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Kreps–Porteus preferences?
Insurance: Mathematics and Economics 88 (2019) 1–6
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How do changes in risk and risk aversion affect self-protection with Selden/Kreps–Porteus preferences? ∗
Jianli Wang a , Hongxia Wang b , , Ho Yin Yick c a
College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China College of Economics and Management, Nanjing Forestry University, Nanjing 210037, China c Department of Finance and Insurance, Lingnan University, Hong Kong, China b
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Article history: Received October 2018 Received in revised form May 2019 Accepted 23 May 2019 Available online 27 May 2019 JEL classification: D81 D91 G11 D61
a b s t r a c t This paper examines the determinants of optimal effort in an intertemporal self-protection model. We separate attitude toward risk and attitude toward intertemporal substitution by adopting Selden/Kreps– Porteus preferences. We not only explore the sufficient conditions on risk preferences for guaranteeing the unambiguous effects of changes in risk on the optimal effort level but also show how a change in risk aversion alone affects the optimal effort level. © 2019 Elsevier B.V. All rights reserved.
Keywords: Self-protection Intertemporal substitution Risk aversion Stochastic dominance Expected-utility
1. Introduction Self-protection means that a decision-maker (DM) can improve her wealth distribution by exerting some effort to reduce the probability of loss. Ehrlich and Becker (1972) first study this issue in the one-period expected-utility framework. Related literature (see e.g., Briys and Schlesinger, 1990; Chiu, 2005; Eeckhoudt and Gollier, 2005; Dionne and Li, 2011; Eeckhoudt et al., 2017) further explores how risk preferences, for example, risk aversion and prudence (see Kimball, 1990), determine the optimal effort level. In these works, the stochastic variable of the DM’s wealth takes only two possible outcomes, namely, non-random loss or no loss. Recently, Chuang et al. (2013), Crainich et al. (2016) and Wong (2017) consider that the two possible loss outcomes are represented by ‘‘better’’ risk and ‘‘worse’’ risk.1 Specifically, Crainich et al. (2016) study the effects of stochastic changes in better and worse risk on the optimal effort. They show some findings inconsistent with intuition by adopting the one-period self-protection ∗ Corresponding author. E-mail address: [email protected] (H. Wang). 1 Chuang et al. (2013) and Wong (2017) analyze optimal effort of one DM with risky targets. https://doi.org/10.1016/j.insmatheco.2019.05.004 0167-6687/© 2019 Elsevier B.V. All rights reserved.
model. For example, intuitively, for the risk-averse DM, the motive of exerting effort to increase the probability of the better risk is strengthened if the worse risk has a mean-preserving spread (see Rothschild and Stiglitz, 1970), whereas they find that the ambiguous effect of mean-preserving increases in the worse risk on effort level. The purpose of this paper is to identify whether the results in Crainich et al. (2016) hold or not when self-protection is a two-period activity, and how changes in risk aversion affect optimal effort. Since Menegatti (2009) proposes an intertemporal model where one DM can improve her future loss risk via spending current effort, the two-period self-protection model is receiving increasing attention.2 However, the two-period selfprotection model in previous works does not separate aversion to risk and aversion to intertemporal substitution.3 This means that the intertemporal substitution elasticity almost varies together with the risk aversion strength, so we cannot show the roles of intertemporal substitution elasticity and changes in risk aversion on the optimal effort level independently. That is, we cannot 2 See, e.g. Courbage and Rey (2012), Eeckhoudt et al. (2012), Hofmann and Peter (2016), Peter (2017) and Wang et al. (2018) etc. 3 The previous model on self-protection includes the same deficiency of the classical precautionary saving model. See, e.g., Kimball and Weil (2009) and Wang and Li (2016).
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completely clarify the motives of self-protection by using the existing two-period self-protection model. In addition, the literature on higher-order risk aversion (risk aversion, risk prudence, etc.)4 contains two kinds of characterizations of comparative risk aversion: the Arrow–Pratt risk aversion measure (see Pratt, 1964; Arrow, 1974) and the Ross risk aversion measure (see Ross, 1981). For example, Jindapon and Neilson (2007) provide a unified framework to reveal both kinds of characterization in a comparative statics problem. Then, the kind of measure for risk aversion strength that can be used to derive the effect of changes in risk aversion on the two-period self-protection activity must be determined. In order to systematically explore the determinants of the optimal self-protection level, we use the Selden/Kreps–Porteus preferences (see Kreps and Porteus, 1978; Selden, 1978, 1979) to construct an intertemporal self-protection model and to separate aversion to risk and aversion to intertemporal substitution. Numerous works explore various economic and financial problems under the Selden/Kreps–Porteus preferences. For example, Kimball and Weil (2009) and Wang and Li (2016) explore the effects of time preference and risk preference on precautionary saving with Selden/Kreps–Porteus preferences. Wang and Li (2015) etc show that both precautionary saving and two-period optimal self-protection with background risks can be neatly related to risk prudence. Then, one may want to know whether the two-period self-protection problem has the same traits as precautionary saving when considering the effects of changes in risk and risk aversion under Selden/Kreps–Porteus preferences. These studies directly motivate us to adopt the Selden/Kreps– Porteus preferences in the optimal decision-making model of self-protection. We use the stochastic dominance criterion to measure the changes in risk. Given the elasticity of intertemporal substitution, we find that higher-order risk aversion determines the direction of change in the optimal current effort level when the better(worse) risk of the second period has a stochastic dominance change; we also show that greater Ross risk aversion can induce more optimal effort when the better risk dominates the worse risk in the sense of nth-order stochastic dominance (n > 1). Furthermore, when the future risk is a Bernoulli distribution, i.e., a non-random loss or no loss, we use the coefficients of both fear of loss and happiness of gain to discuss the effect of risk aversion on optimal effort. Meanwhile, to clarify our conclusions, we provide comparisons to previous works (e.g., Jindapon and Neilson, 2007; Chuang et al., 2013; Crainich et al., 2016). The rest of this paper is organized as follows. Section 2 proposes an intertemporal self-protection model with a Selden/Kreps– Porteus ordinal certainty equivalence representation. Section 3 analyzes how the changes in future risk affect the optimal current effort. Section 4 explores the optimal current effort level under different risk aversion. Section 5 concludes this work. 2. A self-protection model with a Selden/Kreps–Porteus ordinal certainty equivalence representation Throughout this paper, the improvements (deteriorations) in risk are based on the theory of stochastic dominance. Let F (x) and G(x) denote the distribution functions of the random variables x˜ and y˜ with Denote F 1 (x) = F (x), ∫ x support on [a, b], respectively. ∫x F 2 (x) = a F (y)dy and F k (x) = a F k−1 (y)dy, k = 2, 3, . . .. Similar denotations apply to G(x). The well-known nth-order stochastic dominance (nSD) is defined as follows. 4 Higher-order risk aversion is equivalent to signing the higher-order derivative of the utility function in the expected-utility framework, see, e.g., Eeckhoudt and Schlesinger (2006). Menegatti (2014, 2015) further shows the relationships between higher-order risk aversion attitudes (e.g., risk aversion, prudence) by considering the higher-order derivatives of the utility function with standard regularity assumptions.
Definition. y˜ dominates x˜ via nSD if (i) F k (b) ≥ Gk (b) for all k = 1 · · · n, (ii) F n (x) ≥ Gn (x) for all x ∈ [a, b]. Within the expected-utility framework, assume utility function u is continuously differentiable; nSD can be linked to signing the higher-order derivatives of u. Theorem 2.1. y˜ dominates x˜ via nSD if and only if Eu(y˜ ) ≥ Eu(x˜ ) for ∀ u such that (−1)k+1 u(k) ≥ 0, k = 1, . . . , n.5 When y˜ dominates x˜ via nSD, we call y˜ an nSD improvement of x˜ and x˜ an nSD deterioration of y˜ . The nth-order risk increase (Ekern, 1980) is a special case of nSD that requires the equation to hold in condition (i) of the Definition and implies that the first (n − 1)th moments are held constant. For example, the secondorder stochastic dominance (SSD) deterioration with constant mean is a mean-preserving spread in risk. We can refer to Levy (2006) etc. for more properties of nSD. Consider one DM who will receive either the wealth x˜ with a probability p(e) or y˜ with a probability 1 − p(e) in the second period, where e is the current effort level that determines the magnitude of probability p(e). Given y˜ dominates x˜ via nSD, following the terminology of Crainich et al. (2016), we call y˜ the ‘‘better’’ risk and x˜ the ‘‘worse’’ risk. Assume p′ (e) ≤ 0, which means the DM can reduce the probability p(e) of x˜ by increasing effort level e. In this self-protection setting, we assume that the DM’s preferences are additively time separable and characterized by the Selden/Kreps–Porteus ordinal certainty equivalence representation.6 Then, the DM’s optimal self-protection problem is max φ (w0 − e) + U(u−1 (p(e)Eu(x˜ ) + (1 − p(e))Eu(y˜ ))), e
(1)
where w0 is a sure wealth income in the first period. φ (·) and U(·) are the first-period and second-period utility functions, respectively (φ ′ > 0, U ′ > 0). U(·) is defined on the certainty equivalent of the second-period wealth, u−1 (p(e)Eu(x˜ ) + (1 − p(e))Eu(y˜ )), which is computed by the atemporal utility function u(·), u′ > 0. The concavity of u(·) captures the DM’s aversion to risk. U(·) represents attitudes toward intertemporal substitution which is independent of attitude toward risk. That is, model (1) allows risk preferences and intertemporal substitution to vary independently. When U(·) = u(·), model (1) becomes a traditional two-period optimal self-protection model within the expectedutility framework (Menegatti, 2009; Courbage and Rey, 2012; Eeckhoudt et al., 2012; Wang and Li, 2015 etc.). Thus, our model is more general than the previous model, which results in greater complexity for the mathematical analysis of the self-protection problem. It is especially easy to find that the concavity of model (1) in e depends not only on the properties of p(e) but also the concavity of both U(·) and u(·). To achieve our goal, we use the technique of the single crossing difference to provide comparative statics analysis.7 3. Risk changes and optimal effort Assume the worse risk has a change from x˜ 0 to x˜ 1 . For convenience, we call x˜ 0 the initial worse risk. Let W (e, x˜ i ) = φ (w0 − e) + U(u−1 (p(e)Eu(x˜ i )+(1−p(e))Eu(y˜ ))) and denote ei ∈ argmaxW (e, x˜ i ), i = 0, 1. Given y˜ dominates the worse risk x˜0 via nSD, when there is a stochastic change in the worse risk from x˜ 0 to x˜ 1 , we obtain the following conclusion. 5 u(k) denotes the kth derivative of the utility function u; E is the expectation operator. 6 Kreps and Porteus (1978) and Selden (1978, 1979) provide the axiomatic foundations of the Selden ordinal certainty equivalence representation. 7 See Appendix. For more applications of single crossing difference, also see, e.g., Wang and Li (2015) and Nocetti (2016).
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Proposition 3.1. The mSD improvement (deterioration) in the worse risk from x˜ 0 to x˜ 1 decreases (increases) the optimal effort, ′′ ′′ i.e., e1 ≥ e0 (e1 ≤ e0 ) if − UU ′ ≥ − uu′ and (−1)k+1 u(k) > 0, k =
1, . . . , max{m, n}. Specifically, when Eu(x˜1 ) − Eu(y˜ ) ≥ 0, that is, the initial worse risk becomes better than the better risk y˜ under the expected-utility maximization framework, (−1)k+1 u(k) (·) > 0, k = 1, . . . , max{m, n} alone implies that the mSD improvement in the worse risk reduces the optimal effort, i.e. e1 ≤ e0 . Proof. See Appendix.
Proposition 3.1 is consistent with our intuition since, intuitively, the motive of exerting effort to increase the probability of the better risk is strengthened if the initial worse risk becomes worse but is weakened if the worse risk has a stochastic improvement. ′′ ′′ The condition − UU ′ ≥ − uu′ indicates that the resistance to intertemporal substitution is larger than the Arrow–Pratt risk aversion strength. Previous studies, for example, Kimball and Weil (2009) and Wang and Li (2016), also use the condition ′′
′′
− UU ′ ≥ − uu′ to examine the determinants of the precautionary y1−ρ saving motive. If we take the functional forms,8 U(y) = 1−ρ , ρ > 0, ̸= 1 and u(x) = u′′ (x)
x1−γ 1−γ
′′
(x) , γ > 0, ̸= 1, we have − UU ′ (x) = ρ x−1
and − u′ (x) = ηx−1 , then ρ ≥ γ guarantees this condition when the wealth level x > 0. ′′ In terms of model, the condition − UU ′ to the comparison between the ratios
u′′ is related u′ ′ − 1 U (u (p(e)Eu(x˜ i )+(1−p(e))Eu(y˜ ))) u′ (u−1 (p(e)Eu(x˜ )+(1−p(e))Eu(y˜ )))
≥ −
i
evaluated in x˜ 0 and in x˜ 1 (see Eq. (8), Appendix 7.1). These two ratios can also be written as the products of the marginal utility of U and of the derivative of the certainty equivalent, since 1 = (u−1 )′ (p(e)Eu(x˜ i ) + (1 − p(e))Eu(y˜ )). u′ (u−1 (p(e)Eu(x˜ )+(1−p(e))Eu(y˜ ))) i
Therefore, the ratios in Eq. (8) measure the overall effects on utility of the marginal benefits of the effort, i.e. the change in certainty equivalent weighted by the marginal utility of U. Since these changes in utility are evaluated in two different situations U ′ (x) (in x˜ 0 and in x˜ 1 ), the condition of Eq. (8) corresponds to u′ (x) ′′
′′
decreasing in x which is equivalent to − UU ′ ≥ − uu′ . The condition (−1)k+1 u(k) (·) > 0 means the DM is kth-order risk aversion (Ekern, 1980). Eeckhoudt and Schlesinger (2006) clarify the meaning of (−1)k+1 u(k) (·) > 0 by a simple lottery-pairs preference and further stimulate new research on higher-order risk aversion. Similarly, assume that the initial better risk y˜ 0 dominates the worse risk x˜ via nSD, when the better risk undergoes a change from the initial better risk y˜ 0 to another risk y˜ 1 , we obtain the following conclusion by the same approach to Proposition 3.1. Proposition 3.2. The mSD improvement (deterioration) in the better risk from y˜ 0 to y˜ 1 increases (decreases) the optimal effort ′′ ′′ if − UU ′ ≤ − uu′ and (−1)k+1 u(k) > 0, k = 1, . . . , max{m, n}. Specifically, when Eu(x˜ ) − Eu(y˜ 1 ) ≥ 0, that is, the initial better risk becomes worse than the worse risk x˜ under the expected-utility maximization framework, (−1)k+1 u(k) > 0, k = 1, . . . , max{m, n} alone implies that the mSD improvement in the better risk increases the optimal effort. Our conclusions have some differences from those of Chuang et al. (2013) and Crainich et al. (2016). For example, Proposition 9 in Crainich et al. (2016) shows if the risk preference of one DM is (−1)k+1 u(k) ≥ 0, k = 1, . . . , n, then the effect on effort 8 See, e.g., Kimball and Weil (2009).
3
is ambiguous when there is an nth-order decrease(increase) in the worse risk.9 This result is inconsistent with our conclusions since we observe an unambiguous effect of changes in risk on effort. This difference is mainly due to the fact that in our intertemporal self-protection model, effort is spent in the first period but the stochastic change in loss risk occurs in the second period. Interestingly, when U(·) = u(·), our conclusions need only the condition of higher-order risk aversion, i.e., (−1)k+1 u(k) ≥ 0, k = 1, . . . , n. Thus, Propositions 3.1 and 3.2 are analogous to Proposition 1 in Eeckhoudt and Schlesinger (2008), which clarifies the effect of labor–income risk on precautionary saving by signing the higher derivatives of the utility function. That is, higher-order risk aversion can guarantee the unambiguous effects of changes in risk on both optimal saving and effort. Specifically, let x˜ = w1 − L, L > 0, y˜ = w1 , and p(e) = p0 − ϵ (e), ϵ ′ (e) > 0, where p(0) is the initial probability of loss. We can consider the classic loss risk with a Bernoulli distribution. The worse risk x˜ has a first-order stochastic dominance (FSD) deterioration if the magnitude of loss L or the initial probability p(0) of loss increases. Thus, let m = n = 1 in Proposition 3.1, we find that the optimal effort level increases as L or p(0) increases as ′′ ′′ long as − UU ′ ≥ − uu′ holds for all DMs with monotone increasing utility functions, regardless of whether the DMs are risk averse or risk loving. We re-explore the example of Alzheimer’s disease in Crainich et al. (2016). Following their work, consider one DM is unable to change the probability t of Alzheimer’s disease but can improve the severity of the disease by spending current intellectual effort e. Let L(early onset) and M(late onset) denote possible values of the severity of the disease; then, in our intertemporal framework, the DM faces two Bernoulli-distributed random variables: x˜ = (w1 − L, t ; w1 , 1 − t) with probability p(e) and y˜ = (w1 − M , t ; w1 , 1 − t) with probability 1 − p(e), where w1 is the initial level of quality of life in the future. In this simplified model, a change in L or M is equivalent to a change in FSD in the worse(better) risk. As a consequence, our conclusions reveal that the effect of a change in L or M on the optimal intellectual effort is unambiguous for all DM with u′ > 0. This is in contrast to the findings of Crainich et al. (2016), who show that the effect of a change in L or M on effort is ambiguous if the DM is a risk averter(risk lover).10 Overall, our conclusions confirm the intuition of Crainich et al. (2016), who demonstrate “At the beginning of the paper we argued that a plausible conjecture on DM choices may be that an FSD improvement in the better risk (enlarging the ‘gap’ between the two risks) pushes the DM to increase effort and · · ·”11 but are different from their results and those of other previous works. 4. Different degrees of risk aversion and optimal effort This section discusses the effect of a change in risk aversion on the optimal effort. We start with the measures of risk aversion strength. v is more Arrow–Pratt risk averse than u if
(x) (x) − vv′ (x) ≥ − uu′ (x) for all x ∈ [a, b] (see Pratt, 1964; Arrow, 1974). ′′
′′
The literature finds that the Arrow–Pratt characterization of risk aversion strength is often not sufficiently strong for comparing preference orders in risky choice situations. Thus, Ross (1981) v ′′ (x)
u′′ (x)
first uses − v ′ (y) ≥ − u′ (y) for all x, y ∈ [a, b] to state that ‘‘v is more risk averse than u’’. It is evident that the Arrow–Pratt characterization of risk aversion strength is a special example 9 Proposition 2 in Chuang et al. (2013) also produces a similar finding. 10 See Propositions 1, 2, and 3 in Crainich et al. (2016). 11 See the paragraph after Corollary1 in Crainich et al. (2016).
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of Ross’s (1981) characterization of risk aversion strength when x = y. Ross’s (1981) characterization of risk aversion strength is generalized to the case of kth-order(k ≥ 2) risk aversion by Jindapon and Neilson (2007), Li (2009) and Liu and Meyer (2013) etc., which has numerous exploration and application, regardless of its stronger limitation than that of the Arrow–Pratt index. Definition. Given two DMs with utility functions u and v , v is v (k) (x) more kth-order(k ≥ 2) Ross risk averse than is u if (−1)k−1 v ′ (y) ≥ (−1)k−1
u(k) (x) u′ (y)
for all x, y ∈ [a, b].
Consider two DMs with different atemporal utility functions u, v but the same second-period utility function U. Let W (e, u) = φ (w0 − e) + U(u−1 (p(e)Eu(x˜ ) + (1 − p(e))Eu(y˜ ))),
(2)
and W (e, v ) = φ (w0 − e) + U(v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ ))).
(3)
Denote ei ∈ argmaxW (e, i), i = u, v . We explore the sufficient conditions for the unambiguous effect of changes in risk aversion on optimal effort. We obtain the following conclusion. Proposition 4.1. Assume the DMs u, v share the same secondperiod utility function U, when the better risk y˜ dominates the worse risk x˜ via nSD, n ≥ 2, for both DMs with (−1)k+1 u(k) > 0 and ′′ ′′ (−1)k+1 v (k) > 0, k = 1, . . . , n, if − UU ′ ≥ − uu′ and v is more kth-order (k = 2, . . . , n) Ross risk averse than u, then v exerts more effort than u. Proof. See Appendix. Proposition 4.1 indicates that the Ross characterization of risk aversion strength can be used to perform the comparative statics analysis in an intertemporal self-protection model, which means the DM conducts self-protection activity by paying a utility cost of effort. In terms of proof, our sufficient conditions are gotten by ′′
′′
both Eq. (12) and Eq. (13) which are guaranteed by − UU ′ ≥ − uu′ and v being more kth-order (k = 2, . . . , n) Ross risk averse than ′′ ′′ u respectively. Thus, the condition − UU ′ ≥ − uu′ is required, but
≥ − vv′ is not involved. In the setting of ′′ ′′ Proposition 4.1, the similar proof also means that if − UU ′ ≤ − uu′ and v is less kth-order (k = 2, . . . , n) Ross risk averse than u, ′′ ′′ then v exerts less effort than u. That is, holding − UU ′ ≥ − uu′ or ′′ ′′ − UU ′ ≤ − uu′ fixed, our conclusions are determined by changes in ′′ ′′ risk aversion no matter − UU ′ is bigger than − vv ′ or not. ′′
the condition − UU ′
′′
Jindapon and Neilson (2007) show that the Ross characterization of risk aversion strength governs the change in optimal static decision-making behavior when the effort requires monetary cost. They also show more Arrow–Pratt risk averse individual makes more effort after separating the cost and the benefit of the efforts. It seems that Jindapon and Neilson (2007) need less restrictive conditions. This is because that the solution of problem with a utility cost of effort in Jindapon and Neilson (2007) is assumed to be invariant when the utility function u has a positive linear transformation,12 but it is not invariant with positive linear transformations of the utility function (see Liu and Wang, 2017). It is clear that if the aversion to risk and aversion to intertemporal substitution are not separated in this problem, we cannot change risk preferences and intertemporal substitution independently from each other. In our model, we also can hold risk preferences fixed (i.e. u = v ) but assume different secondperiod utility functions. Then, under this circumstance, simple 12 See Eq. (3) in Jindapon and Neilson (2007) and the proof of their Theorem 3.
calculations mean that the margin utility in the second-period determines whether more effort is desirable. Thus, our model with Selden/Kreps–Porteus Preferences is helpful to clarify the different motives of exerting more effort. We return to the classic loss risk: x˜ = w1 − L, L > 0 and y˜ = w1 . Eeckhoudt et al. (2017) propose the concepts of linearly(quadratically)-restricted greater Ross risk aversion,13 to provide a clear-cut comparative statics analysis for the oneperiod self-protection. However, in our setting, we cannot use the Ross characterization of risk aversion strength to provide sufficient conditions for more effort by Proposition 4.1 since this classic loss risk type means that y˜ dominates x˜ via FSD, whereas Proposition 4.1 considers only the case of nSD with the condition n ≥ 2. In terms of proofs, the cause is that the Ross characterization of risk aversion strength cannot be used to address the sufficient condition for inequality (13) in Appendix. Let N = v −1 (p(e)v (w1 − L) + (1 − p(e))v (w1 )); clearly, w1 > N > w1 − L. To show that ev ≥ eu , we need to find the sufficient condition for the inequality (13), which can be rewritten as u(w1 ) − u(w1 − L) v (w1 ) − v (w1 − L) > v ′ (N) u′ (N) v (w1 ) − v (N) v (N) − v (w1 − L) u(w1 ) − u(N) ⇔ + > v ′ (N) v ′ (N) u′ (N) u(N) − u(w1 − L) + ′ u (N)
v (N) − v (w1 − L) u(N) − u(w1 − L) u(w1 ) − u(N) ⇔ − > v ′ (N) u′ (N) u′ (N) v (w1 ) − v (N) − . (4) v ′ (N) Li (2010) shows that if v is more Arrow–Pratt risk averse than u, then v has more fear of loss than does u at the wealth level N v (N)−v (w1 −L) w1 −L) (i.e., > u(N)−u′u((N) ), whereas u has more happiness v ′ (N) u(w1 )−u(N) of gain than does v at the wealth level N (i.e., > u′ (N) v (w1 )−v (N) ). v ′ (N)
This result means that both the right and left sides
of inequality (4) are positive. Thus, we cannot obtain the sufficient condition by the comparison of only the Arrow–Pratt risk aversion strength. Actually, the left side of inequality (4) differs between u’s and v ’s coefficients of fear of loss at wealth level N, which reflects the positive effect of greater Arrow–Pratt risk aversion on the coefficient of fear of loss. However, the right side of inequality (4) reflects its negative effect on the coefficient of happiness of gain. Thus, by inequality (4), we know that when ′′
′′
− UU ′ ((··)) ≥ − uu′ ((··)) , if v is more Arrow–Pratt risk averse than u and the difference in fear of loss at wealth level N between both DMs is larger than the difference in happiness of gain, then v exerts more effort than u. 5. Conclusion This work models the intertemporal self-protection problem with Selden/Kreps–Porteus preferences. We explore the effects of changes in risk and risk aversion on optimal effort. Holding the elasticity of intertemporal substitution constant, we find that higher-order risk aversion can determine the change in effort when the risk improves or deteriorates in the sense of stochastic dominance. We also find that greater Ross risk aversion can result in higher optimal effort. Our conclusions can easily be applied to analyze the case of an nth-order risk increase in a straightforward manner. 13 Both concepts are special cases of Definition 4 i.e., they are stricter than the general Ross characterization of risk aversion strength in Ross (1981), Jindapon and Neilson (2007), Li (2009) and Liu and Meyer (2013) etc.
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Acknowledgments The authors would like to thank the editor and anonymous referees for providing valuable suggestions which have led to significant improvement of this article. This work is supported by the National Natural Science Foundation of China under Grant Number 71401074, MOE (Ministry of Education in China) Project of Humanities and Social Sciences under Grant Number 19YJC790125, and by the Fundamental Research Funds for the Central Universities under Research Project No. NS2018049. Appendix
A.1. Proof of Proposition 3.1 Proof. We first consider the mSD deterioration from x0 to x1 , i.e., x˜ 0 dominates x˜ 1 via mSD. In order to show that e0 ≤ e1 , from Theorem A.1, we know that we need to prove, for all e′′ > e′ , W (e′′ , x˜ 0 ) − W (e′ , x˜ 0 ) ≥ 0 ⇒ W (e′′ , x˜ 1 ) − W (e′ , x˜ 1 ) > 0.
(5)
This can be obtained by W (e′′ , x˜ 1 ) − W (e′ , x˜ 1 ) > W (e′′ , x˜ 0 ) − W (e′ , x˜ 0 )
⇔ φ (w0 − e′′ ) + U(u−1 (p(e′′ )Eu(x˜ 1 ) + (1 − p(e′′ ))Eu(y˜ ))) − φ (w0 − e′ ) − U(u−1 (p(e′ )Eu(x˜ 1 ) + (1 − p(e′ ))Eu(y˜ ))) > φ (w0 − e′′ ) + U(u−1 (p(e′′ )Eu(x˜ 0 ) + (1 − p(e′′ ))Eu(y˜ ))) − φ (w0 − e′ ) − U(u−1 (p(e′ )Eu(x˜ 0 ) + (1 − p(e′ ))Eu(y˜ ))).
(6)
U(u−1 (p(e′′ )Eu(x˜ 1 ) + (1 − p(e′′ ))Eu(y˜ ))) − U(u−1 (p(e′ )Eu(x˜ 1 ) ′
+(1 − p(e ))Eu(y˜ ))) > U(u−1 (p(e′′ )Eu(x˜ 0 ) + (1 − p(e′′ ))Eu(y˜ ))) − U(u−1 (p(e′ )Eu(x˜ 0 ) +(1 − p(e′ ))Eu(y˜ ))). (7) Because e′′ > e′ , we know a sufficient condition for (7) is U (u
(p(e)Eu(x˜ 1 ) + (1 − p(e))Eu(y˜ )))
u′ (u−1 (p(e)Eu(x˜ 1 ) + (1 − p(e))Eu(y˜ )))
>
U ′ (u−1 (p(e)Eu(x˜ 0 ) + (1 − p(e))Eu(y˜ ))) u′ (u−1 (p(e)Eu(x˜ 0 ) + (1 − p(e))Eu(y˜ )))
p′ (e)(Eu(x˜ 1 ) − Eu(y˜ )) p′ (e)(Eu(x˜ 0 ) − Eu(y˜ )). (8)
Note that (−1)k+1 u(k) > 0, k = 1, . . . , max{m, n} means both u−1 (p(e)u(x˜ 0 ) + (1 − p(e))u(y˜ )) > u−1 (p(e)u(x˜ 1 ) + (1 − p(e))u(y˜ )) and Eu(x˜ 1 ) − Eu(y˜ ) ≤ Eu(x˜ 0 ) − Eu(y˜ ) ≤ 0. Then, (8) can be guaranteed by the fact that
U ′ (x) u′ (x)
U ′′ U′ k+1 (k)
is decreasing in x (i.e., −
we show that both −
U (u
−1
U ′′ U′
≥ −
u′′ u′
and (−1)
(p(e)Eu(x˜ 0 ) + (1 − p(e))Eu(y˜ )))
u′ (u−1 (p(e)Eu(x˜ 0 ) + (1 − p(e))Eu(y˜ )))
>
′′
′′
′′
0, k = 1, . . . , max{m, n} imply e0 ≥ e1 . When Eu(x˜ 1 ) − Eu(y˜ ) ≤ 0 (i.e., the ‘worse’ risk becomes better than the ‘better’ risk), (9) clearly holds since Eu(x˜ 0 ) − Eu(y˜ ) ≤ 0 and p′ (e) ≤ 0; thus, (−1)k+1 u(k) > 0, k = 1, . . . , max{m, n}, implies e0 ≥ e1 . A.2. Proof of Proposition 4.1
U ′ (u−1 (p(e)Eu(x˜ 1 ) + (1 − p(e))Eu(y˜ ))) u′ (u−1 (p(e)Eu(x˜ 1 ) + (1 − p(e))Eu(y˜ )))
W (e′′ , u) − W (e′ , u) ≥ (>)0 ⇒ W (e′′ , v ) − W (e′ , v ) ≥ (>)0. (10) Following proofs of Proposition 3.1, we know that (10) can be obtained by U ′ (v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ )))
p′ (e)(E v (x˜ ) − E v (y˜ )) v ′ (v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ ))) U ′ (u−1 (p(e)Eu(x˜ ) + (1 − p(e))Eu(y˜ ))) ′ ≥ ′ −1 p (e)(Eu(x˜ ) − Eu(y˜ )) u (u (p(e)Eu(x˜ ) + (1 − p(e))Eu(y˜ ))) U ′ (v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ ))) ⇔ ′ −1 u (v (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ ))) u′ (v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ ))) ′ × ′ −1 p (e)(E v (x˜ ) − E v (y˜ )) v (v (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ ))) U ′ (u−1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ ))) > ′ −1 u (u (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ ))) × p′ (e)(Eu(x˜ ) − Eu(y˜ )). (11) U ′ (v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ ))) u′ (v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ )))
≥
U ′ (u−1 (p(e)u(w1 − L) + (1 − p(e))u(w1 ))) u′ (u−1 (p(e)u(w1 − L) + (1 − p(e))u(w1 )))
u′ (v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ )))
v ′ (v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ ))) > p′ (e)(Eu(x˜ ) − Eu(y˜ )).
U ′′ (·)
u′′ (·)
(12) since − U ′ (·) ≥ − u′ (·) means that
(13)
U ′ (x) u′ (x)
is decreasing in x.
Let N = v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ )); then, inequality (13) can be rearranged as E v (y˜ ) − E v (x˜ )
v ′ (v −1 (p(e)E v (x˜ ) > ⇔
+ (1 − p(e))E v (y˜ ))) Eu(y˜ ) − Eu(x˜ )
u′ (v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ )))
n−1 ∑ (−1)i v (i) (b)
v ′ (N)
i=1
b
∫
(−1)n
+
′
p (e)(Eu(x˜ 0 ) − Eu(y˜ ))
Note that (−1)k+1 u(k) > 0, k = 1, . . . , max{m, n} means both u−1 (p(e)u(x˜ 0 ) + (1 − p(e))u(y˜ )) < u−1 (p(e)u(x˜ 1 ) + (1 − p(e))u(y˜ )) and
p′ (e)(E v (x˜ ) − E v (y˜ ))
If v is more kth-order (k = 2, . . . , n) Ross risk averse than u, then v is more Arrow–Pratt risk averse than u; thus, v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ )) < u−1 (p(e)Eu(x˜ ) + (1 − p(e))Eu(y˜ )). Then, we have
≥− Thus, u > 0, k =
p′ (e)(Eu(x˜ 1 ) − Eu(y˜ )). (9)
(12)
and
u′′ ). u′
1, . . . , max{m, n} imply that (5) holds. Similarly, when x˜ 0 is dominated by x˜ 1 via mSD, in order to show that e0 ≥ e1 , we need to find sufficient conditions for the inequality ′
′′
(i.e., − UU ′ ≥ − uu′ ). Thus, both − UU ′ ≥ − uu′ and (−1)k+1 u(k) >
This can be guaranteed by both
This is equivalent to
−1
Eu(x˜ 1 ) − Eu(y˜ ) ≥ Eu(x˜ 0 ) − Eu(y˜ ); then, when Eu(x˜ 1 ) − Eu(y˜ ) ≤ 0, U ′ (x) (9) can be guaranteed by the fact that u′ (x) is decreasing in x
Proof. To show that ev ≥ eu , from Theorem A.1, we know that we need to prove that for all e′′ > e′ ,
Theorem A.1 (e.g. Milgrom, 2004, Theorem 4.1). Given sets X and S, {f (·; s)}s∈S obeys strict single crossing difference, i.e., f (x′′ , s′ ) − f (x′ , s′ ) ≥ 0 ⇒ f (x′′ , s′′ ) − f (x′ .s′′ ) > 0 for all x′′ > x′ and s′′ > s′ , if and only if arg maxx∈X f (x; s) is increasing in s, i.e., inf arg maxx∈X f (x; s′′ ) ≥ sup arg maxx∈X f (x; s′ ) for all s′′ > s′ .
′
5
a
>
(Gi+1 (b) − F i+1 (b))
v (n) (x) n (G (x) − F n (x))dx v ′ (N)
n−1 ∑ (−1)i u(i) (b)
u′ (N)
i=1 b
∫
(−1)n
+ a
(Gi+1 (b) − F i+1 (b))
u(n) (x) u′ (N)
(Gn (x) − F n (x))dx
6
⇔
J. Wang, H. Wang and H.Y. Yick / Insurance: Mathematics and Economics 88 (2019) 1–6 n−1 ∑ (−1)i+1 u(i) (b) i+1 (−1)i+1 v (i) (b) − )(F (b) − Gi+1 (b)) ( v ′ (N) u′ (N) i=1 ∫ b v (n) (x) u(n) (x) + ((−1)n+1 ′ − (−1)n+1 ′ ) v (N) u (N) a × (F n (x) − Gn (x))dx > 0.
(14)
The first equivalence is obtained by repeated integration by parts. Moreover, (14) holds since v is more kth-order (k = 2, . . . , n) Ross risk averse than u and G(x) dominates F (x) via nSD. Thus, we complete the proof. References Arrow, K.J., 1974. Essays in the Theory of Risk-Baring. Markham, Chicago, IL. Briys, E., Schlesinger, H., 1990. Risk aversion and the propensities for self-insurance and self-protection?. Southern Econ. J. 57 (2), 458–467. Chiu, W., 2005. Degree of downside risk aversion and self-protection?. Insurance Math. Econom. 36 (1), 93–101. Chuang, O., Eeckhoudt, L., Huang, R., Tzeng, L., 2013. Risky targets and effort. Insurance Math. Econom. 52, 465–468. Courbage, C., Rey, B., 2012. Optimal prevention and other risks in a two-period model. Math. Social Sci. 63, 213–217. Crainich, D., Eeckhoudt, L., Menegatti, M., 2016. Changing risks and optimal effort. J. Econom. Behav. Organiz. 125, 97–106. Dionne, G., Li, J., 2011. The impact of prudence on optimal prevention revisited?. Econom. Lett. 113 (2), 147–149. Eeckhoudt, L., Gollier, C., 2005. The impact of prudence on optimal prevention. Econom. Theory 26, 989–994. Eeckhoudt, L., Huang, R.J., Tzeng, L.Y., 2012. Precautionary effort: a new look. J. Risk Insurance 79, 585–590. Eeckhoudt, L., Liu, L., Meyer, J., 2017. Restricted increases in risk aversion and their application. Econom. Theory 64 (1), 161–181. Eeckhoudt, L., Schlesinger, H., 2006. Putting risks in its proper place. Amer. Econ. Rev. 96, 280–289. Eeckhoudt, L., Schlesinger, H., 2008. Changes in risk and the demand for saving. J. Monetary Econ. 55, 1329–1336. Ehrlich, I., Becker, G., 1972. Market insurance, self insurance and self protection. J. Polit. Econ. 80, 623–648. Ekern, S., 1980. Increasing nth degree risk. Econ. Lett. 6, 329–333. Hofmann, A., Peter, R., 2016. Self-insurance, self-protection and saving on consumption smoothing and risk management. J. Risk Insurance 83 (3), 719–734. Jindapon, P., Neilson, W., 2007. Higher-order generalizations of arrow–pratt and ross risk aversion: A comparative statics approach. J. Econom. Theory 136, 719–728.
Kimball, M., 1990. Precautionary savings in the small and in the large. Econometrica 58, 53–73. Kimball, M.S., Weil, P., 2009. Precautionary saving and consumption smoothing across time and possibilities. J. Money Credit Bank. 41, 245–284. Kreps, D.M., Porteus, E.L., 1978. Temporal resolution of uncertainty and dynamic choice theory. Econometrica 46, 185–200. Levy, H., 2006. Stochastic Dominance: Investment Decision Making under Uncertainty. Springer. Malkiel, B. G., New York. Li, J., 2009. Comparative higher-degree ross risk aversion. Insurance Math. Econom. 45, 333–336. Li, J., 2010. Fear of loss and happiness of win: properties and applications. J. Risk Insurance 77 (4), 749–766. Liu, L., Meyer, J., 2013. Substituting one risk increase for another: A method for measuring risk aversion. J. Econom. Theory 148, 2706–2718. Liu, L., Wang, J., 2017. A note on the comparative statics approach to nth-degree risk aversion. Econom. Lett. 159, 116–118. Menegatti, M., 2009. Optimal prevention and prudence in a two-period model. Math. Social Sci. 58, 393–397. Menegatti, M., 2014. New results on the relationship between risk aversion, prudence and temperance. European J. Oper. Res. 232 (3), 613–617. Menegatti, M., 2015. New results on high-order risk changes. European J. Oper. Res. 243, 678–681. Milgrom, P., 2004. Putting Auction Theory To Work. Cambridge University Press, Cambridge. Nocetti, D.C., 2016. Robust comparative statics under risk. Manage. Sci. 62 (5), 1381–1392. Peter, R., 2017. Optimal self-protection in two periods: On the role of endogenous saving. J. Econ. Behav. Organiz. 137, 19–36. Pratt, J.W., 1964. Risk aversion in the small and in the large. Econometrica 32, 122–136. Ross, S.A., 1981. Some stronger measures of risk aversion in the small and in the large with applications. Econometrica 49, 621–663. Rothschild, M., Stiglitz, J., 1970. Increasing risk: I. a definition. J. Econom. Theory 2, 225–243. Selden, L., 1978. A new representation of preferences over ‘certainuncertain’ consumption pairs: The ‘ordinal certainty equivalent’ hypothesis. Econometrica 46, 1045–1060. Selden, L., 1979. An OCE analysis of the effect of uncertainty on saving under risk independence. Rev. Econom. Stud. 7, 3–82. Wang, J., Li, J., 2015. Precautionary effort: another trait for prudence. J. Risk Insurance 82 (4), 977–983. Wang, J., Li, J., 2016. Lattice-based monotone comparative statics on saving with selden/kreps-porteus preferences. J. Math. Econom. 65, 132–138. Wang, H., Wang, J., Yick, H.Y., 2018. Willingness to pay for stochastic improvements of future risk under different risk aversion. Econom. Lett. 168, 52–55. Wong, K.P., 2017. A note on risky targets and effort. Insurance Math. Econom. 73, 27–30.
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Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime
How do changes in risk and risk aversion affect self-protection with Selden/Kreps–Porteus preferences? ∗
Jianli Wang a , Hongxia Wang b , , Ho Yin Yick c a
College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China College of Economics and Management, Nanjing Forestry University, Nanjing 210037, China c Department of Finance and Insurance, Lingnan University, Hong Kong, China b
article
info
Article history: Received October 2018 Received in revised form May 2019 Accepted 23 May 2019 Available online 27 May 2019 JEL classification: D81 D91 G11 D61
a b s t r a c t This paper examines the determinants of optimal effort in an intertemporal self-protection model. We separate attitude toward risk and attitude toward intertemporal substitution by adopting Selden/Kreps– Porteus preferences. We not only explore the sufficient conditions on risk preferences for guaranteeing the unambiguous effects of changes in risk on the optimal effort level but also show how a change in risk aversion alone affects the optimal effort level. © 2019 Elsevier B.V. All rights reserved.
Keywords: Self-protection Intertemporal substitution Risk aversion Stochastic dominance Expected-utility
1. Introduction Self-protection means that a decision-maker (DM) can improve her wealth distribution by exerting some effort to reduce the probability of loss. Ehrlich and Becker (1972) first study this issue in the one-period expected-utility framework. Related literature (see e.g., Briys and Schlesinger, 1990; Chiu, 2005; Eeckhoudt and Gollier, 2005; Dionne and Li, 2011; Eeckhoudt et al., 2017) further explores how risk preferences, for example, risk aversion and prudence (see Kimball, 1990), determine the optimal effort level. In these works, the stochastic variable of the DM’s wealth takes only two possible outcomes, namely, non-random loss or no loss. Recently, Chuang et al. (2013), Crainich et al. (2016) and Wong (2017) consider that the two possible loss outcomes are represented by ‘‘better’’ risk and ‘‘worse’’ risk.1 Specifically, Crainich et al. (2016) study the effects of stochastic changes in better and worse risk on the optimal effort. They show some findings inconsistent with intuition by adopting the one-period self-protection ∗ Corresponding author. E-mail address: [email protected] (H. Wang). 1 Chuang et al. (2013) and Wong (2017) analyze optimal effort of one DM with risky targets. https://doi.org/10.1016/j.insmatheco.2019.05.004 0167-6687/© 2019 Elsevier B.V. All rights reserved.
model. For example, intuitively, for the risk-averse DM, the motive of exerting effort to increase the probability of the better risk is strengthened if the worse risk has a mean-preserving spread (see Rothschild and Stiglitz, 1970), whereas they find that the ambiguous effect of mean-preserving increases in the worse risk on effort level. The purpose of this paper is to identify whether the results in Crainich et al. (2016) hold or not when self-protection is a two-period activity, and how changes in risk aversion affect optimal effort. Since Menegatti (2009) proposes an intertemporal model where one DM can improve her future loss risk via spending current effort, the two-period self-protection model is receiving increasing attention.2 However, the two-period selfprotection model in previous works does not separate aversion to risk and aversion to intertemporal substitution.3 This means that the intertemporal substitution elasticity almost varies together with the risk aversion strength, so we cannot show the roles of intertemporal substitution elasticity and changes in risk aversion on the optimal effort level independently. That is, we cannot 2 See, e.g. Courbage and Rey (2012), Eeckhoudt et al. (2012), Hofmann and Peter (2016), Peter (2017) and Wang et al. (2018) etc. 3 The previous model on self-protection includes the same deficiency of the classical precautionary saving model. See, e.g., Kimball and Weil (2009) and Wang and Li (2016).
2
J. Wang, H. Wang and H.Y. Yick / Insurance: Mathematics and Economics 88 (2019) 1–6
completely clarify the motives of self-protection by using the existing two-period self-protection model. In addition, the literature on higher-order risk aversion (risk aversion, risk prudence, etc.)4 contains two kinds of characterizations of comparative risk aversion: the Arrow–Pratt risk aversion measure (see Pratt, 1964; Arrow, 1974) and the Ross risk aversion measure (see Ross, 1981). For example, Jindapon and Neilson (2007) provide a unified framework to reveal both kinds of characterization in a comparative statics problem. Then, the kind of measure for risk aversion strength that can be used to derive the effect of changes in risk aversion on the two-period self-protection activity must be determined. In order to systematically explore the determinants of the optimal self-protection level, we use the Selden/Kreps–Porteus preferences (see Kreps and Porteus, 1978; Selden, 1978, 1979) to construct an intertemporal self-protection model and to separate aversion to risk and aversion to intertemporal substitution. Numerous works explore various economic and financial problems under the Selden/Kreps–Porteus preferences. For example, Kimball and Weil (2009) and Wang and Li (2016) explore the effects of time preference and risk preference on precautionary saving with Selden/Kreps–Porteus preferences. Wang and Li (2015) etc show that both precautionary saving and two-period optimal self-protection with background risks can be neatly related to risk prudence. Then, one may want to know whether the two-period self-protection problem has the same traits as precautionary saving when considering the effects of changes in risk and risk aversion under Selden/Kreps–Porteus preferences. These studies directly motivate us to adopt the Selden/Kreps– Porteus preferences in the optimal decision-making model of self-protection. We use the stochastic dominance criterion to measure the changes in risk. Given the elasticity of intertemporal substitution, we find that higher-order risk aversion determines the direction of change in the optimal current effort level when the better(worse) risk of the second period has a stochastic dominance change; we also show that greater Ross risk aversion can induce more optimal effort when the better risk dominates the worse risk in the sense of nth-order stochastic dominance (n > 1). Furthermore, when the future risk is a Bernoulli distribution, i.e., a non-random loss or no loss, we use the coefficients of both fear of loss and happiness of gain to discuss the effect of risk aversion on optimal effort. Meanwhile, to clarify our conclusions, we provide comparisons to previous works (e.g., Jindapon and Neilson, 2007; Chuang et al., 2013; Crainich et al., 2016). The rest of this paper is organized as follows. Section 2 proposes an intertemporal self-protection model with a Selden/Kreps– Porteus ordinal certainty equivalence representation. Section 3 analyzes how the changes in future risk affect the optimal current effort. Section 4 explores the optimal current effort level under different risk aversion. Section 5 concludes this work. 2. A self-protection model with a Selden/Kreps–Porteus ordinal certainty equivalence representation Throughout this paper, the improvements (deteriorations) in risk are based on the theory of stochastic dominance. Let F (x) and G(x) denote the distribution functions of the random variables x˜ and y˜ with Denote F 1 (x) = F (x), ∫ x support on [a, b], respectively. ∫x F 2 (x) = a F (y)dy and F k (x) = a F k−1 (y)dy, k = 2, 3, . . .. Similar denotations apply to G(x). The well-known nth-order stochastic dominance (nSD) is defined as follows. 4 Higher-order risk aversion is equivalent to signing the higher-order derivative of the utility function in the expected-utility framework, see, e.g., Eeckhoudt and Schlesinger (2006). Menegatti (2014, 2015) further shows the relationships between higher-order risk aversion attitudes (e.g., risk aversion, prudence) by considering the higher-order derivatives of the utility function with standard regularity assumptions.
Definition. y˜ dominates x˜ via nSD if (i) F k (b) ≥ Gk (b) for all k = 1 · · · n, (ii) F n (x) ≥ Gn (x) for all x ∈ [a, b]. Within the expected-utility framework, assume utility function u is continuously differentiable; nSD can be linked to signing the higher-order derivatives of u. Theorem 2.1. y˜ dominates x˜ via nSD if and only if Eu(y˜ ) ≥ Eu(x˜ ) for ∀ u such that (−1)k+1 u(k) ≥ 0, k = 1, . . . , n.5 When y˜ dominates x˜ via nSD, we call y˜ an nSD improvement of x˜ and x˜ an nSD deterioration of y˜ . The nth-order risk increase (Ekern, 1980) is a special case of nSD that requires the equation to hold in condition (i) of the Definition and implies that the first (n − 1)th moments are held constant. For example, the secondorder stochastic dominance (SSD) deterioration with constant mean is a mean-preserving spread in risk. We can refer to Levy (2006) etc. for more properties of nSD. Consider one DM who will receive either the wealth x˜ with a probability p(e) or y˜ with a probability 1 − p(e) in the second period, where e is the current effort level that determines the magnitude of probability p(e). Given y˜ dominates x˜ via nSD, following the terminology of Crainich et al. (2016), we call y˜ the ‘‘better’’ risk and x˜ the ‘‘worse’’ risk. Assume p′ (e) ≤ 0, which means the DM can reduce the probability p(e) of x˜ by increasing effort level e. In this self-protection setting, we assume that the DM’s preferences are additively time separable and characterized by the Selden/Kreps–Porteus ordinal certainty equivalence representation.6 Then, the DM’s optimal self-protection problem is max φ (w0 − e) + U(u−1 (p(e)Eu(x˜ ) + (1 − p(e))Eu(y˜ ))), e
(1)
where w0 is a sure wealth income in the first period. φ (·) and U(·) are the first-period and second-period utility functions, respectively (φ ′ > 0, U ′ > 0). U(·) is defined on the certainty equivalent of the second-period wealth, u−1 (p(e)Eu(x˜ ) + (1 − p(e))Eu(y˜ )), which is computed by the atemporal utility function u(·), u′ > 0. The concavity of u(·) captures the DM’s aversion to risk. U(·) represents attitudes toward intertemporal substitution which is independent of attitude toward risk. That is, model (1) allows risk preferences and intertemporal substitution to vary independently. When U(·) = u(·), model (1) becomes a traditional two-period optimal self-protection model within the expectedutility framework (Menegatti, 2009; Courbage and Rey, 2012; Eeckhoudt et al., 2012; Wang and Li, 2015 etc.). Thus, our model is more general than the previous model, which results in greater complexity for the mathematical analysis of the self-protection problem. It is especially easy to find that the concavity of model (1) in e depends not only on the properties of p(e) but also the concavity of both U(·) and u(·). To achieve our goal, we use the technique of the single crossing difference to provide comparative statics analysis.7 3. Risk changes and optimal effort Assume the worse risk has a change from x˜ 0 to x˜ 1 . For convenience, we call x˜ 0 the initial worse risk. Let W (e, x˜ i ) = φ (w0 − e) + U(u−1 (p(e)Eu(x˜ i )+(1−p(e))Eu(y˜ ))) and denote ei ∈ argmaxW (e, x˜ i ), i = 0, 1. Given y˜ dominates the worse risk x˜0 via nSD, when there is a stochastic change in the worse risk from x˜ 0 to x˜ 1 , we obtain the following conclusion. 5 u(k) denotes the kth derivative of the utility function u; E is the expectation operator. 6 Kreps and Porteus (1978) and Selden (1978, 1979) provide the axiomatic foundations of the Selden ordinal certainty equivalence representation. 7 See Appendix. For more applications of single crossing difference, also see, e.g., Wang and Li (2015) and Nocetti (2016).
J. Wang, H. Wang and H.Y. Yick / Insurance: Mathematics and Economics 88 (2019) 1–6
Proposition 3.1. The mSD improvement (deterioration) in the worse risk from x˜ 0 to x˜ 1 decreases (increases) the optimal effort, ′′ ′′ i.e., e1 ≥ e0 (e1 ≤ e0 ) if − UU ′ ≥ − uu′ and (−1)k+1 u(k) > 0, k =
1, . . . , max{m, n}. Specifically, when Eu(x˜1 ) − Eu(y˜ ) ≥ 0, that is, the initial worse risk becomes better than the better risk y˜ under the expected-utility maximization framework, (−1)k+1 u(k) (·) > 0, k = 1, . . . , max{m, n} alone implies that the mSD improvement in the worse risk reduces the optimal effort, i.e. e1 ≤ e0 . Proof. See Appendix.
Proposition 3.1 is consistent with our intuition since, intuitively, the motive of exerting effort to increase the probability of the better risk is strengthened if the initial worse risk becomes worse but is weakened if the worse risk has a stochastic improvement. ′′ ′′ The condition − UU ′ ≥ − uu′ indicates that the resistance to intertemporal substitution is larger than the Arrow–Pratt risk aversion strength. Previous studies, for example, Kimball and Weil (2009) and Wang and Li (2016), also use the condition ′′
′′
− UU ′ ≥ − uu′ to examine the determinants of the precautionary y1−ρ saving motive. If we take the functional forms,8 U(y) = 1−ρ , ρ > 0, ̸= 1 and u(x) = u′′ (x)
x1−γ 1−γ
′′
(x) , γ > 0, ̸= 1, we have − UU ′ (x) = ρ x−1
and − u′ (x) = ηx−1 , then ρ ≥ γ guarantees this condition when the wealth level x > 0. ′′ In terms of model, the condition − UU ′ to the comparison between the ratios
u′′ is related u′ ′ − 1 U (u (p(e)Eu(x˜ i )+(1−p(e))Eu(y˜ ))) u′ (u−1 (p(e)Eu(x˜ )+(1−p(e))Eu(y˜ )))
≥ −
i
evaluated in x˜ 0 and in x˜ 1 (see Eq. (8), Appendix 7.1). These two ratios can also be written as the products of the marginal utility of U and of the derivative of the certainty equivalent, since 1 = (u−1 )′ (p(e)Eu(x˜ i ) + (1 − p(e))Eu(y˜ )). u′ (u−1 (p(e)Eu(x˜ )+(1−p(e))Eu(y˜ ))) i
Therefore, the ratios in Eq. (8) measure the overall effects on utility of the marginal benefits of the effort, i.e. the change in certainty equivalent weighted by the marginal utility of U. Since these changes in utility are evaluated in two different situations U ′ (x) (in x˜ 0 and in x˜ 1 ), the condition of Eq. (8) corresponds to u′ (x) ′′
′′
decreasing in x which is equivalent to − UU ′ ≥ − uu′ . The condition (−1)k+1 u(k) (·) > 0 means the DM is kth-order risk aversion (Ekern, 1980). Eeckhoudt and Schlesinger (2006) clarify the meaning of (−1)k+1 u(k) (·) > 0 by a simple lottery-pairs preference and further stimulate new research on higher-order risk aversion. Similarly, assume that the initial better risk y˜ 0 dominates the worse risk x˜ via nSD, when the better risk undergoes a change from the initial better risk y˜ 0 to another risk y˜ 1 , we obtain the following conclusion by the same approach to Proposition 3.1. Proposition 3.2. The mSD improvement (deterioration) in the better risk from y˜ 0 to y˜ 1 increases (decreases) the optimal effort ′′ ′′ if − UU ′ ≤ − uu′ and (−1)k+1 u(k) > 0, k = 1, . . . , max{m, n}. Specifically, when Eu(x˜ ) − Eu(y˜ 1 ) ≥ 0, that is, the initial better risk becomes worse than the worse risk x˜ under the expected-utility maximization framework, (−1)k+1 u(k) > 0, k = 1, . . . , max{m, n} alone implies that the mSD improvement in the better risk increases the optimal effort. Our conclusions have some differences from those of Chuang et al. (2013) and Crainich et al. (2016). For example, Proposition 9 in Crainich et al. (2016) shows if the risk preference of one DM is (−1)k+1 u(k) ≥ 0, k = 1, . . . , n, then the effect on effort 8 See, e.g., Kimball and Weil (2009).
3
is ambiguous when there is an nth-order decrease(increase) in the worse risk.9 This result is inconsistent with our conclusions since we observe an unambiguous effect of changes in risk on effort. This difference is mainly due to the fact that in our intertemporal self-protection model, effort is spent in the first period but the stochastic change in loss risk occurs in the second period. Interestingly, when U(·) = u(·), our conclusions need only the condition of higher-order risk aversion, i.e., (−1)k+1 u(k) ≥ 0, k = 1, . . . , n. Thus, Propositions 3.1 and 3.2 are analogous to Proposition 1 in Eeckhoudt and Schlesinger (2008), which clarifies the effect of labor–income risk on precautionary saving by signing the higher derivatives of the utility function. That is, higher-order risk aversion can guarantee the unambiguous effects of changes in risk on both optimal saving and effort. Specifically, let x˜ = w1 − L, L > 0, y˜ = w1 , and p(e) = p0 − ϵ (e), ϵ ′ (e) > 0, where p(0) is the initial probability of loss. We can consider the classic loss risk with a Bernoulli distribution. The worse risk x˜ has a first-order stochastic dominance (FSD) deterioration if the magnitude of loss L or the initial probability p(0) of loss increases. Thus, let m = n = 1 in Proposition 3.1, we find that the optimal effort level increases as L or p(0) increases as ′′ ′′ long as − UU ′ ≥ − uu′ holds for all DMs with monotone increasing utility functions, regardless of whether the DMs are risk averse or risk loving. We re-explore the example of Alzheimer’s disease in Crainich et al. (2016). Following their work, consider one DM is unable to change the probability t of Alzheimer’s disease but can improve the severity of the disease by spending current intellectual effort e. Let L(early onset) and M(late onset) denote possible values of the severity of the disease; then, in our intertemporal framework, the DM faces two Bernoulli-distributed random variables: x˜ = (w1 − L, t ; w1 , 1 − t) with probability p(e) and y˜ = (w1 − M , t ; w1 , 1 − t) with probability 1 − p(e), where w1 is the initial level of quality of life in the future. In this simplified model, a change in L or M is equivalent to a change in FSD in the worse(better) risk. As a consequence, our conclusions reveal that the effect of a change in L or M on the optimal intellectual effort is unambiguous for all DM with u′ > 0. This is in contrast to the findings of Crainich et al. (2016), who show that the effect of a change in L or M on effort is ambiguous if the DM is a risk averter(risk lover).10 Overall, our conclusions confirm the intuition of Crainich et al. (2016), who demonstrate “At the beginning of the paper we argued that a plausible conjecture on DM choices may be that an FSD improvement in the better risk (enlarging the ‘gap’ between the two risks) pushes the DM to increase effort and · · ·”11 but are different from their results and those of other previous works. 4. Different degrees of risk aversion and optimal effort This section discusses the effect of a change in risk aversion on the optimal effort. We start with the measures of risk aversion strength. v is more Arrow–Pratt risk averse than u if
(x) (x) − vv′ (x) ≥ − uu′ (x) for all x ∈ [a, b] (see Pratt, 1964; Arrow, 1974). ′′
′′
The literature finds that the Arrow–Pratt characterization of risk aversion strength is often not sufficiently strong for comparing preference orders in risky choice situations. Thus, Ross (1981) v ′′ (x)
u′′ (x)
first uses − v ′ (y) ≥ − u′ (y) for all x, y ∈ [a, b] to state that ‘‘v is more risk averse than u’’. It is evident that the Arrow–Pratt characterization of risk aversion strength is a special example 9 Proposition 2 in Chuang et al. (2013) also produces a similar finding. 10 See Propositions 1, 2, and 3 in Crainich et al. (2016). 11 See the paragraph after Corollary1 in Crainich et al. (2016).
4
J. Wang, H. Wang and H.Y. Yick / Insurance: Mathematics and Economics 88 (2019) 1–6
of Ross’s (1981) characterization of risk aversion strength when x = y. Ross’s (1981) characterization of risk aversion strength is generalized to the case of kth-order(k ≥ 2) risk aversion by Jindapon and Neilson (2007), Li (2009) and Liu and Meyer (2013) etc., which has numerous exploration and application, regardless of its stronger limitation than that of the Arrow–Pratt index. Definition. Given two DMs with utility functions u and v , v is v (k) (x) more kth-order(k ≥ 2) Ross risk averse than is u if (−1)k−1 v ′ (y) ≥ (−1)k−1
u(k) (x) u′ (y)
for all x, y ∈ [a, b].
Consider two DMs with different atemporal utility functions u, v but the same second-period utility function U. Let W (e, u) = φ (w0 − e) + U(u−1 (p(e)Eu(x˜ ) + (1 − p(e))Eu(y˜ ))),
(2)
and W (e, v ) = φ (w0 − e) + U(v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ ))).
(3)
Denote ei ∈ argmaxW (e, i), i = u, v . We explore the sufficient conditions for the unambiguous effect of changes in risk aversion on optimal effort. We obtain the following conclusion. Proposition 4.1. Assume the DMs u, v share the same secondperiod utility function U, when the better risk y˜ dominates the worse risk x˜ via nSD, n ≥ 2, for both DMs with (−1)k+1 u(k) > 0 and ′′ ′′ (−1)k+1 v (k) > 0, k = 1, . . . , n, if − UU ′ ≥ − uu′ and v is more kth-order (k = 2, . . . , n) Ross risk averse than u, then v exerts more effort than u. Proof. See Appendix. Proposition 4.1 indicates that the Ross characterization of risk aversion strength can be used to perform the comparative statics analysis in an intertemporal self-protection model, which means the DM conducts self-protection activity by paying a utility cost of effort. In terms of proof, our sufficient conditions are gotten by ′′
′′
both Eq. (12) and Eq. (13) which are guaranteed by − UU ′ ≥ − uu′ and v being more kth-order (k = 2, . . . , n) Ross risk averse than ′′ ′′ u respectively. Thus, the condition − UU ′ ≥ − uu′ is required, but
≥ − vv′ is not involved. In the setting of ′′ ′′ Proposition 4.1, the similar proof also means that if − UU ′ ≤ − uu′ and v is less kth-order (k = 2, . . . , n) Ross risk averse than u, ′′ ′′ then v exerts less effort than u. That is, holding − UU ′ ≥ − uu′ or ′′ ′′ − UU ′ ≤ − uu′ fixed, our conclusions are determined by changes in ′′ ′′ risk aversion no matter − UU ′ is bigger than − vv ′ or not. ′′
the condition − UU ′
′′
Jindapon and Neilson (2007) show that the Ross characterization of risk aversion strength governs the change in optimal static decision-making behavior when the effort requires monetary cost. They also show more Arrow–Pratt risk averse individual makes more effort after separating the cost and the benefit of the efforts. It seems that Jindapon and Neilson (2007) need less restrictive conditions. This is because that the solution of problem with a utility cost of effort in Jindapon and Neilson (2007) is assumed to be invariant when the utility function u has a positive linear transformation,12 but it is not invariant with positive linear transformations of the utility function (see Liu and Wang, 2017). It is clear that if the aversion to risk and aversion to intertemporal substitution are not separated in this problem, we cannot change risk preferences and intertemporal substitution independently from each other. In our model, we also can hold risk preferences fixed (i.e. u = v ) but assume different secondperiod utility functions. Then, under this circumstance, simple 12 See Eq. (3) in Jindapon and Neilson (2007) and the proof of their Theorem 3.
calculations mean that the margin utility in the second-period determines whether more effort is desirable. Thus, our model with Selden/Kreps–Porteus Preferences is helpful to clarify the different motives of exerting more effort. We return to the classic loss risk: x˜ = w1 − L, L > 0 and y˜ = w1 . Eeckhoudt et al. (2017) propose the concepts of linearly(quadratically)-restricted greater Ross risk aversion,13 to provide a clear-cut comparative statics analysis for the oneperiod self-protection. However, in our setting, we cannot use the Ross characterization of risk aversion strength to provide sufficient conditions for more effort by Proposition 4.1 since this classic loss risk type means that y˜ dominates x˜ via FSD, whereas Proposition 4.1 considers only the case of nSD with the condition n ≥ 2. In terms of proofs, the cause is that the Ross characterization of risk aversion strength cannot be used to address the sufficient condition for inequality (13) in Appendix. Let N = v −1 (p(e)v (w1 − L) + (1 − p(e))v (w1 )); clearly, w1 > N > w1 − L. To show that ev ≥ eu , we need to find the sufficient condition for the inequality (13), which can be rewritten as u(w1 ) − u(w1 − L) v (w1 ) − v (w1 − L) > v ′ (N) u′ (N) v (w1 ) − v (N) v (N) − v (w1 − L) u(w1 ) − u(N) ⇔ + > v ′ (N) v ′ (N) u′ (N) u(N) − u(w1 − L) + ′ u (N)
v (N) − v (w1 − L) u(N) − u(w1 − L) u(w1 ) − u(N) ⇔ − > v ′ (N) u′ (N) u′ (N) v (w1 ) − v (N) − . (4) v ′ (N) Li (2010) shows that if v is more Arrow–Pratt risk averse than u, then v has more fear of loss than does u at the wealth level N v (N)−v (w1 −L) w1 −L) (i.e., > u(N)−u′u((N) ), whereas u has more happiness v ′ (N) u(w1 )−u(N) of gain than does v at the wealth level N (i.e., > u′ (N) v (w1 )−v (N) ). v ′ (N)
This result means that both the right and left sides
of inequality (4) are positive. Thus, we cannot obtain the sufficient condition by the comparison of only the Arrow–Pratt risk aversion strength. Actually, the left side of inequality (4) differs between u’s and v ’s coefficients of fear of loss at wealth level N, which reflects the positive effect of greater Arrow–Pratt risk aversion on the coefficient of fear of loss. However, the right side of inequality (4) reflects its negative effect on the coefficient of happiness of gain. Thus, by inequality (4), we know that when ′′
′′
− UU ′ ((··)) ≥ − uu′ ((··)) , if v is more Arrow–Pratt risk averse than u and the difference in fear of loss at wealth level N between both DMs is larger than the difference in happiness of gain, then v exerts more effort than u. 5. Conclusion This work models the intertemporal self-protection problem with Selden/Kreps–Porteus preferences. We explore the effects of changes in risk and risk aversion on optimal effort. Holding the elasticity of intertemporal substitution constant, we find that higher-order risk aversion can determine the change in effort when the risk improves or deteriorates in the sense of stochastic dominance. We also find that greater Ross risk aversion can result in higher optimal effort. Our conclusions can easily be applied to analyze the case of an nth-order risk increase in a straightforward manner. 13 Both concepts are special cases of Definition 4 i.e., they are stricter than the general Ross characterization of risk aversion strength in Ross (1981), Jindapon and Neilson (2007), Li (2009) and Liu and Meyer (2013) etc.
J. Wang, H. Wang and H.Y. Yick / Insurance: Mathematics and Economics 88 (2019) 1–6
Acknowledgments The authors would like to thank the editor and anonymous referees for providing valuable suggestions which have led to significant improvement of this article. This work is supported by the National Natural Science Foundation of China under Grant Number 71401074, MOE (Ministry of Education in China) Project of Humanities and Social Sciences under Grant Number 19YJC790125, and by the Fundamental Research Funds for the Central Universities under Research Project No. NS2018049. Appendix
A.1. Proof of Proposition 3.1 Proof. We first consider the mSD deterioration from x0 to x1 , i.e., x˜ 0 dominates x˜ 1 via mSD. In order to show that e0 ≤ e1 , from Theorem A.1, we know that we need to prove, for all e′′ > e′ , W (e′′ , x˜ 0 ) − W (e′ , x˜ 0 ) ≥ 0 ⇒ W (e′′ , x˜ 1 ) − W (e′ , x˜ 1 ) > 0.
(5)
This can be obtained by W (e′′ , x˜ 1 ) − W (e′ , x˜ 1 ) > W (e′′ , x˜ 0 ) − W (e′ , x˜ 0 )
⇔ φ (w0 − e′′ ) + U(u−1 (p(e′′ )Eu(x˜ 1 ) + (1 − p(e′′ ))Eu(y˜ ))) − φ (w0 − e′ ) − U(u−1 (p(e′ )Eu(x˜ 1 ) + (1 − p(e′ ))Eu(y˜ ))) > φ (w0 − e′′ ) + U(u−1 (p(e′′ )Eu(x˜ 0 ) + (1 − p(e′′ ))Eu(y˜ ))) − φ (w0 − e′ ) − U(u−1 (p(e′ )Eu(x˜ 0 ) + (1 − p(e′ ))Eu(y˜ ))).
(6)
U(u−1 (p(e′′ )Eu(x˜ 1 ) + (1 − p(e′′ ))Eu(y˜ ))) − U(u−1 (p(e′ )Eu(x˜ 1 ) ′
+(1 − p(e ))Eu(y˜ ))) > U(u−1 (p(e′′ )Eu(x˜ 0 ) + (1 − p(e′′ ))Eu(y˜ ))) − U(u−1 (p(e′ )Eu(x˜ 0 ) +(1 − p(e′ ))Eu(y˜ ))). (7) Because e′′ > e′ , we know a sufficient condition for (7) is U (u
(p(e)Eu(x˜ 1 ) + (1 − p(e))Eu(y˜ )))
u′ (u−1 (p(e)Eu(x˜ 1 ) + (1 − p(e))Eu(y˜ )))
>
U ′ (u−1 (p(e)Eu(x˜ 0 ) + (1 − p(e))Eu(y˜ ))) u′ (u−1 (p(e)Eu(x˜ 0 ) + (1 − p(e))Eu(y˜ )))
p′ (e)(Eu(x˜ 1 ) − Eu(y˜ )) p′ (e)(Eu(x˜ 0 ) − Eu(y˜ )). (8)
Note that (−1)k+1 u(k) > 0, k = 1, . . . , max{m, n} means both u−1 (p(e)u(x˜ 0 ) + (1 − p(e))u(y˜ )) > u−1 (p(e)u(x˜ 1 ) + (1 − p(e))u(y˜ )) and Eu(x˜ 1 ) − Eu(y˜ ) ≤ Eu(x˜ 0 ) − Eu(y˜ ) ≤ 0. Then, (8) can be guaranteed by the fact that
U ′ (x) u′ (x)
U ′′ U′ k+1 (k)
is decreasing in x (i.e., −
we show that both −
U (u
−1
U ′′ U′
≥ −
u′′ u′
and (−1)
(p(e)Eu(x˜ 0 ) + (1 − p(e))Eu(y˜ )))
u′ (u−1 (p(e)Eu(x˜ 0 ) + (1 − p(e))Eu(y˜ )))
>
′′
′′
′′
0, k = 1, . . . , max{m, n} imply e0 ≥ e1 . When Eu(x˜ 1 ) − Eu(y˜ ) ≤ 0 (i.e., the ‘worse’ risk becomes better than the ‘better’ risk), (9) clearly holds since Eu(x˜ 0 ) − Eu(y˜ ) ≤ 0 and p′ (e) ≤ 0; thus, (−1)k+1 u(k) > 0, k = 1, . . . , max{m, n}, implies e0 ≥ e1 . A.2. Proof of Proposition 4.1
U ′ (u−1 (p(e)Eu(x˜ 1 ) + (1 − p(e))Eu(y˜ ))) u′ (u−1 (p(e)Eu(x˜ 1 ) + (1 − p(e))Eu(y˜ )))
W (e′′ , u) − W (e′ , u) ≥ (>)0 ⇒ W (e′′ , v ) − W (e′ , v ) ≥ (>)0. (10) Following proofs of Proposition 3.1, we know that (10) can be obtained by U ′ (v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ )))
p′ (e)(E v (x˜ ) − E v (y˜ )) v ′ (v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ ))) U ′ (u−1 (p(e)Eu(x˜ ) + (1 − p(e))Eu(y˜ ))) ′ ≥ ′ −1 p (e)(Eu(x˜ ) − Eu(y˜ )) u (u (p(e)Eu(x˜ ) + (1 − p(e))Eu(y˜ ))) U ′ (v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ ))) ⇔ ′ −1 u (v (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ ))) u′ (v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ ))) ′ × ′ −1 p (e)(E v (x˜ ) − E v (y˜ )) v (v (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ ))) U ′ (u−1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ ))) > ′ −1 u (u (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ ))) × p′ (e)(Eu(x˜ ) − Eu(y˜ )). (11) U ′ (v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ ))) u′ (v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ )))
≥
U ′ (u−1 (p(e)u(w1 − L) + (1 − p(e))u(w1 ))) u′ (u−1 (p(e)u(w1 − L) + (1 − p(e))u(w1 )))
u′ (v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ )))
v ′ (v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ ))) > p′ (e)(Eu(x˜ ) − Eu(y˜ )).
U ′′ (·)
u′′ (·)
(12) since − U ′ (·) ≥ − u′ (·) means that
(13)
U ′ (x) u′ (x)
is decreasing in x.
Let N = v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ )); then, inequality (13) can be rearranged as E v (y˜ ) − E v (x˜ )
v ′ (v −1 (p(e)E v (x˜ ) > ⇔
+ (1 − p(e))E v (y˜ ))) Eu(y˜ ) − Eu(x˜ )
u′ (v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ )))
n−1 ∑ (−1)i v (i) (b)
v ′ (N)
i=1
b
∫
(−1)n
+
′
p (e)(Eu(x˜ 0 ) − Eu(y˜ ))
Note that (−1)k+1 u(k) > 0, k = 1, . . . , max{m, n} means both u−1 (p(e)u(x˜ 0 ) + (1 − p(e))u(y˜ )) < u−1 (p(e)u(x˜ 1 ) + (1 − p(e))u(y˜ )) and
p′ (e)(E v (x˜ ) − E v (y˜ ))
If v is more kth-order (k = 2, . . . , n) Ross risk averse than u, then v is more Arrow–Pratt risk averse than u; thus, v −1 (p(e)E v (x˜ ) + (1 − p(e))E v (y˜ )) < u−1 (p(e)Eu(x˜ ) + (1 − p(e))Eu(y˜ )). Then, we have
≥− Thus, u > 0, k =
p′ (e)(Eu(x˜ 1 ) − Eu(y˜ )). (9)
(12)
and
u′′ ). u′
1, . . . , max{m, n} imply that (5) holds. Similarly, when x˜ 0 is dominated by x˜ 1 via mSD, in order to show that e0 ≥ e1 , we need to find sufficient conditions for the inequality ′
′′
(i.e., − UU ′ ≥ − uu′ ). Thus, both − UU ′ ≥ − uu′ and (−1)k+1 u(k) >
This can be guaranteed by both
This is equivalent to
−1
Eu(x˜ 1 ) − Eu(y˜ ) ≥ Eu(x˜ 0 ) − Eu(y˜ ); then, when Eu(x˜ 1 ) − Eu(y˜ ) ≤ 0, U ′ (x) (9) can be guaranteed by the fact that u′ (x) is decreasing in x
Proof. To show that ev ≥ eu , from Theorem A.1, we know that we need to prove that for all e′′ > e′ ,
Theorem A.1 (e.g. Milgrom, 2004, Theorem 4.1). Given sets X and S, {f (·; s)}s∈S obeys strict single crossing difference, i.e., f (x′′ , s′ ) − f (x′ , s′ ) ≥ 0 ⇒ f (x′′ , s′′ ) − f (x′ .s′′ ) > 0 for all x′′ > x′ and s′′ > s′ , if and only if arg maxx∈X f (x; s) is increasing in s, i.e., inf arg maxx∈X f (x; s′′ ) ≥ sup arg maxx∈X f (x; s′ ) for all s′′ > s′ .
′
5
a
>
(Gi+1 (b) − F i+1 (b))
v (n) (x) n (G (x) − F n (x))dx v ′ (N)
n−1 ∑ (−1)i u(i) (b)
u′ (N)
i=1 b
∫
(−1)n
+ a
(Gi+1 (b) − F i+1 (b))
u(n) (x) u′ (N)
(Gn (x) − F n (x))dx
6
⇔
J. Wang, H. Wang and H.Y. Yick / Insurance: Mathematics and Economics 88 (2019) 1–6 n−1 ∑ (−1)i+1 u(i) (b) i+1 (−1)i+1 v (i) (b) − )(F (b) − Gi+1 (b)) ( v ′ (N) u′ (N) i=1 ∫ b v (n) (x) u(n) (x) + ((−1)n+1 ′ − (−1)n+1 ′ ) v (N) u (N) a × (F n (x) − Gn (x))dx > 0.
(14)
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