State-Dependent Preferences and Insurance Demand∗
Robert Kremslehner†
Alexander Muermann‡
July 2014
We compare the demand for insurance under state-dependent and state-independent
preferences when allowing for positive premium loading and continuous state space.
We show that monotonicity of marginal utility across states is not sufficient to yield
unambiguous comparative results–as it is under a fair premium. Nevertheless, unambiguous results can be derived for a specified set of state-independent preferences
if both marginal utility and absolute degree of risk aversion are monotonic across
states.
∗
We thank Georges Dionne, Meglena Jeleva, Christophe Courbage, and seminar participants at the EGRIE
meeting for valuable comments and discussion.
†
Department of Finance, Accounting and Statistics, Vienna University of Economics and Business, Welthandelsplatz 1, Building D4, A - 1020 Vienna, AUSTRIA, E-mail: [email protected]
‡
Department of Finance, Accounting and Statistics, Vienna University of Economics and Business, and VGSF
(Vienna Graduate School of Finance), Welthandelsplatz 1, Building D4, A - 1020 Vienna, AUSTRIA, E-mail:
[email protected]
1
1 Introduction
In von Neumann and Morgenstern (1944) expected utility theory, preferences of individuals are
characterized through a state-independent utility function, i.e. the utility in a state depends only
on the consumption in that state but not directly on the state itself. In contrast, Eisner and
Strotz (1961) investigate the demand for flight insurance and argue that the utility function of an
individual is likely to be different when he is alive compared to when he is dead. Similarly, Arrow
(1973) emphasizes that individuals who fell ill may extract less utility from spending money on
vacation. Individuals’ preferences therefore directly depend on the state of health.
Cook and Graham (1977) apply state-dependent preferences to examine the optimal insurance
demand for irreplaceable commodities, such as family heirlooms or a photo album. They show
that less (more) than full coverage of the financial loss is optimal if the irreplaceable commodity
is normal (inferior). Dionne (1982) and Schlesinger (1984) show that the optimal amount of
insurance under state-dependent preferences can be inferred from changes of marginal utility of
consumption across states. More specifically, if insurance is actuarially fair then an individual
with state-dependent preferences optimally purchases less (more) than full insurance coverage
if his marginal utility of consumption decreases (increases) as the irreplaceable commodity is
lost.1,2 Furthermore, Karni (1985) and Nordquist (1985) show that the optimal insurance demand
under state-dependent preferences is increasing in the risk premium. In the context of nonpecuniary background risk, e.g. health, Rey (2003) determines conditions on the sign of the
correlation between insurable losses and background risk such that the results of Dionne (1982)
and Schlesinger (1984) hold. For example, if marginal utility of consumption decreases as health
deteriorates and health is positively correlated with wealth, then less than full insurance is
optimal. There exist few papers that empirically investigate how marginal utility of consumption
changes across states of health. Viscusi and Evans (1990) estimate changes in marginal utility of
consumption between good and bad health using a survey on pay raise that workers demand for
taking on additional risk at work. They conclude that marginal utility is statistically significantly
lower if workers are severely injured.3 However, for minor health losses Evans and Viscusi (1991)
do not find significant evidence that marginal utility is affected by health status. In a recent
1
Mossin (1968) shows that risk-averse individuals with state-independent preferences optimally purchase full
coverage at an actuarially fair premium rate.
2
Eeckhoudt et al. (2007) call individuals whose marginal utility of consumption increases as health deteriorates
correlation averse.
3
Frech III (1994) argues that the tort system and strict liability might cause inefficiencies if marginal utility of
consumers is lower in the accident state.
2
paper, Finkelstein et al. (2013) investigate the impact of chronical diseases on the marginal
utility of consumption. They find that marginal utility declines by about 10 - 25 percent as
health deteriorates, as measured by a one standard deviation increase in the number of chronic
diseases. The authors further argue that the estimated 10 - 25 percent decline in marginal utility
lowers the optimal amount of medical insurance by about 20 to 45 percentage points compared
to the optimal amount of medical insurance under state-independent preferences.4
In this paper, we compare optimal insurance demand between state-independent and statedependent preferences. All literature mentioned above and related to this objective is based on
the assumption that insurance is priced at an actuarially fair rate and that individuals face a
binary distribution, e.g. staying healthy or falling sick with corresponding medical cost. Under these two assumptions, the sign of the change in marginal utility of consumption as health
deteriorates is the only empirically important factor that impacts the demand for insurance.
We contribute to the literature by analyzing the demand for insurance under state-dependent
preferences for positive loading factors and continuously distributed loss distributions. We find
that whether marginal utility of consumption decreases (increases) as health deteriorates is not
sufficient to guarantee that individuals with state-dependent preferences purchase less (more) insurance than individuals with state-independent preferences–as it does under a fair premium and
binary distribution. An additional property of preferences that is crucial for such unambiguous
statements is how the absolute degree of risk aversion of consumption changes as health deteriorates. More specifically, for a fixed loading factor we specify state-independent preferences such
that an individual with such preferences purchases more (less) insurance at this loading factor
than an individual with state-dependent preferences if marginal utility of consumption decreases
(increases) as health deteriorates. If the degree of risk aversion is monotonic in health, then
we derive a bound for the degree of risk aversion such that individuals with state-independent
preferences and degrees of risk aversion that are larger (smaller) than this bound purchase more
(less) insurance at all loading factors than an individual with state-dependent preferences. This
result shows that it is relevant not only how marginal utility but also how the degree of risk
aversion changes across states of health.
The article is structured as follows. In Section 2, we extend the binary insurance model to
positive loading factors. In Section 3, we further extend the model to continuous distributions.
4
Finkelstein et al. (2013) derive the optimal amount of insurance by calibrating their two-period model with
Epstein-Zin preferences (Epstein and Zin (1989)), a binary distribution of health, and actuarially fair premium.
3
We conclude in Section 4. All proofs are in the Appendix.
2 State-Dependent Preferences: Two-State Space
Consider an individual whose preferences depend on wealth ω and health H and are described
by a bi-variate utility function U (ω, H).5 We assume that the individual’s utility function is
increasing in both wealth and health and concave in wealth, i.e. U1 > 0, U2 > 0, and U11 < 0.6
Let ω0 and H0 = 1 denote the initial levels of wealth and health. The individual faces the risk
of falling ill with probability 0 < p < 1. In case of sickness he incurs medical treatment costs
L ≥ 0 and his health state is reduced to H = 0. An insurance company offers medical insurance
coverage I at a premium P (I) = (1 + λ)pI where λ denotes the proportional loading factor.
∗
The optimal level of insurance coverage ISDEU
(λ) that maximizes his state-dependent expected
utility (SDEU) is determined by the following optimization problem
max SDEU (I) = max E[U (ω(I, H̃), H̃)],
I
I
(1)
where ω (I, 1) = ω0 − P and ω (I, 0) = ω0 − P − L + I.
For an actuarially fair premium (λ = 0), Dionne (1982) and Schlesinger (1984) show that the
optimal amount of insurance coverage for individuals with state-dependent preferences relative
to one implied by state-independent preferences depends on whether marginal utility of wealth
increases or decreases in health, i.e. on the sign of U12 . If marginal utility of wealth decreases
(increases) as health deteriorates U12 > (<) 0 less (more) than full coverage is optimal. An
individual with state-dependent preferences therefore purchases less (more) insurance than any
risk-averse individual with state-independent preferences (see Mossin (1968)).
In this section, we extend this comparative result for positive loading factors. For λ = 0, full
insurance is optimal for any risk-averse individual with state-independent preferences. It is thus
possible to compare the optimal insurance demand between state-dependent preferences and all
state-independent preferences that exhibit risk aversion. In contrast, for λ > 0, the optimal
∗ (λ), depends on the degree of
amount of insurance under state-independent preferences, IEU
risk aversion, i.e. on the specific utility function U . When comparing the optimal amount of
5
6
The interpretation of H as the state of health is arbitrary. Other interpretations are possible.
We use the common notation Ui for the partial derivative of U with respect to the i’s argument, e.g. U1 =
∂U
and U2 = ∂H
.
4
∂U
∂ω
∗
insurance under state-dependent preferences, ISDEU
(λ), with the optimal insurance under state∗
independent preferences, it seems natural in a first step to compare ISDEU
(λ) against the optimal
∗
∗
amounts of insurance, IEU(0)
(λ) and IEU(1)
(λ), implied by the two state-independent utility
functions U (·, 0) and U (·, 1) that form the basis for the state-dependent preferences. In the
following proposition, we show that the result of Dionne (1982) and Schlesinger (1984) can be
extended to positive loading factors when comparing against these two specific state-independent
preferences. From hereon, we focus on the case in which marginal utility of wealth is increasing
in health, i.e. U12 > 0.7
Proposition 1. Suppose U12 > 0. An individual with state-dependent preferences optimally
chooses less insurance coverage for all loading factors λ ≥ 0 than an individual with stateindependent preferences characterized by either utility function U ∈ {U (·, 0), U (·, 1)}. That is
n
o
∗
∗
∗
(λ) < min IEU(0)
(λ) , IEU(1)
(λ)
ISDEU
for all λ ≥ 0.
Figure 1 illustrates the result of Proposition 1. The plot shows the optimal insurance coverage
I ∗ for state-dependent and state-independent preferences as function of the loading factor λ.
I∗
∗
ISDEU
(λ)
1
∗
(λ)
IEU(0)
∗
IEU(1)
(λ)
λ
0
Figure 1: Comparison of optimal insurance coverage of individuals with state-dependent and
state-independent preferences at different loading factors λ.
Since the optimal amount of insurance of EU-maximizers8 is increasing in the degree of risk
7
U12 > 0 is consistent with the empirical evidence of Viscusi and Evans (1990) and Finkelstein et al. (2013)
and imply correlation loving preferences between wealth and health (see Eeckhoudt et al. (2007)). We obtain
analogous results for U12 < 0 which is covered in the appendix.
8
We use the term “SDEU-maximizers” for individuals with state-dependent preference and “EU-maximizers” for
individuals with state-independent preferences.
5
aversion, we can extend the result obtained above from the comparison set of the two stateindependent utility functions U (·, 0) and U (·, 1) to the set of all state-independent utility functions which exhibit degrees of risk aversion that are larger than the one implied by either U (·, 0)
or U (·, 1).
Corollary 2. Suppose U12 > 0. An individual with state-dependent preferences optimally chooses
less insurance coverage for all loading factors λ than any individual with state-independent preferences whose degree of risk aversion is higher at all levels of wealth ω than the degree of risk
aversion implied by either of the two utility functions U ∈ {U (·, 0), U (·, 1)}. That is
∗
∗
ISDEU
(λ) < IEU
(λ)
U (0)
for all λ ≥ 0 and U with RaU (ω) ≥ Ra
U (1)
(ω) for all ω or RaU (ω) ≥ Ra
(ω) for all ω where
RaU (ω) denotes the coefficient of absolute risk aversion in wealth under the state-independent
utility function U and U (0) and U (1) denote the utility functions U (·, 0) and U (·, 1), respectively.9
In this section, we have shown a SDEU-maximizer whose marginal utility of wealth decreases
as health deteriorates, i.e. U12 > 0, optimally purchases less insurance coverage at all loading
factors λ than all EU-maximizers whose degree of risk aversion is globally greater than the one
implied by either U (·, 0) or U (·, 1).10
3 State-Dependent Preferences: Continuous State Space
In this section, we examine the case in which both health status H̃ and medical cost L̃ are continuously distributed random variables. As in the previous section, we associate a deterioration
in health with an increase in medical costs. More specifically, we define the function h← : H → L
that assigns to each health status H ∈ H a loss of size L ∈ L and assume that h← is strictly
decreasing and left continuous.11 Furthermore, let h← be the generalized inverse of function h
which is continuous and decreasing due to the properties of h← . We additionally assume that h is
non-constant a.s. and define the individuals utility function as U (ω, H = h(L)) where the medical
9
Risk aversion in wealth is meant in the sense of Pratt (1964) and Arrow (1971).
Equivalently, correlation averse SDEU maximizers, i.e. with U12 < 0, optimally purchase more insurance
coverage at all loading factors λ than all EU-maximizers whose degree of risk aversion is smaller than the one
implied by either U (·, 0) or U (·, 1).
11
H and L are thus the sets of all realizations of health status H and medical cost L, respectively.
10
6
treatment cost L̃ is distributed according to a cumulative distribution function F : [0, L̄] → R+
with F (0) = 0 and F (L̄) = 1, where L̄ denotes the maximum loss.12
The individual is offered a coinsurance contract, i.e. the indemnity schedule I : [0, L̄] → R+ is
given by I(L) ≡ αL where L is a realization of L̃ and α is the coinsurance rate. The premium
is determined by Pα = α P (λ) with a loading factor λ ≥ 0 where P (λ) = (1 + λ)E[L̃]. The
∗
individual chooses the optimal level of insurance coverage αSDEU
(λ) by maximizing expected
state-dependent utility, i.e.
max SDEU (α) = max E[U (ω(α, L̃), h(L̃))],
α
α
(2)
where ω(α, L) = ω0 − (1 + λ)αE[L̃] − (1 − α)L.
In the following proposition we show that the result of Dionne (1982) and Schlesinger (1984) under
their assumption that the premium is actuarially fair, i.e. λ = 0, generalizes to the continuous
state space.
Proposition 3. Suppose U12 > 0 and λ = 0. Then an individual with state-dependent preferences
optimally chooses less than full insurance coverage, i.e.
∗
αSDEU
(λ = 0) < 1.
Next, for positive loading factors λ ≥ 0, we compare the insurance purchase behavior of SDEUmaximizers with the one of EU-maximizers whose preferences are characterized by the stateindependent utility functions of the set U ≡ {U (·, H) : H ∈ H} that constitutes the statedependent preferences. For the two state case, we have shown in Proposition 1 that an individual
with state-dependent preferences optimally purchases less insurance coverage for all loading
factors λ than an individual with state-independent utility function U for all U ∈ U. This
result does not generalize to a continuous state space.13 Nevertheless, we show in the following
proposition that for each loading factor λ there exists a state-independent utility function U (λ) ∈
U such that a SDEU-maximizer purchases less insurance at this loading factor λ than an EUmaximizer with utility function U (λ).
12
Health status H̃ and medical cost L̃ are thus assumed to be countermonotonic. We note that this assumption
does not necessarily imply that H̃ and L̃ are perfectly negatively correlated as measured by linear correlation.
13
We provide an example below to illustrate that an EU-maximizer might purchase less insurance even if marginal
utility is decreasing as health deteriorates.
7
Proposition 4. Suppose U12 > 0. For each loading factor λ ≥ 0, we define the state-independent
utility function U (λ) ≡ U (·, H̄(λ)) ∈ U where H̄(λ) ≡ h((1 + λ)E[L̃]). Then for each loading
factor λ ≥ 0, the SDEU-maximizer optimally purchases less insurance than an EU-maximizer
with utility function U (λ), i.e.
∗
∗
αSDEU
(λ) < αEU(
H̄(λ)) (λ) .
Proposition 4 implies that the higher the loading factor λ the worse the necessary fixed health
status H̄(λ) to ensure that an EU-maximizer with utility function U (·, H̄(λ)) purchases more
insurance than the SDEU-maximizer.
As in the case with two states, the degree of risk aversion can be used to make further statements.
First, suppose that the degree of risk aversion of EU-maximizers with utility function U (·, H) ∈ U
is independent of the fixed health status H, i.e.
∂
a
∂H (RU (·,H) (ω))
= 0 for all ω.14 The optimal
∗
amount of insurance αEU(H)
is therefore also independent of the fixed health status H. In this
case, the result of Proposition 1 generalizes to the continuous state space.
Corollary 5. Suppose U12 > 0 and
∂
a
∂H (RU (H) (ω))
= 0 for all ω. Then an individual with state-
dependent preferences optimally chooses less insurance coverage for all loading factors λ ≥ 0 than
an individual with state-independent preferences characterized by any utility function U (·, H) ∈ U,
i.e.
∗
∗
(λ) < αEU(H)
(λ)
αSDEU
for all λ ≥ 0 and U (·, H) ∈ U. Furthermore, an individual with state-dependent preferences
optimally chooses less insurance coverage for all loading factors λ ≥ 0 than any EU-maximizing
individual who exhibits a degree of risk aversion which, at all levels of wealth, is larger than
a
RU
(H) , i.e.
∗
∗
αSDEU
(λ) < αEU
(λ)
a (ω) ≥ Ra
for all λ ≥ 0 and U with RU
U (H) (ω) for all ω.
This result is illustrated in Figure 2 which depicts the optimal coinsurance rates α∗ as function
of the loading factor λ. For the two loading factors λ1 < λ2 , the two utility functions U (·, H̄(λ1 ))
and U (·, H̄(λ2 )) exhibit identical degrees of risk aversion. The optimal amount of insurance of
14
The utility function used in Finkelstein et al. (2013) satisfies this property.
8
an EU-maximizer with utility function U (·, H̄(λ1 )) thus coincides with the optimal amount of
insurance of an EU-maximizer with utility function U (·, H̄(λ2 )) for all λ ≥ 0. Since this holds
for any health status H̄(λ), Proposition 4 implies Corollary 5. Note that the result that statedependence implies lower insurance coverage as derived in ? only extends to positive loading
factors under the condition that the degree of risk aversion is independent of the health status.
α∗
∗
αSDEU
(λ)
1
∗
αEU
λ, H̄ (λ)
0
∗
αEU
λ, H̄ (λ1 )
∗
αEU
λ, H̄ (λ2 )
λ1
λ2
λ
Figure 2: Comparison of optimal insurance coverage of individuals with state-dependent and
state-independent preferences at different loading factors λ if risk aversion is unaffected
by health state.
Now suppose that the degree of risk aversion of EU-maximizers with utility functions U (·, H) ∈ U
increases as health deteriorates, i.e.
∂
a
∂H (RU (H) (ω))
< 0 for all ω. Then the optimal amount of
∗
insurance αEU(
is increasing in the loading factor λ. Proposition 4 then implies the following
H̄(λ))
corollary.
Corollary 6. Suppose U12 > 0 and
∂
a
∂H (RU (H) (ω))
< 0 for all ω. Then an individual with
state-dependent preferences optimally chooses less insurance coverage for all loading factors
λ ≥ 0 than an individual with state-independent preferences characterized by the utility function U (·, H̄(λmax )) ∈ U, i.e.
∗
∗
αSDEU
(λ) < αEU(
H̄(λmax )) (λ)
∗
for all λ ≥ 0 where λmax is defined through αEU(
(λmax ) = 0. Furthermore, an individual
H̄(λmax ))
with state-dependent preferences optimally chooses less insurance coverage for all loading factors
λ ≥ 0 than any EU-maximizing individual who exhibits a degree of risk aversion which, at all
a
levels of wealth, is larger than RU
, i.e.
(H̄(λmax ))
∗
∗
αSDEU
(λ) < αEU
(λ)
9
a (ω) ≥ Ra
for all λ ≥ 0 and U with RU
(ω) for all ω.
U (H̄(λmax ))
This result is illustrated in Figure 3.
α∗
∗
αSDEU
(λ)
1
∗
αEU
λ, H̄ (λ)
∗
αEU
λ, H̄ (λmax )
∗
αEU
λ, H̄ (λ1 )
∗
αEU
λ, H̄ (0)
0
λmax
λ1
λ
Figure 3: Comparison of optimal insurance coverage of individuals with state-dependent and
state-independent preferences at different loading factors λ if risk aversion increases as
health deteriorates.
∗
(λ) of the SDEU-maximizer as a
The thick solid line plots the optimal coinsurance rate αSDEU
∗ (λ, H̄(λ)) of an EU-maximizer as a function
function of λ. The optimal coinsurance rate αEU
of λ is depicted by the dashed line for all 0 ≤ λ ≤ λmax . The three thin solid lines plot
∗ (λ, H̄) as a function of λ for the three fixed levels of health
the optimal coinsurance rates αEU
status H̄ = H̄(0), H̄(λ1 ), and H̄(λmax ) for all 0 ≤ λ ≤ λmax . Note that for each fixed loading
∗ (·, H̄(·)) intersects with the thin solid line α∗ (·, H̄(λ)). Since the
factor λ, the dashed line αEU
EU
∗ (λ, H̄(0)) ≤ α∗ (λ, H̄(λ )) ≤
degree of risk aversion increases as health deteriorates, we have αEU
1
EU
∗ (λ, H̄(λmax )) and α∗
∗
max .
αEU
SDEU (λ) < αEU(H̄(λmax )) (λ) for all 0 ≤ λ ≤ λ
We obtain the analogous result if the degree of risk aversion of EU-maximizers with utility
functions U (·, H) ∈ U decreases as health deteriorates, i.e.
Corollary 7. Suppose U12 > 0 and
∂
a
∂H (RU (H) (ω))
∂
a
∂H (RU (H) (ω))
> 0 for all ω.
> 0 for all ω. Then an individual with state-
dependent preferences optimally chooses less insurance coverage for all loading factors λ ≥ 0 than
an individual with state-independent preferences characterized by the utility function U (·, H̄(0)) ∈
U, i.e.
∗
∗
αSDEU
(λ) < αEU(
H̄(0)) (λ)
for all λ ≥ 0. Furthermore, an individual with state-dependent preferences optimally chooses less
insurance coverage for all loading factors λ ≥ 0 than any EU-maximizing individual who exhibits
10
a
a degree of risk aversion which, at all levels of wealth, is larger than RU
, i.e.
(H̄(0))
∗
∗
αSDEU
(λ) < αEU
(λ)
a (ω) ≥ Ra
for all λ ≥ 0 and U with RU
(ω) for all ω.
U (H̄(0))
This result is illustrated in Figure 4.
α∗
∗
αSDEU
(λ)
1
∗
αEU
λ, H̄ (λ)
∗
αEU
λ, H̄ (0)
∗
αEU
λ, H̄ (λ1 )
∗
αEU
λ, H̄ (λmax )
0
λmax
λ1
λ
Figure 4: Comparison of optimal insurance coverage of individuals with state-dependent and
state-independent preferences at different loading factors λ if risk aversion decreases as
health deteriorates.
Example. In this example, we illustrate that an EU-maximizer might purchase less insurance
even if marginal utility is decreasing as health deteriorates. Suppose that the set of utility
1
functions is defined as U (ω, H) = −e− 9 H ω . Note that this utility function satisfies the conditions
of Corollary 7, i.e. marginal utility is increasing in health for all H and ω such that H ω < 9
and the degree of risk aversion is increasing in health. Let ω0 = 3 and suppose that the density
distribution function of the loss is given by f (L) =
10
1−e−10
e−10 L , with L ∈ [0, 1]. Further,
suppose that the mapping from losses to health levels is given by h(L) = 2.9 − L. Figure
5 plots the optimal amount for insurance for the SDEU-maximizer (thick line) and for EU
maximizers with fixed health status H = 2.9, H = H̄(0) ≈ 2.8, and H = 1.9 (thin lines). We
observe that EU-maximizers with fixed health status H = 2.9 and H = H̄(0) purchase more
insurance at all loading factors than the SDEU-maximizer. In contrast, the EU-maximizers with
fixed health status H = 1.9 purchase less insurance at high loading factors than the SDEUmaximizer. For λ = 0.02, the optimal amount of insurance for the SDEU-maximizer is given by
∗
αSDEU
(λ = 0.02) = 0.1464. For EU-maximizers with fixed health status H = 2.9, H = H̄(0) ≈
∗
2.8, and H = 1.9, the optimal amounts of insurance are given by αEU(H=2.9)
(λ = 0.02) = 0.3888,
11
∗
∗
αEU(H=
(λ = 0.02) = 0.3670, and αEU(H=1.9)
(λ = 0.02) = 0.0671, respectively.
H̄(0))
α∗
∗
αSDEU
(λ)
1
∗
αEU
(λ, H = 2.9)
0.75
∗
αEU
λ, H = H̄(0)
∗
αEU
(λ, H = 1.9)
0.5
0.25
0
0
0.01
0.02
0.03
λ
Figure 5: Illustration of the example: an individual with preferences characterized by a utility
function U (·, H = 1.9) optimally chooses less insurance coverage for high loading factors
than an individual with state-dependent preferences.
4 Conclusion
We analyze the impact of state-dependent preferences on insurance demand when allowing for
positive loading factors and continuously distributed states. We find that the results do not
only depend on how marginal utility of consumption changes in health–as pointed out by Dionne
(1982) and Schlesinger (1984) under fair premium rates and binary distribution–but also on other
characteristics of preferences such as how absolute degree of risk aversion changes in health.
This has important implications for the conclusions that can be drawn from empirical research
on state-dependent preferences. If insurance includes a loading, then the decline of marginal
utility does not unambiguously imply a decline in the optimal amount of insurance. For such
conclusions it is necessary to jointly estimate how the degree of risk aversion changes as health
deteriorates.
12
Appendix - Proofs
Proof of Proposition 1.
We face the maximization problem stated in equation 1 max SDEU (I) with
I
h
i
SDEU (I) = E U (ω(I, H̃), H̃) = pU (ω (I, 0) , 0) + (1 − p) U (ω (I, 1) , 1) ,
where H = 0 denotes bad health and H = 1 denotes good health. Final wealth levels in bad and
good health are given by ω (I, 0) = ω0 − P − L + I and ω (I, 1) = ω0 − P , respectively, where
P = (1 + λ) pI is the premium at loading factor λ. The FOC is given by
SDEU0 (I) = p (1 − (1 + λ) p) U1 (ω (I, 0) , 0) − (1 − p) (1 + λ) pU1 (ω (I, 1) , 1) = 0.
The SOC
SDEU00 (I) = p (1 − (1 + λ) p)2 U11 (ω (I, 0) , 0) + (1 − p) (1 + λ)2 p2 U11 (ω (I, 1) , 1) < 0
∗
∗
is satisfied. Thus, the unique global maximum ISDEU
(λ) for all λ is determined by SDEU0 (ISDEU
(λ)) =
0.
Now we compare insurance purchase behavior of state-dependent preferences with state-independent
preferences. The FOC for individuals with preferences characterized by U (ω, H̄) with fixed H̄ is
given by
EU0U (·,H̄) (I) = p (1 − (1 + λ) p) U1 (ω (I, 0) , H̄) − (1 − p) (1 + λ) pU1 (ω (I, 1) , H̄) = 0.
∗
The SOC is satisfied. The unique global maximum IEU(
(λ) is therefore determined by
H̄)
∗
EU0U (·,H̄) (IEU(
H̄) (λ)) = 0.
∗
∗
Evaluation of the FOC for ISDEU
(λ) at I = IEU(
(λ) yields
H̄)
∗
∗
SDEU0 (IEU(
H̄) (λ)) = p (1 − (1 + λ) p) U1 (ω(IEU(H̄) (λ) , 0), 0)
∗
− (1 − p) (1 + λ) pU1 (ω(IEU(
H̄) (λ) , 1), 1).
∗
IEU(
(λ) satisfies
H̄)
∗
∗
EU0U (·,H̄) (IEU(
H̄) (λ)) = p (1 − (1 + λ) p) U1 (ω(IEU(H̄) (λ) , 0), H̄)
∗
− (1 − p) (1 + λ) pU1 (ω(IEU(
H̄) (λ) , 1), H̄)
= 0.
Substitution yields
∗
∗
SDEU0 (IEU(1)
(λ)) = p (1 − (1 + λ) p) U1 (ω(IEU(1)
(λ) , 0), 0)
∗
− U1 (ω(IEU(1)
(λ) , 0), 1)
and
∗
∗
SDEU0 (IEU(0)
(λ)) = (1 − p) (1 + λ) p U1 (ω(IEU(0)
, 1), 0)
∗
− U1 (ω(IEU(0)
, 1), 1) .
13
∗
∗
We conclude that if U12 S 0 then ISDEU
(λ) T IEU(
(λ) for all λ ≥ 0.
H̄)
Proof of Proposition 3.
We face the maximization problem stated in equation 2 max SDEU (α) with
α
h
i
SDEU (α) = E U (ω(α, L̃), h(L̃)) ,
where ω(α, L̃) = ω0 − (1 + λ)αE[L̃] − (1 − α)L. The FOC is given by
h
i
SDEU0 (α (λ)) = E (L̃ − P )U1 (ω(α (λ) , L̃), h(L̃)) = 0.
The SOC
h
i
SDEU00 (α (λ)) = E (L̃ − P )2 U11 (ω(α (λ) , L̃), h(L̃)) < 0
∗
∗
is satisfied. The unique global maximum αSDEU
(λ) is therefore determined by SDEU0 (αSDEU
(λ)) =
0. For λ = 0, evaluating the FOC at α = 1 yields
h
i
SDEU0 (1) = E (L̃ − E[L̃])U1 (ω0 − E[L̃], h(L̃)) =
h
i
= Cov L̃, U1 (ω0 − E[L̃], h(L̃))
∗
More than full (full, partial) insurance αSDEU
(0) T 1 is therefore optimal if
h
i
Cov L̃, U1 (ω0 − E[L̃], h(L̃)) T 0
which is equivalent to U12 S 0.
Proof of Proposition 4.
We face the maximization problem stated in equation 2. For λ > 0, we compare insurance
purchase behavior of SDEU preferences with EU preferences. The FOC for individual with
preferences characterized by U (·, H̄) with fixed H̄(λ) ≡ h((1 + λ)E[L̃]) is given by
h
i
EU0U (·,H̄) (α (λ)) = E (L̃ − P )U1 (ω(α (λ) , L̃), H̄ (λ)) .
∗
The SOC is satisfied. The unique global maximum αEU(
(λ) is therefore determined by
H̄)
0
∗
EUU (·,H̄) (αEU(H̄) (λ)) = 0.
∗
∗
Evaluating the FOC for αSDEU
(λ) at αEU(
(λ) yields
H̄)
h
i
∗
∗
SDEU0 (αEU(
(λ))
=
E
(
L̃
−
P
)U
(ω(α
(λ)
,
L̃),
h(
L̃))
1
H̄)
EU(H̄)
ˆP
∗
(L − P ) U1 (ω(αEU(
H̄) (λ) , L), h(L))dF (L)
=
0
ˆL̄
∗
(L − P ) U1 (ω(αEU(
H̄) (λ) , L), h(L))dF (L).
+
P
14
∗
The FOC EU0U (·,H̄) (α) at αEU(
(λ) satisfies
H̄)
h
i
∗
∗
EU0U (·,H̄) (αEU(
(λ))
=
E
(
L̃
−
P
)U
(ω(α
(λ)
,
L̃),
H̄)
1
H̄)
EU(H̄)
ˆP
∗
(L − P ) U1 (ω(αEU(
H̄) (λ) , L), H̄)dF (L)
=
0
ˆL̄
∗
(L − P ) U1 (ω(αEU(
H̄) (λ) , L), H̄)dF (L) = 0.
+
P
Suppose U12 > 0. Then, for L < (1 + λ)E[L̃] we have
∗
∗
U1 (ω(αEU(
H̄) (λ) , L), h (L)) ≥ U1 (ω(αEU(H̄) (λ) , L), h((1 + λ) E[L̃]))
∗
∗
(L − P ) U1 (ω(αEU(
H̄) (λ) , L), h (L)) ≤ (L − P ) U1 (ω(αEU(H̄) (λ) , L), h((1 + λ) E[L̃]))
and for L > (1 + λ)E[L̃] we have
∗
∗
U1 (ω(αEU(
H̄) (λ) , L), h (L)) ≤ U1 (ω(αEU(H̄) (λ) , L), h((1 + λ) E[L̃]))
∗
∗
(L − P ) U1 (ω(αEU(
H̄) (λ) , L), h (L)) ≤ (L − P ) U1 (ω(αEU(H̄) (λ) , L), h((1 + λ) E[L̃]))
∗
∗
with strict inequality for some L. Thus SDEU0 (αEU(
(λ)) < EU0U (·,H̄) (αEU(
(λ)) = 0. SimH̄)
H̄)
ilar reasoning applies to U12 < 0 and U12 = 0. Hence, we conclude that if U12 S 0 then
∗
∗
∗
∗
(λ) T αEU(
(λ) for all
SDEU0 (αEU(
(λ)) T EU0U (·,H̄) (αEU(
(λ)) = 0 and therefore αSDEU
H̄)
H̄)
H̄)
λ ≥ 0 with H̄ ≡ h((1 + λ) E[L̃]).
15
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