Willingness to pay for stochastic improvements of future risk under different risk aversion

Accepted Manuscript Willingness to pay for stochastic improvements of future risk under different risk aversion Hongxia Wang, Jianli Wang, Yick Ho Yin

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S0165-1765(18)30134-4 https://doi.org/10.1016/j.econlet.2018.04.005 ECOLET 8004

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Economics Letters

Received date : 2 February 2018 Revised date : 3 April 2018 Accepted date : 4 April 2018 Please cite this article as: Wang H., Wang J., Yin Y.H., Willingness to pay for stochastic improvements of future risk under different risk aversion. Economics Letters (2018), https://doi.org/10.1016/j.econlet.2018.04.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Highlights (for review)

Highlights We explore the willingness to pay for stochastic improvements of future risks. We provide a comparative statics analysis using Ross risk aversion strength. We study the effects of changes in risk aversion on saving and self-protection.

*Manuscript Click here to view linked References

Willingness to Pay for Stochastic Improvements of Future Risk under Different Risk Aversion Hongxia Wanga , Jianli Wangb,∗, Yick Ho Yinc a

b

College of Economics and Management, Nanjing Forestry University, Nanjing, China

College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, China c Department

of Finance and Insurance, Lingnan University, Hong Kong, China.

April 3, 2018

Abstract Within a general intertemporal decision-making framework, this work shows one individual with more kth-degree(k = 2, · · · , n) Ross risk aversion always chooses more current paying to improve his future payoff distribution when such stochastic improvement satisfies the nthdegree mean-preserving stochastic dominance. Moreover, when stochastic improvement of the future payoff distribution is not mean-preserving, the notions of linearly(quadratically)restricted more Ross risk aversion proposed by Eeckhoudt, Liu and Meyer (2017) can help provide a clear-cut comparative statics analysis.

Key words: Ross risk aversion; Stochastic dominance; Self-protection; Saving JEL classification: D81

∗

Corresponding author. E-mail address: [email protected] (J. Wang).

1

Introduction

The essence of two-period economic and financial decision-making problems under risks(e.g. insurance policy purchase, precautionary saving, self-protection) is to determine the optimal current paying of the decision maker(DM) for improving stochastic distribution of future wealth variables. Related works provide some solutions and comparative static results on this line of research. For example, Diamond and Stiglitz (1974) and Jewitt (1989) explore whether a DM with more Arrow-Pratt risk aversion is more willing to pay for stochastic improvement under a single-crossing condition. Chiu (2005) develops a choice-theoretic framework to interpret ArrowPratt risk aversion and Ross risk aversion by considering the desirability of different types of stochastic changes. Jindapon and Neilson (2007) study the roles of both these risk aversion measures by introducing a linear form of stochastic improvement which satisfies an nth-degree risk increase(Ekern, 1980). Chiu (2012) demonstrates the willingness to pay for stochastic improvements under a double-crossing condition depends on the DM’s risk aversion strength and downside risk aversion strength(Menezes et al., 1980). These works mainly focus on the relationship between risk aversion strength and optimal paying for some specific forms of stochastic improvement in the one-period model. In a recent paper, based on the studies on both precautionary saving(Kimball, 1990, Eeckhoudt and Schlesinger, 2008 etc.) and precautionary effort(Eeckhoudt et al., 2012, Wang and Li, 2015 etc.), Wang et al. (2015) consider an intertemporal precautionary paying for stochastic improvements of future risks and examine the impact of stochastic change of background risk on optimal current paying. Inspired by these previous literature, in an intertemporal framework, this note explores the effect of DM’s risk aversion strength on the willingness to pay for stochastic improvements of future risk by adopting various orders of stochastic dominance to characterize stochastic improvements and applying the notions of Ross risk aversion, linearly(quadratically)restricted more Ross risk aversion(Eeckhoudt et al., 2017) to compare the risk aversion strength. In this note, we firstly give a general intertemporal decision-making model under risks. We then explore the optimal current paying level under different Ross risk aversion strengths. At last, we provide some simple applications on precautionary saving and self-protection.

1

2

Model

Consider a DM with sure current income w0 and future income w1 who faces a risk x ˜ which occurs in the second period. He can improve the stochastic distribution of x ˜ by paying a monetary cost π in advance. We use x ˜(π) to denote the future risk conditional on current paying π. Let F (x|π) and f (x|π) with support contained in [x, x ¯] be x ˜(π)’s distribution function and probability density function conditional upon the level of π respectively. Following the literature, intertemporal preferences of the DM are considered as the form of “exponential discounting”1 . The objective problem of the DM is2 max u(w0 − π) + π

Where, u is the utility function;

1 1+δ

1 Eu(w1 + x ˜(π)). 1+δ

(1)

is the discount factor, δ ≥ 0. We assume u0 > 0, u00 ≤ 0

and the optimal objective function is concave in π. The model (1) means the current paying can improve the stochastic distribution of future wealth by reducing the first-period utility level, which models a general intertemporal decision-making setting.3

3

Optimal Current Paying and Ross risk aversion

Denote F 0 (x|π) = f (x|π), F 1 (x|π) = F (x|π) and F k (x|π) =

Rx x

F k−1 (y|π)dy, k = 1, 2, · · ·. We

provide the following concept to characterize stochastic changes of future risk. Definition 3.1 If4 Fπ2 (¯ x|π) = 0, Fπk (¯ x|π) ≤ 0 for k = 3, · · · n, and Fπn (x|π) ≤ 0 for all x ∈ [x, x ¯], then x ˜(π) has a stochastic improvement in the sense of the nth-degree mean-preserving stochastic dominance (nMPSD) when π increases. x(π)) The condition Fπ2 (¯ x|π) = 0 means E(˜ x(π)) is constant, since Fπ2 (¯ x|π) = − dE(˜ . When n = 2, dπ

Definition 3.1 implies that x ˜(π1 ) is mean-preserving spread(Rothschild and Stiglitz, 1970) of x ˜(π2 ), where π2 ≥ π1 ; when n = 3, Definition 3.1 is weaker than mean-variance-preserving spread(Menezes et al., 1980). More generally, stochastic improvement in the sense of nMPSD is stricter than the nth-degree stochastic dominance, but weaker than the nth-degree risk increase. Next, we recall concepts and property of the nth-degree Ross risk aversion measure. 1

For example, see Chapter VI in Gollier (2001) etc.. E(·) is the expectation operator. 3 For more details, please refer to Wang et al. (2015) and the following Applications. 2

4

Let Fπ (x|π) denote

dF (x|π) . dπ

2

Definition 3.2 Given the two DMs with utility functions u and v, v is more nth-degree Ross (n)

(n)

(x) (x) ¯], n ≥ 2. risk averse than u, if (−1)n−1 vv0 (y) ≥ (−1)n−1 uu0 (y) for all x, y ∈ [x, x

Theorem 3.3 For the nth-degree risk averse functions5 u and v, v is more nth-degree Ross risk averse than u if and only if there exists one function φ(x) with φ0 (x) ≤ 0 and (−1)n−1 φ(n) (x) ≥ 0 for n ≥ 2 and λ > 0 such that v(x) = λu(x) + φ(x). This characterization of risk aversion strength is firstly proposed by Ross (1981) and generalized to the case of nth-degree risk aversion by Li (2009) and Liu and Meyer (2013) etc.. Let πi denote the optimal solution for the DM with utility function i, i = u, v. We can show Proposition 3.4 When additional current paying improves the stochastic distribution of future risk in the sense of nMPSD(n ≥ 2), πv ≥ πu if and only if v is more kth-degree(k = 2, 3, · · · , n) Ross risk averse than u. Proof See Appendix. Proposition 3.4 indicates Ross risk aversion determines the optimal current paying level when an increase of current paying brings an improvement of nMPSD in future risk. Given two stochastic distribution functions G(x) and H(x), Jindapon and Neilson (2007) consider a linear form of stochastic change by exploring the stochastic distribution function πG(x) + (1 − π)H(x) with π ∈ [0, 1], where, H(x) has more nth-degree risk than G(x). We find π1 G(x)+(1−π1 )H(x) is an nth-degree risk increase of π2 G(x)+(1−π2 )H(x) for any π2 > π1 . This means stochastic change presented by Jindapon and Neilson (2007) is a special example of nMPSD. Thus, Proposition 3.4 can be used to examine the stochastic change in Jindapon and Neilson (2007) when letting the distribution function of future risk x ˜(π) be πG(x) + (1 − π)H(x). However, when stochastic improvement is a first-order stochastic dominance(FSD) change, Proposition 3.4 does not hold, since FSD is not mean-preserving.

4

Optimal Current Paying and Restricted Increases in Ross Risk Aversion

This section addresses stochastic changes by relaxing the mean-preserving assumption, i.e. we allow Fπ2 (¯ x|π) 6= 0. It is straightforward to find that a risk neutral DM chooses the optimal current 5

If (−1)n+1 un (·) ≥ 0, the DM u is nth degree risk averse. For example, see Ekern (1980).

3

1 paying π = 0 if − 1+δ Fπ2 (¯ x|π) ≤ 1 for all π. We take this condition as a benchmark of stochastic

improvements to give an unambiguous analysis using the notions of linearly(quadratically)restricted more Ross risk aversion. Definition 4.1 (Eeckhoudt et al., 2017) v(x) is linearly-restricted more risk averse than u(x) if there are λ and some linear form a + bx, where λ > 0, b ≤ 0, such that v(x) = λu(x) + a + bx. The linearly-restricted more Ross risk aversion is stricter than more Ross risk aversion, since it needs that φ is a linear function a + bx.

6

We get

Proposition 4.2 If v is linearly-restricted more Ross risk averse than u, then πv ≥ πu holds 1 when stochastic improvement satisfies − 1+δ Fπ2 (¯ x|π) ≤ 1 for all π.

Proof See Appendix. 1 Fπ2 (¯ x|π) ≤ 1 means that the intertemporal self-protection always reduces the The condition − 1+δ 1 1 Fπ2 (¯ x|π) ≤ 1 is equivalent to −1 + 1+δ (Ex(π))0π ≤ 0. expected value of lifetime wealth since − 1+δ

Thus, under this condition a risk neutral DM always chooses zero as the optimal current paying level. Proposition 4.2 demonstrates linearly-restricted more Ross risk aversion means more current paying if zero is the optimal current paying level for the risk neutral DM. Eeckhoudt et al. (2017) also propose the concept of quadratically-restricted more Ross risk aversion which includes linearly-restricted more Ross risk aversion as a special case. Definition 4.3 (Eeckhoudt et al., 2017) v(x) is quadratically-restricted more risk averse than u(x) if there are λ and some quadratic form a + bx + cx2 , where λ > 0, b + 2cx ≤ 0 and c ≤ 0, such that v(x) = λu(x) + a + bx + cx2 . To explore the effect of changes in risk aversion on optimal current paying, we get Proposition 4.4 If v is quadratically-restricted more Ross risk averse than u, then πv ≥ πu 1 holds when stochastic improvement satisfies both − 1+δ Fπ2 (¯ x|π) ≤ 1 and Fπ3 (¯ x|π) ≤ 0 for all π. 1 The condition − 1+δ Fπ2 (¯ x|π) ≤ 1 represents how the change of π affects the mean value of

future risk; Fπ3 (¯ x|π) ≤ 0 means an increase of π decreases the second-order moment of future risk about the origin since Fπ3 (¯ x|πu ) =

¯ 2 1 Rx 2 x x fπ (x|π)dx.

Proposition 4.4 clarifies the trade-off

between mean value of future risk and its second-order moment about the origin when the DM 6

See also Eeckhoudt et al. (2017) for more details.

4

prepares to improve the distribution of future risk. Specially, it is easy to find Fπ2 (¯ x|π) ≥ 0 is 1 sufficient for − 1+δ Fπ2 (¯ x|π) ≤ 1. Thus, when stochastic improvement satisfies Fπ2 (¯ x|π) ≥ 0 and

Fπ3 (¯ x|π) ≤ 0 for all π, we can show πv ≥ πu by Proposition 4.4. This means that quadraticallyrestricted more Ross risk aversion induces more current paying if an increase of π decreases both the mean value of future risk and its second-order moment about the origin respectively. This finding is consistent with the intuitive meaning of quadratically-restricted more Ross risk aversion in Proposition 4 of Eeckhoudt et al. (2017).

5 5.1

Applications Self-protection and Changes in Risk aversion

Menegatti (2009) studies one two-period self-protection model max u(w0 − e) + p(e)Eu(w1 − L) + (1 − p(e))Eu(w1 ). e

(2)

Where, L > 0, p0 (e) ≤ 0. Model (2) means more current effort can decrease the loss probability p(e). For this self-protection problem (2), let π = e and x ˜(e) = [p(e), −L; 1 − p(e), 0] in model (1), then Proposition 4.4 implies Corollary 5.1 For the two-period self-protection model (2), if the optimal effort level for the risk neutral DM is zero, i.e. −p0 (e)L < 1 for all e, then quadratically-restricted more Ross risk aversion guarantees more optimal effort. Eeckhoudt et al. (2017) study how quadratically-restricted more Ross risk aversion affects the self-protection in the one-period framework. We explore the effect of quadratically-restricted more Ross risk aversion in the two-period self-protection model. More generally, we can consider the self-protection model of Eeckhoudt et al. (2012) with general future risk as following: max u(w0 − e) + p(e)Eu(˜ a) + (1 − p(e))Eu(˜b). e

(3)

Where, a ˜ is an nth degree risk increase of ˜b. This implies that p(e2 )˜ a + (1 − p(e2 ))˜b is preferred to p(e1 )˜ a + (1 − p(e1 ))˜b, e2 > e1 . When n ≥ 2, E(p(e)˜ a + (1 − p(e))˜b) is constant since E˜ a = E˜b. This implies additional current paying can bring a stochastic improvement of future risk p(π)˜ a + (1 − p(e))˜b in the sense of nMPSD. As a direct result of Proposition 3.4, we have Corollary 5.2 For the two-period self-protection model (3), more nth-degree Ross risk aversion guarantees more optimal effort. 5

5.2

Precautionary Saving and Changes in Risk aversion

Eeckhoudt and Schlesinger (2008) etc. explore the effect of the changes in risk on precautionary saving model as following: max EW = u(w0 − s) + π

1 Eu(w1 + s + ˜). 1+δ

(4)

Where, ˜ is a zero-mean risk. For this precautionary saving problem, let π = s and x ˜(s) = s + ˜ in model (1), then the increase of s improves the stochastic distribution of future risk x ˜(s) in the 1 sense of FSD. Actually, for one risk neutral DM, the first order condition is −1 + 1+δ (Ex(s))0s =

−1 +

1 1+δ

≤ 0 when δ ≥ 0, thus Proposition 4.2 implies

Corollary 5.3 For the precautionary saving model (4), linearly-restricted more Ross risk aversion guarantees more optimal precautionary saving. Kimball (1990) uses the coefficient of risk prudence to examine precautionary saving premium; Liu (2014) generalizes Kimball’s (2009) work to the case of the nth-degree stochastic dominance. However, they assume different DMs have the same saving level in the absence of future risk. Corollary 5.3 doesn’t consider the precautionary saving premium. Instead, we pay attention to the relationship between the optimal precautionary saving level with different risk aversion strengths and find linearly-restricted more Ross risk aversion induces more optimal precautionary saving.

6

Acknowledgements

The authors would like to thank the editor and an anonymous referee for their helpful comments and suggestions. This work is supported by the National Natural Science Foundation of China with grant numbers 71401074, 71231005, the Fundamental Research Funds for the Central Universities under Research Project No. NS2018048.

7

Appendix

Via repeatedly integration by parts, we can re-express model (1) as Z x¯ X 1 n−1 1 i (i) i+1 n max u(w0 − π) + (−1) u (w1 + x ¯)F (¯ x|π) + (−1) u(n) (w1 + x)F n (x|π)dx. π 1 + δ i=0 1+δ x

6

Then πu satisfies the first-order condition: −u0 (w0 − πu ) +

7.1

Z x¯ X 1 n−1 1 (−1)n u(n) (w1 + x)Fπn (x|πu )dx =(5) 0. (−1)i u(i) (w1 + x ¯)Fπi+1 (¯ x|πu ) + 1 + δ i=1 1+δ x

The proof of Proposition3.4

If v is more kth degree Ross risk averse than u, k = 2, 3, · · · , n, then by considering the first order condition, we know πv ≥ πu ⇔ −v 0 (w0 − πu ) +

Z x¯ X 1 n−1 1 v (n) (w1 + x)Fπn (x|πu )dx ≥ 0 (−1)i v (i) (w1 + x ¯)Fπi+1 (¯ x|πu ) + (−1)n 1 + δ i=1 1+δ x

⇔ −λu0 (w0 − πu ) + −φ0 (w0 − πu ) + ⇔ −φ0 (w0 − πu ) +

Z x¯ X 1 n−1 1 (−1)n λu(n) (w1 + x)Fπn (x|πu )dx (−1)i λu(i) (w1 + x ¯)Fπi+1 (¯ x|πu ) + 1 + δ i=1 1+δ x

Z x¯ X 1 1 n−1 (−1)i φ(i) (w1 + x ¯)Fπi+1 (¯ x|πu ) + (−1)n φ(n) (w1 + x)Fπn (x|πu )dx ≥ 0 1 + δ i=1 1+δ x

Z x¯ X 1 n−1 1 (−1)i φ(i) (w1 + x ¯)Fπi+1 (¯ x|πu ) + (−1)n φ(n) (w1 + x)Fπn (x|πu )dx ≥(6) 0. 1 + δ i=1 1+δ x

In (6), the second equivalence is gotten by Theorem 3.3; the last equivalence is gotten by (5). It is evident to show both Definition 3.1 and Theorem 3.3 guarantee (6). Thus, if v is more kth-degree Ross risk averse than u, k = 2, 3, · · · , n, then πv ≥ πu . We show “only if” by constructing a contradiction. Suppose πv ≥ πu , but there exists (m)

(m)

(x) (x) ¯] such that (−1)n−1 uu0 (y) > (−1)n−1 v v0 (y) , ∀x, y ∈ [α, β] for some an interval[α, β] ⊂ [x, x

m(i.e. (−1)m φ(m) (x) > 0 for x ∈ [α, β]). We may choose x ˜(π) such that Fπm (x|π) < 0 on [α, β], Fπm (x|π) = 0 elsewhere and Fπk (x|π) = 0 for all k 6= m and x ∈ [x, x ¯]. Then the above inequality(6) is incorrect. Thus, we prove “only if”.

7.2

The proof of Proposition4.2

When v(x) is linearly-restricted more Ross risk averse than u(x), we have πv ≥ πu holds if Z

x ¯ 1 v 0 (w1 + x)Fπ (x|πu )dx 1+δ x Z x¯ Z x¯ b 1 0 0 u (w1 + x)Fπ (x|πu )dx − b − Fπ (x|πu )dx = −λu (w0 − πu ) − λ 1+δ x 1+δ x b = −b − x|π) ≥ 0. F 2 (¯ 1+δ π

−v 0 (w0 − πu ) −

1 This can be gotten by both − 1+δ Fπ2 (¯ x|π) ≤ 1 and b ≤ 0.

7

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7.3

The proof of Proposition4.4

When v(x) is quadratically-restricted more Ross risk averse than u(x), we know πv ≥ πu holds if 1 −v (w0 − πu ) − 1+δ 0

Z x¯ x

v 0 (w1 + x)Fπ (x|πu )dx Z x¯

Z

x ¯ 2c xFπ (x|πu )dx 1+δ x x 1 2c = −b − 2c(w0 − πu ) − (b + 2c¯ x) F 2 (¯ x|πu ) + F 3 (¯ x|πu ) 1+δ π 1+δ π 1 x ¯ 2c = −b[1 + Fπ2 (¯ x|πu )] − 2c(w0 − πu + Fπ2 (¯ x|πu )) + F 3 (¯ x|πu ) ≥ 0. 1+δ 1+δ 1+δ π

= −b − 2c(w0 − πu ) −

In the inequality (8), −b[1 + by 1 +

1 2 x|π) 1+δ Fπ (¯

b 1+δ

Fπ (x|πu )dx −

1 2 x|π )] − 2c(w u 0 1+δ Fπ (¯

− πu +

x ¯ 2 x|π )) u 1+δ Fπ (¯

≥ 0 can be guaranteed

x|π) ≤ 0, (8) holds since c ≤ 0. ≥ 0 and b + 2cx ≤ 0 for all x. Then, if Fπ3 (¯

Thus we complete the proof.

8

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