In this section, we discuss comparative statics of the model with regard to the types of equilibrium examined in Section 3. In our model, the type of market equilibrium varies with cost of investing in self-protection, F , the intensity of regret of type R individuals, as measured by the convexity of g, and the fraction of type R individuals in the population, λ.
4.1 Cost of investment in self-protection
If the cost of investing in self-protection, F , is extremely high or low both types of individuals optimally do not invest or invest in self-protection. Individuals are therefore heterogeneous only
regarding their preferences but not regarding their risk type. Thus, type N individuals optimally obtain full coverage, whereas type R individuals optimally obtain partial coverage, as shown by Braun and Muermann (2004).
Case 1 For very small (high) levels of F , it is optimal for both type R and type N individuals to invest (to not invest) in self-protection and the unique equilibrium is a separating equilibrium in which both types receive the optimal amount of coverage at the rate c = pF (c = p0), i.e. full coverage for type N and partial coverage for type R individuals.
Suppose that the cost of investing in self-protection is in a range such that we obtain equilibria under which type R individuals do not invest in self-protection but type N individuals do. As the cost F increases, the set of contracts under which it is optimal for type N individuals to invest in self-protection decreases, i.e. the locus of contracts ∆N(I, c) = 0 shifts down (see Figure 10 with F1< F2 < F3). We then derive the following comparative statics.
Case 2 Suppose that the cost of investing in self-protection, F , is high enough such that ∆R(I, c) <
0 for all I and c with pF ≤ c ≤ p0.
1. For low levels of F (e.g. F1 in Figure 10), condition 1b in Proposition 3 is satisfied - i.e. type R indifference curve crosses ∆N(I, p0) = 0 line above the pricing line PF - and a separating equilibrium with advantageous selection as in Figure 7 is obtained.
2. For medium levels of F (e.g. F2 in Figure 10), condition 2c in Proposition 3 is satisfied -i.e. type R indifference curve crosses ∆N(I, p0) = 0 line below the pricing line PF - and a separating equilibrium with advantageous selection as in Figure 8 is obtained.
3. For high levels of F (e.g. F3 in Figure 10), either a separating equilibrium as in Figure 9 or a pooling equilibrium as in Figure 5 is obtained.
4.2 Intensity of regret
We measure the intensity of regret by the convexity of the function g. From the slope of type R individuals’ indifference curve (see equation 6) we deduce that the more convex the g function is the steeper the indifference curves of type R individuals are. Furthermore, a higher convexity of the function g implies a lower level of optimal insurance coverage for type R individuals.2 Figure 11 illustrates the comparative statics with respect to the convexity of g (g3 is more convex than g2 which is more convex than g1 - thus R3< R2 < R1).
Case 3 Suppose that the cost of investing in self-protection, F , is high enough such that ∆R(I, c) <
0 for all I and c with pF ≤ c ≤ p0.
1. For highly convex functions g (e.g. g3 in Figure 11), condition 1b in Proposition 3 is satisfied - i.e. type R indifference curve crosses ∆N(I, p0) = 0 line above the pricing line PF - and a separating equilibrium with advantageous selection as in Figure 7 is obtained.
2. For medium levels of convexity of g (e.g. g2 in Figure 11), condition 2c in Proposition 3 is satisfied - i.e. type R indifference curve crosses ∆N(I, p0) = 0 line below the pricing line PF - and a separating equilibrium with advantageous selection as in Figure 8 is obtained.
3. For low levels of convexity of g (e.g. g1 in Figure 11), either a separating equilibrium as in Figure 9 or a pooling equilibrium as in Figure 5 is obtained.
4.3 Fraction of type R individuals in the population
In Rothschild and Stiglitz (1976), even the separating equilibrium does not exist if the fraction of high risk type individuals in the population is too low. The reason behind this non-existence result is that a pooling contract not only attracts high risk individuals but also low risk individuals as the pooling premium rate is just slightly above the fair premium rate for low risk individuals. The same
2Alternatively, Braun and Muermann (2004) propose a “regret coefficient” k in the utility function of type R individuals such that uR(W ) = u (W ) − kg (u (Wmax) − u (W )). They show that the higher the regret coefficient k the lower the optimal amount of insurance coverage under a fair premium.
reasoning applies to the separating equilibrium with adverse selection in Figure 9 if the fraction λ of type R individuals is too low. Then both types of individuals are be better off under a pooling contract. This pooling contract, however, does satisfy condition 2b in Proposition 1, which implies that it cannot be an equilibrium. Thus, as in Rothschild and Stiglitz (1976), there does not exist any equilibrium.
Interestingly, the same result does not hold under the conditions for the existence of a separating equilibrium with advantageous selection.
Lemma 1 The existence of the separating equilibrium with advantageous selection as in Figure 7 does not depend on the level of λ.
Proof. Note that all conditions in 1 of Proposition 3 are independent of λ. Furthermore, for any level of λ, no pooling contract attracts type N individuals (see Figure 7).
5 Conclusion
Economic models of moral hazard and adverse selection predict a positive correlation between the amount of insurance coverage individuals purchase and their claim frequency. The empirical evi-dence on this prediction is mixed. In some markets, e.g. in the annuity and health insurance market, such positive correlation is confirmed, whereas in other markets, e.g. in the life and long-term care insurance market, the opposite relation holds. In this paper, we propose heterogeneous, hidden de-grees of aversion towards anticipatory regret as a rationale for self-selection in insurance markets.
In our equilibrium analysis, we have shown that both pooling and separating equilibria can exist.
Furthermore, there exist separating equilibria of both types, advantageous and adverse selection.
We have characterized the conditions for each type of equilibrium and examined the comparative statics with respect to the model’s parameter. Understanding the reasons behind advantageous and adverse selection is highly relevant for the design of governmental policies aimed at reducing
populations as the markets for annuities, long-term care insurance, and Medigap insurance become increasingly important for them.
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*
EUN
D
B
45 0
PF
A WL
WNL
0
w L
w−
*
EUR
P0
P
=0 ΔN
Figure 1: No Pooling Equilibrium for I < ˆI: type N individuals invest in self-protection, whereas type R individuals do not. Indifference curve of type R individuals is flatter at B than that of type N individuals. Contract D attracts type N individuals but not type R individuals.
*
EU N
B D
45 0
P F
A W L
WNL
0
w L
w−
*
EU R
P 0
P
=0 ΔN
Figure 2: No Pooling Equilibrium for I = ˆI = ¯I: type N individuals invest in self-protection, whereas R type individuals do not. Indifference curve of type R and type N individuals have the same slope but are more convex at B . Contract D attracts type N individuals but not type R individuals.
*
EU N
B D
45 0
P F
A W L
WNL
0
w L
w−
*
EU R
P 0
P
=0 ΔN
Figure 3: No Pooling Equilibrium for I = ˆI < ¯I: type N individuals invest in self-protection, whereas type R individuals do not. Indifference curve of type R and type N individuals have the same slope but are more convex at B . Contract D attracts type N individuals but not type R individuals.
*
EU R
D B
45 0
P F
A W L
WNL
*
EU N
0
=0 ΔN
w L
w−
P 0
P
Figure 4: No Pooling Equilibrium at ˆI < I < ¯I: type N individuals invest in self-protection, whereas type R individuals do not. Indifference curve of type R individuals is steeper at B than that of type N individuals. Contract D attracts type N individuals but not type R individuals.
*
EU N
B
45 0
P F
A W L
WNL
0
w L
w−
*
EU R
P 0
P
=0 ΔN
X
Figure 5: Pooling Equilibrium ( ¯I, ¯p): type N individuals are indifferent between investing and not investing in self-protection under B, type R individuals do not invest in self-protection. Indifference curve of type R individuals is steeper at B than that of type N individuals and type R prefer contract B over any contract on P0. Note that contract D cannot be offered to attract type N individuals and induces them to not invest in self-protection and the company offering D would make losses.
*
EU N
N
45 0
P F
A W L
WNL
0
w L
w−
*
EU R
P 0
P
=0 ΔN
R
Figure 6: Separating Equilibrium: both types of individuals do not invest in self-protection and receive their respectively optimal amount of insurance coverage.
*
EU N
N
45 0
P F
A W L
WNL
0
w L
w−
*
EU R
P 0
P ΔN =0
R X
Figure 7: Separating Equilibrium with advantageous selection 1: type N individuals invest in self-protection, whereas type R individuals do not. Type N individuals obtain more insurance coverage than type R individuals.
*
EU N
N
45 0
P F
A W L
WNL
0
w L
w−
*
EU R
P 0
P ΔN =0
R X
' N
Figure 8: Separating Equilibrium with advantageous selection 2: type N individuals invest in self-protection, whereas type R individuals do not. Type N individuals obtain more insurance coverage than type R individuals.
*
EU N
N
45 0
P F
A W L
WNL
0
w L
w−
*
EU R
P 0
P ΔN =0
R X
' N
Figure 9: Separating Equilibrium with adverse selection: type N individuals invest in self-protection, whereas type R individuals do not. Type N individuals obtain less insurance coverage than type R individuals.
0 ) ( 1 = Δ FN
45 0
P F
A W L
WNL
) (F1 EUN
0
w L
w−
P 0
P
*
EU R
0 ) ( 2 = Δ FN
0 ) ( 3 = Δ FN
) (F2 EUN
) (F3 EUN
Figure 10: Comparative statics with respect to F - F1 < F2< F3.
R 3
) ( 2
* g EUR
) ( 3
* g
EUR )
( 1
* g EUR
45 0
P F
A W L
WNL
0
w L
w−
P 0
=0 ΔN P
R 2
R 1
Figure 11: Comparative statics with respect to intensity of regret - g3is more convex than g2 which is more convex than g1.