On relative and partial risk attitudes : theory andimplications

- Henry CHIU (University of Manchester, Economics school of social sciences, UK)- Louis EECKHOUDT (IESEG, School of management, LEM, Lille)- Béatrice REY (Université Lyon 1, Laboratoire SAF)

2010.29

Laboratoire SAF – 50 Avenue Tony Garnier - 69366 Lyon cedex 07 http://www.isfa.fr/la_recherche

On Relative and Partial Risk Attitudes:

Theory and Implications ∗

W. Henry Chiu

Economics, School of Social Sciences

University of Manchester, Manchester, M13 9PL. U.K.

E-mail: henry.chiu@manchester.ac.uk

Louis Eeckhoudt

IESEG School of Management, LEM, Lille, France

and CORE, 34 Voie du Roman Pays,

1348 Louvain-la-Neuve, Belgium

Beatrice Rey

Actuarial Institute of Lyon,

University Claude Bernard,

50 avenue Tony Garnier, 69007 Lyon, France

April 19, 2010

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 characteriza-

tions 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: D81.

∗We are grateful to an anonymous referee for very helpful comments on earlier drafts of this paper.

1

1 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 −xu00(x)/u0(x) and

−xu00(x+w)/u0(x+w) are defined respectively by Pratt (1964) and Menezes and Hanson (1970) to

be measures of relative and partial risk aversion. Apart from their well-known properties in relation

to the risk premium demonstrated by Pratt (1964) and Menezes and Hanson (1970), 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 (1976), Cheng, Magill, and Shafer (1987), Hadar and Seo (1990)). 1 A number of

authors further show that whether the relative prudence measure −xu000(x)/u00(x) is uniformly larger

or smaller than 2 is important in determining another disparate set of comparative statics results

(Rothschild and Stiglitz (1971), Hadar and Seo (1990), Chiu and Madden (2007), Choi, Kim, and

Snow (2001)). Most recently, Eeckhoudt and Schlesinger (2008) 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 (1980) encompasses the concepts

of a first-degree stochastic dominant deterioration, a mean-preserving increase in risk (Rothschild

and Stiglitz (1970)), a downside risk increase (Menezes, Geiss, and Tressler (1980)), and an outer

risk increase (Menezes and Wang (2005)) as special cases with N being 1, 2, 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 1 and 2 respectively.

The existing theoretical results thus suggest that for the relative N -th degree risk aversion mea-

sure, 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

1For a review of issues on empirically estimating the magnitude of relative risk aversion and available evidence, seeMeyer and Meyer (2005).

2

used by Eeckhoudt and Schlesinger (2006) for interpreting the signs of successive derivatives of a

von-Neuman-Morgenstern utility function, Eeckhoudt, Etner, and Schroyen (2009) give interpreta-

tions of the benchmark values of the relative risk aversion and relative prudence measures that are

pertinent to the analysis of risky situations. 2 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 2 sets out the basic definitions and well-

known 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.

2 Preliminaries and Partial and Relative Risk Attitude Measures

Throughout the paper, random variables x, y, etc. are assumed positive unless indicated otherwise

and bounded above with probability one and their (cumulative) distribution functions are denoted

by Fx, Fy, etc. For a distribution function Fx(x), define F1x (x) = Fx(x) and

Fn+1x (x) =

R x0 Fn

x (y)dy for all x > 0 and all n ∈ {1, 2, . . .}.

2Relative risk aversion has often been interpreted as the elasticity of marginal utility with respect to wealth. Suchan interpretation, however, seems more pertinent to a risk-free situation and lacks relevant intuition.

3

The standard notions of Nth-degree stochastic dominance and N -th degree risk increases (Ekern

(1980) are defined as follows.

Definition 1

(i) x dominates y by N th-degree stochastic dominance (NSD) if FNx (x) ≤ FN

y (x) for all x > 0 where

the inequality is strict for some x.

(ii) y is an N-th degree risk increase of x if FNx (x) ≤ FN

y (x) for all x > 0 where the inequality is strict

for some x and Fnx (M) = Fn

y (M) for n = 2, . . . ,N where M > 0 is such that F 1x (M) = F 1y (M) = 1.

As pointed out by Ekern (1980), the condition Fnx (M) = Fn

y (M) for n = 2, . . . , N (where M is

a number no smaller than the upper bounds of the supports of x and y) means the first (N − 1)

moments of x and y are equal. Thus y being a first degree risk increase of x is clearly the same

as x dominating y 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 (1970). A

third-degree risk increase, on the other hand, is equivalent to a downside risk increase as defined

by Menezes, Geiss and Tressler (1980), 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 (2005) 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) frame-

work are summarized as follows. 3

Lemma 1

(i) For all x and y such that y is an N-th degree risk increase over x, Eu(y) ≤ (≥) Eu(x) if and

only if (−1)Nu(N)(x) ≤ (≥) 0 for all x > 0.

(ii) For all x and y such that x dominates y via NSD, Eu(y) ≤ (≥) Eu(x) if and only if (−1)nu(n)(x) ≤

(≥) 0 for all x > 0 and n = 1, 2, . . . , N .

3A proof of the result can be found in Ingersoll (1987) for example.

4

Given a Von Neumann-Morgenstern utility function u( ), we shall term the functions

−xu(N+1)(x)

u(N)(x)and − x

u(N+1)(x+ w)

u(N)(x+w)

respectively “relative N -th degree risk aversion measure” (following Eeckhoudt and Schlesinger

(2008)) and “partial N -th degree risk aversion measure”. Clearly, the measures of relative and

partial risk aversion defined respectively by Pratt (1964) and Menezes and Hanson (1970) corre-

spond to the case where N = 1 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 (2008) 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

−xu(N+1)(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 1, 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+1) in determining

choice between options involving two random prospects where one is an N -th degree risk increase

over the other? Eeckhoudt and Schlesinger (2006) define prudence in terms of preferences over the

lotteries A0 and B0

4That is, we aim to give an interpretation of the condition that is in terms of preferences over lotteries and is thusindependent of any specific decision context.

5

©©©©

©©©

HHHHHHH

©©©©

©©©

HHHHHHH12

12

(x−Ex)− k

0

−k

(x−Ex)

rrA0 B0

12

12

where k is a positive constant and show that in the EU framework A0 º B0 if and only if u000 ≥ 0.

That is, u000 ≥ 0 means a preference to bear the harm of a zero-mean risk (x−Ex) 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+1) 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 y is an N -th degree risk increase over x and w2 dominates w1

via FSD.

©©©©

©©©

HHHHHHH

©©©©

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HHHHHHH12

12

x+ w2

y + w1

x+ w1

y + w2

rrA1 B1

12

12

An individual preferring A1 to B1 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, A1 º B1 if and only if

1

2Eu(y + w2) +

1

2Eu(x+ w1) ≥

1

2Eu(y + w1) +

1

2Eu(x+ w2)

Equivalently

Eu(x+ w1)−Eu(y + w1) ≥ Eu(x+ w2)−Eu(y + w2)

which is true if and only if Eu0(x)− Eu0(y) ≤ 0, which in turn, given that y is an N -th degree risk

increase over x, is true if and only if (−1)N+1u(N+1) ≤ 0. That is, we have the following.

6

Theorem 1 5 In the EU framework,

(i) Given that y is an N-th degree risk increase over x and w2 dominates w1 via FSD, A1 º B1 if

and only if (−1)N+1u(N+1)(x) ≤ 0 for all x > 0.

(ii) Given that x dominates y via NSD and w2 dominates w1 via FSD, A1 º B1 if and only if

(−1)n+1u(n+1)(x) ≤ 0 for all x > 0 and n = 1, 2, . . . , N .

Thus, just as u000 ≥ 0 means prudence or a preference for bearing a second-degree risk increase with

a higher level of wealth, (−1)N+1u(N+1) ≤ 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 (1968) original result on precautionary savings

in Eeckhoudt and Schlesinger (2008, Corollary 1).

Now consider the pair of lotteries, A2(w) and B2(w), where w ≥ 0, k1 < k2 and y is an N -th

degree risk increase over x.

©©©©

©©©

HHHHHHH

©©©©

©©©

HHHHHHH12

12

k2x+ w

k1y + w

k1x+w

k2y + w

rrA2(w) B2(w)

12

12

Should an individual who dislikes an N -th degree risk increase prefer A2(w) or B2(w)? Since k2/k1 >

1, the “more N -th degree risky” y rather than x is “scaled up” in A2(w). Intuitively, this appears

to mean that with A2(w) the individual gets a larger increase in risk which he dislikes and hence

B2(w) should be preferred.6 Perhaps less obviously, however, since x and y 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 A2(w), the individual would be bearing the greater risk when

he is richer. Therefore, if he also prefers to bear a greater N -th degree risk when he is richer,

the choice between A2(w) and B2(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 A2(w) less

5The result is of course a special case of the main result in Eeckhoudt, Schlesinger, and Tsetlin (2009), which dealswith the case where w2 dominates w1 via any degree of stochastic dominance.

6In the case where N = 1, y being “more first-degree risky” than x simply means x is an FSD improvement over yand we can hence equivalently say that the first-degree stochastic dominant x rather than y is scaled up in B2(w), whichintuitively should make B2(w) more attractive assuming FSD improvements are desirable or equivalently first-degreerisk increases are undesirable.

7

attractive because the more N -th degree risky y rather than x has been magnified. The second can

be called “apportionment effect” which works to make A2(w) more attractive because with A2(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 A2(w)

to B2(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 2 In the EU framework,

(i) Assuming (−1)nu(n)(x) ≤ 0 for all x > 0 and n = N,N +1, given w ≥ 0, y being an N-th degree

risk increase over x and k1 < k2,

A2(w) º (¹) B2(w) if and only if − xu(N+1)(x+ w)

u(N)(x+ w)≥ (≤) N for all x > 0.

(ii) Assuming (−1)nu(n)(x) ≤ 0 for all x > 0 and n = 1, 2, . . . N + 1, given w ≥ 0, x dominating y

via NSD and k1 < k2,

A2(w) º (¹) B2(w) if and only if − xu(N+1)(x+ w)

u(N)(x+ w)≥ (≤) N for all x > 0 and n = 1, 2, . . . ,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 y via NSD if y 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 A2(0) vis a vis B2(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

−xu(N+1)(x)

u(N)(x)≤ N for all x > 0.

8

But it is clear from their utility characterizations that A2(0) ¹ B2(0) actually implies A2(w) ¹ B2(w)

for w ≥ 0 since for w ≥ 0

−xu(N+1)(x)

u(N)(x)≤ N for all x > 0

implies

−(x+ w)u(N+1)(x+w)

u(N)(x+ w)≤ N for all x > 0

which in turn implies

−xu(N+1)(x+ w)

u(N)(x+ w)≤ N for all x > 0

provided that u(N) and u(N+1) are of opposite signs. On the other hand, A2(w) ¹ B2(w) for all

w ≥ 0 clearly implies A2(0) ¹ B2(0). We can therefore claim the following.

Lemma 2 In the EU framework,

(i) Assuming (−1)nu(n)(x) ≤ 0 for all x > 0 and n = N,N + 1,

−xu(N+1)(x)

u(N)(x)≤ N for all x > 0 if and only if − xu(N+1)(x+ w)

u(N)(x+ w)≤ N for all x > 0 and w ≥ 0.

(ii) Assuming (−1)nu(n)(x) ≤ 0 for all x ≥ 0 and n = 1, 2, . . . ,N + 1,

−xu(N+1)(x)

u(N)(x)≤ N for all x > 0 if and only if − xu(N+1)(x+ w)

u(N)(x+ w)≤ N for all x > 0 and w ≥ 0.

So far Theorem 1 gives the utility characterization for the preference to bear a greater risk in

the presence of higher levels of wealth while Theorem 2 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 A3 versus B3 and A2(w1) versus B2(w1) that follow where

y is an N -th degree risk increase over x and k1 < k2.

9

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HHHHHHH

©©©©

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HHHHHHH12

12

k2x+ w2

k1y + w1

k1x+ w1

k2y + w2

rrA3 B3

12

12

©©©©

©©©

HHHHHHH

©©©©

©©©

HHHHHHH12

12

k2x+ w1

k1y + w1

k1x+ w1

k2y + w1

rrA2(w1) B2(w1)

12

12

As discussed previously, whether A2(w1) is preferred to B2(w1) is determined by the desirability of

scaling up the more N -th degree risky y rather than x, which has an apportionment effect working in

A2(w1)’s favor and a counteracting risk aversion effect. By contrast, supposing first w2 dominates w1

via FSD, whether A3 is preferred to B3 is determined by the desirability of scaling up the more N -th

degree risky y rather than x in the presence of higher levels of initial wealth, i.e., w2 rather than w1.

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 y rather than x, there is an additional apportionment effect

working in A2(w1)’s favor: the scaling up of y in A3 and x in B3 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, w1 dominates w2 via FSD, the additional apportionment effect works in B3’s

favor. We can thus state the following result.

Lemma 3 In the EU framework,

(i) Assuming (−1)nu(n)(x) ≤ 0 for all x > 0 and n = N,N + 1, given that y is an N-th degree risk

increase over x and k1 < k2, if w1 dominates w2 via FSD, then A2(w1) ¹ B2(w1) implies A3 ¹ B3,

and if w2 dominates w1 via FSD, then A3 ¹ B3 implies A2(w1) ¹ B2(w1).

(ii) Assuming (−1)nu(n)(x) ≤ 0 for all x > 0 and n = 1, 2, . . . , N + 1, given that x dominates y via

NSD and k1 < k2, if w1 dominates w2 via FSD, then A2(w1) ¹ B2(w1) implies A3 ¹ B3, and if w2

dominates w1 via FSD, then A3 ¹ B3 implies A2(w1) ¹ B2(w1).

(A formal proof can be found in the Appendix.)

10

Since A2(w) º (¹) B2(w) for all w ≥ 0 clearly implies A2(w1) º (¹) B2(w1) for w1 being a

non-negative random variable, Theorem 2 and Lemmas 2 and 3 imply the following.

Theorem 3 In the EU framework,

(i) Given that y is an N-th degree risk increase over x, w1 dominates w2 via FSD, and k1 < k2,

A3 ¹ B3 if −xu(N+1)(x)

u(N)(x)≤ N and (−1)nu(n)(x) ≤ 0 for all x > 0 and n = N,N + 1

(ii) Given that x dominates y via NSD y, w1 dominates w2 via FSD, and k1 < k2,

A3 ¹ B3 if −xu(n+1)(x)

u(n)(x)≤ n and (−1)nu(n)(x) ≤ 0 for all x > 0 and n = 1, 2, . . . , N + 1

The result can also be stated in terms of the partial N -th degree risk aversion measure given the

equivalence shown in Lemma 2 of the conditions

−xu(N+1)(x)

u(N)(x)≤ N for all x > 0 and − xu(N+1)(x+ w)

u(N)(x+ w)≤ N for all x > 0 and w ≥ 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

4.1 Interest Rate Risk and Precautionary Savings

Consider a consumer who has a two-period planning horizon and receives a certain income stream of

w0 at date t = 0 and w1 at date t = 1. At date 0, the consumer must decide how much to consume

and how much to save, for consumption at date t = 1. Any amount saved earns a rate of interest r,

where we assume r > −1. The consumer chooses s ≥ 0 to maximize

u(w0 − s) +1

1 + δEu(s(1 + r) + w1). (1)

11

The optimal s is assumed to be unique and internal. Rothschild and Stiglitz (1971) consider a

mean-preserving increase in risk in the distribution of (1 + r). Now consider a generalization of the

problem where there is an N -th degree risk increase in (1 + r). Intuitively, the relationship between

s and (1 + r) in the optimization problem (1) 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 (1 + r) (in the presence

of the initial wealth w1). Lemma 4 in the Appendix formally establishes this by showing that a

preference of A2(w1) over B2(w1) in the previous section for y being an N-th degree risk increase

over x and k1 < k2 implies a higher level of savings when there is an N -th degree risk increase in

(1 + r). This, together with Theorem 2, in turn implies a variant of the result in Eeckhoudt and

Schlesinger (2008) as stated in what follows.

Proposition 1 Let s∗1 and s∗2 be the optimal levels of savings when the gross interest rate, (1 + r),

is equal to x and y respectively.

(i) Suppose y is an N-th degree risk increase over x and (−1)nu(n)(x) ≤ 0 for all x > 0 and

n = N,N + 1. Then s∗1 ≤ (≥) s∗2 if

−xu(N+1)(x+ w1)

u(N)(x+ w1)≥ (≤) N for all x > 0.

(ii) Suppose x dominates y by NSD and (−1)nu(n)(x) ≤ 0 for all x > 0 and n = 1, 2, . . .N +1. Then

s∗1 ≤ (≥) s∗2 if

−xu(n+1)(x+ w1)

u(n)(x+ w1)≥ (≤) n for all x > 0 and n = 1, 2, . . . , N.

Eeckhoudt and Schlesinger (2008) considers the special case where w1 = 0 while the result of

Rothschild and Stiglitz (1971) is a special case with w1 = 0 and N = 2. Proposition 1 is thus a slight

generalization of Proposition 2 in Eeckhoudt and Schlesinger (2008), but its derivation from Theorem

2 in Section 3 provides a formal and yet more intuitive explanation for the conditions involving risk

aversion measures, which is previously unavailable.

4.2 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, he thus

12

chooses k to maximize

Eu(kx+ (w − k)r) (2)

and the optimal k is assumed to be unique and internal. In the special case where r ≡ 1, Cheng,

Magill, and Shafer (1987) consider the effect on the optimal portfolio choice of an FSD improvement

in x. For r being a random variable independent of x and y, Hadar and Seo (1990), 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 (2) that if there is an N -th degree risk

increase in x, choosing a larger k 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 B3 to A3 in the previous section for

k1 < k2, w1 dominating w2 via FSD, and y 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 2 Suppose y and r are stochastically independent and Eu(kx+(w−k)r) and Eu(ky+

(w − k)r) are maximized at k∗1 and k∗2 respectively. Then

(i) k∗1 ≥ k∗2 given y being an N-th degree risk increase of x if

−xu(N+1)(x)

u(N)(x)≤ N and (−1)nu(n)(x) ≤ 0 for all x > 0 and n = N,N + 1.

(ii) k∗1 ≥ k∗2 given x dominating y via NSD if

−xu(n+1)(x)

u(n)(x)≤ n and (−1)nu(n)(x) ≤ 0 for all x > 0 and n = 1, 2, . . . , N + 1.

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 2.

13

Proposition 2 thus not only subsumes the results of Cheng, Magill, and Shafer (1987) and Hadar

and Seo (1990) 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 (1980) 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 xu0000(x)/u000(x), termed “relative temperance” by Eeckhoudt

and Schlesinger (2008), 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 (2000)), 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 (1971). Cheng, Magill, and Shafer (1987) 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 B3 and A3 in the

previous section and thus implies the following result.

Proposition 3 Suppose Eu(xk − c(k) − B + w) and Eu(yk − c(k) − B + w) are maximized at k∗1

14

and k∗2 respectively. Then

(i) k∗1 ≥ k∗2 given y being an N-th degree risk increase of x if

−xu(N+1)(x)

u(N)(x)≤ N and (−1)nu(n)(x) ≤ 0 for all x > 0 and n = N,N + 1.

(ii) k∗1 ≥ k∗2 given x dominating y via NSD if

−xu(n+1)(x)

u(n)(x)≤ n and (−1)nu(n)(x) ≤ 0 for all x > 0 and n = 1, 2, . . . , N + 1.

The result of Cheng, Magill, and Shafer (1987, Proposition 7) is a special case of Proposition

3 with N = 1: If x is an FSD improvement of y, i.e., y is a first-degree risk increase of x, then

k∗1 ≥ k∗2 whenever relative risk aversion is no larger than 1. As another example, the output price

may become more risky in the sense of a mean-preserving spread defined by Rothschild and Stiglitz

(1970). This is then the special case with N = 2: 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 char-

acterizations 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 (2007) 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’

7Sandmo (1970) only considers the effect on output when the initially non-random price becomes uncertain.

15

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 (1992), Dionne and Gollier

(1992), and Hadar and Seo (1992)) can be generalized to deal with cases where the insurable loss

undergoes an N-th degree stochastic deterioration.

APPENDIX

Proof of Theorem 2. In the EU framework, A2(w) º (¹) B2(w) if and only if

Eu(k1x+w)−Eu(k1y + w)− [Eu(k2x+ w)−Eu(k2y + w)] ≥ (≤) 0 (4)

Defining Q(k, x, y) = Eu(kx+w)−Eu(ky +w) and denoting the partial derivative of Q respect to

k by Qk, (4) can be further written as Q(k1, x, y)−Q(k2, x, y) ≥ (≤) 0, which is clearly true if and

only if Qk(k, x, y) ≤ (≥) 0.

Let φ(x) = xu0(kx + w). By Lemma 1, Qk(k, x, y) = Exu0(x + w) − Eyu0(y + w) = Eφ(x) −

Eφ(y) ≤ (≥) 0 for all x and y such that y is an N -th degree risk increase over x if and only if

(−1)Nφ(N)(x) ≥ (≤)0, which, given x > 0 and (−1)nu(n) ≤ 0 for n = N,N + 1, is equivalent to

−xu(N+1)(x+w)

u(N)(x+ w)≥ (≤) N for all x > 0

Also by Lemma 1, Eφ(x) − Eφ(y) ≤ (≥) 0 for all x and y such that x dominate y via NSD if

and only if (−1)nφ(n)(x) ≥ (≤) 0 for all n = 1, 2, . . . , N , which, given x > 0 and (−1)nu(n) ≤ 0 for

n = 1, 2, . . .N + 1, is equivalent to

−xu(n+1)(x+ w)

u(n)(x+ w)≥ (≤) n for all x > 0 and n = 1, 2, . . . , N. 2

Proof of Theorem 3. We only prove A3 ¹ B3 implies A2(w1) ¹ B2(w1) for y being an N -th degree

risk increase over x and w2 dominating w1 via FSD. The other parts of the Theorem can be proved

analogously.

16

In the EU framework, A3 ¹ B3 if and only if

Eu(k2x+ w2)−Eu(k2y + w2) ≥ Eu(k1x+ w1)−Eu(k1y + w1) (5)

Since y is an N-th degree risk increase of x (and hence k2y is an N -th degree risk increase of

k2x) and (−1)N+1u(N+1)(x) ≤ 0, we have Eu0(k2x)−Eu0(k2y) ≤ 0, which, given w2 dominating w1

via FSD, implies

Eu(k2x+ w1)−Eu(k2y + w1) ≥ Eu(k2x+ w2)−Eu(k2y + w2) (6)

(5) and (6) thus imply

Eu(k2x+ w1)−Eu(k2y + w1) ≥ Eu(k1x+ w1)−Eu(k1y + w1)

which is equivalent to A2(w1) ¹ B2(w1). 2

Lemma 4 Let s∗1 and s∗2 be the optimal levels of savings when the gross interest rate, (1 + r), is

equal to x and y respectively, where y is an N-th degree risk increase over x or x dominates y via

NSD. Then s∗1 ≤ (≥) s∗2 if A2(w1) º (¹) B2(w1) for k1 < k2.

Proof. Let s∗1 and s∗2 be the optimal levels of savings given the gross interest rate, (1 + r), being

equal to x and y respectively and y being an N -th degree risk increase over x. Clearly, since s∗1 and

s∗2 are the optimal values under x and y respectively and the maxima are unique,

u(w0 − s∗1) +1

1 + δEu(s∗1x+ w1) > u(w0 − s∗2) +

1

1 + δEu(s∗2x+ w1)

and

u(w0 − s∗2) +1

1 + δEu(s∗2y + w1) > u(w0 − s∗1) +

1

1 + δEu(s∗1y +w1)

Summing the two sides of the inequalities and simplifying, we have

Eu(s∗1x+ w1) +Eu(s∗2y + w1) > Eu(s∗2x+ w1) +Eu(s∗1y +w1) (7)

17

Suppose s∗1 < s∗2. Then A2(w1) ¹ B2(w1) for k1 < k2 means

Eu(s∗1x+ w1) +Eu(s∗2y + w1) ≤ Eu(s∗2x+ w1) +Eu(s∗1y +w1)

which contradicts (7). That is, A2(w1) ¹ B2(w1) implies s∗1 ≥ s∗2. We can similarly show that

A2(w1) º B2(w1) implies s∗1 ≤ s∗2. 2

Lemma 5 Suppose A4 and B4 are given by

©©©©

©©©

HHHHHHH

©©©©

©©©

HHHHHHH12

12

k2x+ (w − k2)r

k1y + (w − k1)r

k1x+ (w − k1)r

k2y + (w − k2)r

rrA4 B4

12

12

where x, y and r are independent and y is an N-th degree risk increase of x or x dominates y via

NSD, and Eu(kx+(w− k)r) and Eu(ky+(w− k)r) are maximized at k∗1 and k∗2 respectively. Then

k∗1 ≥ (≤) k∗2 if A4 ¹ (º) B4 for k1 < k2.

Proof. Since Eu(kx + (w − k)r) and Eu(ky + (w − k)r) are maximized at k∗1 and k∗2 respectively

and the maxima are unique, by the same argument used in Lemma 4, we must have

Eu(k∗1x+ (w − k∗1)r) +Eu(k∗2 y + (w − k∗2)r) > Eu(k∗2x+ (w − k∗2)r) +Eu(k∗1 y + (w − k∗1)r) (8)

If k∗1 < (>)k∗2, then A4 ¹ (º) B4 for k1 < k2 means

Eu(k∗1x+ (w − k∗1)r) +Eu(k∗2y + (w − k∗2)r) ≤ Eu(k∗2x+ (w − k∗2)r) +Eu(k∗1 y + (w − k∗1)r)

which contradicts (8). 2

Proof of Proposition 2. Since (w − k1)r dominates (w − k2)r via FSD for k1 < k2 in A4 and B4

in Lemma 5, the result is implied by the lemma and Theorem 3. 2

18

Lemma 6 Suppose A5 and B5 are given by

©©©©

©©©

HHHHHHH

©©©©

©©©

HHHHHHH12

12

k2y − c(k2)−B + w

k1x− c(k1)−B +w

k1y − c(k1)−B + w

k2x− c(k2)−B + w

rrA5 B5

12

12

where y is an N-th degree risk increase of x or x dominates y via NSD, and Eu(xk− c(k)−B+w)

and Eu(yk−c(k)−B+w) are maximized at k∗1 and k∗2 respectively. Then k∗1 ≥ (≤) k∗2 if A5 ¹ (º) B5

for k1 < k2.

(The proof is analogous to that of Lemma 5.)

Proof of Proposition 3. Since [−c(k1)−B + w] > [−c(k2)−B + w] for k1 < k2 in A5 and B5 in

Lemma 6, Theorem 3 and the lemma imply the result. 2

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