On relative and partial risk attitudes : theory and implications

Transcription

1 On relative and partial risk attitudes : theory and implications - Henry CHIU (University of Manchester, Economics school of social sciences, UK) - Louis EECKHOUDT (IESEG, School of management, LEM, Lille) - Béatrice REY (Université Lyon, Laboratoire SAF) 00.9 Laboratoire SAF 50 Avenue Tony Garnier Lyon cedex 07

2 On Relative and Partial Risk Attitudes: Theory and Implications W. Henry Chiu Economics, School of Social Sciences University of Manchester, Manchester, M3 9PL. U.K. Louis Eeckhoudt IESEG School of Management, LEM, Lille, France and CORE, 34 Voie du Roman Pays, 348 Louvain-la-Neuve, Belgium Beatrice Rey Actuarial Institute of Lyon, University Claude Bernard, 50 avenue Tony Garnier, Lyon, France April 9, 00 Abstract This paper develops context-free interpretations for the relative and partial N-th degree risk attitude measures and show that various conditions on theses measures are utility characterizations of the effects of scaling general stochastic changes in different settings. It is then shown that these characterizations can be applied to generalize comparative statics results in a number of important problems, including precautionary savings, optimal portfolio choice, and competitive firms under price uncertainty. Key Words: N-th degree risk, stochastic dominance, relative risk aversion, comparative statics, risk apportionment. JEL Classification: D8. We are grateful to an anonymous referee for very helpful comments on earlier drafts of this paper.

3 Introduction For many years in numerous theoretical and empirical papers in finance as well as in economics, the level and the behavior of the relative and partial risk aversion measures have played a central role. Specifically, letting u(x) be a utility function for wealth, the functions xu 00 (x)/u 0 (x) and xu 00 (x + w)/u 0 (x + w) aredefined respectively by Pratt (964) and Menezes and Hanson (970) to be measures of relative and partial risk aversion. Apart from their well-known properties in relation to the risk premium demonstrated by Pratt (964) and Menezes and Hanson (970), whether the values of these measures are bounded above or below by unity has been shown to be important in deriving definitive comparative statics results under uncertainty in a variety of settings (Fishburn and Porter (976), Cheng, Magill, and Shafer (987), Hadar and Seo (990)). Anumberof authors further show that whether the relative prudence measure xu 000 (x)/u 00 (x) is uniformly larger or smaller than is important in determining another disparate set of comparative statics results (Rothschild and Stiglitz (97), Hadar and Seo (990), Chiu and Madden (007), Choi, Kim, and Snow (00)). Most recently, Eeckhoudt and Schlesinger (008) show that whether an individual will save more in response to an N-th degree risk increase in future interest rate depends on whether what they term the relative N-th degree risk aversion measure is uniformly larger or smaller than N. The concept of an N-th degree risk increase introduced by Ekern (980) encompasses the concepts of a first-degree stochastic dominant deterioration, a mean-preserving increase in risk (Rothschild and Stiglitz (970)), a downside risk increase (Menezes, Geiss, and Tressler (980)), and an outer risk increase (Menezes and Wang (005)) as special cases with N being,, 3, and 4 respectively. Correspondingly, the measures of relative risk aversion and relative prudence are the relative N-th degree risk aversion measures with N being and respectively. The existing theoretical results thus suggest that for the relative N-th degree risk aversion measure, N is of significance as a kind of benchmark value and a better understanding of conditions relating the relative N-th degree risk aversion measure to its benchmark value can lead not only to a unifying interpretation of existing comparative statics results in various contexts but also to their possible generalization to cases involving higher-order risk changes. Adopting an approach For a review of issues on empirically estimating the magnitude of relative risk aversion and available evidence, see Meyer and Meyer (005).

4 used by Eeckhoudt and Schlesinger (006) for interpreting the signs of successive derivatives of a von-neuman-morgenstern utility function, Eeckhoudt, Etner, and Schroyen (009) give interpretations of the benchmark values of the relative risk aversion and relative prudence measures that are pertinent to the analysis of risky situations. Broadly following the same approach, this paper shows that to understand the meanings and significance of the benchmark values for the relative N-th degree risk aversion measure and the analogous partial N-th degree risk aversion measure is to understand the effects of scaling a general N-th degree risk increase. More specifically, we show first (in Section 3) that various conditions on the relative and partial N-th degree risk aversion measures are utility characterizations of the effects of scaling general stochastic changes in different settings. We then demonstrate (in Section 4) the usefulness of these context-independent interpretations of the risk aversion measures by showing that in a number of important economic problems how the optimal choice changes in response to a general risk change in a key parameter is determined by the effect of scaling the risk change in different settings and thus straightforward applications of these results lead to significant generalizations of existing comparative statics results in these problems. The specific economic problems we study in detail include those of precautionary savings, optimal portfolio choice, and competitive firms under price uncertainty. The rest of the paper is organized as follows. Section sets out the basic definitions and wellknown results on N-degree risk increases and stochastic dominance. Section 3 presents the utility characterizations of scaling stochastic changes and their interpretations. Section 4 shows that the utility characterizations can be readily applied to generalize comparative statics results. Section 5 concludes with brief remarks on other applications. Preliminaries and Partial and Relative Risk Attitude Measures Throughout the paper, random variables x, ỹ, etc. are assumed positive unless indicated otherwise and bounded above with probability one and their (cumulative) distribution functions are denoted by F x,fỹ, etc. For a distribution function F x (x), define F x (x) =F x(x) and F x n+ (x) = R x 0 F x ñ (y)dy for all x>0andalln {,,...}. Relative risk aversion has often been interpreted as the elasticity of marginal utility with respect to wealth. Such an interpretation, however, seems more pertinent to a risk-free situation and lacks relevant intuition. 3

5 The standard notions of Nth-degree stochastic dominance and N-th degree risk increases (Ekern (980) are defined as follows. Definition (i) x dominates ỹ by Nth-degree stochastic dominance (NSD) if F Ñ x (x) F Ñ y (x) for all x>0 where the inequality is strict for some x. (ii) ỹ is an N-thdegreeriskincreaseof x if F Ñ x (x) F Ñ y (x) for all x>0 where the inequality is strict for some x and Fx ñ(m) =F y ñ(m) for n =,...,N where M>0 is such that F x (M) =F ỹ (M) =. As pointed out by Ekern (980), the condition Fx ñ(m) =F y ñ (M) forn =,...,N (where M is a number no smaller than the upper bounds of the supports of x and ỹ) meansthefirst (N ) moments of x and ỹ are equal. Thus ỹ being a first degree risk increase of x is clearly the same as x dominating ỹ by first-degree stochastic dominance (FSD), and a second-degree risk increase is equivalent to a mean-preserving increase in risk as defined by Rothschild and Stiglitz (970). A third-degree risk increase, on the other hand, is equivalent to a downside risk increase as defined by Menezes, Geiss and Tressler (980), which corresponds to a dispersion transfer from higher to lower wealth levels and implies a decrease in skewness as measured by the third central moment. A fourth-degree risk increase is further equivalent to what Menezes and Wang (005) define to be an increase in outer risk, which corresponds to a dispersion transfer from the center of a distribution to its tails while maintaining its mean, variance and skewness (i.e., third central moment). The well-known characterizing properties of these concepts in the Expected Utility (EU) framework are summarized as follows. 3 Lemma (i) For all x and ỹ such that ỹ is an N-thdegreeriskincreaseover x, Eu(ỹ) ( ) Eu( x) if and only if ( ) N u (N) (x) ( ) 0for all x>0. (ii) For all x and ỹ such that x dominates ỹ via NSD, Eu(ỹ) ( ) Eu( x) if and only if ( ) n u (n) (x) ( ) 0for all x>0 and n =,,...,N. 3 A proof of the result can be found in Ingersoll (987) for example. 4

6 Given a Von Neumann-Morgenstern utility function u( ), we shall term the functions x u(n+) (x) u (N) (x) and x u(n+) (x + w) u (N) (x + w) respectively relative N-th degree risk aversion measure (following Eeckhoudt and Schlesinger (008)) and partial N-th degree risk aversion measure. Clearly, the measures of relative and partial risk aversion defined respectively by Pratt (964) and Menezes and Hanson (970) correspond to the case where N = or measures of relative and partial first degree risk aversion, and what has been termed the relative prudence measure corresponds to the measure of relative second degree risk aversion. Eeckhoudt and Schlesinger (008) have shown that whether an individual will save more in response to an N-th degree risk increase in future interest rate depends on whether x u(n+) (x) u (N) (x) N. In the sections that follow, we will seek to give a context-free interpretation to such a condition 4 and to analogous conditions on the partial N-th degree risk aversion measure and show that a better understanding of these conditions leads to immediate generalizations of a range of comparative statics results under uncertainty. 3 The Effects of Scaling Stochastic Changes As is summarized in Lemma, the equivalence between the sign of u (N) and an EU maximizer s preferences over two random prospects where one is an N-th degree risk increase over the other is well-established and well-understood. But what is the significance of the sign of u (N+) in determining choice between options involving two random prospects where one is an N-th degree risk increase over the other? Eeckhoudt and Schlesinger (006) define prudence in terms of preferences over the lotteries A 0 and B 0 4 That is, we aim to give an interpretation of the condition that is in terms of preferences over lotteries and is thus independent of any specific decisioncontext. 5

7 ( x E x) 0 A H H 0 B 0 HH k HH ( x E x) k where k is a positive constant and show that in the EU framework A 0 º B 0 if and only if u That is, u means a preference to bear the harm of a zero-mean risk ( x E x) and that of a sure loss k separately. Since a zero-mean random variable is a second-degree risk increase over 0, we may similarly interpret the significance of the sign of u (N+) in determining choice between options involving two random prospects where one is an N-th degree risk increase over the other. Consider the following pair of lotteries where ỹ is an N-th degree risk increase over x and w dominates w via FSD. ỹ + w A H B ỹ + w H HH x + w HH x + w An individual preferring A to B means that he prefers bearing the greater N-th degree risk when he is richer (which can be in the non-stochastic as well as the stochastic sense), or equivalently he prefers to disaggregate the harm of a greater N-th degree risk and that of lower wealth. In the EU framework, A º B if and only if Eu(ỹ + w )+ Eu( x + w ) Eu(ỹ + w )+ Eu( x + w ) Equivalently Eu( x + w ) Eu(ỹ + w ) Eu( x + w ) Eu(ỹ + w ) which is true if and only if Eu 0 ( x) Eu 0 (ỹ) 0,whichinturn,giventhatỹ is an N-th degree risk increase over x, is true if and only if ( ) N+ u (N+) 0. That is, we have the following. 6

8 Theorem 5 In the EU framework, (i) Given that ỹ is an N-thdegreeriskincreaseover x and w dominates w via FSD, A º B if and only if ( ) N+ u (N+) (x) 0 for all x>0. (ii) Given that x dominates ỹ via NSD and w dominates w via FSD, A º B if and only if ( ) n+ u (n+) (x) 0 for all x>0 and n =,,...,N. Thus, just as u means prudence or a preference for bearing a second-degree risk increase with a higher level of wealth, ( ) N+ u (N+) 0 can be interpreted as prudence with respect to an N-th degree risk increase or a preference for bearing a N-degree risk increase with a higher level of wealth. This explains the generalization of Leland s (968) original result on precautionary savings in Eeckhoudt and Schlesinger (008, Corollary ). Now consider the pair of lotteries, A (w) andb (w), where w 0, k <k and ỹ is an N-th degree risk increase over x. A H (w) k ỹ + w k ỹ + w B H (w) HH k x + w HH k x + w Should an individual who dislikes an N-thdegreeriskincreasepreferA (w) orb (w)? Since k /k >, the more N-th degree risky ỹ rather than x is scaled up in A (w). Intuitively, this appears to mean that with A (w) the individual gets a larger increase in risk which he dislikes and hence B (w) should be preferred. 6 Perhaps less obviously, however, since x and ỹ are positive random variables, a scaling up of either of them causes a shift in the distribution upwards, which is equivalent to an increase in wealth. That is, with A (w), the individual would be bearing the greater risk when he is richer. Therefore, if he also prefers to bear a greater N-thdegreeriskwhenheisricher, the choice between A (w) andb (w) will be governed by the relative strengths of two opposing effects. The first can be called (N-th degree) risk aversion effect which works to make A (w) less 5 The result is of course a special case of the main result in Eeckhoudt, Schlesinger, and Tsetlin (009), which deals with the case where w dominates w via any degree of stochastic dominance. 6 InthecasewhereN =,ỹ being more first-degree risky than x simply means x is an FSD improvement over ỹ and we can hence equivalently say that the first-degree stochastic dominant x rather than ỹ is scaled up in B (w), which intuitively should make B (w) more attractive assuming FSD improvements are desirable or equivalently first-degree risk increases are undesirable. 7

9 attractive because the more N-th degree risky ỹ rather than x has been magnified. The second can be called apportionment effect which works to make A (w) more attractive because with A (w) the harm of a greater risk and that of a lower wealth are disaggregated, i.e., the greater risk is better apportioned. The following main result demonstrates that whether an EU maximizer prefers A (w) to B (w) is indeed determined by the relative strength of the two effects, which is encapsulated in the magnitude of the partial N-th degree risk aversion measure: The risk aversion effect is dominated by the apportionment effect if and only if the partial N-th degree risk aversion measure is larger than a benchmark value. This result will turn out to play a central role in many applications as shown in Section 4. Theorem In the EU framework, (i) Assuming ( ) n u (n) (x) 0 for all x>0 and n = N,N +,givenw 0, ỹ being an N-th degree risk increase over x and k <k, A (w) º (¹) B (w) if and only if x u(n+) (x + w) u (N) (x + w) ( ) N for all x>0. (ii) Assuming ( ) n u (n) (x) 0 for all x>0 and n =,,...N +,givenw 0, x dominating ỹ via NSD and k <k, A (w) º (¹) B (w) if and only if x u(n+) (x + w) u (N) (x + w) ( ) N for all x>0 and n =,,...,N. (A formal proof can be found in the Appendix.) The intuitive interpretation for part (ii) of the theorem is apparent given our preceding discussion and the fact that x dominates ỹ via NSD if ỹ can be obtained from x via any sequence of increases in n-th degree risk, for all positive integers n N. The characterization of the comparative attractiveness of lotteries A (0) vis a vis B (0) in terms of the utility function analogously gives interpretation to the relative N-th degree risk aversion measure and in particular to a condition such as x u(n+) (x) u (N) (x) N for all x>0. 8

10 But it is clear from their utility characterizations that A (0) ¹ B (0) actually implies A (w) ¹ B (w) for w 0sinceforw 0 x u(n+) (x) u (N) (x) N for all x>0 implies (x + w) u(n+) (x + w) u (N) (x + w) N for all x>0 whichinturnimplies x u(n+) (x + w) u (N) (x + w) N for all x>0 provided that u (N) and u (N+) are of opposite signs. On the other hand, A (w) ¹ B (w) for all w 0 clearly implies A (0) ¹ B (0). We can therefore claim the following. Lemma In the EU framework, (i) Assuming ( ) n u (n) (x) 0 for all x>0 and n = N,N +, xu(n+) (x) u (N) (x) N for all x>0 if and only if xu(n+) (x + w) u (N) (x + w) N for all x>0 and w 0. (ii) Assuming ( ) n u (n) (x) 0 for all x 0 and n =,,...,N +, xu(n+) (x) u (N) (x) N for all x>0 if and only if xu(n+) (x + w) u (N) (x + w) N for all x>0 and w 0. So far Theorem gives the utility characterization for the preference to bear a greater risk in the presence of higher levels of wealth while Theorem provides conditions for the (un)desirability of scaling up a greater risk. But as will be seen in the sequel, to better understand a number of important problems of optimal choice under uncertainty, it is important to understand the effects of scaling stochastic changes in the presence of differing initial (or background) wealth. Consider two pairs of lottery options A 3 versus B 3 and A ( w )versusb ( w ) that follow where ỹ is an N-th degree risk increase over x and k <k. 9

11 k ỹ + w k ỹ + w A H H 3 B 3 HH k x + w HH k x + w k ỹ + w A H ( w ) B ( w ) k ỹ + w H HH k x + w HH k x + w As discussed previously, whether A ( w ) is preferred to B ( w ) is determined by the desirability of scaling up the more N-th degree risky ỹ rather than x, which has an apportionment effect working in A ( w ) s favor and a counteracting risk aversion effect. By contrast, supposing first w dominates w via FSD, whether A 3 is preferred to B 3 is determined by the desirability of scaling up the more N-th degree risky ỹ rather than x inthepresenceofhigherlevelsofinitialwealth,i.e., w rather than w. Hence in addition to the opposing risk aversion effect and apportionment effect brought about by the scaling up of the more N-th degree risky ỹ rather than x, there is an additional apportionment effect working in A ( w ) s favor: the scaling up of ỹ in A 3 and x in B 3 is accompanied by higher initial levels of wealth, which reinforces the effect of the upward shift in distribution due to the up-scaling. If, on the other hand, w dominates w via FSD, the additional apportionment effect works in B 3 s favor. We can thus state the following result. Lemma 3 In the EU framework, (i) Assuming ( ) n u (n) (x) 0 for all x>0 and n = N,N +, given that ỹ is an N-th degree risk increase over x and k <k,if w dominates w via FSD, then A ( w ) ¹ B ( w ) implies A 3 ¹ B 3, and if w dominates w via FSD, then A 3 ¹ B 3 implies A ( w ) ¹ B ( w ). (ii) Assuming ( ) n u (n) (x) 0 for all x>0 and n =,,...,N +, given that x dominates ỹ via NSD and k <k,if w dominates w via FSD, then A ( w ) ¹ B ( w ) implies A 3 ¹ B 3,andif w dominates w via FSD, then A 3 ¹ B 3 implies A ( w ) ¹ B ( w ). (A formal proof can be found in the Appendix.) 0

12 Since A (w) º (¹) B (w) for all w 0 clearly implies A ( w ) º (¹) B ( w )for w being a non-negative random variable, Theorem and Lemmas and 3 imply the following. Theorem 3 In the EU framework, (i) Given that ỹ is an N-thdegreeriskincreaseover x, w dominates w via FSD, and k <k, A 3 ¹ B 3 if xu(n+) (x) u (N) (x) N and ( ) n u (n) (x) 0 for all x>0 and n = N,N + (ii) Given that x dominates ỹ via NSD ỹ, w dominates w via FSD, and k <k, A 3 ¹ B 3 if xu(n+) (x) u (n) (x) n and ( ) n u (n) (x) 0 for all x>0 and n =,,...,N + The result can also be stated in terms of the partial N-thdegreeriskaversionmeasuregiventhe equivalence shown in Lemma of the conditions xu(n+) (x) u (N) (x) N for all x>0and xu(n+) (x + w) u (N) (x + w) N for all x>0andw 0. As will be shown in the next section, Theorem 3 implies generalizations of comparative statics results in a number of settings. Our derivation of the theorem thus provides intuitive explanation for these results. 4 Applications 4. Interest Rate Risk and Precautionary Savings Consider a consumer who has a two-period planning horizon and receives a certain income stream of w 0 at date t =0andw at date t =. At date 0, the consumer must decide how much to consume and how much to save, for consumption at date t =. Any amount saved earns a rate of interest r, where we assume r >. The consumer chooses s 0 to maximize u(w 0 s)+ +δ Eu(s( + r)+w ). ()

13 The optimal s is assumed to be unique and internal. Rothschild and Stiglitz (97) consider a mean-preserving increase in risk in the distribution of ( + r). Now consider a generalization of the problem where there is an N-thdegreeriskincreasein( + r). Intuitively, the relationship between s and ( + r) in the optimization problem () indicates that whether this will lead to a higher level of savings depends on whether it is desirable to scale up the risk increase in ( + r) (inthepresence of the initial wealth w ). Lemma 4 in the Appendix formally establishes this by showing that a preference of A (w )overb (w ) in the previous section for ỹ being an N-th degree risk increase over x and k <k implies a higher level of savings when there is an N-th degree risk increase in ( + r). This, together with Theorem, in turn implies a variant of the result in Eeckhoudt and Schlesinger (008) as stated in what follows. Proposition Let s and s be the optimal levels of savings when the gross interest rate, ( + r), is equal to x and ỹ respectively. (i) Suppose ỹ is an N-thdegreeriskincreaseover x and ( ) n u (n) (x) 0 for all x > 0 and n = N,N +.Thens ( ) s if x u(n+) (x + w ) u (N) (x + w ) ( ) N for all x>0. (ii) Suppose x dominates ỹ by NSD and ( ) n u (n) (x) 0 for all x>0 and n =,,...N+.Then s ( ) s if x u(n+) (x + w ) u (n) (x + w ) ( ) n for all x>0 and n =,,...,N. Eeckhoudt and Schlesinger (008) considers the special case where w =0whiletheresultof Rothschild and Stiglitz (97) is a special case with w =0andN =. Proposition is thus a slight generalization of Proposition in Eeckhoudt and Schlesinger (008), but its derivation from Theorem in Section 3 provides a formal and yet more intuitive explanation for the conditions involving risk aversion measures, which is previously unavailable. 4. Optimal Portfolio Choice Consider an investor who has to allocate his wealth w between two assets whose returns are given by (non-negative) random variables x and r. Letting k be the amount invested in asset x, hethus

14 chooses k to maximize Eu(k x +(w k) r) () and the optimal k is assumed to be unique and internal. In the special case where r, Cheng, Magill, and Shafer (987) consider the effect on the optimal portfolio choice of an FSD improvement in x. For r being a random variable independent of x and ỹ, Hadar and Seo (990), on the other hand, consider the effects on the choice of k of stochastic changes in x, including first-degree and second-degree stochastic dominant improvements and a mean-preserving spread. Our results in the previous section imply significant generalizations of these results as we will be able to determine the effects on k of an N-th degree risk increase in x and a stochastic improvement of any degree in x. To see this intuitively, observe in the optimization problem () that if there is an N-th degree risk increase in x, choosingalargerk means scaling up the risk increase in the presence of an inferior background risk (w k) r in the sense of FSD. Thus an individual will choose to invest less in asset x in response to an N-th degree risk increase in x if he prefers B 3 to A 3 in the previous section for k <k, w dominating w via FSD, and ỹ being an N-th degree risk increase over x. Lemma 5 in the Appendix establishes this formally and thus implies the following generalization of the earlier results. Proposition Suppose ỹ and r are stochastically independent and Eu(k x +(w k) r) and Eu(kỹ + (w k) r) are maximized at k and k respectively. Then (i) k k given ỹ being an N-thdegreeriskincreaseof x if xu(n+) (x) u (N) (x) N and ( ) n u (n) (x) 0 for all x>0 and n = N,N +. (ii) k k given x dominating ỹ via NSD if xu(n+) (x) u (n) (x) n and ( ) n u (n) (x) 0 for all x>0 and n =,,...,N +. As with Theorem 3, the result can clearly also be stated in terms of the partial N-th degree risk aversion measure given the equivalence of the conditions shown in Lemma. 3

15 Proposition thus not only subsumes the results of Cheng, Magill, and Shafer (987) and Hadar and Seo (990) but also provides conditions for higher-order risk increases in the return of one of the assets to induce a lower investment in the asset. For concreteness, consider the case with N =3. This is the case where the asset return undergoes an increase in third-degree risk or downside risk as defined by Menezes, Geiss and Tressler (980) who show that such an increase implies a decrease in the skewness of the distribution as measured by its third central moment and leaves the mean and variance constant. Our result says that the investment in the asset will decrease as a result of such an increase provided that the measure xu 0000 (x)/u 000 (x), termed relative temperance by Eeckhoudt and Schlesinger (008), does not exceed three. While the importance of asset return skewness in determining asset prices has been increasingly recognized in the finance literature (see for example Harvey and Siddique (000)), the condition on the utility function for a downside risk increase in asset return to imply a lower investment has thus far not been identified. 4.3 Competitive Firm under Price Uncertainty Let w be the initial wealth of a risk averse competitive firm owner, k the firm s output, c(k)+b the cost of producing k where c(k) is increasing and strictly concave, c(0) = 0, B 0, and w B. The owner then chooses k to maximize Eu( pk c(k) B + w). (3) The optimal k is assumed to be unique and internal. This is thus the problem first considered by Sandmo (97). Cheng, Magill, and Shafer (987) consider the effect on the optimal output choice when the output price undergoes an FSD improvement. In view of Theorem 3 in the previous section, their result can be significantly generalized since intuitively if there is an N-th degree risk increase in p in the optimization problem (3), choosing a larger k means scaling up the risk increase in the presence of lower background wealth [ c(k) B + w]. Lemma 6 in the Appendix formalizes the relationship between the choice of k and the preferences over special versions of B 3 and A 3 in the previous section and thus implies the following result. Proposition 3 Suppose Eu( xk c(k) B + w) and Eu(ỹk c(k) B + w) are maximized at k 4

16 and k respectively. Then (i) k k given ỹ being an N-thdegreeriskincreaseof x if xu(n+) (x) u (N) (x) N and ( ) n u (n) (x) 0 for all x>0 and n = N,N +. (ii) k k given x dominating ỹ via NSD if xu(n+) (x) u (n) (x) n and ( ) n u (n) (x) 0 for all x>0 and n =,,...,N +. The result of Cheng, Magill, and Shafer (987, Proposition 7) is a special case of Proposition 3withN =: If x is an FSD improvement of ỹ, i.e.,ỹ is a first-degree risk increase of x, then k k whenever relative risk aversion is no larger than. As another example, the output price may become more risky in the sense of a mean-preserving spread defined by Rothschild and Stiglitz (970). This is then the special case with N = : the optimal choice of output will decrease as a result of such a risk increase in price provided that relative prudence does not exceed two. While the effect of a riskier price on the choice of output is clearly relevant and important in the theory of firm under price uncertainty, it has hitherto not been considered in the literature. 7 Proposition 3 also provides conditions for higher-order risk increases in the output price to induce a lower output. 5 Concluding Remarks In this paper, we develop context-free interpretations for the measures of relative and partial N-th degree risk aversion and show that various conditions on theses measures are utility characterizations of the effects of scaling general stochastic changes in different settings. We then apply these characterizations to generalize comparative statics results in a number of important problems, including precautionary savings, optimal portfolio choice, and competitive firms under price uncertainty. Other applications not explicitly presented can also be analogously obtained. For example, the results of Chiu and Madden (007) can be generalized to show that under appropriate restrictions on the relative N-th degree risk aversion, N-th degree stochastic dominant improvements in individuals 7 Sandmo (970) only considers the effect on output when the initially non-random price becomes uncertain. 5

17 background risks can reduce overall crime rate in a simple general equilibrium model with property crime. Similarly, the results on the optimal choice of coinsurance (Meyer (99), Dionne and Gollier (99), and Hadar and Seo (99)) can be generalized to deal with cases where the insurable loss undergoes an N-th degree stochastic deterioration. APPENDIX ProofofTheorem.In the EU framework, A (w) º (¹) B (w) ifandonlyif Eu(k x + w) Eu(k ỹ + w) [Eu(k x + w) Eu(k ỹ + w)] ( ) 0 (4) Defining Q(k, x, ỹ) =Eu(k x + w) Eu(kỹ + w) and denoting the partial derivative of Q respect to k by Q k, (4) can be further written as Q(k, x, ỹ) Q(k, x, ỹ) ( ) 0, which is clearly true if and only if Q k (k, x, ỹ) ( ) 0. Let φ(x) =xu 0 (kx + w). By Lemma, Q k (k, x, ỹ) =E xu 0 ( x + w) Eỹu 0 (ỹ + w) =Eφ( x) Eφ(ỹ) ( ) 0forall x and ỹ such that ỹ is an N-th degree risk increase over x if and only if ( ) N φ (N) (x) ( )0,which,givenx>0and( ) n u (n) 0forn = N,N +,isequivalentto xu(n+) (x + w) u (N) (x + w) ( ) N for all x>0 Also by Lemma, Eφ( x) Eφ(ỹ) ( ) 0forall x and ỹ such that x dominate ỹ via NSD if and only if ( ) n φ (n) (x) ( ) 0foralln =,,...,N,which,givenx>0and( ) n u (n) 0for n =,,...N +,isequivalentto xu(n+) (x + w) u (n) (x + w) ( ) n for all x>0andn =,,...,N. ProofofTheorem3.We only prove A 3 ¹ B 3 implies A ( w ) ¹ B ( w )forỹ being an N-th degree risk increase over x and w dominating w via FSD. The other parts of the Theorem can be proved analogously. 6

18 In the EU framework, A 3 ¹ B 3 if and only if Eu(k x + w ) Eu(k ỹ + w ) Eu(k x + w ) Eu(k ỹ + w ) (5) Since ỹ is an N-th degree risk increase of x (and hence k ỹ is an N-th degree risk increase of k x) and( ) N+ u (N+) (x) 0, we have Eu 0 (k x) Eu 0 (k ỹ) 0, which, given w dominating w via FSD, implies Eu(k x + w ) Eu(k ỹ + w ) Eu(k x + w ) Eu(k ỹ + w ) (6) (5) and (6) thus imply Eu(k x + w ) Eu(k ỹ + w ) Eu(k x + w ) Eu(k ỹ + w ) which is equivalent to A ( w ) ¹ B ( w ). Lemma 4 Let s and s be the optimal levels of savings when the gross interest rate, ( + r), is equal to x and ỹ respectively, where ỹ is an N-th degree risk increase over x or x dominates ỹ via NSD. Then s ( ) s if A (w ) º (¹) B (w ) for k <k. Proof. Let s and s be the optimal levels of savings given the gross interest rate, ( + r), being equal to x and ỹ respectively and ỹ being an N-thdegreeriskincreaseover x. Clearly, since s and s are the optimal values under x and ỹ respectively and the maxima are unique, u(w 0 s )+ +δ Eu(s x + w ) >u(w 0 s )+ +δ Eu(s x + w ) and u(w 0 s )+ +δ Eu(s ỹ + w ) >u(w 0 s )+ +δ Eu(s ỹ + w ) Summing the two sides of the inequalities and simplifying, we have Eu(s x + w )+Eu(s ỹ + w ) >Eu(s x + w )+Eu(s ỹ + w ) (7) 7

19 Suppose s <s.thena (w ) ¹ B (w )fork <k means Eu(s x + w )+Eu(s ỹ + w ) Eu(s x + w )+Eu(s ỹ + w ) which contradicts (7). That is, A (w ) ¹ B (w )impliess s. We can similarly show that A (w ) º B (w )impliess s. Lemma 5 Suppose A 4 and B 4 are given by k ỹ +(w k ) r A H 4 B 4 k ỹ +(w k ) r H HH k x +(w k ) r HH k x +(w k ) r where x, ỹ and r are independent and ỹ is an N-thdegreeriskincreaseof x or x dominates ỹ via NSD, and Eu(k x +(w k) r) and Eu(kỹ +(w k) r) are maximized at k and k respectively. Then k ( ) k if A 4 ¹ (º) B 4 for k <k. Proof. Since Eu(k x +(w k)r) andeu(kỹ +(w k)r) are maximized at k and k respectively and the maxima are unique, by the same argument used in Lemma 4, we must have Eu(k x +(w k )r)+eu(k ỹ +(w k )r) >Eu(k x +(w k )r)+eu(k ỹ +(w k )r) (8) If k < (>)k,thena 4 ¹ (º) B 4 for k <k means Eu(k x +(w k )r)+eu(k ỹ +(w k )r) Eu(k x +(w k )r)+eu(k ỹ +(w k )r) which contradicts (8). ProofofProposition.Since (w k ) r dominates (w k ) r via FSD for k <k in A 4 and B 4 in Lemma 5, the result is implied by the lemma and Theorem 3. 8

20 Lemma 6 Suppose A 5 and B 5 are given by k ỹ c(k ) B + w A H 5 B 5 k ỹ c(k ) B + w H HH k x c(k ) B + w HH k x c(k ) B + w where ỹ is an N-thdegreeriskincreaseof x or x dominates ỹ via NSD, and Eu( xk c(k) B + w) and Eu(ỹk c(k) B+w) are maximized at k and k respectively. Then k ( ) k if A 5 ¹ (º) B 5 for k <k. (The proof is analogous to that of Lemma 5.) ProofofProposition3.Since [ c(k ) B + w] > [ c(k ) B + w] fork <k in A 5 and B 5 in Lemma 6, Theorem 3 and the lemma imply the result. REFERENCES Cheng, H., M. J. Magill, and W. Shafer. (987): Some Results on Comparative Statics under Uncertainty, International Economic Review, 8 (), Chiu, W. Henry and Paul Madden (007): Crime, Punishment, and Background Risks, Journal of Economic Behavior and Organization, 6, Choi, G., I. Kim and A. Snow (00): Comparative Statics Predictions for Changes in Uncertainty in the Portfolio and Savings Problem, Bulletin of Economic Research, 53,6-7. Diamond, Peter A. and Joseph E. Stiglitz. (974): Increases in Risk and in Risk Aversion, Journal of Economic Theory, 8, Dionne, Georges and Christian Gollier (99): Comparative Statics Under Multiple Sources of Risk with Applications to Insurance Demand, Geneva Papers on Risk and Insurance Theory 7 (), -33. Eeckhoudt, Louis, Johanna Etner and Fred Schroyen (009): The Value of Relative Risk Aversion and Prudence: A Context-Free Interpretation, Mathematical Social Sciences, 58(), -7. 9

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