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DIRECTION AND INTENSITY OF RISK PREFERENCE AT THE THIRD ORDER.

INTRODUCTION

In expected utility theory, higher-order risk preferences, such as prudence at the third order and temperance at the fourth, are becoming increasingly important in establishing comparative statics predictions for behavioral responses to risk and in framing empirical studies of the links between risk and risk preferences. Considerable interest focuses on the third order, where the early results of Leland (1968) and Sandmo (1970), tying prudence, or equivalently downside risk aversion, to the precautionary motive for saving, are now augmented by more recent studies showing that prudence implies precautionary self-protection in a temporal context (Wang and Li, 2015) and revealing the importance of prudence for predicting the effect of risk on the ranking of monitoring systems when randomized monitoring is practicable (Fagart and Sinclair-Desgagne, 2007), on patience in bargaining (White, 2008), on precautionary bidding in auctions (Kocher, Pahlke, and Trautmann, 2015), and on the private supply of public goods (Bramoulle and Treich, 2009; Eichner and Pethig, 2015), among many other applications.

In the terminology of Eeckhoudt (2012), the linking of prudence to precautionary saving and precautionary self-protection is governed by the "direction" of third-order risk preference, as the precautionary motive hinges on a positive sign for the third derivative of the von Neumann--Morgenstern utility function. In contrast, the remaining predictions cited concern the "intensity" of third-order risk preference, as they turn on the magnitude of the index of prudence introduced by Kimball (1990). In this article we reexamine direction and intensity of risk preference at the third order, and establish the Schwarzian index, introduced by Keenan and Snow (2002, 2012), as a complementary measure of third-order preference intensity identified with substitution effects of downside risk.

We introduce the various indicators of direction and intensity in the second section, and in the third section, as a heuristic in the manner of Pratt (1964), we use Taylor series for small risks to show that each of these indices measures the willingness to trade off a distinct pair of orders of risk. In the fourth section, we summarize known results and apply them to establish that the substitution effect of an increase in downside risk on the choice of an optimal control reduces the degree of absolute downside risk aversion as measured by the Schwarzian. In the fifth section, applications in the context of saving relate the Schwarzian to the strength of the precautionary response to risk through the substitution effect of downside risk and the response of the marginal rate of time preference to background risk. Conclusions are offered in the sixth section.

INDICATORS OF DIRECTION AND INTENSITY

It is well known that a negative second derivative for a utility function u(y) implies a positive willingness to pay for the elimination of risk about income y, while a positive third derivative implies positive precautionary saving in response to a background risk to future income. Eeckhoudt and Schlesinger (2006) equate these directional attitudes with consistent preference for combining good with bad in simple 50:50 lotteries, providing a choice-theoretic foundation for identifying nth-degree risk aversion with a negative (positive) sign for the nth even (odd) derivative of utility, as suggested by Ekern (1980).

A measure of intensity in risk preference is, in contrast, described by Eeckhoudt (2012) as one that indicates when, and the extent to which, one decision maker is more averse to bearing risk than another. At the second order, the index of absolute risk aversion [R.sub.u](y) [equivalent to] -u"(y)/u'(y) introduced by Arrow (1965) and Pratt (1964) serves this purpose, providing a measure that yields transitive rankings of utility functions and intuitive comparative statics predictions for decisions involving portfolio choice, demand for insurance, and in many other contexts. (1)

At the third order, Menezes, Giess, and Tressler (1980) show that dislike of any increase in downside risk, defined as a mean-and-variance preserving spread (MVPS) in the distribution for income, is unique to decision makers with u"' > 0, thereby characterizing downside risk aversion. (2) Hence, the direction of third-order risk preference for u(y) is indicated by the sign of its third derivative, or equivalently by the sign of the index

[d.sub.u](y) [equivalent to] u"'(y)/u'(y). (1)

With respect to intensity, Crainich and Eeckhoudt (2008) show that the marginal rate of time preference decreases in response to a mean-zero background risk to future income more strongly when the magnitude of [d.sub.u] is greater, implying that a greater decline in the interest rate would be needed to offset the precautionary motive implied by [d.sub.u] > 0. (3) However, this result applies only to the singular class of cubic utility functions, for which changes in risk at orders above the third are irrelevant.

Recognizing that -u'(y) has the properties of a utility function when u(y) is risk averse, Kimball (1990) shows that a simple reinterpretation of Pratt's analysis of risk aversion leads to the index of prudence,

[P.sub.u](y) [equivalent to] -u"'(y)/u"(y), (2)

as a measure of the strength of the precautionary motive for saving. In contrast, Keenan and Snow (2002) adapt to the third order of risk preference the method of characterizing risk aversion developed by Diamond and Stiglitz (1974), and conclude that the Schwarzian index,

[S.sub.u](y) [equivalent to] [d.sub.u](y) - (3/2)[R.sub.u][(y).sup.2], (3)

measures the intensity of absolute downside risk aversion.

In this article we show that in the small, the indices [R.sub.u], [d.sub.u], [P.sub.u], and [S.sub.u] measure the willingness to trade off distinct orders of risk, and that in the large, they measure preference intensity precisely insofar as they provide global characterizations of the mapping [phi] that transforms one utility u into another v = [phi](u).

At the second order, where [R.sub.u] is both an indicator of direction and a measure of intensity, Pratt (1964, Theorem 1) shows that [R.sub.v](y) > [R.sub.u](y) for all y is equivalent to [phi] being risk averse, that is, [R.sub.[phi]](u) [equivalent to] -[phi]"(u)/[phi]'(u) > 0. Moreover, Diamond and Stiglitz (1974, Theorem 2) show that, with decreasing absolute risk aversion, that is, [partial derivative][R.sub.u](y)/[partial derivative]y < 0 for all y, risk has a positive substitution effect on the optimal choice of a control variable such as saving.

At the third order, however, no single index characterizes each of these properties, and more than one index measures intensity of risk preference.

While the sign of [d.sub.u] unambiguously indicates direction, Kimball (1990) links greater prudence, that is, [P.sub.v](y) > [P.sub.u](y) for all y, to greater precautionary saving in response to future income risk. In contrast, we link a greater value for the Schwarzian, that is, [S.sub.v](y) > [S.sub.u](y) for all y, to the strength of the precautionary motive for saving, not as measured by the increase in saving, but instead as measured by the decline in the marginal rate of time preference. In particular, we show that the magnitude of this decline is greater for v than for u, for any utility functions u and v = [phi](u),if

[S.sub.v](y) > [S.sub.u](y) and [R.sub.v](y) > [R.sub.u](y) for all y. (C)

Finally, we show that when the Schwarzian is decreasing in income, the substitution effect of greater downside risk on saving is positive. (4) Thus, when u exhibits decreasing absolute risk aversion and a decreasing Schwarzian, that is, [partial derivative][R.sub.u](y)/[partial derivative]y < 0 and [partial derivative][S.sub.u](y)/[partial derivative]y < 0 for all y, a decline in endowed wealth results in utility v satisfying condition (C), thereby relating the decline in the marginal rate of time preference dictated by (C) to substitution effects of risk at the second and third orders. (5)

TAYLOR SERIES APPROXIMATIONS

In this section we analyze a small change in a small income risk by positing a shift from random variable [[??].sub.1] to [[??].sub.2], each representing a distinct absolute risk to a sure endowed income [bar.y], and with both risks resulting in the same expected utility. We examine Taylor series, first for expected utility to expose distinctions among [R.sub.u], [d.sub.u], and [P.sub.u] as measures of the willingness to trade off alternative orders of risk, and then for transformations of utility to relate [S.sub.u] to the intensity of skewness preference.

Trade-Offs Involving Orders of Income Risk

Following Eeckhoudt, Gollier, and Schlesinger (2005, p. 11), we can express the expected utilities E[u([bar.y] + [[??].sub.i])] as Taylor series, and then obtain for the shift from [[??].sub.1] to [[??].sub.2]: (6)

[mathematical expression not reproducible] (4)

Adopting Pratt's specifications and truncating this series at the second order yields

[mathematical expression not reproducible] (5)

showing that in the small, [R.sub.u] measures the subjective trade-off between the difference in the means and the difference in the second moments of the random incomes, with greater income variation requiring a higher expected return, to a degree governed by the intensity of absolute risk aversion.

When we truncate the series at the third order, and stipulate that the second moments are the same, we obtain

[mathematical expression not reproducible] (6)

showing that in the small, [d.sub.u] measures the trade-off between the difference in the means and the difference in the third moments, or skewness of the random incomes, with greater negative skewness requiring a higher expected return to a degree governed by the magnitude of [d.sub.u] > 0. (7)

When, instead, it is the means that are the same in the Taylor series, we obtain

[mathematical expression not reproducible] (7)

showing that, in the small, [P.sub.u] measures the trade-off between the difference in the second moments and the difference in skewness of the random incomes, with greater negative skewness requiring a lower variance to maintain expected utility given [P.sub.u] > 0. (8)

One implication of greater prudence is obtained by evaluating Equation (4) with utility v(y), truncating at the third order, assuming equal means for income, and exploiting Equation (7), to arrive at

[mathematical expression not reproducible] (8)

An unfavorable change in skewness must be offset by a lower variance to maintain expected utility for u as determined by (7), but the more prudent decision maker v places greater weight on the deterioration in skewness than on the second-order improvement in risk and therefore dislikes this shift in the distribution for income.

Similar results for increases in [R.sub.u] and [d.sub.u] follow by applying the same logic using (5) and (6). In each case, when u is indifferent to a trade-off of higher- to lower-order risk, v dislikes the trade-off if the relevant measure of risk preference is higher for v. Since each of the measures [R.sub.u], [d.sub.u], and [P.sub.u] is expressed as a trade-off between distinct orders of risk, each indicates both direction of preference with respect to the higher-order risk, and intensity of aversion to an increase in the higher-order risk, although [P.sub.u], as a trade-off of variance to skewness, indicates dislike of negative skewness only if u is risk averse.

Trade-Offs Involving Orders of Utility Risk

The Schwarzian [S.sub.u] lacks an interpretation in terms of trading off orders of income risk, and so its sign is not an indicator of aversion to downside risk, but it does have an interpretation in terms of trading off orders of utility risk, signaling its relevance as a measure of preference intensity at the third order. To develop this interpretation, we apply the approach exploited in the preceding section to a transformation of utility v = [phi](u) such that v([bar.y] + [[??].sub.i]) = [phi]([bar.u] + [[??].sub.i]), with [bar.u] = u([bar.y]) and [[??].sub.i] = u([bar.y] + [[??].sub.i]) - u([bar.y]) for i = 1, 2. Given this change of variables, and maintaining the assumption that expected utility for u is the same under both income risks, we have

[mathematical expression not reproducible]. (9)

Truncating this series at the second order yields

[mathematical expression not reproducible]. (10)

This expression is negative when the shift from [[??].sub.1] to [[??].sub.2] increases the variance of utility exactly insofar as the transformation [phi] is risk averse at u([bar.y]), so that [R.sub.[phi]]([bar.u]) is positive.

Advancing to the third order, and assuming that the variance as well as the mean of u is the same under both income risks, we obtain

[mathematical expression not reproducible]. (11)

With an unfavorablechangeinskewness, E[[[??].sub.2.sup.3] - [[??].sub.1.sup.3]] is negative and v = [phi](u) dislikes these mean-and-variance preserving shifts in the distribution for u(y) precisely when the transformation [phi] is downside risk averse and [d.sub.[phi]]([bar.u]) is positive.

Given that in the small, (10) links [R.sub.[phi]]([bar.u]) > 0 with v = [phi](u) disliking shifts in the distribution for income that leave the mean of utility u(y) unchanged while increasing its variance, it would be useful to have in hand an index, expressed in terms of utility, that implies [R.sub.[phi]]([bar.u]) > 0 when the index increases. The index [R.sub.u](y) serves this role because the measure R satisfies a 1-cocycle condition converting composition of functions, such as v = [phi](u), into the addition operation through the relation

[R.sub.v](y) - [R.sub.u](y) = u'(y)[R.sub.[phi]](u(y)). (12)

This condition equates risk-averse transformations of utility, for which [R.sub.[phi]](u) is positive, with increases in the degree of risk aversion at y, measured by the Arrow-Pratt index [R.sub.u](y). (9) Condition (12) thus ensures that a ranking of utility functions by [R.sub.[phi]] > 0 is transitive and therefore capable of producing meaningful comparative statics predictions. By extending the condition [R.sub.[phi]](u([bar.y])) > 0 to hold at all incomes, one obtains a globally concave transformation [R.sub.[phi]](u(y)) > 0 for all y, and the implications of [R.sub.u](y) increasing in the small extend globally as well, since mean preserving spreads (MPSs) in the distribution for utility u(y) are disliked by v = [phi](u) if [R.sub.[phi]](u) is uniformly positive, as shown by Keenan and Snow (2009, Lemma 1).

In like manner, since (11), in the small, links [d.sub.[phi]]([bar.u]) > 0 with v = [phi](u) disliking shifts in the distribution for income that leave the mean and variance of utility u(y) unchanged, while inducing an unfavorable shift in its skewness, it would be useful to have in hand a utility index that implies [d.sub.[phi]]([bar.u]) > 0 when the index increases. However, at the third order it is the Schwarzian measure S that exhibits the critical 1-coclycle property owned by R at the second, namely,

[S.sub.v](y) - [S.sub.u](y) = [[u'(y)].sup.2][S.sub.[phi]](u(y)), (13)

linking [S.sub.[phi]] > 0 to [S.sub.u](y) increasing at y. (10) Since [S.sub.[phi]] > 0 implies [d.sub.[phi]] > 0, it follows from condition (13) that [S.sub.v] > [S.sub.u] implies [d.sub.[phi]] > 0, thus indicating that [S.sub.u](y) is the desired measure. Unfortunately, the parallel with [R.sub.u](y) is incomplete: Whereas [R.sub.u](y) increasing is equivalent to [R.sub.[phi]]([bar.u]) > 0, [S.sub.u](y) increasing is only sufficient for [d.sub.[phi]](u) > 0, except when the change in risk preferences is infinitesimal. (11)

To demonstrate equivalence for infinitesimal changes, we posit a smooth family of transformation functions [phi](u, [alpha]) parameterized by [alpha] and giving rise to a family of utility functions u(y, [alpha]) [equivalent to] [phi](u(y), [alpha]). Assume that [phi](u, [alpha]) is the identity mapping when [alpha] = 0, and define [S.sub.[phi]](u(y), [alpha]) to be the Schwarzian for [phi](u(y), [alpha]). (12)

Lemma 1: A transformation of utility that induces an infinitesimal change in risk preferences has a positive Schwarzian if and only if it is downside risk averse, that is, [partial derivative][S.sub.[phi]](u(y), 0)/[partial derivative][alpha] > 0 if and only if [partial derivative][d.sub.[phi]](u(y), 0)/[partial derivative][alpha] > 0.

Proof: Adapting definition (3) to [S.sub.[phi]](u(y), [alpha]) and differentiating with respect to [alpha] yields

[[partial derivative][S.sub.[phi]](u(y), [alpha])/[partial derivative][alpha]] = [[partial derivative][d.sub.[phi]](u(y), [alpha])/[partial derivative][alpha]] - 3[R.sub.[phi]](u(y), [alpha]) [[partial derivative][R.sub.[phi]](u(y), [alpha])/[partial derivative][alpha]]. (14)

When evaluated with [alpha] = 0, Equation (14) identifies the effect of an infinitesimal change in risk preferences. Using primes to denote utility derivatives of [phi](u(y), [alpha]),we have [phi]'(u(y), 0) = 1, while all higher derivatives vanish. Hence, with [alpha] = 0, [R.sub.[phi]](u(y), 0) = 0 and (14) reduces to

[[partial derivative][S.sub.[phi]](u(y), 0)/[partial derivative][alpha]] = [[partial derivative][d.sub.[phi]](u(y), 0)/[partial derivative][alpha]]. (15)

Hence, the Schwarzian for the transformation is positive if and only if the transformation is downside risk averse.

Adapting the 1-cocycle condition (13) to [S.sub.[phi]](u(y), [alpha]) and differentiating with respect to [alpha] yields

[[partial derivative][S.sub.[phi]](u(y), 0)/[partial derivative][alpha]] = [[1/u'(y)].sup.2] [[partial derivative][S.sub.u](y, 0)/[partial derivative][alpha]]. (16)

By doing the same with approximation (11), while exploiting Equations (15) and (16), we arrive at (13)

[mathematical expression not reproducible]. (17)

Hence, for small risks, a small change in risk preferences that increases the Schwarzian [S.sub.u]([bar.y]) results in a utility function that dislikes any shift in the distribution for u that induces an unfavorable change in skewness while preserving the mean and variance. By extending this condition on the Schwarzian to all incomes, we obtain [S.sub.v](y) > [S.sub.u](y), and so [S.sub.[phi]](u) > 0 by the 1-cocycle condition (13), thereby characterizing large changes in third-order risk preferences by a global property of the transformation [phi]. The implications of Equation (17) extend to global changes as well, since v = [phi](u) dislikes any MVPS in the distribution for u if [S.sub.[phi]](u) > 0 uniformly, as shown by Keenan and Snow (2009, Theorem 2).

Approximation (8) for the cost in expected utility occasioned by an MPS in the distribution for income can also be extended from infinitesimal to global changes in risk preferences, since the index of absolute prudence obeys a 1-cocycle condition for -u'. In particular, [P.sub.v](y) [equivalent to] [R.sub.-u'](y) implies that the 1-cocycle condition (12) for [R.sub.[phi]] applies to the transformation [psi] such that -v' = [psi](-u'), so that

[P.sub.v](y) - [P.sub.u](y) = -u'(y)[R.sub.[psi]](-u(y)). (18)

By exploiting this condition, Chiu (2005) establishes the global analogue to (17).

Unlike [R.sub.u], [P.sub.u], and [S.sub.u], the index [d.sub.u] does not satisfy a 1-cocycle condition. Indeed, evaluating [d.sub.v] for utility v = [phi](u) yields

[d.sub.v](y) - [d.sub.u](y) = [[u'].sup.2]([d.sub.[phi]] + 3[R.sub.u][R.sub.[phi]]/u'), (19)

making clear that [d.sub.v](y) > [d.sub.u](y) does not typically imply that [d.sub.[phi]] is positive. (14) We conclude that the lack of a 1-cocycle condition for [d.sub.u] limits its usefulness as a measure of preference intensity to small risks and the trade-off (6) between mean income and increased skewness.

INCREASES IN RISK AND IN RISK AVERSION

Henceforth our analysis encompasses large risks. We begin this section with a review of the fundamental results relating increases in risk and in risk aversion at the second and third orders. Following Diamond and Stiglitz (1974), we posit a utility function u(y, c) that depends on a control variable c, with risk about income represented by the nondegenerate cumulative distribution function F(y, [theta]) parameterized by an index [theta], whose support set for y is contained in the positive interval [a, b]. For utility u,a compensated increase in risk is a shift in the distribution for income [F.sub.[theta]](y, [theta]) that induces an MPS in the distribution for u(y, c), characterized by the integral conditions

[[integral].sub.a.sup.y][u.sub.sub.y](t, c)[F.sub.[theta]](t, [theta])dt [greater than or equal to] 0[for all]y [member of] [a, b] with equality at y = b (20)

developed by Diamond and Stiglitz. When utility is risk neutral, these are the conditions for an MPS in the distribution for income introduced by Rothschild and Stiglitz (1970) to characterize an increase in risk. In their fundamental result, Diamond and Stiglitz show that in response to a compensated increase in risk, the optimal control changes in a direction that reduces the degree of absolute risk aversion. (15)

Proposition 1A: Diamond and Stiglitz (1974, Theorem 2). Let c([theta]) be the value of c that maximizes [integral] u(y, c)dF(y, [theta]). If an increase in [theta] induces an MPS in the distribution for utility u(y, c), then dc([theta])/d[theta] and [partial derivative][R.sub.u](y, c)/[partial derivative]c have opposite signs.

A compensated increase in downside risk is a spread in the distribution for income [F.sub.[theta]](y, [theta]) that induces an MVPS in the distribution for utility, characterized by the integral conditions introduced by Keenan and Snow (2002),

[[integral].sub.a.sup.b] [u.sub.y](y, c)[F.sub.[theta]](y, [theta])dy = 0 and [[integral].sub.a.sup.y] [u.sub.y](t, c) [[integral].sub.a.sup.t] [u.sub.y](s, c)[F.sub.[theta]](s, [theta])ds dt [greater than or equal to] 0[for all]y [member of] [a, b] with equality at y = b. (21)

When utility is risk neutral, these are the conditions for an MVPS in the distribution for income developed by Menezes, Giess, and Tressler (1980) to characterize an increase in downside risk. Keenan and Snow (2002) extend Proposition 1A to the third order of risk preference, showing that in response to a compensated increase in downside risk, the change in the optimal control reduces the degree of absolute downside risk aversion. (16)

Proposition 1B: Keenan and Snow (2002, Theorem 2). Let c([theta]) be the value of c that maximizes [integral] u(y, c)dF(y, [theta]). If an increase in [theta] induces an MVPS in the distribution for utility u(y, c), then dc([theta])/d[theta] and [partial derivative][S.sub.u](y, c)/[partial derivative]c have opposite signs.

These results treat risk as exogenous, with the control c chosen to maximize the expected value of utility u(y, c) given the risk F(y, [theta]). As Diamond and Stiglitz (1974, pp. 344-345) point out, the logic can be reversed, with c as an exogenous preference parameter and the risk parameter [theta] as the control variable. Noting that the signs of the comparative statics effects dc([theta])/d[theta] and d[theta](c)/dc are the same as the sign of the cross partial derivative [integral] [u.sub.c](y, c)d[F.sub.[theta]](y, [theta]), Diamond and Stiglitz further observe that the optimal control [theta](c) satisfies the first-order condition [integral] u(y, c)d[F.sub.[theta]](y, [theta](c)) = 0, and therefore locally the shift [F.sub.[theta]](y, [theta](c)) is mean preserving for the distribution of utility u(y, c). To obtain a prediction for the sign of d[theta](c)/dc, they thus need only to impose a condition ensuring that the distributional shift satisfies the first part of condition (20) for an MPS in the distribution for utility. We obtain a similar result at the third order by imposing the condition that an increase in [theta] induces a variance preserving spread (VPS) in the distribution for utility, one that satisfies the second of the integral conditions stated at (21).

Theorem 1: Let [theta]([c.sub.1]) be the interior value of [theta] that uniquely maximizes [integral] u(y, [c.sub.1])dF(y, [theta]), and assume that an increase in [theta] induces a VPS in the distribution for utility u(y, [c.sub.1]).

(a) In the small: [d.sub.[theta]]([c.sub.1])/dc and [partial derivative][S.sub.u](y, [c.sub.1])/[partial derivative]c have opposite signs.

(b) In the large: Assuming [theta](c) is continuous on an interval [[c.sub.1], [c.sub.2]] where [partial derivative][S.sub.u](y, c)/[partial derivative]c > 0 throughout, one has [theta](c) < [theta]([c.sub.1]) for all c [member of] ([c.sub.1], [c.sub.2]].

Proof: With the variance of utility held constant, the argument of Diamond and Stiglitz (1974) transfers in a straightforward fashion to the third order to establish part (a) for infinitesimal changes in risk preferences. In part (b), it follows from part (a) that [theta](c) is decreasing at [c.sub.1] and thereafter can never cross [theta]([c.sub.1]) continuously and so by assumption cannot cross [theta]([c.sub.1]) at all. To see this, note that as c increases from [c.sub.1], [S.sub.u](y, c) rises above [S.sub.u](y, [c.sub.1]), implying that u(y, c) is a downside risk-averse transformation of u(y, [c.sub.1]), and therefore, by Keenan and Snow (2009, Theorem 1), u(y, c) dislikes any MVPS of the distribution for u(y [c.sub.1]), and by part (a) prefers a value of [theta] below [theta]([c.sub.1]). Hence, [partial derivative][S.sub.u](y, c)/[partial derivative]c > 0 throughout implies [theta](c) < [theta]([c.sub.1]) for all c [member of] ([c.sub.1], [c.sub.2]].

Keenan and Snow (2017, Theorem 2) show that when [S.sub.[??]] < [S.sub.u], there is a smooth path of utility functions from u to [??] along which [S.sub.u] is decreasing, not increasing. By part (a) of Theorem 1, [theta](c) will initially increase above [theta]([c.sub.1]), not decrease. Now, assuming [partial derivative][S.sub.u](y, c)/[partial derivative]c < 0 throughout in part (b) does not imply [theta](c) > [theta]([c.sub.1]) throughout, since [S.sub.u] decreasing is consistent with u(y c) being a downside risk-averse transformation of u(y, [c.sub.1]) for c > [c.sub.1], in which event one can have [theta](c) = [theta]([c.sub.1]) and then, again by part (a) of Theorem 1, values of [theta] will fall below [theta]([c.sub.1]) rather than always remaining above this initial value. Nonetheless, it is evident that one can choose among the many distributions available at least one where matters reverse themselves globally relative to part (b), given that [S.sub.u] is decreasing throughout, and so the optimal [theta] will be greater for all c under this chosen distribution, not just for those c near [c.sub.1].

This last observation applies whenever [S.sub.[??]] < [S.sub.u], and so will hold when one also has [??] = [phi](u) with [phi]"' > 0. As a consequence, one can never conclude on the basis of [phi]"' > 0 alone that the optimal value for [theta] is lower, no matter the distribution (given only a VPS at the optimal [theta] for u), not even if the solution path is continuous. This last, though, would be the corresponding result to that proven above in Theorem 1(b) for [S.sub.u] increasing, but where that assumption is replaced by [phi]"' > 0 between u and v, the former condition implying the latter, but not the latter the former. We thus conclude that [S.sub.u] increasing plays a role separate from [phi]"' > 0 in obtaining comparative statics results, and that while [S.sub.u] increasing can always replace the condition [phi]"' > 0, it can also be used to obtain results not possible were it replaced by the weaker assumption [phi]"' > 0.

PRECAUTIONARY SAVING

The saving problem with time-separable preferences provides a context in which the indices [S.sub.u] and [P.sub.u] play contrasting roles as measures of the intensity of the precautionary motive. Consider a two-period decision problem in which the control variable c is saving, and endowed income is [bar.y] > 0 in the first period and [??] [greater than or equal to] 0 in the second. Define intertemporal utility for the realized value y of the random component of second-period income by

U(y, c) [equivalent to] u([bar.y] - c)+ [1/[1 + [rho]]] u([??] + (1 + r)c + y), (22)

where r is the interest rate and [rho] > 0 is the subjective rate of time preference. Saving is chosen to maximize expected intertemporal utility [integral] U(y, c)dF(y, [theta]), where F(y, [theta]) represents a mean-zero background risk to future income. (17)

The problem is particularly interesting since, in deciding to save, there is always the trade-off between subjectively discounting the future versus the reward received for awaiting the return r, together with concern for the inherent riskiness that the future brings, represented by the background risk.

Given the subjective rate of time preference [rho], let [c.sub.u](r, [theta]) denote the optimal choice for saving that maximizes criterion (22) and satisfies the first-order condition

-u' ([bar.y] - [c.sub.u](r, [theta])) + [[1 + r]/[1 + [rho]]] [integral] u'([??] + (1 + r)[c.sub.u](r, [theta])+ y)dF(y, [theta]) = 0, (23)

so that optimal saving equates the marginal rate of time preference to the market rate of interest,

(1 + [rho]) [[u'([bar.y] - [c.sub.u](r, [theta]))]/[[integral] u'([??] + (1 + r)[c.sub.u](r, [theta]) + y)dF(y, [theta])]] -1 = r. (23')

It has been known since Leland (1968) and Sandmo (1970) that introducing a background risk to future income stimulates additional, precautionary saving when [d.sub.u] is positive. (18) In that event, precautionary saving is the difference between [c.sub.u](r [theta]) and [c.sub.u](r 0), where the latter denotes optimal saving in the absence of background risk.

Substitution Effects of Greater Risk and Greater Downside Risk

Since the expected utility for u is held constant by the distributional spreads postulated in Propositions 1A and 1B, the behavioral consequences predicted are substitution effects. Hence, we conclude that substitution effects associated with greater risk and greater downside risk act to reduce the intensity of absolute risk aversion as measured by [R.sub.u] at the second order and as measured by [S.sub.u] at the third.

In the following corollaries, we apply these results to the saving decision by using U(y c) in place of u(y c) in the two propositions. Once it is recognized that [U.sub.y](y, c) = (1 + [rho])u'([??] + (1 + r)c + y), so that, with u(y + (1 + r)c + y) replacing u(y c), we have [R.sub.U](y c) [equivalent to] [R.sub.u](y c) and [partial derivative][R.sub.u]/[partial derivative]c = [partial derivative][R.sub.u]/[partial derivative]y, along with [S.sub.U](y c) [equivalent to] [S.sub.u](y c) and [partial derivative][S.sub.u]/[partial derivative]c = [partial derivative][S.sub.u]/[partial derivative]y, the two corollaries follow immediately from the corresponding propositions. (19)

Corollary 1A: Saving increases with a compensated increase in income risk if [partial derivative][R.sub.u]/[partial derivative]y < 0.

This corollary shows that the substitution effect of greater income risk entails an increase in saving if utility exhibits decreasing absolute risk aversion (DARA), for which the necessary and sufficient condition is [P.sub.u] > [R.sub.u], given [R.sub.u] > 0.

Corollary 1B: Saving increases with a compensated increase in downside income risk if [partial derivative][S.sub.u]/[partial derivative]y < 0.

This corollary shows that the substitution effect of greater downside income risk entails an increase in saving if utility exhibits decreasing absolute downside risk aversion (DADSRA). Hence, while the degree of prudence bears on the extent of precautionary saving, the substitution effect of downside risk is not governed by decreasing absolute prudence (DAP), that is, [partial derivative][P.sub.u]/[partial derivative]y < 0. Instead, DAP and DARA imply that background risk increases aversion to bearing a contemporary foreground risk. (20) By contrast, background risk in the context of saving represents a future peril, and the substitution effect of an increase in its downside risk is dictated by DADSRA. Carrying out the differentiation, we obtain

[partial derivative][S.sub.u]/[partial derivative]y = [R.sub.u][[P.sub.v]([R.sub.u] - [T.sub.u])+ 3[R.sub.u]([P.sub.u] - [R.sub.u])], (24)

which is negative if [T.sub.u] > 2[P.sub.u] > 0 and [R.sub.u] > 0, where [T.sub.u] =-[u.sup.iv]/u"' is the index of temperance. (21) Thus, whereas DARA depends on the magnitude of the third-order index of prudence relative to the index of risk aversion, DADSRA depends on the magnitude of the fourth-order index of temperance relative to (twice) the index of prudence.

Intensity of the Precautionary Motive

In this section we distinguish between two measures of the intensity of the precautionary motive in the context of saving. One is the magnitude of the additional, precautionary saving stimulated by a background risk, and the other is the magnitude of the reduction in the marginal rate of time preference caused by the same background risk.

Precautionary Saving and the Index of Prudence. Intensity of demand for precautionary saving is linked by Kimball (1990) to the index of absolute prudence [P.sub.u](y) by exploiting the fact that -u'(y) is itself a risk-averse utility function when u(y) is both risk averse and downside risk averse. This observation allows for the mathematics of greater risk aversion to be applied to the marginal condition for optimal saving, and identifies the index of absolute prudence for utility u(y) as the appropriate measure of intensity, since [P.sub.v](y) [equivalent to] [R.sub.-u](y). (22) To see the consequence, let [c.sub.u](r, [theta]) and [c.sub.v](r, [theta]) be the values of c that maximize the expected value of (22) relative to the prudent utility functions u(y) and v(y), respectively. Greater saving by v than by u in the presence of background risk implies greater precautionary saving by v if both utility functions save the same amount in the absence of background risk. When u and v have identical endowments, they save the same amount in the absence of risk when r = [rho], since then perfect consumption smoothing is optimal for any risk-averse utility function.

Theorem 2: With interest rate r = [rho], the introduction of any background risk F(y, [theta]) induces greater precautionary saving by v than by u; that is, we have [c.sub.v]([rho], [theta]) - [c.sub.v]([rho], 0) > [c.sub.u]([rho], [theta]) - [c.sub.u]([rho], 0), if and only if [P.sub.v](y) > [P.sub.v](y) for all y.

Proof: Given any interest rate r, saving by v exceeds saving by u in the presence of risk F(y, [theta]), that is, [c.sub.v](r, [theta]) > [c.sub.u](r, [theta]), if and only if the marginal value of saving for v is positive when evaluated at optimal saving for u,

-v'([bar.y] - [c.sub.u](r, [theta])) + [[1 + r]/[1 + [rho]]] v'([??] + (1 + r)[c.sub.u](r, [theta])+ y)dF(y, [theta]) > 0. (25)

Using [psi] to denote the mapping such that -v' = [psi](-u') < 0, inequality (25) can be written as

[psi](-u'([bar.y] - [c.sub.u](r, [theta]))) - [[1 + r]/[1 + [rho]]][integral] [psi](-u'([??] + (1 + r)[c.sub.u](r, [theta]) + y))dF(y, [theta]) > 0. (26)

Setting r equal to [rho], we have [c.sub.v]([rho], 0) = [c.sub.u]([rho], 0), so that precautionary saving is greater for v than for u if and only if [c.sub.v]([rho], [theta]) > [c.sub.u]([rho], [theta]).

Introducing [??](-u', [theta]) to represent the distribution for (minus) second-period marginal utility -u'([??] + (1 + [rho])[c.sub.u]([rho], [theta])+ y) induced by F(y, [theta]), and using first-order condition (23), inequality (26) with r = [rho] can be written as

[psi](-[integral] u'd[??]) - [integral] [psi](-u')d[??] > 0. (27)

By Jensen's inequality, (27) is satisfied for all nondegenerate risks if and only if [psi] is strictly concave, which is equivalent to [R.sub.-v'] > [R.sub.-u']. It follows that precautionary saving is greater for v than for u if and only if [P.sub.v] > [P.sub.u].

As inequality (26) suggests, if r [not equal to] [rho], then the tight relationship between greater prudence and greater precautionary saving established in Theorem 2 need not hold if r is not close to [rho]. (23) Bauer and Buchholz (2009, Proposition 2) show, for example, that when r < [rho], greater prudence assures greater saving in the presence of background risk but only with the additional assumption of greater risk aversion, that is, [P.sub.v] > [P.sub.u] combined with [R.sub.v] > [R.sub.u] implies [c.sub.v](r, [theta]) > [c.sub.u] (r, [theta]). Nonetheless, precautionary saving may or may not be greater since, with greater risk aversion, v has greater initial saving.

The Marginal Rate of Time Preference and the Schwarzian Index. By assuming that the interest rate remains constant, Theorem 2 addresses the strength of the demand for precautionary saving, linking this aspect of third-order preference intensity to the magnitude of absolute prudence. In this section, we examine two thought experiments in which it is an individual's optimal saving that remains constant. In the first, the interest rate adjusts to offset the precautionary motive.

Recall that [c.sub.u](r 0) represents optimal saving by u in the absence of background risk. For saving to remain constant once background risk is introduced, the interest rate must satisfy first-order condition (23) when evaluated with saving equal to [c.sub.u](r, 0). We shall refer to this value as the compensating interest rate since it neutralizes the precautionary motive. Denoting this value by [r.sub.u](r, [theta]), Equation (23') implies

[mathematical expression not reproducible], (28)

which equates the compensating interest rate to the marginal rate of time preference conditional on optimal saving in the absence of background risk, [c.sub.u](r, 0). If the background risk were removed (or if [d.sub.u] [equivalent to] 0), then the ratio of marginal utilities on the right-hand side of (28) would equal (1 + r)/(1 + [rho]), and the compensating interest rate would then equal r. By contrast, with background risk present and [d.sub.u] > 0, the ratio of marginal utilities on the right-hand side of (28) is less than (1 + r)/(1 + [rho]), implying [r.sub.u](r, [theta]) < r. (24)

The magnitude of the requisite decline in the interest rate, r - [r.sub.u](r, [theta]), measures the responsiveness of the marginal rate of time preference to the introduction of background risk. (25) The following result shows that, if v(y) is a risk-averse and downside risk-averse transformation of utility u(y), then the compensating interest rate is lower for v than for u provided saving in the absence of risk is the same for both, which requires that the initial rate of interest is equal to the subjective rate of time preference.

Theorem 3: Assume [R.sub.u] > 0 and r = [rho]. The compensating interest rate in response to a background risk is lower for v = [phi](u) than for u, that is, [r.sub.v]([rho], [theta]) < [r.sub.u]([rho], [theta]), if [R.sub.[phi]]([bar.u]) > 0 and [d.sub.[phi]](u) > 0.

Proof: Restating Equation (28) for utility v, substituting [phi]'u' for v', and rearranging terms, we find that [r.sub.v](r, [theta]) < [r.sub.u](r, [theta]) if and only if

[mathematical expression not reproducible]. (29)

Notice that we have [c.sub.v](r, 0) = [c.sub.u](r, 0) when r = [rho], allowing us to eliminate the marginal utilities in the numerators and to restate inequality (29) as

[integral] u'([x.sub.u] + y)[[phi]'(u([x.sub.u] + y)) - [phi]'(u([x.sub.u]))]dF > 0, (29')

where [x.sub.u] = [??] + (1 + [rho])[c.sub.u]([rho], 0) = [bar.y] - [c.sub.u]([rho], 0), the second equality expressing the fact that perfect consumption smoothing is optimal when r = [rho]. Given risk aversion for u, [phi]' < 0 implies that both u'([x.sub.u] + y) and [phi]'(u([x.sub.u] + y)) are decreasing functions of y, and therefore the covariance rule implies

[mathematical expression not reproducible]. (30)

Thus, inequality (29') is satisfied if

[integral] [phi]'(u([x.sub.u] + y))dF > [phi]'(u([x.sub.u])). (31)

By Jensen's inequality, [phi]"' > 0 implies [integral] [phi]' (u([x.sub.u] + y))dF > [phi]'([integral] u([x.sub.u] + y)dF), which in turn implies inequality (31) given u' < 0 and [phi]' < 0.

The implication of Theorem 3 is that the marginal rate of time preference becomes more responsive to background risk when utility undergoes a risk-averse and downside risk-averse transformation. Keenan and Snow (2016) refer to members of this family of transformation functions as strong increases in downside risk aversion, emphasizing that this family is transitive, and therefore produces meaningful rankings of utility with respect to aversion to risk and downside risk. Since a transformation is downside risk averse if its Schwarzian is positive, the following is an immediate consequence of Theorem 3.

Corollary 2: Assume [R.sub.u] > 0. Then [r.sub.v]([rho], [theta]) < [r.sub.u]([rho], [theta]) if condition (C) is satisfied, that is, if [S.sub.v](u) > [S.sub.u](y) and [R.sub.v](y) > [R.sub.u](y) for all y.

Thus, the intensity of absolute downside risk aversion bears on the strength of the precautionary motive through its influence on the responsiveness of the marginal rate of time preference to background risk. Greater prudence is clearly not a necessary condition for [r.sub.v]([rho], [theta]) < [r.sub.u]([rho], [theta]), since it is not implied by [S.sub.v] > [S.sub.u] and [R.sub.v] > [R.sub.u]. Further, when u is risk neutral, inequality (29) in the proof of Theorem 3 is satisfied if and only if (31) is satisfied, and we now see that greater prudence is also not sufficient in general, since introducing prudence when [R.sub.u] = 0 will not typically ensure that inequality (31) is satisfied.

We conclude this section with a second thought experiment in which saving remains constant, in this instance because the shift in the distribution for income preserves expected marginal utility.

Theorem 4: Assume that utility functions u(y) and v(y) are related by a transformation of marginal utility such that -v' = [psi](-u'), and that u is risk averse and exhibits constant absolute prudence. If an increase in [theta] induces an MVPS in the distribution for -u'(y), then [partial derivative][c.sub.u](r, [theta])/[partial derivative][theta] = 0, but [partial derivative][c.sub.v](r, [partial derivative])/[partial derivative][theta] > 0 if [psi]"' > 0.

Proof: By assumption, the shift [F.sub.[theta]](y, [theta]) has no effect on saving by u since it preserves the mean of second-period marginal utility and therefore leaves first-order condition (23) unaffected, as [integral]u'([x.sub.u] + y)d[F.sub.[theta]](y, [theta]) = 0, where we now define [x.sub.u] = [??] + (1 + r)[c.sub.u](r, [theta]) without requiring that r = [rho]. Saving by v increases if and only if [integral]v'([x.sub.v] + y)d[F.sub.[theta]](y, [theta]) is positive, where [x.sub.v] = [??] + (1 + r)[c.sub.v](r, [theta]). Substituting -[psi](-u') for v' and applying integration by parts twice yields

[mathematical expression not reproducible], (32)

where [u".sub.v] = u"([x.sub.v] + y), [u".sub.u] = u"([x.sub.u] + y), and

[mathematical expression not reproducible]. (33)

With constant absolute prudence for u, the second term on the right-hand side of (33) vanishes, and then by applying integration by parts to the final term at (32), we arrive at

[mathematical expression not reproducible]. (34)

The partial integrals being nonnegative, Equation (34) is positive and saving by v increases if and only if the derivative multiplying these integrals is negative. Taking account of constant absolute prudence, we obtain

[mathematical expression not reproducible]. (35)

This equation is negative and v increases saving if [psi]"' > 0.

When the decision criterion is reformulated so that dissaving z is chosen to maximize the expected future value of utility, first-order condition (23) is replaced by

[[1 + [rho]]/[1 + r]]u'([bar.y] + z/(1 + r)) - [integral] u'([??] - z + y)dF(y, [theta]) = 0, (36)

and an MPS in the distribution for -u'([??] - z + y) leaves the marginal value of dissaving unchanged while increasing its spread. When this change is induced by an increase in [theta], the partial integrals in the second line of (32) are nonnegative, and the change [F.sub.[theta]] (y, [theta]) induces a mean decreasing spread in the distribution for -v' if [psi]' < 0, given (33) and constant absolute prudence for u. In this case, an increase in the mean value of future income just sufficient to compensate -u' for the increase in spread, thereby leaving saving by u unchanged, undercompensates a more prudent utility v, one for which -v' = [psi](-u') with [psi]' < 0, and the marginal value of dissaving declines, leading to an increase in saving by v. Theorem 4 shows that this prediction extends to the third order, where an MVPS for -u' leaves saving by u unaffected, but leads to an increase in saving by v if [psi]"' > 0.

Just as comparisons of utility based solely on [phi]"' > 0 can cycle, with v being a downside risk-averse transformation of u and vice versa, so could comparisons based for the condition [psi]"' > 0 were it not for the fact that u and v cannot simply switch roles in Theorem 4, the distributional shift being specific to u. Using a cycling condition, though, is perhaps not good policy in general, and is one of the reasons we have focused on measures in this article, since expressing conditions in terms of a change in a measure automatically prevents cycling of the transformations involved. In the case of Theorem 4, the obvious measure is [H.sub.u](y) = [S.sub.-u'](y), where [H.sub.v](y) > [H.sub.u](y) for all y implies [psi]"' > 0, with the converse holding only for infinitesimal changes in risk preferences.

Applications. To place our results on the intensity of the precautionary motive in an economic context, consider an exchange economy with identical consumers, each with identical endowments, and assume that [??] = [bar.y]. Since the equilibrium interest rate must sustain saving equal to zero as the optimal choice for the representative consumer, the equilibrium interest rate must equal the subjective rate of time preference in the absence of background risk, so that initially r = [rho]. If the representative consumer is downside risk averse, then the introduction of a future background risk stimulates demand for precautionary saving, and the equilibrium interest rate must decline to equal the compensating rate [r.sub.u]([rho], [theta]) in order to sustain saving equal to zero. When u exhibits DARA and DADSRA, and the transformation [phi] is the result of a reduction in endowed incomes [bar.y] = [??], condition (C) is satisfied, and by Corollary 2 the compensating rate is lower for v than for u. Additionally, by Corollaries 1A and 1B, the substitution effects of risk and of downside risk are positive. In this manner, the responsiveness of the marginal rate of time preference to background risk is closely linked to compensated responses to risk.

The implications of greater absolute prudence, on the one hand, and greater absolute downside risk aversion, on the other, are illustrated in Figures 1 and 2. The former depicts the trade-off between present consumption, [x.sub.1] = [bar.y] - c, and the sure component of future consumption, [x.sub.2] = [bar.y] +(1 + r)c. Two indifference curves are illustrated for the expected utility function u([x.sub.1]) + [1/(1 + [rho])] [integral]u([x.sub.2] + y)dF(y, [theta]). The marginal rate of substitution is (1 + [rho])u'([x.sub.1])/ [integral] u'([x.sub.2] + y)dF, or one plus the marginal rate of time preference, and the relative price of [x.sub.2] is 1/(1 + r), so budget lines have a slope of 1 + r in absolute value. The interest rate r = [rho] yields the budget line tangent to the solid indifference curve at the 45-degree line, and there is neither borrowing nor lending.

Given downside risk aversion, the introduction of background risk stimulates precautionary saving, [c.sub.u]([rho], [theta]) > 0, and with greater prudence, [P.sub.v] > [P.sub.u], v has a stronger demand for precautionary saving (Theorem 2). At the same time, the marginal rate of substitution declines as illustrated by the dash-lined indifference curve. To maintain zero saving, the interest rate must decline as indicated by the flatter slope at the 45-degree line. With greater downside risk aversion and greater risk aversion, [S.sub.v] > [S.sub.u] and [R.sub.v] > [R.sub.u], the compensating interest rate is lower for v than for u, [r.sub.v]([rho], [theta]) < [r.sub.u]([rho], [theta]) (Corollary 2). Finally, with background risk present and undergoing a shift that induces an MVPS in the distribution for -u', there would be no change in saving by u, but with [H.sub.v] > [H.sub.u] saving would be greater for v, provided u exhibits constant absolute prudence (Theorem 4).

Figure 2 illustrates the effect of background risk on the saving schedule of a representative consumer. In the absence of background risk, the schedule is depicted as the curve labeled [theta] = 0, and the interest rate [rho] sustains saving equal to zero. Assuming downside risk aversion, background risk stimulates demand for precautionary saving, the schedule shifts rightward to the curve labeled [theta] > 0, and [c.sub.u]([rho], [theta]) is positive. Greater absolute prudence, [P.sub.v] > [P.sub.u], implies that v desires more saving than [c.sub.u]([rho], [theta]) in response to the background risk (Theorem 2), whereas greater absolute downside risk aversion and greater absolute risk aversion, [S.sub.v] > [S.sub.u] and [R.sub.v] > [R.sub.u], imply that v requires an equilibrium, compensating interest rate lower than [r.sub.u]([rho], [theta]) to offset the precautionary motive (Corollary 2). As the arbitrary rightward shift of the saving schedule in Figure 2 suggests, a greater increase in desired saving does not imply a greater decline in the compensating interest rate, or vice versa.

It follows that, starting from an equilibrium with no background risk, and therefore r = [rho], greater downside risk aversion, coupled with greater risk aversion, yields a ranking of utility functions by the strength of the precautionary motive associated with a given background risk, as measured by the magnitude of the decline in the equilibrating interest rate needed to offset the precautionary motive. On the other hand, the result of Bauer and Buchholz (2009, Proposition 2) discussed earlier shows that in the presence of background risk, saving by v exceeds saving by u at any interest rate r < [rho] if v is more risk averse and more prudent than u, again implying that the equilibrium interest rate is lower for v than for u. Thus, greater downside risk aversion and greater prudence, when coupled with greater risk aversion, yield distinct rankings of utility functions as measured by the decline in the equilibrium interest rate required to neutralize the precautionary motive.

Finally, if an existing background risk shifts by an MVPS of -u', then the compensating interest rate for u remains unchanged, but declines for v when [H.sub.v] > [H.sub.u] and u exhibits constant absolute prudence. For this class of utility functions, [H.sub.u] provides a third measure of the intensity of the precautionary motive.

CONCLUSIONS

The Taylor series with which we began show that [R.sub.u], [d.sub.u], and [P.sub.u] reflect the willingness to trade off various pairs of changes in the first three moments of a random income, revealing their usefulness as indicators of direction in risk preference, and suggesting that they are similar with respect to their usefulness as indicators of preference intensity for small risks. In particular, for each index, a higher value implies that a more favorable change in the lower order is needed to offset an unfavorable change in the higher order. However, only [R.sub.u] and [P.sub.u] satisfy 1-cocycle conditions, which allows them to serve usefully as measures of preference intensity in the large. Lack of a 1-cocycle condition limits the usefulness of [d.sub.u] as a measure of the trade-off between mean income and its skewness to small changes in small risks.

In contrast to the preceding measures, the Schwarzian index [S.sub.u] lacks an interpretation in terms of a trade-off between orders of risk, and its sign therefore does not indicate direction of third-order risk preference. However, in the small, a change in the utility function u(y) that increases the Schwarzian measure results in a utility function that dislikes any change in the distribution for income that induces an increase in downside risk for the distribution of u(y), while inducing an unfavorable change in its skewness. Further, unlike [d.sub.u], the Schwarzian [S.sub.u] satisfies a 1-cocycle condition, and therefore its usefulness as a measure of preference intensity extends to large changes in risk and global comparisons of risk preferences.

To illustrate behavioral distinctions among the measures of intensity [R.sub.u], [P.sub.u], and [S.sub.u], we analyze the saving problem. We first show that, just as decreasing absolute risk aversion, [partial derivative][R.sub.u](y)/[partial derivative]y < 0, implies that greater risk has a positive substitution effect on saving, decreasing absolute downside risk aversion, [partial derivative][S.sub.u](y)/[partial derivative]y < 0, implies that greater downside risk also has a positive substitution effect.

Since a background risk to future income introduces a precautionary motive for saving when utility is downside risk averse, the magnitude of precautionary saving provides a natural measure of third-order preference intensity, and is linked by Kimball (1990) to the degree of absolute prudence measured by [P.sub.u]. The effect of background risk on the marginal rate of time preference provides an alternative measure of preference intensity that we link to the degree of absolute downside risk aversion as measured by [S.sub.u]. We conclude that absolute prudence [P.sub.u] and absolute downside risk aversion [S.sub.u] measure distinct and complementary aspects of risk preference intensity at the third order.

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Donald C. Keenan, Department of Economics and Management, Universite de Cergy-Pontoise & THEMA, Cergy-Pontoise, France 95011. Keenan can be contacted via e-mail: dkeenan@uga.edu. Arthur Snow, Department of Economics, University of Georgia, Athens, GA 30602. Snow can be contacted via e-mail: snow@uga.edu.

(1) Throughout, we assume that utility functions are strictly increasing, real-valued functions of income y > 0 and that transformations of utility are strictly increasing, real-valued functions defined on a subset of the real numbers. We use primes to denote the first three derivatives of u(y) and denote the fourth derivative by [u.sub.iv](y).

(2) Experimental evidence of downside risk aversion is provided by Deck and Schlesinger (2010, 2014) and by Ebert and Wiesen (2011).

(3) The marginal rate of time preference Fisher (1954, pp. 117-119) is the marginal rate of substitution measuring the willingness to forego sure future income for sure present income, minus one. When saving is personally optimal, the marginal rate of time preference equals the interest rate as in Equation (23') below.

(4) This result, stated below in Corollary 1B, ties the Schwarzian to a comparative statics prediction without relying on side conditions regarding second-order risk aversion. In contrast, an important limitation of the prediction concerning the marginal rate of time preference, stated formally below in Corollary 2, is the tie to condition (C), requiring not only that S increases, but that R increases as well.

(5) Throughout, our emphasis is on the parallels, and divergences, between the measures of risk aversion at the second and third orders, which brings to the fore the sign of substitution effects in the context of saving. In other economic environments, the second- and third-order measures jointly have an uncompensated effect on the optimal control. For example, condition (C) implies that demand is greater by v than by u for a self-protection activity that induces a contraction in downside risk about future income (Keenan and Snow, 2017).

(6) Throughout this section we use E[*]to denote the expected value of the term within brackets.

(7) Modica and Scarsini (2005) consider a similar experiment, one allowing expected utility to vary while stipulating that the risks have the same first and second moments. Under these conditions, (4) implies E[[u.sub.2] - [u.sub.1]]/u'([bar.y]) [approximately equal to] (1\6) [d.sub.u]([bar.y])E[[[??].sub.2] - [[??].sub.1]], where [u.sub.i] = u([bar.y] + [[??].sub.i]),implyingthatan unfavorable shift in skewness imposes a greater cost the higher is [d.sub.u][bar.y]. Modica and Scarsini express this as being willing to pay a higher premium to avoid higher (negative) skewness.

(8) As Brockett and Kahane (1992) emphasize, basing the choice between two risks on relative skewness and skewness preference, say, when the mean and variance are the same, does not yield a reliable guide to their ranking by expected utility and the infinite Taylor series (4). In contrast, we use the series to isolate the specific trade-offs between orders of risk governed by each of the indices of risk preference.

(9) Pratt (1964) establishes Equation (12) by showing that a concave transformation of u is equivalent to [R.sub.u] increasing.

(10) Keenan and Snow (2012) discuss the 1-cocycle properties for R and S.

(11) Specifically, for finite changes in risk preferences, [S.sub.v] (y) > [S.sub.u](y) implies [d.sub.[phi]](u) > 0, and (11) shows that v(y) dislikes changes that leave the mean and variance of v(y) unchanged, while inducing an unfavorablechangeinskewness. However, [S.sub.v](y) < [S.sub.u](y) does not necessarily reverse this preference, since [d.sub.[phi]](u) < 0 is not implied by [S.sub.v](y) < [S.sub.u](y). In contrast, [R.sub.v](y) > [R.sub.u](y) implies [R.sub.[phi]]([bar.u]) > 0, and (10) shows that v(y) dislikes changes that leave the mean of u (y) unchanged while increasing the variance, and this preference can be reversed by changing the ranking of utility functions by R,since [R.sub.v](y) < [R.sub.u](y) implies [R.sub.[phi]]([bar.u]) < 0, reversing the sign of (10).

(12) Lemma 1 is proved by a different argument in Keenan and Snow (2002, Theorem 1).

(13) Adapting (11) to incorporate the preference parameter [alpha], note that differentiating with respect to [alpha] and evaluating the result at [alpha] = 0 yields the expression on the left-hand side of (17), since [phi]'(u([bar.y]), 0) = 1, plus a second term in which [partial derivative][phi]'(u([bar.y]), 0)/[partial derivative][alpha] is multiplied by E[[phi]([bar.u] + [[??].sub.u], 0)] - E([phi]([bar.u] + [[??].sub.u], 0)] = 0.

(14) The exception arises when u is risk neutral, for then [R.sub.u] = 0 = [d.sub.u] and (19) reduces to [d.sub.v] = [[u'].sup.2][d.sub.[phi]].

(15) Throughout we assume that second-order sufficient conditions are satisfied.

(16) Propositions 1A and 1B are local results in that they are stated only for infinitesimal changes in risk preferences. The first readily extends to the large by ensuring that [R.sub.u] always increases along a path of preferences from u to v when [R.sub.v] > [R.sub.u], for then the final change in c must be negative; reversing the direction reverses the conclusion. The second extends in the same way as shown by Keenan and Snow (2017).

(17) Intertemporal expected utility (22) does not separate preferences concerning variation in consumption across states of nature and variation across time. The Epstein-Zin-lognormal model, employed by Martin (2013) to show how the successive cumulants of wealth affect the equilibrium interest rate, effects a separation of these preferences, with constant relative risk aversion (CRRA), that is, R(y) = [gamma]/y, characterizing the attitude toward risk bearing. CRRA preferences satisfy decreasing absolute risk aversion and exhibit the useful property that the Schwarzian has the same sign for all incomes, since S(y) = [gamma][1 - (1/2)g]/[y.sup.2]. It follows that, with CRRA preferences, S is first increasing and then decreasing in the degree of relative risk aversion, attaining a maximum at [gamma] = 1.

(18) Crainich et al. (2013) and Nocetti (2016) show that [d.sub.u] > 0 implies precautionary saving by risk lovers as well as risk averters.

(19) The caveats noted earlier regarding Propositions 1A and 1B apply equally to their corollaries.

(20) Gollier and Pratt (1996) establish that DAP and DARA are sufficient for this trait they call risk vulnerability.

(21) Given [R.sub.u] > 0, Equation (24) implies [partial derivative][S.sub.u]/[partial derivative]y < 0 if (a) 4[P.sub.u][R.sub.u] - [P.sub.u][T.sub.u] - 3[R.sub.u.sup.2] < 0. Observe that -2[([P.sub.u] - [R.sub.u]).sup.2] [less than or equal to] 0 implies (b) 4[P.sub.u][R.sub.u] - 2[R.sub.u.sup.2] [less than or equal to] 2[P.sub.U.sup.2], while [T.sub.u] > 2[P.sub.u] > 0 implies (c) -[P.sub.u][T.sub.u] + 2[P.sub.u.sup.2] < 0. Together, inequalities (b) and (c) imply (d) -[P.sub.u][T.sub.u] + 4[P.sub.u][R.sub.u] - 2[R.sub.u.sup.2] < 0. Subtracting [R.sub.u.sup.2] from the left-hand side of inequality (d) yields inequality (a).

(22) We assume [R.sub.u](y) > 0 so that -u'(y) has a positive first derivative and therefore exhibits nonsatiation, ensuring that [R.sub.-u'](y) is a well-defined measure of risk preference intensity for -u'(y) at the second order.

(23) For example, assume [psi](-u') is linear so that [P.sub.v] = [P.sub.u]. Under this assumption, if the left-hand side of (26) is a decreasing function of r when r = [rho], then as r rises above [rho] we have [c.sub.v](r, [theta]) > [c.sub.u] (r, [theta]) even though [P.sub.v] = [P.sub.u]. On the other hand, as r falls below [rho], we have [c.sub.v] (r, [theta]) < [c.sub.u](r, [theta]) with [P.sub.v] = [P.sub.u] and, by continuity, this inequality also holds with a sufficiently small increase in prudence, in which event greater prudence does not imply greater saving. In neither case, however, does the prediction address the difference in precautionary saving since if r [not equal to] [rho], differences in the initial saving by u and v could reverse these predictions.

(24) When [d.sub.u] > 0, marginal utility is risk loving and therefore the denominator on the right-hand side of (28) exceeds the numerator.

(25) Crainich and Eeckhoudt (2008) show that 1 + [r.sub.u] = (1 + [rho])/[1 + ([[sigma].sup.2]/2)[d.sub.u]] for a cubic utility function, with [[sigma].sup.2] denoting the variance of the background risk. In this special case of cubic utility, [r.sub.u] is inversely proportional to [d.sub.u].

DOI: 10.1111/jori.12232
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Author:Keenan, Donald C.; Snow, Arthur
Publication:Journal of Risk and Insurance
Date:Jun 1, 2018
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