in ase single rossing is violated. We onlude with the impliations
of our model for so alled advantageous seletion. Proofs are relegated to the appendix.
5
2. Insurane model
This setion introdues a general model of health insurane that allows us to onsider both the ase
where single rossing (SC) is satised and the ase where it is not satised (NSC). The setion onludes
with a model where SC is not satised beause of inome dierenes between onsumer types. Whereas
the RS model predits that people with high expeted health are expenditures have generous overage
in their insurane, we allow for the ase that these people annot aord suh generous insurane.
Following RS, we onsider an agent with utility funtion
6
or generosity of her insurane ontrat,
θ ∈ {θ l , θ h }
with
θh > θl > 0
u(q, p, θ) where q ∈ [0, 1]
p ≥ 0 denotes the prie of insurane (insurane
denotes the type of onsumer.
q h = q l = 1;
sense of higher expeted osts (in ase
denotes overage
7 Higher
θ
premium) and
denotes a higher risk in the
see below). This ould, for instane, be the ase
due to hroni illness or higher risk due to a geneti preondition. We make the following assumptions
on the utility funtion.
Assumption 1
0.
The utility funtion
We dene the indierene urve
u(q, p, θ) is ontinuous
p(q, u, θ)
and dierentiable. It satises
uq > 0, up <
as follows:
u(q, p(q, u, θ), θ) ≡ u
We assume that these indierene urves
p(q, u, θ)
(1)
are dierentiable in
q
and
u
with
pq = −uq /up >
0, pu = 1/up < 0.
Further, the rossing at
q=1
satises:
pq (1, uh , θ h ) > pq (1, ul , θ l )
for all
ul ≥ ūl = u(0, 0, θ l ), uh ≥ ūh = u(0, 0, θ h ).
In words, utility
For given type
θ
u
is inreasing in overage
and utility level
yield the same utility.
u,
q
and dereasing in the premium
the indierene urve
Beause higher
q
requires a lower prie. Thus, raising
Apart from literal overage where
1/(1 + dedutible).
u
1−q
p(q, u, θ)
leads to higher utility,
Hene, indierene urves are upward sloping in
6
(C1)
(q, p)
p
p
paid for insurane.
maps out ombinations
(q, p)
that
inreases to keep utility onstant.
spae (pq
> 0).
Inreasing
shifts an indierene urve downwards (pu
denotes the agent's opayments
q
u
(for given
q)
< 0).
ould, for example, be interpreted as
Note that in models without moral hazard both parameters are similar in the sense that high risk
types dislike o-payments and dedutibles more relative to low risk types.
7
We follow RS and muh of the risk adjustment literature in assuming that there are only two types. For an analysis
of a violation of single rossing with a ontinuum of types
θ,
6
see Araujo and Moreira (2010) and Shottmüller (2011).
k ∈ {h, l}
Type
buys insurane if it leads to a higher utility than her outside option
outside option is given by the empty insurane ontrat:
This
q = p = 0.
q = 1 a marginal redution in q should be ompensated by a bigger derease in p for θ h ompared
At
to
ūk .
θl.
This reets the fat that the
θh
type faes higher expeted health are expenditures. At
q = 1,
i.e. at full overage, other fators like willingness to pay for treatment (whih ould be dierent for
dierent types) play no role. In this sense, this assumption denes what higher
higher
θ
means: at
q = 1,
types fae higher expeted osts. With the same idea we assume that expeted osts for the
insurer of a ontrat with
all
θ
uh ≥ ūh , ul ≥ ūl .
q=1
is higher for the
u
Intuitively,
θh
than for the
θl
type:
c(1, uh , θ h ) > c(1, ul , θ l )
for
should not matter for health are onsumption at full overage and
the high risk type will use the insurane more.
To allow for inome eets (for instane, in treatment hoie; see below) the ost funtion depends
on
u.
However, we assume two regularity onditions.
Assumption 2
For eah type
• cu (q, uk , θ k ) ≥ 0
for
k ∈ {h, l}
and
q ∈ [0, 1]
we assume that
uk ≥ ūk ,
• c(1, uk , θ k ) = c(1, ũk , θ k ),
for
uk , ũk ≥ ūk .
In words, as the inome of the agent inreases (whih eteris paribus leads to higher utility), the
agent has more money to spend on treatment. As the insurer pays a fration
q ≥ 0 of these treatments,
this leads to (weakly) higher osts for the insurer. Seond, osts at full overage (q
in utility.
Intuitively, if
q = 1
uk ≥ ūk .8
Beause of (C1), the single rossing ondition reads
9
pq (q, uh , θ h ) > pq (q, ul , θ l ) > 0
uh ≥ ūh , ul ≥ ūl
suh that
an indierene urve of type
do not vary
treatments are for free for the agent and there is no reason to forgo
treatments, irrespetive of the level of
and
= 1)
θh
for all
p(q, uh , θ h ) = p(q, ul , θ l ).
q ∈ [0, 1]
The intuition is the following.
intersets with an indierene urve of type
Then (SC) implies that the slope of the
θh
(SC)
θl
Suppose
in some point
(p, q).
indierene urve will be higher. It follows that these two
indierene urves an interset only one.
We onsider both the ase where (SC) is satised and the ase where it is violated (denoted by
NSC). In both the SC and NSC ases, we maintain the assumption that
level (EI) for eah type
8
q = 1 is the eient
insurane
θ ∈ {θ l , θ h }.
By assumption 3 full overage is soially desirable.
Hene we do not onsider the ase where insurane leads to
ineieny by induing over-onsumption of treatments.
9
This is also alled sorting, onstant sign or Spene-Mirrlees ondition (Fudenberg and Tirole (1991, pp. 259)).
7
Assumption 3
For a given utility level
uk ,
welfare (and therefore prots) are maximized at full ov-
erage, i.e.
max p(q, uk , θ k ) − c(q, uk , θ k )
(EI)
q∈[0,1]
is maximized by
q=1
for eah
k ∈ {h, l}
and
uk ≥ ūk .
This basially means that the insurane motive, i.e. transferring risk from a risk averse agent to a
risk neutral insurer, is not overruled by other onsiderations. To illustrate, we do not assume that the
low inome agent's preferene for health/treatment is so low that foregoing insurane would be soially
optimal. Put dierently, we assume that full insurane is soially desirable. Underinsuranewith no
insurane as extreme aseresults therefore not from rst best but from informational distortions and
prie disrimination motives.
Our motivation for making this assumption is twofold. First, this assumption simply normalizes
the soially eient insurane level in the same way as in RS. Hene, we only deviate from the RS
set up by allowing for both SC and NSC. Seond, we want to argue that under realisti assumptions,
θh
types have less than full insurane. If the optimal insurane level is atually below one, than this
result would follow rather trivially.
10 We show that with the
The literature on insurane models onsiders mostly perfet ompetition.
assumptions made so far, perfet ompetition implies
market power on the insurane side is needed to get
qh = 1
q h < 1.
(even if (SC) is not satised). Hene,
Following the RS denition of the perfet
ompetition equilibrium, we require that (i) eah oered ontrat makes nonnegative prots and (ii)
given the equilibrium ontrats there is no other ontrat yielding positive prots.
Proposition 1
If an equilibrium exists under perfet ompetition then
q h = 1.
As is well known, existene of equilibrium in the RS framework is not guaranteed. Equilibrium does
not exist if the only possible (separating) equilibrium is broken by a pooling ontrat. If the fration
of
θh
type agents in the population is high enough, then suh a deviation to a pooling ontrat is not
protable and an equilibrium exists. If an equilibrium exists, it has
q h = 1.
The proposition shows that even with violations of single rossing, high risk types will get (weakly)
higher overage than low risk types. Hene we need to deviate from perfet ompetition to get
qh < ql .
This proposition is in some sense reminisent of Wambah (2000), Smart (2000) and Netzer and Sheuer
(2010): these papers analyze perfetly ompetitive insurane markets and a reverse order, i.e. riskier
types have less overage, is impossible in these papers.
10
See Jak (2006) and Olivella and Vera-Hernández (2007) for exeptions using a Hotelling model to formalize market
power on the insurer side of the market. These papers assume that (SC) is satised and hene nd eient insurane
for the
θh
type.
8
Corollary 1
Whenever
qh < ql
is observed, insurers have market power.
Although previous models of insurane markets assume perfet ompetition, reent researh for
the US (see Dafny (2010)) shows that health insurers do have market power. More generally, in most
ountries where health insurane is provided by private ompanies, these rms tend to be big (due to
eonomies of sale in risk diversiation). Hene one would expet them to have some market power.
In order to analyse the ontrats oered by insurers with market power, we expliitly introdue the
inentive ompatibility (IC) onstraints for eah type
p(q l , ul , θ l ) ≥ p(q l , uh , θ h )
(ICh )
p(q h , uh , θ h ) ≥ p(q h , ul , θ l )
The rst onstraint implies that the ontrat intended for
θh
(weakly) lower indierene urve for
That is, the inequality implies
(ICl )
θh
(q h , p(q h , uh , θ h )))
(i.e.
than the ontrat that is meant for the
u(q h , ph , θ h ) ≥ u(q l , pl , θ h )
Similarly, the seond inequality implies that
where
θl
type
pi = p(q i , ui , θ i )
lies on a
(q l , p(q l , ul , θ l )).
i ∈ {h, l}.
with
u(q l , pl , θ l ) ≥ u(q h , ph , θ l ).
Irrespetive of the mode of ompetition and whether (SC) holds, we have the following result that
we use below.
Lemma 1
sheme
At least one type has full overage. If the types are separated under the optimal ontrat
(q l , pl ), (q h , ph )
with
q l 6= q h ,
then at most one inentive onstraint binds.
We onlude this setion with a model where SC is violated due to dierenes in inome between
types. The idea of the model is that for
q < 1 people have
to nane part of the osts of treatment out
of their own poket and low inome agents may deide to hoose heaper treatment or forgo treatment
altogether. This eet is doumented in the medial literature, see for example Piette, Heisler, and
Wagner (2004b), Piette, Heisler, and Wagner (2004a) or Goldman, Joye, and Zheng (2007).
In partiular, we assume that a type
(density) funtion
F (s|θ)(f (s|θ)).
no treatment. Lower health states
worse health than the
low
s
θl
We take
s
set
H(s)
where
is ompat and
s=1
H(s)
s ∈ [0, 1]
as the state in whih the agent is healthy and needs
θh
then for the
s<1
θl
F (s|θ h ) > F (s|θ l )
for eah
for eah
s ∈ h0, 1i.
she an inrease her health by treatment
h ∈ H(s)
lead to higher health states than not falling ill. If
and eah
H(s)
9
θh
type has
In words,
type.
denotes the set of possible treatments in state
s+h ≤ 1
with distribution
orrespond to worse health. The assumption that the
One an agent reeives a health shok
s + h,
onsumer faes health shok
type an now be stated as
states are more likely for the
health level
θ
s ∈ [0, 1].
s.
h ∈ H(s)
to
We assume that the
That is, treatment annot
is a singleton, the onsumer has no treatment
hoie. If the set
q < 1,
H(s)
has more than one element, low inome onsumers with partial insurane, i.e.
may deide to hoose heaper treatment than if they have full insurane, i.e.
h̄(s) = max{h ∈ H(s)}
as the best possible treatment and assume that
This means that a less aited agent (high
an agent who is more seriously ill (low
Let
w(θ)
s).
s < 1)
If
We dene
is non-inreasing in
s.
annot inrease his health by treatment more than
0 ∈ H(s),
denote the wealth (or inome) of a type
u(q, p, θ) =
h̄(s)
q = 1.
θ
an agent an forgo treatment altogether.
agent. Then we write
1
Z
{v(w(θ) − p − (1 − q)h(s, q, θ), s + h(s, q, θ))}dF (s|θ)
0
where
h(s, q, θ)
(2)
is dened as:
h(s, q, θ) = arg max v(w(θ) − p − (1 − q)h, s + h)
h∈H(s)
where
v(y, h)
is the utility funtion of an agent whih depends on onsumption of other goods (y ) and
health (h). We assume that
vhy ≥ 0.
v(y, h)
satises
vy , vh > 0, vyy , vhh < 0
and that health is a normal good:
That is, utility inreases in both health and onsumption of other goods at a dereasing rate.
As inome inreases, people's preferene for health inreases as well. Further, we assume that inome
and health status are negatively orrelated:
w(θ h ) ≤ w(θ l ).
This negative orrelation is empirially
doumented, for example, in Frijters, Haisken-DeNew, and Shields (2005), Gravelle and Sutton (2009)
or Munkin and Trivedi (2010).
Using this notation we an write
c(q, u, θ) = q
Z
1
h(s, q, θ)dF (s|θ)
(3)
0
The rst order ondition for an interior solution
h(s, q, θ) ∈ H(s)
an be written as
(1 − q)vy (w(θ) − p − (1 − q)h(s, q, θ), s + h(s, q, θ)) = vh (w(θ) − p − (1 − q)h(s, q, θ), s + h(s, q, θ))
(4)
To see the impliations of this model for single rossing, onsider the slope of the indierene urves
in
(q, p)-spae:
uq
pq (q, u, θ) = −
=
up
In words, the slope
on state
s
pq
R1
0
vy (w(θ) − p − (1 − q)h(s, q, θ), s + h(s, q, θ))h(s, q, θ)dF (s|θ)
R1
0 vy (w(θ) − p − (1 − q)h(s, q, θ), s + h(s, q, θ))dF (s|θ)
equals the weighted average of treatment
h(s, q, θ)
over the states
vy (w(θ) − p − (1 − q)h(s, q, θ), s + h(s, q, θ))f (s|θ)
R1
0 vy (w(θ) − p − (1 − q)h(s, q, θ), s + h(s, q, θ))dF (s|θ)
(where the weights integrate to 1).
10
s
(5)
with weight
(6)
s + h̄(s) = 1
To illustrate (C1) assume that
(treatment makes a patient healthy again),
11 then it is
routine to verify that
R1
0
pq (1, u, θ) =
vy (w(θ) − p, 1)h̄(s)dF (s|θ)
=
R1
0 vy (w(θ) − p, 1)dF (s|θ)
where the last equality follows from the fat that
h(s, 1, θ) = h̄(s)
that
for both types. If treatment is free (q
is high) ompared to
θl.
1
(1 − s)dF (s|θ)
0
vy (w(θ) − p, 1) is onstant in s.
= 1)
θh
(h̄(s)). The stohasti dominane assumption implies that
h̄(s) = 1 − s
Z
Note that we use here
eah agent uses the highest treatment
puts more weight on low
s
states (where
Hene under these assumptions, ondition (C1) is satised.
(SC) is satised if there are no wealth dierenes between types, i.e.
w(θ h ) = w(θ l ),
and
H(s)
satises some regularity ondition.
The idea is that without wealth dierenes, (4) yields for both
types the same optimal treatment.
Put dierently,
h(s, q, θ)
is independent of
θ.
If patients hoose
θh
more treatment in worse health states, single rossing will be satised: due to stohasti dominane,
s states with high h(s).
types have higher weight (6) on low
types for all
q ∈ [0, 1].
s
is onvex for eah
Treatment
h(s, q, θ)
and non-inreasing in
Hene
pq
in (5) is higher for
is indeed non-inreasing in
s.12
s
if
H(s)
θh
than for
is well behaved:
θl
H(s)
It then follows from equation (4) using the impliit
funtion theorem that
(−(1 − q)2 vyy + 2(1 − q)vyh − vhh )
Hene from the assumptions on
v
it follows that
dh
= vhh − (1 − q)vyh
ds
h(s, q, θ)
(7)
is non-inreasing in
s.
As
H(s)
is non-
inreasing, this also holds true for boundary solutions where the impliit funtion theorem annot be
used.
However, if
w(θ h ) < w(θ l )
then
q <1
an imply that
h(s, q, θ h ) < h(s, q, θ l ).
This follows from
equation (4), sine
−(1 − q)vyy + vhy
dh
=
>0
dw
−(1 − q)2 vyy + 2(1 − q)vyh − vhh
Hene, if
h(s, q, θ l ) ∈ H(s)
In words, sine a fration
θh
is an interior maximum, the
1−q
type tends to hoose lower treatment
Alternatively, we an assume that
h̄′ (s) ∈ h−1, 0i
suh that
θl
type (as health is a normal good). Sine
s + h̄(s)
is inreasing in
ill, treatment does not bring bak full health. Then a suient ondition for (C1) is
that
′
h̄ (s) = 0,
h̄(s2 ) ≤ h̄(s1 )
then
for
pq (1, ·)
would be the same for both types.
s1 ≤ s2 ). vyyh ≥ 0
12
We say that the set
exists
′
h ∈ H(s1 )
guarantees that
vy
′
h̄ (s) < 0
s.
In words, if an agent falls
vyyh ≥ 0:
Suppose it were the ase
will now put more weight on low states (as
is inreasing less in
l
s
for the high type. Hene, putting more
h
vy is low aets the θ type less than the θ type. Consequently, (C1) is satised.
H(s) is non-inreasing in s if for eah s1 , s2 with s1 ≤ s2 we have that for eah h ∈ H(s2 )
weight on low states where
suh that
′
h ≥ h.
h.
of the treatment ost has to be paid by the insured, a low inome
patient may hoose heaper treatment than the riher
11
θh
(8)
As a speial ase this inludes the possibility that
′
h (s) < 0.
11
H(s) = h(s)
there
is a singleton, with
he does not utilize the insurane as muh as the (rih) low risk type, type
willingness to pay for insurane overage (for
q
q
θh
has a lower marginal
lose to zero). However, for high levels of overage, i.e.
lose to 1, wealth dierenes matter less in the treatment hoie beause the patient does not have
to pay (muh) for the treatment. Consequently, although (C1) is satised with
w(θ h ) < w(θ l ),
(SC)
an be violated.
h ∈ H(s)
Hene, this model where agents dier in inome and treatment hoie
is endogenous
an generate the violation of (SC) mentioned above. In the next setion, we give a numerial example
where (SC) is indeed violated.
3. Example
As an example of an utility funtion that satises the assumptions (C1) and (EI) above and violates
(SC), onsider the following mean-variane utility set up.
There are two states of the world:
of falling ill is denoted by
F h = 0.07 > 0.05 = F l .
13
An agent either falls ill or stays healthy.
F h (F l < F h )
for type
θ h (θ l ).
The probability
In the numerial example, we hoose
One an agent falls ill, the set of possible treatments is denoted by
The utility of an agent of type
i = h, l
with treatment hoie
h ∈ {h, h̄}
H = {h, h̄}.
is written as:
u(q, p, θ i ) = F i (v(h, θ i ) − (1 − q)h) + (1 − F i )v(1, θ i ) − p
− 12 r i F i (1
where
v(h, θ i )
denotes the utility for type
i
i
− F )(v(1, θ ) − v(h, θ ) + (1 − q)h)
i = h, l
of having health
h
and
ri > 0
risk aversion. Hene an agent's utility is given by the expeted utility minus
u
denotes the degree of
1 i
2 r times the variane in
the agent's utility. This is a simple way to apture that the agent is risk averse.
Along an indierene urve where
(9)
2
i
14
is xed, we nd the following slope:
dp
= F i h(q, θ i ) + r i F i (1 − F i )(v(1, θ i ) − v(h(q, θ i ), θ i ) + (1 − q)h(q, θ i ))h(q, θ i )
dq
13
For the Python ode used to generate this example, see:
http://sites.google.om/site/janboonehomepage/home/webappendies .
14
When an agent of type i buys a produt at prie p that gives utility v ,
utility of inome for agent
i
α
(10)
i.
First, overall utility an be written as
an dier. Low inome types are then modelled to have high
an write
i
v − αp
formalization with
where
α = 1.
α
i
α
there are two ways to apture the marginal
i
v−αp
where
v
is the same for eah type
i
; high marginal utility of inome. Alternatively, one
is the same for all types. Then low inome types have low
vi .
We have hosen the latter
The assumption that treatment is a normal good is then implemented by assuming that
v(h̄, θh ) − v(h, θh ) < v(h̄, θl ) − v(h, θl ).
12
and
where
h(q, θ i )
h
is the solution for
solving
max v(h, θ i ) − (1 − q)h
h∈{h,h̄}
In words, one an agent falls ill, she deides whih treatment to hoose based on the benet
and the out-of-poket expenses
v(h, θ i )
(1 − q)h.
With the parameter values that we onsider below, it is the ase that
r h F h (1 − F h )(v(1, θ h ) − v(h̄, θ h ))h̄ = r l F l (1 − F l )(v(1, θ l ) − v(h̄, θ l ))h̄
In words, at
dp/dq
q=1
(where both types hoose the highest treatment
type equal
θh
h̄ = 0.6, h = 0.2
v(1, θ h ) = 0.9, v(h̄, θ h ) = 0.7, v(h, θ h ) = 0.45
1.1, v(h̄, θ l ) = 0.9, v(h, θ l ) = 0.5.
the
θl
θl
and the assoiated utilities for the
and similarly for the
type hooses
h̄
even if
θl
q=0
for low values of
the value for
q
q.
suh that the
In partiular, for
θh
θl
type as
q
q=0
type ompared
θh
type. Sine
q > 0).
This
does not aet treatment hoie and higher
we have
θh
type hooses
h̄
v(h̄, θ h ) − h̄ < v(h, θ h ) − h.
type is indierent between treatment
h̄
and
if
q =1
Let
q̃
h(q, θ h ) = h̄.
θh
Then inreasing overage
hene (EI) is satised for
q > q̃ .
q
h:
(12)
type, we proeed in two steps. First, onsider
q > q̃
redues the variane in utility for the risk averse
Now onsider
q < q̃
suh that
h(q, θ h ) = h.
but
denote
v(h̄, θ h ) − (1 − q̃)h̄ = v(h, θ h ) − (1 − q̃)h
To verify that (EI) is satised for the
θh
v(1, θ l ) =
type:
(and the inequality is strit for
leads to more insurane (provided by a risk neutral insurer). The
h
θl
type is willing to spend more on treatment than the
implies that ondition (EI) is satised for the
prefers
F h h̄ > F l h̄.
Hene, having high health is more important for the
type. This implies that
0.9 − 0.6 ≥ 0.5 − 0.2
q
the variane terms in the slope
(equation (10)) are equalized. Hene assumption (C1) is satised beause
For the numerial example we assume
to the
h̄)
(11)
θh
suh that
type and
In order to satisfy (EI), it
must be the ase that prots (prie minus expeted osts) when oering full overage are higher than
prots when oering a partial overage ontrat yielding the same utility. This an be written as:
F h (v(h̄, θ h ) − h̄) + (1 − F h )v(1, θ h ) − uh − 12 r h F h (1 − F h )(v(1, θ h ) − v(h̄, θ h )2 ≥
F h (v(h, θ h ) − h) + (1 − F h )v(1, θ h ) − uh − 21 r h F h (1 − F h )(v(1, θ h ) − v(h, θ h ) + (1 − q)h)2
Note that the right hand side of this inequality inreases in
example, we hoose
satised for all
15
rh
q ≤ q̃
Given this value of
q and hene is highest at q̃ .
suh that the inequality holds with equality at
and hene (EI) is satised.
rh , rl
is hosen to satisfy equation (11).
13
q = q̃ .15
In our numerial
This implies that it is
With the parameter values above, it is routine to verify that (SC) is violated. Figure 1 below shows
two indierene urves for the
indierene urve for the
θl
θl
type (in red) and one for the
type is steeper than for the
type buys the expensive treatment
the
θh
type happens at
q̃
where the
while the
θh
θh
type (in blue). Indeed, for
type buys
h.
θl
type swithes from the heap to the more expensive treatment.
θh
are worth more to the
fat, the gure shows that for
q > q̃ ,
the indierene urve for the
θl
the
The kink in the indierene urve for
q > q̃
the
q < q̃
type. This is due to the fat that the
for
Hene small inreases in
q
h̄
θh
θh
type than small inreases in
θh
q < q̃ .
In
type is steeper than the one for
type. This is the violation in single rossing.
Hene in a simple mean-variane utility framework, it is straightforward and intuitive to generate
a violation of (SC).
4. Insurane market monopoly
This setion derives a simple result in a monopoly setting within the general redued form framework
of setion 2. The motivation for analyzing the monopoly ase is proposition 1: An equilibrium with
qh < ql
an only exist if insurane ompanies have market power.
Although the extreme ase of a
monopoly is not very realisti, it allows to derive a lear ut result and gives some intuition for the
more realisti ase of an oligopoly whih will be analyzed in the following setion.
Proposition 2
highest
The type with the highest willingness to pay for full overage, i.e. the type
p(1, ūk , θ k ),
obtains a full overage ontrat in an insurane monopoly.
θk
with
Either his inentive
ompatibility or his individual rationality onstraint is binding (or both). The other type's individual
rationality onstraint is binding.
Let
θk
denote the type with the highest willingness to pay for full overage. It follows from the
proposition that
θk
obtains a ontrat
(q, p) = (1, pk )
outome is now pinned down by the hoie of
onstraints are binding.
If
pk = p(1, ū−k , θ −k ),
separates the types and
In this ase, type
pk .
θ −k
both types are pooled.
θ −k
If
for some
pk ≤ p(1, ūk , θ k ).
pk = p(1, ūk , θ k ),
then both individual rationality
might be exluded, i.e.
If
The monopoly
θ −k
gets the ontrat
pk ∈ hp(1, ū−k , θ −k ), p(1, ūk , θ k )i,
(0, 0).
the equilibrium
gets an insurane ontrat with partial overage. The optimal level of
is determined by the share of
θk
pk
types in the population.
A diret impliation of proposition 2 is that high risk types will always have full overage if single
rossing is satised.
To see this, note that the indierene urve orresponding to
individual rationality onstraint) goes through the origin
14
(p, q) = (0, 0)
ūk
(that is the
for both types. With (SC) the
indierene urve of the high risk type is steeper and lies therefore above the individual rationality
onstraint of the low risk type for all overage levels.
Without single rossing this is no longer the ase. Figure 1 shows indeed that in our numerial
example the low risk type has a higher willingness to pay for full overage than the high risk type.
Therefore, a monopolist will give full overage to low risk types in our example.
separated, we nd that
If the types are
q h < q l = 1.
5. Insurane market oligopoly
This setion haraterizes equilibrium in a duopoly insurane market. Again the general redued form
model introdued in setion 2 is used. We illustrate with the example from setion 3 that equilibria
with
qh < ql
exist if single rossing is violated.
There are three reasons why we hoose to onsider an oligopoly insurane market. First, proposition
1 shows that with perfet ompetition it is impossible to have
q h < 1.
Seond, as mentioned above,
assuming a monopoly market is not very realisti. We are not aware of a ountry where health insurane
is provided privately by a monopolist. Third, in ountries like Germany or the Netherlands, insurane
is mandatory and the government runs a risk adjustment sheme. The next setion analyzes the eets
of risk adjustment on eieny and solidarity. If there is mandatory insurane, risk adjustment has
no eet on the outome at all in ase of monopoly. The reason is that the monopolist serves all types
and hene risk adjustment is a lump sum transfer for a monopolist insurer. As we show below, risk
adjustment (even if the whole market is served, as with mandatory insurane) does aet the market
outome in ase of oligopoly.
The demand share of insurer
[0, 1].
j ∈ {a, b}'s
produt on market
i ∈ {h, l}
is written as
D(uij , ui−j ) ∈
This funtion is the same for both types. That is, agents only dier in expeted health are
osts and inome, not in their preferenes over insurers.
16 We make the following natural assumption
on demand.
Assumption 4
D1 (uij , ui−j ) > 0, D2 (uij , ui−j ) < 0
An insurer gains market share if it oers onsumers higher utility and loses market share if its opponent
oers higher utility. Let
16
φ ∈ [0, 1]
denote the share of
θh
See Bijlsma, Boone, and Zwart (2011) for an analysis where
15
θh
types in the population.
and
θl
have dierent demand elastiities.
When insurer
b
hooses
uhb , qbh , ulb , qbl ,
the maximization problem for insurer
a
an be written as
max φD(uh , uhb )(p(q h , uh , θ h ) − c(q h , uh , θ h ))
uh ,q h ,ul ,q l
+ (1 − φ)D(ul , ulb )(p(q l , ul , θ l ) − c(q l , ul , θ l ))
l
l
l
l
h
(Puh ,ul )
b
h
b
+ µh (p(q , u , θ ) − p(q , u , θ ))
+ µl (p(q h , uh , θ h ) − p(q h , ul , θ l ))
We fous on a symmetri pure strategy equilibrium of this problem where
ulb = ul , qah = qbh = q h
(Puh ,ul ) with
b
b
uhb = uh
and
and
qal = qbl = q l .
ulb = ul .
That is,
(uh , q h , ul , q l )
We assume that problem
is haraterized by the rst order onditions for
uha = uhb = uh , ula =
is a solution to
(Puh ,ul )
(Puh ,ul ):
problem
is onave suh that a solution
q h , q l , uh , ul :17
∂p(q h , uh , θ h ) ∂p(q h , ul , θ l )
∂(p(q h , uh , θ h ) − c(q h , uh , θ h ))
+
µ
(
−
)=0
l
∂q h
∂q h
∂q h
∂(p(q l , ul , θ l ) − c(q l , ul , θ l ))
∂p(q l , ul , θ l ) ∂p(q l , uh , θ h )
(1 − φ)D(ul , ul )
+
µ
(
−
)=0
h
∂q l
∂q l
∂q l
∂(p(q h , uh , θ h ) − c(q h , uh , θ h ))
φD1 (uh , uh )(p(q h , uh , θ h ) − c(q h , uh , θ h )) + φD1 (uh , uh )
∂uh
∂p(q l , uh , θ h )
∂p(q h , uh , θ h )
−µh
+
µ
=0
l
∂uh
∂uh
∂(p(q l , ul , θ l ) − c(q l , ul , θ l ))
(1 − φ)D1 (ul , ul )(p(q l , ul , θ l ) − c(q l , ul , θ l )) + (1 − φ)D(ul , ul )
∂ul
l
l
l
∂p(q , u , θ )
∂p(q l , uh , θ l )
+µh
−
µ
=0
l
∂ul
∂ul
φD(uh , uh )
(13)
(14)
(15)
(16)
It follows from lemma 1 together with the assumption that (Puh ,ul ) is onave that in any separating
b
b
equilibrium, exatly one IC onstraint is binding. Hene either


µh > 0

µ = 0
l
or


µh = 0

µ > 0
l
and
p(q l , ul , θ l ) = p(q l , uh , θ h )
(17)
h h h
and p(q , u , θ )
≥
p(q h , ul , θ l )
and
p(q l , ul , θ l ) ≥ p(q l , uh , θ h )
and
p(q h , uh , θ h ) = p(q h , ul , θ l )
(18)
We dene the following three ases as solutions to equations (13)-(16):
h
l
l
(H) (uh
H , qH , uH , qH )
h
l
l
(L) (uh
L , q L , uL , q L )
17
solves (13)-(16) with
solves (13)-(16) with
µh > 0, µl = 0,
µh = 0, µl > 0
and
The onavity assumption an be stated in terms of the bordered Hessian of problem (Puh ,ul ). However, it is more
b
b
onveniently stated after some more notation is introdued. This is done in equations (30) and (33) below.
16
h
l
l
(P) (uh
P , q P , uP , q P )
solves (13)-(16) with
µh = µl = 0.
It is straightforward to verify that in ase of pooling (P) we have
are interested in the ase where
q h 6= q l
q < 1.
However, we
(as this is the relevant ase in pratie). Further, below we
onsider the role of risk adjustment to enhane eieny.
for one type we have
qPh = qPl = 1.
This is only relevant for the ase where
Hene we will ignore the pooling ase from here onwards and fous on
separating equilibria (H) and (L).
The following proposition derives suient onditions for an (H) and (L) equilibrium.
Proposition 3
With single rossing, solution (H) is a symmetri equilibrium and we have
qh = 1 > ql .
If single rossing is not satised and
•
the solution
h , ul , q l )
(uhH , qH
H H
p(1, uhH , θ h )
under (H) above satises
p(1, ulH , θ l )
−
=
1
Z
(pq (q, uhH , θ h ) − pq (q, ulH , θ l ))dq ≥ 0
l
qH
then this solution is a symmetri equilibrium and we have
•
the solution
(uhL , qLh , ulL , qLl )
p(1, ulL , θ l )
(19)
qh = 1 > ql
under (L) above satises
−
p(1, uhL , θ h )
=
Z
1
h
qL
(pq (q, ulL , θ l ) − pq (q, uhL , θ h ))dq ≥ 0
then this solution is a symmetri equilibrium and we have
(20)
ql = 1 > qh.
The ondition (19) [(20)℄ in the (H) [(L)℄ outome makes sure that (ICl ) [(ICh )℄ is satised as
well. If (SC) is satised, equation (ICl ) is automatially satised for the following reason. Beause of
inentive ompatibility, the two equilibrium indierene urves have to ross somewhere. Under (SC)
the high type's indierene urve is steeper there (and onsequently above the low type's urve for
slightly higher
q ).
By (SC) there is no other intersetion and therefore the high type's indierene
urve is above the low type's also at
the
for
θh
q
q = 1.
Hene ondition (19) is satised. Without single rossing,
indierene urve is less steep than the
θl
indierene urve for low
q
and the other way around
lose to 1. In that ase, we need to hek expliitly in the solution whether the IC onstraint that
was ignored is indeed satised.
To prove existene of the (H) and (L) equilibria in an oligopoly framework, it is suient to give
an example for eah. The existene of (H) equilibria in oligopoly is already established in the literature
through models satisfying the single rossing assumption, see for example Olivella and Vera-Hernández
(2007).
Here we give an example using the utility setup of setion 3 to demonstrate existene of a
(L) equilibrium with oligopoly. Reall from proposition 1 that there is no (L) equilibrium with perfet
ompetition.
17
Example
On the supply side we assume that there are two insurers loated at the end
(ont.)
points 0 and 1 of a Hotelling line. Agents are uniformly distributed over the
at position
x ∈ [0, 1]
inurs transportation ost
insurer oers a menu of ontrats
θh
type, the seond for the
θl
xt ((1 − x)t)
{(q h , ph ), (q l , pl ), (0, 0)}
[0, 1]
interval. An agent
when buying from insurer
a (b).
Eah
where the rst ontrat is intended for the
type and the third ontrat denotes the agent's outside option of not
buying insurane at all (whih will not be used in equilibrium).
We will show that it is straightforward to nd examples with a (L) equilibrium. The easiest way to do
this is to nd parameter values suh that the individual rationality (IR) urve (that is, the indierene
urve
p(q, ū, θ))
for the
θl
type lies everywhere above the IR urve for the
θh
type. As shown in gure 1
this is the ase for the parameter values hosen in setion 3. Clearly the Hotelling equilibrium ontrats
have to lie on or below the relevant IR urves.
φ = 0.
First, assume that
θl
In words, there are only
(beause of assumption 3) and the Hotelling equilibrium prie on the
This ontrat is denoted
θ l 's
ontrat lies below
φ = 0)
in the gure) and
θ l -market
(1, pl )
ontrat:
ulhotel. = u(1, pl , θ l ).
Contrat
is dened by the intersetion of indierene urve
θ h 's
equals
in gure 1 for the parameter values in setion 3 and
IR urve it is, indeed, the equilibrium outome. Let
level assoiated with the
by anyone as
(1, pl )
ql = 1
types. Then it is routine to verify that
IR urve. This is the best ontrat on
θ h 's
ulhotel.
(q h , ph )
pl = F l h̄ + t.18
t = 0.018.
denote
As this
θ l 's
utility
(although not bought
p(q, ulhotel. , θ l )
(dashed urve
θ l 's
IR urve that satises
inentive
ompatibility onstraint.
φ
Now inrease
ql = 1 > qh.
slightly to
φ > 0
(but small).
For this to be an equilibrium we need that the indierene urve for the
lies above the indierene urve for the
for the
θh
urve at
θ l 's indierene
q = 1.
small hanges in
above
θh
type at
h
h
type (p(q, uhotel , θ )) annot lie above
equilibrium is that
IR
We laim that this results in a (L) equilibrium with
θ h 's IR
φ
urve
q = 1.
θ h 's
θl
type at
q=1
Note that the equilibrium indierene urve
IR urve. Hene a suient ondition for a (L)
p(q, ulhotel. , θ l )
at the new Hotelling equilibrium lies above
θ h 's
We formally show that this is the ase in lemma 3 in the appendix. Intuitively,
will lead to small hanges in the indierene urve
urve at
q=1
in ase
φ = 0,
it will be above
θ h 's IR
p(q, ulhotel. , θ l ).
As this urve is
urve for small positive values of
φ.
Hene a straightforward way to generate (L) equilibria is to nd examples where the
for the
and
θl
φ>0
type lies above the
IR
onstraint for the
θh
type for eah
q ∈ h0, 1].
IR onstraint
Then there exist
t>0
suh that the example has an (L) equilibrium.
Having proved the existene of a (L) equilibrium, we move to the poliy impliations of suh an
18
Reall that in a Hotelling model with onstant marginal osts
instane, Tirole (1988, pp. 280).
18
c,
the equilibrium prie is given by
c + t.
See, for
0.07
h
p(q, ū
h
,
)
l
l
p(q, ū ,
l
l
(1,p )
)
p(q,uhotel.,
l
)
p
0.00
h
(q
0.0
h
,p
0.2
)
0.4
0.6
0.8
1.0
q
Figure 1: Example with parameter values in setion 3 and
t = 0.018
equilibrium.
6. Risk adjustment
Above we have shown that an equilibrium an exist where
θh
types have less than full insurane. This
is in line with the empirial ndings mentioned in the introdution where people with low inome and
low health status have less generous insurane than people with high inome and good health. In this
setion, we show what the poliy impliations are of an equilibrium with
q h < 1.
In partiular, risk
adjustment is often presented as a win-win poliy: Starting from a separating equilibrium it inreases
both the utility of people with bad health (solidarity with those unluky enough to be born this way)
19
and it inreases equilibrium overage (eieny).
That is, when it omes to risk adjustment, there is no trade o between eieny and distributional
objetives for the government. The papers and ountry poliies surveyed in Van de Ven and Ellis (2000)
see risk adjustment as serving both eieny and fairness or solidarity. We show below that under a
19
The underlying assumption is that the
θh
type was born with, say a hroni disease like diabetes. Hene
θh 's
high
expeted health are osts are exogenously given. Then fairness or solidarity onsiderations an lead the planner to give
a higher weight than
λ
to the
θh
type in the objetive funtion. Alternatively, high health are osts an be endogenous
due to, for example, smoking behavior, food habits, drug use et. See Van de Ven and Ellis (2000) for a disussion of
the limits to solidarity due to suh moral hazard eets.
19
20 However, in the NSC
relatively mild assumption, this result holds in an oligopoly model with SC.
ase, the same assumption implies that there is a trade o between eieny and solidarity. In that
qh
ase it is not possible for risk adjustment to inrease both
ase (with
q h < 1),
and
uh .
Hene in the empirially relevant
the sponsor has to hoose expliitly whether the goal is eieny or solidarity.
Risk adjustment annot promote both goals.
We model risk adjustment as follows. The sponsor, say the government, of the health insurane
sheme pays an insurer
ρh (ρl )
for eah of its ustomers that are of type
government an perfetly observe eah ustomer's type.
In ase (H), we write the prots of insurer
a
θ h (θ l ).
We assume that the
21
as follows:
Π(uha , ula , ρ, uhb , ulb ) = φD(uha , uhb )(π(uha , θ h ) + ρh ) + (1 − φ)D(ula , ulb )(π(ula , uha , θ l ) + ρl )
where we use the following denition of the funtions
prot margins on the
θh
and
θl
π
(21)
(with a slight abuse of notation) apturing the
markets resp.:
π(uh , θ h ) = p(1, uh , θ h ) − c(1, uh , θ h )
(22)
π(ul , uh , θ l ) = p(q l (ul , uh ), ul , θ l ) − c(q l (ul , uh ), ul , θ l )
with
q l (ul , uh ) the value of q l that solves equation (ICh ) with equality.
If (SC) is satised, dierentiating
the binding (ICh ) yields
dq l
pq (q l , ul , θ l ) − pq (q l , uh , θ h )
= −pu (q l , ul , θ l )
dul
and hene
dq l
<0
dul
(23)
sine the left hand side is negative by (SC) and the right hand side is positive by assumption 1 (pu
< 0).
Using a similar derivation, one an show in this ase that
dq l
>0
duh
In words, inreasing
Inreasing
20
21
ul
uh
(24)
relaxes the (ICh ) onstraint thereby allowing for higher
does the opposite and hene leads to lower
ql
(for given
ql
(for given
ul ).
uh ).
Selden (1998) makes this point using a model with perfet ompetition where single rossing holds.
This assumption is usually justied by assuming that the government has (ex post) more information than the insurer
ex ante. Hene the insurer is not able to (expliitly) selet risks ex ante but the government an perfetly risk adjust ex
post. Alternatively, the insurer has the relevant information but is prevented by law to at upon this information. To
illustrate, in the Netherlands an insurer annot refuse a ustomer who wants to buy a ertain insurane ontrat. As
mentioned, Glazer and MGuire (2000) show how optimal risk adjustment an work with imperfet signals of ustomers'
types. In the latter ase, the government annot perfetly observe eah ustomer's type.
20
In ase (L) we have:
Π̃(uha , ula , ρ, uhb , ulb ) = φD(uha , uhb )(π̃(uha , ula , θ h ) + ρh ) + (1 − φ)D(ula , ulb )(π̃(ula , θ l ) + ρl )
where we use the following denition of the funtions
(25)
π̃ :
π̃(ul , θ l ) = p(1, ul , θ l ) − c(1, ul , θ l )
(26)
h
l
h
h
h
l
h
h
h
h
l
h
h
π̃(u , u , θ ) = p(q (u , u ), u , θ ) − c(q (u , u ), u , θ )
with
q h (uh , ul )
being the value of
urve is steeper than the
θh
qh
that solves equation (ICl ) with equality.
indierene urve at
qh,
If the
θl
indierene
we nd with a similar derivation as above that
dq h
<0
duh
dq h
>0
dul
In this ase, raising
Let
Πi
ul
(27)
qh
relaxes the (ICl ) onstraint and hene allows for higher
denote the derivative of
Π
with respet to its
i-th
argument.
(for given
uh ).
Then for the (H) ase, a
symmetri equilibrium (fousing on an interior maximum) is haraterized by
Π1 = Π2 = 0
whih we
write as
φD1 (uh , uh )(π(uh , θ h ) + ρh ) + φD(uh , uh )π1 (uh , θ h ) + (1 − φ)D(ul , ul )π2 (ul , uh , θ l ) = 0
(1 − φ) D1 (ul , ul )(π(ul , uh , θ l ) + ρl ) + D(ul , ul )π1 (ul , uh , θ l ) = 0
(28)
(29)
together with the seond order onditions:
Π11 < 0, Π22 < 0
(30)
Π11 Π22 − Π212 > 0
The interpretation of the rst order ondition with respet to
inreasing
uha
slightly, insurer
earns the margin
π h + ρh .
leads to lower prots (π1
the rm an raise
ql
a
On the inframargin, however, inreasing
< 0).
Finally, there is the eet of
The rst order ondition with respet to
θl
(equation (28)) is as follows.
inreases the demand for its produt on the
and still satisfy (ICh ). Hene
marginal eets on the
uha
ul
uh
uh
uh
on the
θh
By
market on whih it
(or equivalently reduing
ph )
θl
uh ,
market. By inreasing
aets the prots on the
θl
market.
(equation (29)) only features the marginal and infra-
market.
In the (L) ase, we have
Π̃1 = Π̃2 = 0
whih we write as
φD1 (uh , uh )(π̃(uh , ul , θ h ) + ρh ) + φD(uh , uh )π̃1 (uh , ul , θ h ) = 0
(1 − φ) D1 (ul , ul )(π̃(ul , θ l ) + ρl ) + D(ul , ul )π̃1 (ul , θ l ) + φD(uh , uh )π̃2 (uha , ula , θ h ) = 0
21
(31)
(32)
together with the seond order onditions:
Π̃11 < 0, Π̃22 < 0
(33)
Π̃11 Π̃22 − Π̃212 > 0
The interpretation of the rst order onditions with respet to
equilibrium. The only dierene is that in this ase
ul
uh
and
ul
is the same as in the (H)
has the additional eet on the
θh
market via
the (ICl ) onstraint.
We use the following assumption that we disuss below.
Assumption 5
Assume that both markets are fully overed:
D(uij , ui−j ) + D(ui−j , uij ) = 1
for
i = h, l.
Consider hanges in risk adjustment with a xed budget:
φ
dρh = dρ, dρl = − 1−φ
dρ
suh
that
φdρh + (1 − φ)dρl = 0
Then we assume that in symmetri equilibrium
uh (ρ), ul (ρ)
(34)
it is the ase that
duh (ρ) dul (ρ)
<0
dρ
dρ
(35)
Health insurane is a produt of whih one buys one unit or zero. In fat, it is often not allowed
to buy more than one insurane ontrat due to moral hazard problems (see Pauly (1974)). We follow
the literature on ompetition with prie disrimination and assume that both markets are fully overed
(this is alled full sale ompetition by Shmidt-Mohr and Villas-Boas (1999) and pure ompetition
by Stole (1995)).
When risk adjustment is implemented at the ountry level, it usually goes hand
in hand with mandatory insurane (e.g.
as in the Netherlands).
Indeed, if insurane would not be
mandatory, a ross subsidy (due to risk adjustment) may indue the net payers to forgo insurane
thereby reduing the eieny of the insurane market.
If risk adjustment is organized at the rm
level (as, for instane, in the US with big employers) the sheme is usually so attrative (due to tax
breaks) that the assumption of full sale ompetition is not an unreasonable one. The advantage of
making this assumption is that the budget onstraint for hanges in risk adjustment an simply be
written as (34).
Assumption (35) states that if an inrease in
assumption: as
less for
θl
ρh
ρ
l
raises
uh ,
then it dereases
ul .
inreases (ρ dereases) one expets insurers to ompete more for
types) and hene
uh
inreases (u
This is a natural
θh
types (ompete
l dereases). We are espeially interested here in the (L)
22
equilibrium. Lemma 4 in the appendix veries that for the example above this assumption is indeed
22
satised.
We further motivate this assumption in two ways. The rst is to assume that the planner hoosing
ρ
has an objetive funtion
f (uh (ρ), ul (ρ))
that is inreasing in both
solution to this optimization problem with respet to
f1
hene
ρ
uh
and
ul
(i.e.
A
is haraterized by
duh (ρ)
dul (ρ)
+ f2
=0
dρ
dρ
duh (ρ)
dul (ρ)
and
have opposite signs. In words, the planner hooses a value of
dρ
dρ
insurane on the Pareto frontier in
f1 , f2 > 0).
(uh , ul )
spae. Put dierently, if a hange in
ρ
ρ
that plaes the
an raise the utility
of both types, the planner exhausts suh possibilities. Then, at the margin, there is trade o between
uh
and
ul .
The seond way to motivate inequality (35) is in terms of equilibrium seletion.
assumption 5 rules out that there exists another equilibrium lose by in whih both
In partiular,
uh
and
ul
are
lower.
Lemma 2
Assume that
D1 (u, u)
Let
uh , ul
is independent from the level of
u
denote a symmetri equilibrium. Dene the following set
Bε (uh , ul ) =
q
ũh < uh , ũl < ul | (uh − ũh )2 + (ul − ũl )2 < ε
If assumption 5 holds, then for
ε > 0
small enough, the set
Bε (uh , ul )
(36)
does not ontain another
symmetri equilibrium.
The assumption implies that in symmetri equilibrium the slope of a rm's demand funtion is not
aeted by the level of
u.
This assumption is satised in the Hotelling example that we use in this
paper.
The lemma exludes the possibility that lose to the symmetri equilibrium
another equilibrium with lower values for both
uh
and
ul .
uh , ul
there would be
Clearly, suh an equilibrium would be more
protable for both insurers beause a rst order Taylor expansion implies that
Π(ũh , ũl , ρ, ũh , ũl ) = Π(uh , ul , ρ, uh , ul ) + (Π1 (uh , ul , ρ, uh , ul ) + Π4 (uh , ul , ρ, uh , ul ))(ũh − uh )
+ (Π2 (uh , ul , ρ, uh , ul ) + Π5 (uh , ul , ρ, uh , ul ))(ũl − ul )
= Π(uh , ul , ρ, uh , ul ) − Π4 (uh , ul , ρ, uh , ul )(uh − ũh ) − Π5 (uh , ul , ρ, uh , ul )(ul − ũl )
> Π(uh , ul , ρ, uh , ul )
22
The proof of lemma 4 is given after the proof of lemma 2 below to avoid dupliation. That is, we rst derive some
general results in the ontext of lemma 2 and then apply these to the example in lemma 4.
23
ũh,l
for
lose enough to
Π4 , Π5 < 0.
uh,l ;
Π1 = Π2 = 0
where we have used the stationarity onditions
And similarly for prot funtion
Π̃.
and
In this sense, assumption 5 rules out a loal multipliity
of equilibria leading to a oordination problem between insurane providers.
Now we an show the following.
Proposition 4
Let
uh , ul
denote the utility levels for type
p(q, uh , θ h )
that the indierene urves
between
uh
and
p(q, ul , θ l )
θh, θl
in a symmetri equilibrium. Assume
have one point of intersetion. Then the relation
and eieny is as follows:
•
if
q h = 1,
a hange in
ρ
whih raises
ql
also inreases
•
if
q l = 1,
a hange in
ρ
whih raises
qh
redues
uh ;
uh .
The assumption that the indierene urves orresponding to the equilibrium values of
one is by denition satised if (SC) holds.
uh , ul
ross
SC implies that any indierene urves ross at most
one. In the NSC ase, we need a bit more struture. In gure 1 the assumption is satised. In fat,
the following reasoning shows that a model with more than one intersetion point will not be very
intuitive.
that
In the (L) equilibrium, (20) implies that
pq (1, ul , θ l ) < pq (1, uh , θ h ).
pq (q h , uh , θ h ).
pq (q, uh , θ h )
p(1, ul , θ l ) > p(1, uh , θ h ).
Hene, at the rst intersetion point (at
for
q < qh.
Hene, in the ase where
θl
q
lose to zero, the low inome
θh
qh = 1
deides to put relatively more weight on the
in suh a way that both
uh
and
23
we nd that (at the margin) solidarity (with the unluky
l
ql
θh
type (i.e.
f1
f (uh , ul )
θh
type)
above: if the planner
inreases relatively to
f2 ),
she will hange
inrease.
However, if single rossing is violated and we have
q h < 1,
then it is not possible to raise both
uh
In other words, in this ase there is a trade o between solidarity and eieny.
The violation of (SC) is vital to get this result: At
steeper than the slope of the indierene urve of
23
pq (q, ul , θ l ) <
type deides to spend more on
type. This does not seem very reasonable.
and eieny (q ) go hand in hand. In terms of the objetive funtion
qh.
pq (q h , ul , θ l ) >
As the slope is related to the treatment hoie in our model with inome
treatment than the high inome
and
we have
To generate another intersetion point, we need again a swith in slopes
dierenes, this would imply that for
ρ
qh)
Further, (C1) implies
θh.
(ph , q h )
the slope of
one intersetion point and then strengthen assumption 3 to read that
in an (H) equilibrium and
q.
indierene urve is
qh
(eieny) is valued
Therefore, an inrease in
Another way to put more struture on outomes in the NSC ase is the following.
rms will hoose the intersetion point with highest
θ l 's
k
k
One an allow for more than
k
p(q, u , θ ) − c(q, u , θk )
It is routine to verify that this gives
pq (q h , ul , θl ) > pq (q h , uh , θh )
is inreasing in
l
h
h
q.
Then
l
pq (q , u , θ ) > pq (q , ul , θl )
in an (L) equilibrium. Under this assumption, the results in
proposition 4 hold as well.
24
1
A
B
p
q > 0
0
0
Figure 2:
higher by the
θl
1
q
A hange in risk adjustment whih inreases
type. Consequently, the utility of
θh
q.
qh
an only be inreased if
is redued as otherwise
inentive ompatibility fails.
Figure 2 illustrates this graphially. The lines
and
θh
A
and
B
denote parts of indierene urves of a
type. The intersetion of these lines gives the overage
q
of the type that has less than full
overage. Given assumption (35), a hange in risk adjustment that inreases
downwards and the
urve of the
determines
θh
ql .
B
urve upwards. In the (H) equilibrium, the (steeper)
type. The
B
urve is the indierene urve of the
Hene inreasing
ql
and
uh
qh
A
θh
A
has to shift the
A urve
urve is the indierene
type. The point of intersetion
urve is the indierene urve of the
then requires that the indierene urve
utility for the
θl
q
go hand in hand.
However, in the (L) equilibrium the steeper
reasing
θl
B
for the
θh
θl
type. In-
type shifts upwards. That is, the
type falls.
7. Conlusion
Standard insurane models, e.g.
Rothshild and Stiglitz (1976) or Stiglitz (1977), predit higher
overage for agents with higher risks. We show that this predition no longer holds if single rossing
is violated and rms have market power.
In the health are setor agents with higher inome have lower risks and more insurane.
Put
dierently, the preditions of the standard insurane model with single rossing are ontradited by
25
the data.
We show that the negative orrelation between inome and risk an ause a violation of
single rossing. With a violation of single rossing, the empirial ndings in the health literature an
be reoniled with a standard insurane model.
We show the poliy impliation of suh a violation of single rossing for risk adjustment.
The
traditional insurane model (with single rossing) views risk adjustment as a measure inreasing eieny as well as solidarity, i.e. overage levels are loser to rst best and low health agents are better
o. Without single rossing there is a trade o between these two poliy goals. This implies that the
sponsor of the health insurane sheme has to be expliit about the goals of risk adjustment.
From an empirial point of view, our paper asts doubt on the positive orrelation test: Given our
result that separating equilibria exist in whih agents with higher risk have less overage (negative
orrelation), it is evident that the results of suh a test have to be interpreted with are. In partiular
suh a test annot be used to test for the presene of asymmetri information when single rossing is
violated.
We onlude with a disussion of advantageous seletion. This an be modeled by assuming that
people dier in their preferenes for risks.
If high risk individuals are less risk averse than low risk
people, it an happen that onsumers who are willing to pay the most for health insurane are people
with low expeted health are osts. Hene oering health insurane with high overage is espeially
attrative for agents with low expeted osts: advantageous seletion.
The impliation of some ad-
vantageous seletion models is that poliies that stimulate insurane overage are welfare reduing. In
fat, there may be over-insurane in equilibrium. See Einav and Finkelstein (2011) for a reent review
of advantageous seletion and empirial papers doumenting this in health are markets.
In our model (in the (L) equilibrium), we also see that at the margin low risk types are willing
to pay more for insurane than high risk types. This is aused by the fat that at less than perfet
overage, low inome, high risk types tend to redue expenditure on treatments. Basially, they annot
aord the treatments that they need. Hene, although the equilibrium is an advantageous seletion
equilibrium, in our model stimulating insurane overage (e.g. through mandatory insurane at full
overage) is eient (beause of assumption 3).
26
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30
A. Proof of results
Proof of proposition 1
qh < 1
Suppose to the ontrary that
leads to nonnegative prots; otherwise it would not be oered in equilibrium.
utility level
θh
(q h , ph )
derives from
and by
ontrat. By assumption 3, the ontrat
p(q, uh , θ h )
ε > 0 small enough the ontrat (1, p(1, uh , θ h ) − ε)
For
higher prots than
(q h , ph ).
the indierene urve of
(1, p(1, uh , θ h ))
If the ontrat
for type
θh
the ase where
and
uh .
and
p(q, uh , θ h )
q h , q l < 1.
θl
and
θh
the
assoiated with his
to
(q h , ph ).
(q h , ph ) and yields
(1, p(1, uh , θ h ) − ε) is a protable deviation, i.e.
qh < 1
uh
types, prots will remain
a ontrat
Q.E.D.
annot be an equilibrium.
We start with the proof of the seond statement.
onstraints were binding, i.e.
θh
Denote by
yields higher prots than
is stritly preferred by
with stritly positive prots and demand. Consequently,
Proof of lemma 1
θh
(1, p(1, uh , θ h ) − ε) also attrats θ l
positive as those are better risks. Therefore,
(q h , ph )
in equilibrium. The ontrat
Suppose both inentive
are both indierent between the two ontrats. First, look at
Call the utility levels of the two types under the equilibrium ontrats
Now take the indierene urves orresponding to these utility levels and all them
and dene
ι = arg maxk∈{l,h} p(1, uk , θ k ).
will inrease prots by assumption 3. By the denition of
qk = 1
Seond, take the ase where
and
q −k < 1
Changing
ι,
θ ι 's
menu point to
ul
p(q, ul , θ l )
(1, p(1, uι , θ ι ))
this hange is also inentive ompatible.
for some
k ∈ h, l
and suppose again that both
inentive onstraints were binding. But aording to assumption 3 pooling on the ontrat of
θk
would
lead to higher prots. Hene, at most one inentive onstraint is binding.
qι = 1
follows from the argument in the rst step and therefore at least one type has to have full
Q.E.D.
overage.
Proof of proposition 2
overage.
Suppose that
Dene
qι < 1
and therefore
individual rationality onstraint of
the denition of
ι.
ι = arg maxk∈{h,l} p(1, ūk , θ k ).
θι
qκ = 1
with
pι
by pooling both types on
θ κ 's
annot be binding as otherwise
rationality onstraint of
θκ
θι
Note that the
θκ
by
has to be binding as the
will be binding.
q ι = 1,
q ι < 1.
(q, p) = (1, p(1, ūκ , θ κ ))
and the individual
If the types are separated, the inentive ompatibility
pooling on
θ ι 's
ontrat would lead to higher prots by
assumption 3 if the inentive onstraint was binding. As inreasing
θι,
κ 6= ι.
would misrepresent as
ontrat. This ontradits the optimality of
annot bind: Sine
bility onstraint of
θι
and
otherwise. By assumption 3, the monopolist ould ahieve a higher prot
If both types are pooled, the optimal ontrat will be
θκ
κ ∈ {h, l}
But then the inentive ompatibility onstraint of
monopolist ould inrease
onstraint of
By lemma 1, one type has full
the individual rationality onstraint of
θκ
pκ
relaxes the inentive ompati-
has to bind: Otherwise, inreasing
pκ
would inrease prots.
Last note that inreasing
pι
would be feasible and inrease prots if neither the inentive ompat-
31
ibility nor the individual rationality onstraint of
Proof of proposition 3
θι
Q.E.D.
was binding.
With single rossing, the solution (H) also satises (ICl ) beause (SC)
implies that
p(1, uhH , θ h ) −
p(1, ulH , θ l )
=
Z
1
l
qH
(pq (q, uhH , θ h ) − pq (q, ulH , θ l ) > 0
(37)
By the onavity assumption on problem (Puh ,ul ), the solution (H) is a symmetri equilibrium.
b
b
If single rossing is not satised, we need to hek that (ICl ) in ase (H) ((ICh ) in ase (L)) is
Q.E.D.
satised. Using an equation similar to (37) this is the assumption given in eah ase.
Lemma 3
In the example depited in gure 1 (L) equilibria exist for
Proof of lemma 3
φ > 0.
We start with two straightforward observations. First, for
the unique equilibrium illustrated in gure 1: By assumption 3
ql = 1
φ=0
the game has
is optimal and then the game is
a standard Hotelling game with exogenous loation. Seond, an equilibrium will exist for eah positive
φ.
This follows immediately from the existene theorem in Gliksberg (1952).
Now take a sequene of
where
φ = φn ,
25
ulj
n.
by
ul
e.g.
the sequene
θl
{1/n}n=1,2,... .
that are oered by rm
ula
n.
subsequene of
ulb
n
ulb
n
j ∈ {a, b}
With a slight abuse of notation we denote the elements of this subse-
as well and ontinue to work with this subsequene only. To eah
equilibrium value
In the game
is hosen from a losed and bounded interval (see footnote 24), there is a on-
verging subsequene of
ula
n
0,
onverging to
denote expeted equilibrium utilities of type
Sine
quene by
φn > 0
24
(assoiated with the game where
beause
we an nd a subsequene
ul
φ = φn ).
orresponds an
Again there will be a onverging
is taken from a losed and bounded interval.
hb la lb ha hb la lb
(uha
n , un , un , un , qn , qn , qn , qn )
ula
n
Continuing in this way
of equilibria onverging to some values
(ũha , ũhb , ũla , ũlb , q̃ ha , q̃ hb , q̃ la , q̃ lb ).
(ũha , ũhb , ũla , ũlb , q̃ ha , q̃ hb , q̃ la , q̃ lb )
To prove the lemma it is suient to show that
for
φ = 0.
for
φ>0
From the uniqueness of the equilibrium for
where
But then
qh
ulj
is arbitrarily lose to
qh = 1
24
and
θh
where
ul0
θl
is the unique equilibrium utility of
it is an (L) equilibrium): Type
θl
φ = 0.
ould ahieve
the inentive ompatibility onstraint of
It is evident from gure 1 that partial insurane is possible for
θl
θh
is violated if
types and it is
The highest relevant utility level is the utility resulting if an agent gets full overage for free. The individual rationality
onstraint gives a minimum utility level. Consequently, the ation spae is ompat (this is obvious for
prot funtions are ontinuous, the theorem applies.
25
for
in a ontrat with full overage where the individual rationality
holds (see gure 1), i.e.
ul ≈ ul0 .
φ = 0, it will then follow that there are equilibria
ul0
has to be stritly less than 1 (i.e.
a disretely higher utility than
onstraint of
ul0
is an equilibrium
If there are several equilibria, one an hoose an arbitrary one.
32
q ∈ [0, 1]).
As
routine to hek that these partial insurane ontrats are protable. Consequently,
0 < qnhj < 1
for
n
high enough.
The last step showing that
(ũha , ũhb , ũla , ũlb , q̃ ha , q̃ hb , q̃ la , q̃ lb )
is an equilibrium for
φ = 0
follows
Fudenberg and Tirole (1991, pp. 30) and is only skethed: Suppose, it was not an equilibrium. Then
there is a deviation yielding a stritly higher prot for one rm. By the ontinuity of the prot funtion,
however, this deviation would also inrease prots for
hb la lb ha hb la lb
(uha
n , un , un , un , qn , qn , qn , qn ) lose enough to
hb la lb ha hb la lb
(ũha , ũhb , ũla , ũlb , q̃ ha , q̃ hb , q̃ la , q̃ lb ) ontraditing that (uha
n , un , un , un , qn , qn , qn , qn ) is an equilibrium
under
Π,
φ = φn .
Q.E.D.
Proof of lemma 2
To see what assumption 5 implies in terms of derivatives of the prot funtion
we dene the matrix
A
(and analogously

A=
Ã)
as
Π11 + Π14 Π12 + Π15
Π12 + Π24 Π22 + Π25

(38)

This allows us to linearize equations (28) and (29) as follows (and similarly for equations (31) and (32)
with
Ã)

A
duh
dul

=0
Then it is straightforward to verify that
Π24 = 0
Π15 < 0
(39)
Π̃15 = 0
Π̃24 < 0
Inverting matrix
A
gives
A−1


1 Π22 + Π25 −(Π12 + Π15 )
=
∆
−Π12
Π11 + Π14
(40)
where
∆ ≡ (Π11 + Π14 )(Π22 + Π25 ) − Π12 (Π12 + Π15 )
(41)
Hene, we nd


duh
 dρ 
with
dul
dρ

= A−1 
Π13 = φD1 (uh , uh ) > 0, Π23 = −φD1 (ul , ul ) < 0.
dent of
u,
−Π13
−Π23

(42)

Using the assumption that
D1 (u, u)
is indepen-
we write
−(Π22 + Π25 ) − (Π12 + Π15 )
duh
= 12 φ
D1 (uh , uh )
dρ
∆
33
(43)
Similarly, we have
dul
Π12 + Π11 + Π14
= 12 φ
D1 (uh , uh )
dρ
∆
Assumption 5 that
duh /dρ
and
dul /dρ
(44)
have opposite signs an be written as
sign(Π11 + Π14 + Π21 + Π24 ) = sign(Π12 + Π15 + Π22 + Π25 )
(45)
Suppose by ontradition that suh a seond symmetri equilibrium would exist.
δh = uh − ũh , δl = ul − ũl
where
a rst order Taylor expansion of
δh , δl > 0
Π1 , Π2
are small sine by assumption
We write
(ũh , ũl ) ∈ Bε (uh , ul ).
Using
resp. we nd
Π1 (uh − δh , ul − δl , ρ, uh − δh , ul − δl ) = Π1 (uh , ul , ρ, uh , ul ) + (Π11 (uh , ul , ρ, uh , ul ) + Π14 (uh , ul , ρ, uh , ul ))δh
+ (Π12 (uh , ul , ρ, uh , ul ) + Π15 (uh , ul , ρ, uh , ul ))δl
Π2 (uh − δh , ul − δl , ρ, uh − δh , ul − δl ) = Π2 (uh , ul , ρ, uh , ul ) + (Π21 (uh , ul , ρ, uh , ul ) + Π24 (uh , ul , ρ, uh , ul ))δh
+ (Π22 (uh , ul , ρ, uh , ul ) + Π25 (uh , ul , ρ, uh , ul ))δl
Adding both equations and using the assumption that both
(uh , ul )
and
(ũh , ũl )
are an equilibrium,
we nd that
δh (Π11 + Π14 + Π21 + Π24 ) + δl (Π12 + Π15 + Π22 + Π25 ) = 0
However, it follows from equation (45) above that the expression
sign as
(Π12 + Π15 + Π22 + Π25 ).
Together with
Similar argument an be given in ase of
Lemma 4
δh , δl > 0
(46)
(Π11 + Π14 + Π21 + Π24 ) has the same
this leads to a ontradition.
Π̃.
Q.E.D.
Consider the example with mean-variane utility and Hotelling ompetition. Then ondition
(35) is satised in (L) equilibrium.
Proof of lemma 4
tributed on the line
With Hotelling ompetition,
[0, 1].26
Hene
D1 (u, u) = 1
as onsumers are uniformly dis-
D1 (u, u) is independent of u and we an use equation (45).
Following
equation (45), we need to show that
sign(Π̃11 + Π̃14 + Π̃21 + Π̃24 ) = sign(Π̃12 + Π̃22 + Π̃25 )
(47)
In our example, the equilibrium indierene urves at their point of intersetion an be written as
p(q h , ul , θ l ) = F l (v(h̄, θ l ) − (1 − q h )h̄) + (1 − F l )v(1, θ l ) − ul
− 21 r l F l (1 − F l )(v(1, θ l ) − v(h̄, θ l + (1 − q h )h̄)2
p(q h , uh , θ h ) = F h (v(h, θ h ) − (1 − q h )h) + (1 − F h )v(1, θ h ) − uh
− 21 r h F h (1 − F h )(v(1, θ h ) − v(h, θ h ) + (1 − q h )h)2
26
More generally, if onsumers are distributed with symmetri density funtion
g( 12 ) is independent of
u.
34
g on [0, 1], it is also true that D1 (u, u) =
Further,
q h (uh , ul )
is dened as
p(q h (uh , ul ), uh , θ h ) ≡ p(q h (uh , ul ), ul , θ l )
(48)
From these observations it follows that
pu (q, u, θ) = −1
pqu (q, u, θ) = 0
dq h (uh , ul )
dq h (uh , ul )
=
−
<0
duh
dul
d2 q h (uh , ul )
d2 q h (uh , ul )
=
−
(duh )2
duh dul
Using this we an write
φ
(p(q h (uha , ula ), uha , θ h ) − q h (uha , ula )hF h + ρh )
2t
h h l
uha − uhb
h h l
h h
h dq (ua , ua )
1
+φ 2 +
−1
(pq (q (ua , ua ), ua , θ ) − hF )
2t
duha
ula − ulb
1
l
l
l
l
1
= (1 − φ)
(p(1, ua , θ ) − F h̄ + ρ ) − 2 +
2t
2t
h
h
u − ub
dq h (uha , ula )
(pq (q h (uha , ula ), uha , θ h ) − hF h )
+φ 21 + a
2t
dula
Π̃1 =
Π̃2
Now it is routine to verify that
Π̃11 + Π̃12 + Π̃14 = −
with
Π̃24 < 0
φ
2t
1 + (pq (q h , uh , θ h ) − hF h )
dq h (uh , ul )
dul
<0
from equation (39). Moreover
Π̃12 + Π̃22 + Π̃25 = −
φ
dq h (uh , ul ) 1 − φ
(pq (q h , uh , θ h ) − hF h )
−
2t
duh
2t
whih we will shown to be negative in every (L) equilibrium. Suppose this were not the ase, then
φ(pq (q h , uh , θ h ) − hF h ) ≥ −(1 − φ)
where
duh
dq h (uh ,ul )
<0
is the amount
uh
hanges if
qh
pl
uh
inreasing
qh
θh
ul
onstant.
types when marginally inreasing
qh
x. The right hand side of (49) denotes the marginal loss in prots if one redues
suh that a marginal inrease of
pu (q, u, θ) = −1
(49)
is inreased marginally while keeping
The left hand side of (49) denotes the additional prots from
while keeping
duh
dq h (uh , ul )
qh
with xed
uh
is inentive ompatible for the
for both types, the neessary redution in
keeping
uh
xed and adjusting
pl
pl
is given by
duh
θl
− dqh (uh ,ul ) > 0.
types:
As
But then
to keep inentive ompatibility is a protable deviation
whenever (49) holds. It is stritly protable as the derease in
35
pl
will attrat additional ustomers from
the ompeting insurer. This ontradits that the original situation is an equilibrium and therefore (49)
annot hold.
Q.E.D.
Hene equation (47) has to be satised in every (L) equilibrium.
Proof of proposition 4
point of intersetion (at
pq (q l , ul , θ l ).
uh
ql )
If
q h = 1 and given that the equilibrium indierene urves have only one
and are ontinuous, equations (19) and (C1) imply that
It follows then from equations (23) and (24) that a hange in
while reduing
ul .
Given assumption 5, the other possibility is that
then equations (23) and (24) imply that
Similarly, if
ql = 1
ql
and a fall in
inreases while
uh
ql
raises
falls. But
falls. Hene the latter ase an be ruled out.
q h ),
equations (20) and (C1) imply that
follows then from equation (27) that a hange in
ul
whih inreases
and given that the equilibrium indierene urves are ontinuous and only have
one point of intersetion (at
in
ul
ρ
pq (q l , uh , θ h ) >
ρ whih raises q h
uh .
pq (q h , ul , θ l ) > pq (q h , uh , θ h ).
It
must be aompanied by an inrease
Q.E.D.
36