Higher order risk attitudes and prevention under different timings of loss

Abstract

This paper provides experimental evidence of the role of higher order risk attitudes—especially prudence—in prevention behavior. Prudence, under an expected utility framework, increases (decreases) self-protection effort compared to the risk neutral level when the risk of losing part of an income exists in a future (the same) period. Motivated by these predictions that give the exact test on prudence, an experiment was designed where subjects go through higher order risk attitude elicitation and make a self-protection decision. In contrast to the expected utility theory, the observed efforts are less than the risk neutral level, regardless of the timing of loss. This violation of expected utility predictions can be explained by probability weighting.

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Notes

  1. 1.

    See also Gollier et al. (2013) for a survey of higher order risk theory.

  2. 2.

    A few studies note how prudence affects behavior in experimental games (Kocher et al. 2015; Cardella and Kitchens 2017; and Krieger and Mayrhofer 2017).

  3. 3.

    Gollier (2004) also explains this role of prudence in Chapter 16.

  4. 4.

    The same assumption is made by Menegatti (2009) and Eeckhoudt and Gollier (2005).

  5. 5.

    1 ECU=20 KRW in the Seoul National University sessions.

  6. 6.

    Through the comparative statics presented in Sect. 2 (Propositions 1 and 2), neither result restricts the functional form of the probability of loss. Our functional form \(p = 1/(\,1 + ke)\)\(,\,k > 0\) has two features. First, the odds against loss proportionally increase in effort (Note \(\,ke = \,(1 - p)/p\)). Second, a simple solution can be derived for the risk-neutral optimal effort \(e_{n} = 1/k\) and the size of loss \(\,d = 4/k\).

  7. 7.

    We provide the screenshots for feedback depending on treatments in “Online Appendix A”.

  8. 8.

    In “Online Appendix”, we explain the details of the process in Part 3 and its instructions.

  9. 9.

    We also tested whether Osaka and Seoul subjects differ in risk attitudes, as reported in “Online Appendix B”. There is no significant difference in risk attitudes between Osaka and Seoul subjects.

  10. 10.

    Instructions and an explanation of the time discount variable are in “Online Appendix A”.

  11. 11.

    Regarding the time discount variable, subjects who discount more have a tendency to exert more effort in C treatment, although the mechanism is unclear because C treatment is not an intertemporal choice problem.

  12. 12.

    “Online Appendix C” reports the distribution for our elicited time preference and how the risk neutral level was recomputed considering time preference. We also confirm that Menegatti’s (2009) theoretical result holds after considering time discounting.

  13. 13.

    See Stott (2006) and Schmidt et al. (2008) for existing variations of the value and probability weighting functions.

  14. 14.

    Eeckhoudt et al. (2017) also assume linear utility.

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Acknowledgements

We sincerely thank Charles Noussair for his continuous encouragement and helpful suggestions, which improved our paper significantly. We thank Soo Hong Chew, Syngjoo Choi, Nobuyuki Hanaki, Stefan Trautmann, Songfa Zhong, the participants of 2017 ESA North American Meeting, the University of Arizona experimental reading group, and the SURE workshop at Seoul National University. Sebastian Ebert kindly shared with us their z-Tree programs. Masuda completed most of this study during his visit to the Economic Science Laboratory at the University of Arizona. We gratefully acknowledge the financial support received from JSPS KAKENHI Grant Number 17K13701, 15H05728, the Keihanshin Consortium for Fostering the Next Generation of Global Leaders in Research (K-CONNEX), the John-Mung Overseas Program, and the Murata Science Foundation. Lee acknowledges funding from the BK21Plus Program of the Ministry of Education and National Research Foundation of Korea (NRF-21B20130000013).

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Correspondence to Takehito Masuda.

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Appendix A. Proofs

Appendix A. Proofs

Proof of Proposition 3

(1) Future loss. Take any \(\upgamma \in (0,1]\). An \(\upgamma\)-Prelec player chooses effort e to maximize

$$\,\,\,\,y - e + w^{\upgamma } (q)z + (1 - w^{\upgamma } (q))(z - d) \equiv P^{f,\upgamma } (e),$$

where \(q = q(e) = 1 - p(e) = ke/(1 + ke)\). First, we check the first order condition.

$$\frac{{dP^{f,\upgamma } }}{de} = - 1 + d\frac{{dw^{\upgamma } }}{dq}\frac{dq}{de}.$$

By substituting \(dw^{\upgamma } /dq = (\upgamma ( - \ln q)^{\upgamma - 1} /q) \cdot w^{\upgamma } (q)\), \(p(e_{n} ) = 1/2,\) and \(- p^{\prime}(e_{n} ) = 1/d,\)

$$\begin{aligned} \frac{{dP^{f,\upgamma } }}{de}|_{{e = e_{n} }} = \,\, - 1 + d \cdot \upgamma \frac{{( - \ln (1/2))^{\upgamma - 1} }}{1/2} \cdot w^{\upgamma } (1/2) \cdot \frac{1}{d} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - 1 + \cdot 2\upgamma (\ln 2)^{\upgamma - 1} \cdot \exp ( - (\ln 2)^{\upgamma } ) \hfill \\ \end{aligned}$$

Since

$$\frac{d}{d\upgamma }\{ \upgamma (\ln 2)^{\upgamma - 1} \cdot \exp ( - (\ln 2)^{\upgamma - 1} \} = { \exp }\left( { - (\ln 2)^{\upgamma } } \right)(\ln 2)^{ - 1 + \upgamma } \left( {2 - 2\upgamma \left( { - 1 + (\ln 2)^{\upgamma } } \right)\ln (\ln 2)} \right) > 0$$

by \(0 < \upgamma \le 1,0 < (\ln 2)^{\upgamma } \le 1\,{\text{and - 1}} < \ln (\ln 2) < 0\), we have

$$\frac{{dP^{f,\upgamma } }}{de}|_{{e = e_{n} }} \le \frac{{dP^{f,1} }}{de}|_{{e = e_{n} }} = 0\,{\rm for\,any}\, \upgamma \in (0,1)$$

Second, we check the second order condition. Since

$$\frac{{d^{2} P^{f,\upgamma } }}{{de^{2} }} = \frac{{d \cdot { \exp }\left( { - \left( { - \ln q} \right)^{\upgamma } } \right)\upgamma \left( { - \ln q} \right)^{ - 2 + \upgamma } \left( {1 + \upgamma \left( { - 1 + \left( { - \ln q} \right)^{\upgamma } } \right) + \left( {1 + 2ke} \right)\ln q} \right)}}{{e^{2} \left( {1 + ek} \right)^{2} }}$$

and ke = q/(1-q), it suffices to show \(h(q,\upgamma ) = 1 + \upgamma \left( { - 1 + \left( { - \ln q} \right)^{\upgamma } } \right) + \frac{1 + q}{1 - q}\ln q \le 0\) for all \(q \in (0,1]\) and \(\upgamma \in (0,1]\).

Let e be Napier’s constant.

Case 1

\(q \in (0,{\mathbf{e}}^{ - 1} ]\)

Since

$$\frac{\partial h}{\partial \upgamma }(q,\upgamma ) = - 1 + \left( { - \ln q} \right)^{\upgamma } + \upgamma \left( { - \ln q} \right)^{\upgamma } \ln \left( { - \ln q} \right) \ge 0\quad{\text{for}}\,{\text{all}}\quad q \in (0,q*],$$

\(h(q,\upgamma ) \le h(q,1) = 2q\ln q/(1 - q)\) for all \(q \in (0,{\mathbf{e}}^{ - 1} ]\) and \(\upgamma \in (0,1]\). Hence,\(h(q,\upgamma ) \le h(q,1) < \mathop {\lim }\limits_{q \to 0} h(q,1) = \mathop {\lim }\limits_{q \to 0} \frac{2q\ln q}{1 - q} = 2 \cdot 1 \cdot \mathop {\lim }\limits_{q \to 0} q\ln q = 2 \cdot \left( { - \mathop {\lim }\limits_{r \to \infty } \frac{\ln r}{r}} \right) = 0\quad {\text{for}}\,{\text{all}}\,\)\(q \in (0,q*]\) and \(\upgamma \in (0,1]\).

Case 2

\(q \in ({\mathbf{e}}^{ - 1} ,1]\)

Since

$$\frac{\partial h}{\partial \upgamma }(q,\upgamma ) < 0 \quad {\rm for\,all } q \in [q*,1],$$

\(h(q,\upgamma ) \le h(q,0) = 1 + \frac{1 + q}{1 - q}\ln q\) for all \(q \in ({\mathbf{e}}^{ - 1} ,1]\) and \(\upgamma \in (0,1]\). Hence,

$$h(q,\upgamma ) \le h(q,0) \le \mathop {\lim }\limits_{q \to 1} h(q,0) = 1 + \mathop {\lim }\limits_{q \to 1} \frac{1 + q}{1 - q}\ln q = 1 - \mathop {\lim }\limits_{q \to 1} (1 + q)\mathop {\lim }\limits_{s \to 0} \frac{\ln (s + 1)}{s} = 1 - 2 = - 1.$$

Hence, the objective function is concave.

(2) Current loss. The same argument holds by linearity of valuation function of money. ||

Proof of observation

Take any a, b, z with a > b. \(L = (a,.50;b + z,.25;b - z),R = (a + z,.25;a - z,.25;b).\)

Suppose a > b + z.

$$\begin{aligned} & V(R) = w(.25)(a + z) + \{ w(.5) - w(.25)\} (a - z) + \{ w(1) - w(.5)\} b \\ & V(L) = \,w(.5)a + \{ w(.75) - w(5)\} (b + z) + \{ w(1) - w(.75)\} (b - z) \end{aligned}$$

Then

$$\begin{aligned} V(R) - V(L) &= w(.25)\{ (a + z) - (a - z)\} + w(.5)\{ (a - z) - b - a + (b + z)\} \\ &\quad + w(.75)\,\{ (b - z) - (b + z)\} + w(1)\{ b - (b - z)\} \\ & = (2w(.25) - 2w(.75)\, + 1)z \ge 0. \end{aligned}$$

The last inequality holds by \(w(.75) - w(.25)\, \le 0.5\) for any Prelec function.

Suppose next a < b + z.

$$\begin{aligned} & V(R) = w(.25)(a + z) + \{ w(.75) - w(.25)\} b + \{ w(1) - w(.75)\} (a - z) \\ & V(L) = \,w(.25)(b + z) + \{ w(.75) - w(.25)\} a + \{ w(1) - w(.75)\} (b - z) \end{aligned}$$

Then

$$\begin{aligned} V(R) - V(L) &= w(.25)\{ (a + z) - (b + z) - (b - a)\} + w(.75)\{ b - (a - z) - a + (b - z)\} \\&\quad + w(1)\{ (a - z) - (b - z)\} \\ &= (2w(.25) - 2w(.75) + 1)(a - b) \ge 0.\end{aligned}$$

The last inequality holds by \(w(.75) - w(.25)\, \le 0.5\) for any Prelec function.||

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Masuda, T., Lee, E. Higher order risk attitudes and prevention under different timings of loss. Exp Econ 22, 197–215 (2019). https://doi.org/10.1007/s10683-018-9588-x

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Keywords

  • Higher order risk attitudes
  • Prudence
  • Prevention
  • Timing of loss
  • Probability weighting

JEL Classification

  • C91
  • C92
  • D81
  • E21